x. List of Tables. Table 1-1 Listed Features of Four Major Classes of CAMs . ..... of the immune system, blood coagulation, thrombosis, extracellular matrix deposition and ..... We aim to answer the following questions: 1) what conformations will appear and ...... The reason they are called pseudo-constants is that they are first-.
Copyright by Tianyi Yang 2009
The Dissertation Committee for Tianyi Yang Certifies that this is the approved version of the following dissertation:
Free Energy Calculations of Biopolymeric Systems at Cellular Interface
Committee:
Muhammad H. Zaman, Supervisor Ernst-Ludwig Florin, Co-Supervisor Manfred Fink Tomio Petrosky Jack B. Swift
Free Energy Calculations of Biopolymeric Systems at Cellular Interface
by Tianyi Yang, B.S.
Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
The University of Texas at Austin August, 2009
Dedication
To: My Father: Songquan Yang My Mother: Gumei Zhou My Wife: Chunyan Li
Acknowledgements
I would like to thank Dr. Muhammad H. Zaman, my advisor, without whose support and guidance this work could not be finished. I would also like to express my thanks to the members of the thesis committee: Dr. Ernst-Ludwig Florin, Dr. Manfred Fink, Dr. Tomio Petrosky and Dr. Jack B. Swift for their advice and direction. The life in the graduate school would not be so fulfilling without the kind, honest, and friendly people around the department of physics. I would also like to express my thanks to the people in the Zaman lab, Ge Wang, Leandro Forciniti, Rajagopal Rangarajan, Dewi Harjanto and Erin Baker. I would like to express my deep gratitude to my father Songquan Yang, my mother Gumei Zhou and my wife Chunyan Li for their unselfish love and support.
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Free Energy Calculations of Biopolymeric Systems at Cellular Interface
Publication No._____________
Tianyi Yang, Ph.D The University of Texas at Austin, 2009
Supervisors: Muhammad H. Zaman Co-Supervisor: Ernst-Ludwig Florin Cells interact with both tethered and motile ligands in their extra-cellular environment, which mediates, initiates and regulates a series of cellular functions, such as cell adhesion, migration, morphology, proliferation, apoptosis, bi-directional signal transduction, tissue homeostasis, wound healing among others. A fundamental understanding of the thermodynamics of receptor-mediated cell interaction is necessary not only from the aspect of physiology, but also for bioengineering applications, e.g. drug discovery, tissue engineering and biomaterial fabrication. Our models on free energy calculations of receptor mediated cell-matrix interactions supplement computational endeavors based on continuum mechanics. By incorporating conformational, entropic, solvation, steric effect, implicit and explicit interactions of receptors and extra-cellular ligand molecules, we can predict free energy, chemical equilibrium constant of binding, spatial and conformational distributions of biopolymers, adhesion force as functions of a set of key variables, e.g. surface coverage of receptor, interaction distance between cell and substrate, specific binding energy, implicit interaction strength, constraint in ligand’s conformation, size of motile nano-ligand, aggregation of receptors, sliding velocity
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relative to fluid. Our work has improved understanding of phenomena in cell-matrix interactions at both cellular and the molecular scales.
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Table of Contents List of Tables ...........................................................................................................x List of Figures ........................................................................................................ xi Chapter 1 Introduction .............................................................................................1 1.1 Cell and Extra-Cellular Matrix .................................................................1 1.2 Cell Adhesion Molecules ..........................................................................6 1.3 Application of Cell Adhesion in Therapeutics and Engineering ............10 1.4 Modeling of Cell-Matrix Interaction ......................................................13 Chapter 2 Model I: The Interaction of Receptors with Tethered Ligands .............19 2.1 Introduction .............................................................................................19 2.2 Model Development................................................................................24 A) Conformations and Probability Density Function (pdf) .................24 B) Constraint Equation ........................................................................28 C) Components of Free Energy ...........................................................31 D) Minimization of Free Energy..........................................................35 E) Numerical Solution .........................................................................38 2.3 Results .....................................................................................................41 A) Interaction with Polymer Coated Surface .......................................41 B) Interaction with Protein Coated Surface .........................................44 C) Interaction with Mixed Surface ......................................................46 2.4 Conclusion ..............................................................................................50 Chapter 3 Model II: Cell Adhesion to Motile Homogeneous Nano Ligands: Effects of Ligand Size and Concentration .....................................................................52 3.1 Introduction .............................................................................................52 3.2 Model Development................................................................................54 3.3 Results .....................................................................................................65 A) Free Energy and Binding Status for Dilute Systems ......................66 B) Free Energy and Binding Status for Concentrated Systems ...........70 viii
C) Comparison of Probability Density Functions ................................80 3.4 Conclusion ..............................................................................................82 Chapter 4 Model III: Comparison of Clustered and Unclustered System: Effect of Receptor Aggregation ...................................................................................84 4.1 Introduction .............................................................................................84 4.2 Model Development................................................................................87 A) Monte Carlo Simulation to Find Distribution of Receptor Clusters87 B) Calculation of Free Energy .............................................................91 4.3 Results and Conclusion ...........................................................................96 Chapter 5 Model IV: Regulation of Cell Adhesion Free Energy and Force by External Sliding Velocity ..........................................................................................101 5.1 Introduction ...........................................................................................101 5.2 Model Development..............................................................................102 5.3 Results ...................................................................................................108 5.4 Discussion and Conclusion ...................................................................114 Chapter 6 Model V: Estimation of Cellular Adhesion Forces Using Mean Field Theory .........................................................................................................116 6.1 Introduction ...........................................................................................116 6.2 Model Development..............................................................................118 6.3 Results and Conclusion .........................................................................122 Chapter 7 Comparison with Experiments ............................................................127 Chapter 8 Concluding Remarks ...........................................................................131 References ............................................................................................................134 Vita…...................................................................................................................144
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List of Tables Table 1-1 Listed Features of Four Major Classes of CAMs ............................................... 8 Table 3-1 Contribution to Helmholtz Free Energy for 1.0 nm Ligand System ................ 78 Table 3-2 Contribution to Helmholtz Free Energy for 2.0 nm Ligand System ................ 78 Table 6-1 Components to Free Energy in the Transition Region ................................... 123
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List of Figures Figure 1.1 Electron Micrograph and Schematic of Cell ..................................................... 2 Figure 1.2 Fibroblast Cells and Extracellular Matrices ...................................................... 5 Figure 1.3 Illustrations of Four Major Classes of Cell Adhesion Molecules ..................... 7 Figure 2.1 Schematic of the System ................................................................................. 19 Figure 2.2 Generic Amino Acid Backbone....................................................................... 26 Figure 2.3 Ramachandran Plot.......................................................................................... 26 Figure 2.4 Free Energy of Receptor-Polymer System ...................................................... 42 Figure 2.5 Free Energy of Receptor-Peptide System........................................................ 45 Figure 2.6 Receptor-Mixed Ligands System .................................................................... 47 Figure 2.7 Effect of Changes in Receptor-Ligand Interaction.......................................... 49 Figure 3.1 Cartoon Depicting Motile Ligand System ....................................................... 52 Figure 3.2 Free Energies in Dilute Systems...................................................................... 67 Figure 3.3 Landscapes and Contours of Binding Free Energy ......................................... 69 Figure 3.4 Free Energies at High Concentration Region .................................................. 71 Figure 3.5 Spatial Distribution of Modified Osmotic Pressure π(z) ................................. 74 Figure 3.6 Probability Density Functions at Low and High Concentration ..................... 81 Figure 4.1 Interactions between Cluster Receptors and Ligands ...................................... 86 Figure 4.2 Receptor Cluster Size Distribution .................................................................. 96 Figure 4.3 Free Energy Difference for Clustered and Unclustered Systems .................... 97 Figure 4.4 Landscape of Free Energy Difference ............................................................. 98 Figure 5.1 Cartoon Depicting the System under Sliding Force ...................................... 102 Figure 5.2 Binding Probability Regulated by Velocity and Binding Energy ................. 109 Figure 5.3 Free Energy of Different Binding Energy and Sliding Velocity ................... 110 Figure 5.4 Free Energy Landscape for Binding Energy and Velocity ............................ 112 Figure 5.5 Results for Low Unbinding Rate ................................................................... 113 Figure 6.1 Tethered Receptor and Ligand System to Calculate Adhesion Force ........... 117 Figure 6.2 Adhesion Force versus Distance ................................................................... 123 Figure 7.1 Comparison of Force-Distance Curve with Experiment ............................... 128
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Chapter 1 Introduction 1.1 CELL AND EXTRA-CELLULAR MATRIX Cell is the structural and functional unit of all known living organisms. It is the smallest unit of an organism that is classified as living, and is often called the building block of life (Alberts, Johnson et al. 2002). Some organisms, such as most bacteria, are unicellular. Other organisms, such as humans, are multicellular. Humans have an estimated 100 trillion cells. All cells are prokaryotic or eukaryotic. Prokaryotic cells are simpler than eukaryote cells (Lodish, Berk et al. 2004) and consist of a single closed compartment that is surrounded by the plasma membrane, lack a defined nucleus, and have a relatively simple internal organization. There are two kinds of prokaryotes: bacteria and archaea. Bacteria, the most common prokaryotes, are single-celled organisms. A single Escherichia coli bacterium has a dry weight of about 2.5 x 10-13 g. Bacteria account for an estimated 1–1.5 kg of the average human’s weight. Eukaryotic cells (see Figure 1.1), unlike prokaryotic cells, contain a defined membrane-bound nucleus and extensive internal membranes that enclose other compartments, the organelles, e.g. mitochondrion, endoplasmic reticulum, golgi apparatus, lysosome, peroxisome (as shown in Figure 1.1), chloroplasts in plants, and so on. The region of the cell lying between the plasma membrane and the nucleus is the cytoplasm, comprising the cytosol (aqueous phase) and the organelles. Eukaryotes comprise all members of the plant and animal kingdoms,
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Figure 1.1 Electron Micrograph and Schematic of Cell Electron micrograph of a white blood cell that secretes antibodies is shown here. The defining characteristic of eukaryotic cells is segregation of the cellular DNA within a defined nucleus, which is bounded by a double membrane. The outer nuclear membrane is continuous with the rough endoplasmic reticulum, a factory for assembling proteins. Golgi vesicles process and modify proteins, mitochondria generate energy, lysosomes digest cell materials to recycle them, peroxisomes process molecules using oxygen, and secretory vesicles carry cell materials to the surface to release them. From (Lodish, Berk et al. 2004). Originally from (Cross and Mercer 1993). including the fungi, which exist in both multicellular forms (e.g. molds) and unicellular forms (e.g. yeasts), and the protozoans (proto, primitive; zoan, animal), which are exclusively unicellular. Eukaryotic cells are commonly about 10–100 µm across, generally much larger than the bacteria. A typical human fibroblast–a connective tissue
2
cell, may be about 15 µm across with a volume and dry weight some thousands of times those of an E. coli bacterial cell (Lodish, Berk et al. 2004). Cell is also the basic functional unit for living organisms. A cell is simply a compartment in essence with watery interior that is separated from surface membrane. The plasma membrane is typically composed of a phospholipid bilayer, with both hydrophobic ends pointing to each other toward the interior of the membrane and hydrophilic ends to the outer and inner of the cell. The entire cell compartment is a factory (different types of cells can be different factories) to sustain the well-being of the organism. The cellular work is performed by molecular machines, some housed in cytosol and other in organelles. Some of the significant cellular functions are (Lodish, Berk et al. 2004): 1) Production and degradation of numerous molecules and structures. For example, ATP, the universal “energy currency” for organism is generated by the organelles mitochondria and chloroplast. Proteins are one of the most important structural and functional molecules. They are made on ribosome with the help of endoplasmic reticulum (ER) and transported to Golgi apparatus for further modification before being sent to destinations. 2) Growth and division. Reproduction is considered one of the imprints of life. Cell division includes mitosis, referring to a parent cell divides into two homo- or hetero-daughters (stem cell), and miosis in sexual production, in which each daughter cell contains half the full number of chromosomes. 3) Cells die from aggravated assault or an internal program. Cells in multicellular organisms die when badly damaged or infected with a virus, resulting in release of potentially toxic constituents that can damage surrounding cells. The natural, programmed death of cell is called apoptosis, 3
which is critical to the proper development and functioning of life. An interesting example is that of cancer cells lose the capability of programmed death and their ceaseless multiplication can be deadly. 4) Cells regulate gene expression to meet changing internal and external needs. Living organisms may endure widely varying environmental conditions that require changes of cellular structure and function as a consequence. Control of gene activity in eukaryotic cells usually involves a balance between actions of transcriptional activators and repressors. 5) To perform specialized functions, cells must generate and utilize genetic information, synthesize, sort, store and transport biomolecules, convert different forms of energy, transduce signals, maintain internal structures, and respond to external environment. Cells constantly monitor their surroundings and adjust their activities accordingly. Cells communicate with each other by sending signals receivable and interpretable by other cells. Signal transduction is executed by numerous transmembrane receptor proteins, which will be given more details in section 1.2. 6) Cells produce their own external environment and interact. The “environment” here refers to extracellular matrix. Both 5) and 6) are related to “cell adhesion”, which includes cell-cell adhesion and cell-matrix adhesion. In general, spatial and temporal coordination of cell–cell and cell–matrix adhesion is critical for maintaining the control of migration, proliferation, survival, and differentiation of the individual cells within a tissue. In multicellular organisms, cells are embedded in a jelly-like mixture of proteins and polysaccharides called extracellular matrix (ECM) (see Figure 1.2). Cells themselves produce and secrete these molecules and create their immediate environment. 4
There are many cell types that contribute to the development of extracellular matrices found in different tissue types. Fibroblasts are the most common cell in connective tissue (see Figure 1.2), where they synthesize ECM, maintain and provide a structural framework. ECM makes up the skeleton of a tissue, with cells filled inside. The ECM is a
Figure 1.2 Fibroblast Cells and Extracellular Matrices Contractile fibroblasts secrete and organize extracellular matrix fibers (blue) that are loaded with growth factor complexes (green), resulting in a turquoise overlay color. The contractile fibers inside the cells are visualized by detecting a smooth muscle protein (red). The cells' nuclei are visualized in yellow. From (Wipff 2007) complex mixture of matrix molecules of various sizes and structure, including the interstitial matrix and the basement membrane (Alberts, Johnson et al. 2002). Interstitial matrix is present between various cells (i.e. in the intercellular spaces) and the interstitial space is filled by gels of polysaccharides and fibrous proteins, which act as a compression buffer against the stress placed on the ECM. Basement membranes are sheet-like depositions of ECM on which various epithelial cells reside. Due to its diverse 5
composition, the ECM serves many functions, such as providing support and anchorage for cells, segregating tissues from one another, regulating intercellular communications, etc. A direct consequence of cell interaction with ECM is the formation of discrete cell surface structures within the matrix (Martinez-Lemus, Sun et al. 2005). As cell-matrix interactions are essential to tissue organization, signaling, migration, differentiation and so on, they play central roles in embryonic development, remodeling, and homeostasis of tissue and organ systems (Boudreau and Bissell 1998; Simian, Hirai et al. 2001; Sternlicht and Werb 2001). Also cell adhesion to matrix signals cooperation with other pathways to regulate biological processes such as cell survival, proliferation, wound healing and tumorigenesis (Egeblad and Werb 2002). Hence elucidating the details of cell-matrix interactions is a critical step toward the understanding of eukaryotic cellular behavior in vivo.
1.2 CELL ADHESION MOLECULES Transmembrane receptor proteins are the key players in the cellular signal transmission and adhesion. Some receptors, e.g. epidermal growth factor receptor (EGFR) play a central role in signal induction. However, a large number of membrane receptors participate in both adhesion and signaling. These receptors are also called cell adhesion molecules (CAMs). Figure 1.3 illustrates structures of four families of important CAMs. Table 1-1 lists some of their key characteristics. The extracellular domain of a CAM interacts with either other molecules of the same family (homophilic binding), other species, or the 6
extracellular matrix (the latter two are called heterophilic binding). They can be classified according to calcium dependency. Among these classes, cadherins and selectins are calcium dependent CAMs, while integrins and immunoglobulins are calcium independent.
Figure 1.3 Illustrations of Four Major Classes of Cell Adhesion Molecules (a) Cadherin; (b) Immunoglobulin; (c) Selectin; (d) Integrins. The picture is from (Hynes 1999).
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Table 1-1 Listed Features of Four Major Classes of CAMs Family
Function
Ligands
Miscellaneous
Cadherens
Key in cell-cell adhesion Adherens junctions Desmosomes
(Ex) other cadherin (In) actin filaments & Intermediate filaments
Require Ca
Immunoglobulin Superfamily (IgSF) Selectins
Immunity
Integrins, other IgSF
Another name: Antibody
Vertebrate circulating cells Endothelial cells, blood cells Leukocytes to endothelial
Carbohydrates-specific
Require Ca
Key in cell-matrix adhesion (focal) Hemi-desmosomes, Leukocyte (“Spreading”)
(Ex) fibronectin, collagen, laminin, immunoglobulins, (In) actin filaments
Heterodimer, α and β subunits
Integrins
2+
2+
Among the class of cell adhesion molecules, “integrin” is the most important cellmatrix adhesion molecule. The cell-matrix interface is regulated by integrins (DeSimone, Stepp et al. 1987; Hynes 1987). Integrins have two basic functions: linking the attachment of the cell to the ECM and transducing signals from ECM to the cell (O'Toole, Katagiri et al. 1994; Shattil and Ginsberg 1997; Boudreau and Bissell 1998). Integrins are transmembrane obligate heterodimers containing two distinct chains (α and β) that require divalent cations for their noncovalent association. To date, 19 α and 8 β subunits have been identified, and these can combine to form at least 25 distinct integrins that form binding with various ECM ligands at different affinities. Some integrin subunits are ubiquitously expressed, while other subunits are expressed in a tissue- or stagerestricted manner (Berrier and Yamada 2007). The ligands of integrins are type I and 8
type IV collagen, laminin, fibronectin, vitronectin and fibrinogen. The extra-celluar domain of integrins can bind to the ligands in ECM through the specific amino acid sequence in the ligands. The most well studied sequence among the ligands is a tripeptide that features in the integrin-interaction site of many ECM proteins, known as arginineglycine-aspartic acid (RGD). The extra-cellular domain of integrins can also attach laterally with other proteins at the cell surface, e.g., tetraspanins, growth factor receptors, matricellular proteins, and matrix proteases or their receptors. The other end of integrin, the intra-cellular, or cytoplasmic domain, which typically is a short region of 50 amino acids in length except integrin β4 (~1,000 monomers), forms multi-molecular complexes with proteins involved in cell signaling and adaptor proteins that assist their connection to the cytoskeletons. Integrins present a bi-directional channel for mechano-chemical information transferring across the cell membrane. Cell adhesion to the ECM allows transmission of information via integrin receptors that regulates intracellular signaling via “outside-in signaling” during cell migration and cell deformation. On the other hand, intracellular signals can induce changes in integrin conformation and activation resulting in the alteration of its ligand-binding activity, which is the so-called “inside-out signaling” (Mizejewski 1999; Honore, Pichard et al. 2000; Petit and Thiery 2000; Hynes 2002; Hynes, Lively et al. 2002). Evidence from a number of in vitro studies demonstrates that receptor clustering is frequently involved in the regulation of adhesion (Sanchez-Mateos, Campanero et al. 1993; Hato, Pampori et al. 1998; Friedl and Brocker 2000; Gottschalk and Kessler 2004; Li, Bennett et al. 2004). After the binding of the extra-cellular domain of an integrin to its 9
ligand, an aggregate of integrins and other intracellular proteins forms an adhesion complex, which enhances the localized intracellular concentration of signaling molecules, connects the integrins to the actin cytoskeleton and activates many signaling pathways involved in adhesion. For example, integrin clustering can trigger tyrosine phosphorylation (Hato, Pampori et al. 1998). Clustered integrins can form multivalent bonds with surfaces bearing ligand molecules of a sufficient density, which can regulate the protein interactions and enzymatic activity of the substrates. Other evidence includes dimerization of selectins (Li, Steeber et al. 1998) which may be important to ensure the functional efficiency of these adhesion receptors. Recent studies have also reported on the correlation between functional activation and membrane clustering of CD11aCD18/LFA1, another ß-2 integrin (Stewart, McDowall et al. 1998; van Kooyk and Figdor 2000) etc. In this thesis, we will focus only on the mechano-chemical study of the extra-cellular domain of receptors and the interactions with respect to ECM and the signaling issues will largely be ignored, however, issues related to clustering that often act as the trigger for integrin based signaling will be addressed.
1.3 APPLICATION OF CELL ADHESION IN THERAPEUTICS AND ENGINEERING Cell-matrix interactions play a critical role in cellular functions not only for healthy cells, but also for diseased cells, and the role of these interactions in cancer cannot be underestimated. The initiation and progression of cancer is accompanied by alterations in the expression profile of various adhesion receptors (Olumi, Grossfeld et al. 1999; Hanahan and Weinberg 2000; Hynes 2004; Brakebusch and Fassler 2005). 10
Transformation by certain oncogenes requires the appropriate integrin profile suggesting that integrin signaling may control cancer initiation. Furthermore, integrin-mediated migration and interactions with other cell types, such as platelets or endothelial cells, contributes to tumor cell invasion and metastasis. Cell adhesion, matrix assembly, cell migration, survival, proliferation and signaling, all contribute to modulate cancer development. Blocking integrin function could potentially interfere with tumor growth and metastasis. Epithelial-mesenchymal transition (EMT) is a program of development of biological cells characterized by loss of cell adhesion, repression of E-cadherin expression, and increased cell mobility. EMT is essential for numerous developmental processes including mesoderm formation and neural tube formation. It was found that during EMT, reduced cell-cell adhesion occurs simultaneously with changes in cellmatrix adhesion that promote individual cell migration and scattering (Hynes 2002). This process occurs in a highly controlled manner during embryonic development in order to prevent the unregulated dissemination of cells to inappropriate tissues. On the other hand it occurs in an uncontrolled manner during malignant carcinoma progression to promote metastasis. The other key application is in the rapidly growing field of Tissue Engineering. Tissue engineering is an interdisciplinary field focused on developing therapeutic strategies aimed at the replacement, repair, maintenance, and/or enhancement of tissue function, for example, the manufacture of artificial liver, artificial pancreas, artificial bone. Cells are often implanted into an artificial structure capable of supporting threedimensional tissue formation. These structures, typically called scaffolds, are often 11
critical, both ex vivo as well as in vivo, to recapitulating the in vivo milieu and allowing cells to influence their own microenvironments. Scaffolds sometimes serve the following purposes of allowing cell attachment and migration, exerting certain mechanical and biological influences to modify the behavior of the cell phase. The scaffold in essence is artificial extracellular matrix. In construction of biocompatible artificial implants, there are two main strategies for modulating the cell-material interactions. One is to create an inert surface not allowing the adsorption of proteins and adhesion of cells, and thus preventing activation of the immune system, blood coagulation, thrombosis, extracellular matrix deposition and other interactions between material and surrounding environments. This type of biomaterial has been used for construction of vesicles for therapeutic drug delivery (Cook, HRKACH et al. 1997). The other, more general and advanced strategy aims at creation of materials promoting attachment, migration, proliferation, differentiation, longterm viability and cell functioning (such as contraction or secretion of extracellular matrix) in a controllable manner, if possible. The most advanced recent trend in tissue engineering aims at creation of so-called “hybrid bioartificial organs”. This strategy is used for construction of artificial vessels, bone, cartilage and parenchymatous organs like pancreas or liver. The artificial component of these constructs is designed as a threedimensional scaffold promoting controlled ingrowth and maturation of cells. It could be colonized under in vitro conditions with patient’s own cells obtained by biopsy prior to the planned surgery, or even with stem cells guided to a certain differentiation pathway. In ideal case, the artificial support should be reorganized and absorbed by growing cells 12
and gradually replaced by the newly formed extracellular matrix and differentiated cells, i.e. fully functional native tissue existing in the organ prior to damage (Park, Wu et al. 1998; Park, Tirelli et al. 2003). In summary, because cell adhesion plays a paramount role in a series of physiological processes of living organism, it is a topic of profound importance in biomedical engineering and biomaterial science. Therefore, a quantitative understanding of the fundamentals of the mechanics and chemistry of adhesion benefits a wide variety of fileds in biomedicine and biology. Next we will give a brief review of some important theoretical/computational endeavors to date that have focused the mechano-chemical properties of receptor mediated cellular adhesion.
1.4 MODELING OF CELL-MATRIX INTERACTION To date, majority of studies to quantify cell-matrix interactions have been experimental and due to the complexity of the system and difficulty in modeling, not many theoretical and computational endeavors have been put forth to quantify these interactions. The current theoretical/computational models in the study of cellular interactions can be roughly divided into two categories according to the methods utilized: chemical kinetic-mechanic models and statistical thermodynamics models. All these models try to tackle cell-matrix interaction from different angles and address different key issues. The largest part belongs to the models based on kinetics and mechanics. The major advantage is the capability in accounting for various non-equilibrium scenarios of cellular processors such as cell migration and cell adhesion under shear flows. However, 13
a major shortcoming using this approach is that these are macroscopic models, therefore lack molecular level details and require experimental data, e.g. reaction kinetic rates, diffusion coefficients as input. The conformations of a receptor (sometimes ligands as well) are of key importance in understanding receptor ligand binding. Historically, there were famous “key-lock” theory and induced fit theory that depicted the conformational specificity of the binding. However, this conformation effect is barely embodied explicitly in the macroscopic kinetic mechanic models (considered implicitly in parameters like kinetic reaction rates). These models can be considered as coarse-grained ones. While detailed atomistic molecular dynamics (MD) or molecular mechanics (MM) simulations on the whole dynamic process of a receptor molecule binding to a ligand to contain more molecular details would benefit the field tremendously, such a study on large scale adhesion currently does not exist. This is partly due to the lack of experimental measurement of receptors’ crystal structures. Integrins are macromolecular, with an extracellular domain typically ranging from 700 to 1100 residues. It is difficult to use nuclear magnetic resonance (NMR) spectroscopy to determine the high resolution structure of a molecule as large as integrin. Also, integrins are transmembrane molecules which are difficult to purify. Therefore, to date, the crystal structure of only one in the twenty-five found has been reported (Xiong, Stehle et al. 2001). The integrin with known crystal structure is αvβ3. The other issue with detailed molecular dynamics simulation is the time scale of a conformation change of receptor molecules, which is macroscopic, much longer than the current maximal simulated time ~µs. Even if there were atomistic 14
simulations, which are dynamic simulations of one receptor binding to ligand(s), it will ignore the collectivity of cell adhesion, which is a consequence of the synergy of many receptors. Given these limitations of current strategies, our models, the statistical thermodynamic models at the intermediate level, are trying to include more molecular details than kinetic-mechanic models and at the same time bridge the molecular scale to the bulk. In the last part of this chapter, we will briefly review some models belonging to the first category, and start presenting our statistical thermodynamic models from chapter 2. G. I. Bell in his seminal publication in 1978 (Bell 1978) suggested a theoretical framework for the analysis of cell-cell adhesion that is mediated by bonds between specific molecules. From the knowledge of reaction rates for reactants in solution together with the diffusion constants of reactants in solution and on membrane, the reaction rates for reactants bound to membranes could be estimated. Force equals energy divided by distance. This was used to deduce the macroscopic forces required to separate cells from microscopic bond properties. Although Bell overestimated the strength of force (~40 pN/bond), later kinetic and mechanical kinetic hybrid models were all based on the chemical reaction kinetics he suggested.
Another calculation by Bell (Bell,
Dembo et al. 1984) considered the thermodynamic tendency of two cells aggregate, each had receptors as well as a surface of glycocalyx. The glycocalyx surfaces perform electrostatic and steric repulsions. Bell calculated the Gibbs free energy change for pulling the two cells from infinity to a finite distance and found the equilibrium 15
distance(s). He determined three distance dependent phases: no adhesion, intermediate adhesion and complete adhesion. Experimentally, cell adhesion is often measured as the detachment force of a cell from the substrate. Evan Evans (Evans 1985; Evans 1985) developed hybrid models of reaction kinetics and mechanics and attempted to relate the detachment force to the functional parameters of receptors and ligands by addressing the following two questions: First, how the macroscopic stresses are translated to microscopic region of the cellsubstrate interface, this is a mechanic problem; Second, how the strength of adhesion in microscopic region is related to parameters that depict the molecular function, this is a kinetic problem. A critical tension was found in mechanical equilibrium to statically detach the cell, above which the cell starts to peel. Evans elucidated the factors controlling the critical tension for the detachment. Two values for the adhesion energy must be used, one for spreading and the other for peeling. Therefore, there are two critical tensions in accordance. One implies the disruption of the contact zone is irreversible and dissipative—the energies to form and disrupt the contacts are different. The other leads to an evolution to different states that may be not be equilibrium. Hammer et al. related a key quantity in mechanical models, the adhesion energy density, to the work done by external forces and energy stored in and dissipated by the deformable cell in observance of the conservation of the energy (Hammer and Lauffenburger 1987; Dembo, Torney et al. 1988; Hammer and Lauffenburger 1989). The adhesion energy density, on the other hand, has to be related to the interactions of receptors, their binding molecules and mediated material, which are assumed to be driven 16
by different chemical potentials of the receptors, ligands and bond. In addition, the strain energy term is included in the chemical potential of the bonds to counteract the effect of applied external forces. The adhesive interaction is treated as a reactive rate process of receptor ligand bond formation in kinetic models, incorporation of which into thermodynamic equilibrium enables the solution of bond density. Zhu studied the biophysics of cell adhesion interactions at the level of individual molecular pairs (Zhu 2000; Zhu, Bao et al. 2000). The model uses a master-equation for the kinetics of a receptor-ligand system along with Monte Carlo simulations and constitutive equations coupling chemical reactions and mechanics describe several interesting biophysical aspects of cell matrix interactions such as competitive binding between soluble and surface-bound ligands, adhesion lifetime under constant force, detachment by ramp force, binding rate and rolling velocity, long-term evolution of adhesion among other quantities. Another series of studies in the continuum category were carried out by Mogilner et al. (Mogilner and Oster 1996; Mogilner and Oster 2003; Mogilner and Oster 2003; Mogilner and Oster 2003) developed a “tethered ratchet” model based on previous “Brownian ratchet” and “elastic ratchet” models to calculate force generation by actin polymerization that is applicable under non-isothermally situations. The motion of many intracellular pathogens is driven by the polymerization of actin filaments. The propulsive force developed by the polymerization process is thought to arise from the thermal motions of the polymerizing filament tips. Shen and Wolynes (Shen and Wolynes 2005) developed a theoretical model of motorized particles to study non-equilibrium features of cytoskeletal networks, e.g., the effects of non-equilibrium energy pumping as well as 17
equilibrium adhesion to the stability of systems consisted of spherical particles and found variational solutions to many-body master equation for motorized particles for the thermally induced Brownian motion. In the following chapters we turn our attention to intermediate-scale equilibrium statistical thermodynamic models using mean field theories (Yang and Zaman 2007; Yang and Zaman 2008; Yang and Zaman 2008; Yang and Zaman 2009; Yang and Zaman 2009). These studies are able to incorporate some of the molecular conformations and scale these interactions to the bulk level and are useful in quantifying adhesion, cellmatrix interactions and receptor clustering.
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Chapter 2 Model I: The Interaction of Receptors with Tethered Ligands 2.1 INTRODUCTION In this chapter, we present a computational model to study the thermodynamics of cell interaction with ligand coated surfaces. These surfaces are often observed in experimental cell adhesion assays. We take into account the chemical and statistical properties of both the receptors and the surface ligands. Effects of heterogeneous polymer/protein ligand species (i.e. multi-species ligands) are also studied for the first time. Finally, as a number of extra-cellular matrix proteins show a strong preference for constrained conformations, we study the effect of conformational confinement on overall
Figure 2.1 Schematic of the System Receptors (type A molecules) on the cell membrane (top plane) are 220 amino acids long. The substrate (bottom surface) is covered with either a 20 amino acid peptide (type B molecule), a 20 monomer long polymer molecule (type C molecule), or a combination of both. 106 unique samples of receptors, peptide, and the polymer were generated to study the cell-matrix adhesion. 19
cell matrix interactions. Our work compliments both coarse-grain mechano-chemical adhesive dynamics and studies on polymer gels. This provides a framework to bridge these rich areas in biological adhesion (Huang, Szleifer et al. 2001; Huang, Szleifer et al. 2002). Free energy is a key quantity in the investigation of thermodynamic systems (Chipot and Pohorille 2007). In this simple model, we study the free energy of a system containing receptor, tethered ligand, and solvent. This mimics receptor-mediated cellmatrix interactions. We investigated the regulation of free energy by a set of significant parameters. These included the surface coverage (area density) of receptor, interaction distance between membrane and substrate, the ratio(s) of ligand(s) to receptor, adhesive energy between receptor and ligand, and the confinement of ligand conformations. As illustrated in Figure 2.1, the system of interest is between a small portion on a cell's membrane where adhesive molecules are tethered and a ligand coated substrate. Here we define “substrate” as either a surface against which the cell is crawling or the membrane of another cell in the cell-cell adhesion. Our system consists of up to three types of chain molecules (or polymers) with receptor (type A) tethered from the cell surface (the top plane), synthetic polymer ligand (type B) and matrix ligand protein (type C) attached to the substrate (the bottom one). Besides chain molecules, the rest of the space between the two planes is filled with solvent, which is common both in real biological and biomimetic systems. The area of either boundary is Ar. The number of the chain molecules A, B, or C, is NA, NB or NC, respectively. Surface coverage (i.e. area density) for A, is defined as σ:= NA / Ar. Whereas the ratio of the number of B or C to that 20
of A is defined as rB:= NB / NA, rC:= NC / NA. The two surface boundaries, top and bottom, are parallel to each other, with the bottom one at z=0 and the top one at z=D. A mean field approach is used for the x-y planes and as such any quantity that has spatial distribution varies only in the z direction. We give an analysis of all possible interactions at the cell-substrate interface. In the receptor mediated interaction systems, interactions are classified according to the specificity. The receptor ligand binding is a specific interaction, which is the dominant factor in cellular adhesion. Remaining interactions are nonspecific interactions. These include Coulombic (electrostatic) interactions, which are between charges and permanent dipoles, van der Waals interactions, generated due to either induction between a permanent dipole and polarizable but non-permanent dipole or London dispersion forces between two non-permanent dipoles due to the temporal fluctuation of dipole moments (Maitland, Rigby et al. 1981), the osmotic pressure of solvent originating from the spatial heterogeneity of solvent molecules, and the steric effects which are due to the repulsive atomic electric orbits. These interactions can be broadly classified into attractive and repulsive categories. Specific receptor-ligand binding and van der Waals belong to the former while Coulombic, steric, and osmotic belong to the latter. We note that electrostatic forces are weakened by increasing the concentration of salt in the intervening medium (Koller and Papoutsakis 1995). In biological systems, electric charges arise from mobile ions, principally Na+, K+, Cl-, Ca2+, Mg2+, H+, and OH, and from fixed chemical moieties that are covalently bound to macromolecules, such as PO43-, COO-, and NH3+ that undergo acidic or basic dissociation or association in 21
aqueous solution. The free motion and adoption of these charges play an essential role in the reduction of electrostatic interactions. Coulombic combined with van der Waals interactions give rise to the Derjaguin, Landau, Verwey and Overbeek (DLVO) theory (Derjaguin and Landau 1941; Verwey and Overbeek 1948). According to this theory, the Coulumbic interaction between two parallel planes decays exponentially with separation. Assuming the distance between cell and substrate (tens of nanometers) is not small compared to Debey-Huckel screening length (typically from sub-nanometer to several nanometers) (Derjaguin and Landau 1941; Verwey and Overbeek 1948), the effect from electrostatic interactions can be considered of minimal significance in our model. Theoretical calculations have shown that the electrostatic force is quite small in comparison to the other physical forces (Parsegian 1973; Bongrand and Bell 1984) and is therefore not the major force involved in the attachment of cells to substrates. This conclusion has been demonstrated experimentally as well. Experiments have shown that under most conditions, intact cells (Sherbet, Lakshimi et al. 1972) and substrates (i.e., tissue culture plastic, glass) have an overall net negative charge. The magnitude of this repulsive electrostatic force is not very large since it is known that cells do in fact adhere to substrates with charges of the same type (Barngrover 1986). On the other hand, negatively charged BHK cells were unable to attach to positively charged polylysine under serum-free conditions in vitro (Aplin and Hughes 1981). In summary, in cellular adhesion, Coulombic interactions are ignorable. Therefore, in our models it is neglected. All the remaining interactions, including specific interactions, van der Waals interactions, osmotic pressure and steric effect are considered. 22
The following assumptions are made to simplify the model. First both surfaces are assumed to be impenetrable. In our model, ions are not explicitly present. They can play a role in the screening of Coulombic interactions, as we adopt their effect by ignoring the electrostatic energy. Except that, in a dilute biological system, we assume ions exhibit similar behavior as solvent. Also there are no channels for water molecules, hence the membrane, a lipid bilayer, is considered as impenetrable for solvent. While the receptor is a trans-membrane molecule consisting of both intra- and inter-cellular domains, our focus is only the extra-cellular part. Therefore, as a first step approximation the receptor molecules are modeled as molecules tethered from the outer side of the cell's membrane. The second assumption is that the adhesion surfaces are assumed to be planar (Figure 2.1). Consequently the mechanical energy associated with membrane is ignored. From both experimental and theoretical studies, the cell's membrane has curvature on the order of a micron leading to mechanical energy component of the free energy associated with the two principal curvatures (Helfrich 1973). However, since the region of our interest in our study is on the macro-molecular scale rather than the micrometer and our purpose is to decouple the pure “adhesion” free energy from that associated with the deformation of the membrane, we assume a planar geometry. The third assumption is that we treat adhesion to be at thermodynamic equilibrium to satisfy a quasi-steady state. In most of the kinetic models, e.g. Bell’s, equilibrium is also implicitly assumed (Bell 1978; Bell, Dembo et al. 1984).
23
2.2 MODEL DEVELOPMENT
A) Conformations and Probability Density Function (pdf) Proteins can have a huge number of conformations. The key cell-matrix adhesion molecules, integrins are complex structured proteins with long extracellular domains from 700 to 1100 amino acids (Humphries 2000). X-ray crystal structure has been reported for just a few of the twenty five known integrins (Xiong, Stehle et al. 2001). Even the primary structures of most integrins are not completely known. In cellular adhesion, sometimes more than one species of receptors participate, which makes the situation more difficult to handle. In solution, the 3D structure of protein can be even more complex because of different solvent conditions and solute concentrations. Although the binding sites of integrins are thought to be located in the N-terminus, i.e. the distal few nanometers (Humphries 2000), the precise locations still wait to be clarified. In spite of the lack of experimental data and complexity of molecular structure and the environment, we still hope to include some conformational imprints of the adhesion molecules in our models. We aim to answer the following questions: 1) what conformations will appear and what will not; 2) what is the ratio of a certain conformation to be present in our system. In order to clarify this confusion, we resort to the statistical method and calculus of variation. In our models, the conformations are oversampled, which means some conformations generated from our simulation may not be found in realistic receptors. This is due to the limited restrictions enforced to the “receptor” being simulated. For the 24
condition of the system (i.e. here it is the equilibrium), the Helmholtz free energy is at its minimum. The probability of the appearance of each conformation is determined by the minimization of free energy. By executing the variation of free energy with respect to the variables (i.e. the probability distribution function), the ratio of each conformation, or the probability distribution function (pdf) of a species of polymer can be derived as a straightforward result. By doing this, we hope that the probabilities for those conformations that are less akin to receptor will be minimized. This will be further elucidated later in the thesis. Now we focus on generating the superset of all possible conformations of a certain polymer species, for example, receptor protein. This has been done with the assist of computer simulation. In principle, the more knowledge of the molecular structure and the more characteristics being addressed in simulation, the more reliable will be our superset. As a first step, we incorporate the protein torsion angles. In our simulation, the difference among conformations relies on the torsion angles between neighboring monomers. The last question is how many conformations we should simulate for each species. As we know, the torsion angles vary continuously, therefore in principle there could be infinite numbers of unique conformations potentially in the simulation. The set of any finite number of conformations is only a “representative set”. Using our simulations, we found that the behavior of the equilibrium system (say, the spatial distribution of each species, the free energy calculated, etc.) saturates with increasing the number of conformations. Therefore, we selected a big number for the count of conformations, which ranges from 1 to 10 million, depending on system and species.
25
Figure 2.2 Generic Amino Acid Backbone In the simulation of receptors, the two dihedral angles are chosen randomly from the allowed regions of Ramachandran plot (the phi/psi map) while the side chain is described using a non-interacting hard sphere.
Figure 2.3 Ramachandran Plot The Ramachandran plot shows the sterically allowed region, the blue-highlighted for the dihedral angles ψ-φ of the primary structure of polypeptide. The second quadrant is the famous β sheet and the third quadrant is α basin. 26
The following are the details about simulation of polymers. To simulate proteins, we use “generic” amino acids with a backbone containing amino group, Cα and carboxyl group (Figure 2.2) with conformations chosen randomly from the sterically allowed regions of the ψ-φ map, Ramachandran plot, (Figure 2.3). The torsion angle ω in Figure 2.2 has a unique value 180°. The regions in the ψ-φ map used to choose the dihedral angles are alpha basin-third quadrant, beta sheet-second quadrant, and “other” basin. The “other” basin accounts for PP-II type conformations (Zaman, Shen et al. 2003). The side chains are simulated using non-interacting, impenetrable hard-spheres. The synthetic polymer ligand conformations are simulated as polyethylene glycol (PEG).
Three
possible torsion angles, namely Trans (180°), Gauche+ (60°) and Gauche- (-60°) (Huang, Szleifer et al. 2001) were used. The adhesion receptor is modeled to capture the essential feature of ligand binding I domain of integrins (~200-300 amino acids). The adhesion receptor is simulated in two steps. In the first step we simulate the first 200 amino acids, and in the second step we simulate the distal portion, last 20 amino acids, which are most likely to interact with the molecules on the surface of the target substrate. For simplicity, the conformations of the first 200 amino acids is considered to be an identical-length “base” for all the receptor molecules whereas the last 20 amino acids have distinct conformations, which were sampled with 106 unique conformations. We used the above simplification only for our initial study and in later studies the full length of the extracellular domain of receptor is explicitly simulated. The surface coating protein as well as the polymer was also sampled with 106 unique conformations each chosen by randomly sampling from sterically allowed regions of their corresponding conformational spaces. 27
Further increasing the number of sampled conformations had negligible changes to the end results. In the sampling of any conformation, the “self-exclusiveness”, i.e. nonoverlapping of different atomic groups is examined. For conformations belong to the same species, the uniqueness of conformation is also examined. From our coarse-grained simulation, the most important information being generated is the spatial distribution of monomers/segments of each conformation. That is
n Aα (z ) , which is the linear density of segments of the α conformation of the receptor in (z, z+dz) layer and is derived by straightforward counting process after the conformations are generated. This quantity will also be mentioned in the next two sections (B and C).
B) Constraint Equation In this section, we focus on the steric effect and constraint equation derived from the former. To account for the steric repulsive effect, we introduce the packing constraint (or so called volume-filling constraint, or incompressibility condition), based on previous studies on gel-polymer adhesion (Szleifer and Carignano 1996). This packing constraint prohibits any different segments to occupy the same volume in space, which is rooted from the steric repulsive interaction. On the other hand, no macroscopic vacuum is allowed. The spatially dependent constraint, in the layer (z, z+dz) (the "z layer" here refers to differential volume bounded by the planes z and
z+dz), in equation, is
formulated as follows, which states that the volume fractions of all the species including polymers and solvent are summed up to be one:
ϕ A (z ) + ϕ B ( z ) + ϕ C ( z ) + ϕ S ( z ) = 1 28
(2.1)
In which, φ(z) is the volume fraction of corresponding molecule in the layer z, which is of course spatially dependent. Subscripts A, B, C and S denote molecules of receptors (A), polymer ligand on substrate (B), protein ligand on the substrate surface (C), and solvent (S), respectively.
denotes ensemble averages in the corresponding
conformational spaces. The volume fraction can be defined as:
ϕ A (z ) =
N A n A ( z ) dzv = σ ∑ P Aα n Aα ( z )v Ar dz {α }
(2.2)
where n A ( z ) dz is the number of segments/monomers of receptor (molecule A) in the layer (z, z+dz) averaged by the ensemble of conformations of molecule A. As mentioned previously, NA is the number of receptors, Ar is the area of the membrane with receptors, v is the volume of one segment, therefore, in the middle formula, the denominator is the volume of the (z, z+dz), the numerator is the conformation-averaged number of segments multiplied by the volume of one segment, i.e. the total volume in the z layer occupied by receptor, hence the quotient is the volume fraction of receptor in the z layer. In the last formula of Eq. (2.2), as mentioned before σ:= NA / Ar, is the surface coverage or area density of receptor, n Aα (z ) is the linear density (i.e., n Aα ( z ) dz is the number) of segments of the α conformation of receptor in (z, z+dz) layer. P Aα , which is an unknown quantity to be solved, is the probability distribution function (pdf, which is mentioned in the section “Conformations and Probability Density Function” above and is determined by free energy minimization, which will be introduced in section 2.2 D) of receptor (molecule A). The meaning of P Aα is the probability of finding α conformation of 29
receptor (species A). {α} is the ensemble of conformations of A. Therefore, due to this probability interpretation, we have an instant constraint for P Aα , i.e. ∑ P Aα = 1 . P Bβ and {α }
P Cγ have similar interpretations and equations for their corresponsive molecules, i.e.
synthetic polymer (PEG) and protein ligands. Assuming that the volume of one segment of molecule A, B, C and solvent to be identical for simplicity (this is not a critical assumption which is removable) the constraint equation can be written as:
σ ∑ P Aα n Aα ( z )v + r B σ ∑ P Bβ n Bβ (z )v + r C σ ∑ PCγ nCγ ( z )v + ϕ S ( z ) = 1 {α }
{β }
(2.3)
{γ }
in which, rB and rC are the number ratios of the two types of ligands to receptor. The nature of receptor-ligand binding is hydrogen bonding. The strength of a hydrogen bond varies from a few to around 15 kBT (Israelachvili 1992). Due to the fact that multiple bonds can form, the receptor-ligand binding strength can be higher than that of a hydrogen bond, ranging from the typical value of a hydrogen bond to as big as 35
kBT (Helm, Knoll et al. 1991). In this first model, in order to address receptor-ligand interactions, we only consider pair-wise interactions between different species of molecules, i.e. such as receptor-protein ligand, receptor-polymer ligand and between protein ligand and polymer ligand to account for the situation where different ligand species cross interact with each other. Interactions among molecules that belong to the same species are ignored due to their orders of magnitude difference compared to receptor-ligand binding energy. In polymer mean field calculations, the adhesive energy between polymers can be represented by a van der Waals type interaction (Carignano and Szleifer 1994; Huang, Szleifer et al. 2001). The parameterization of van der Waals type is 30
empirical and the magnitude of the potential chosen in this work is much bigger (from 15 to 30 kBT) than that originated from the electrostatic interactions between permanent and polarizable dipoles or London dispersion (Israelachvili 1992). Therefore, the binding energy between receptor and ligand is considered in the van der Waals interaction for simplicity. However, as we know receptor-ligand binding is a specific interaction. This specificity issue will be improved in the next model presented in chapter 3, where we separate the adhesive energy into two parts: the explicit receptor-ligand bonding and a much weaker van der Waals interactions between any molecules to account for implicit interactions. More details about intermolecular van der Waals interactions will be presented in the “components of free energy” section (section C).
C) Components of Free Energy Our system of interest satisfies the conditions of an enclosed system, in which the temperature, the volume and the particle number of each of the species are unchanged. Therefore, according to statistical thermodynamics, the Helmholtz free energy, defined as
A := TS − E , is minimized at equilibrium, in which T is the temperature of the system and E is the internal energy. Based on the previous studies on the statistical thermodynamic theory of grafted polymeric layers (Carignano and Szleifer 1993; Szleifer and Carignano 1996), the free energy A of the system of interest can be divided into the entropic and energetic contributions listed in the following: 1.
The component contributed by the conformational entropic of chains of
each of the three molecules, e.g. 31
T S A, conf = − k B T N A ∑ P Aα log P Aα
(2.4)
{α }
in which SA,conf is the conformational entropy from receptor (species A). 2.
That contributed by the translational entropy of the solvent: T SS = −
k B T Ar dz ϕ S (z ) log ϕ S (z ) v ∫
(2.5)
in which SS is the translational (mixing) entropy of solvent. 3.
That contributed by translational entropy of polymers, e.g.
TS A, trans = − N A T log(sσ ) .
(2.6)
SA,conf is the translational entropy of receptor. s is a constant, the area covered by the tethering sites of a receptor molecule on the membrane. 4.
The intermolecular attractions between chain segments of disparate
species, e.g.
E AB = ∫∫ χ 'AB ( z − z ' ) ∑ N A P Aα n Aα ( z )dz ∑ N B P Bβ n Bβ ( z ')dz ' (2.7) {β } {α } The terms in the two brackets are the counts of segments, e.g. in the first bracket is the conformation-averaged number of segments of receptor in the z layer. χ 'AB ( z − z ' ) is the averaged interaction energy between a segment of receptor (molecule A) in the layer (z,
z+dz) and a segment (or monomer) of synthetic polymer ligand (B) in (z’, z’+dz’), assuming uniform probability of appearance of both molecules in their individual allowed regions. Therefore the bar on top of χ' refers to a mean-field average. Because 1) in the spatially discretized representation (which will be discussed in section E) χ 'AB ( z − z ' ) is 32
a matrix with subscripts denoted by z and z', 2) it describes interactions between a pair of molecules each in a layer, we call it “two-layer interaction strength matrix”. Under the philosophy stated above, χ 'AB ( z − z ' ) can be calculated as follows:
________________
χ 'AB ( z − z ' ) =
ρ ρ dxdy dx' dy 'V (r ; r ') ∫∫ ∫∫ ) ( )
( x , y ∈ plane
AB
z
x ', y ' ∈ plane z '
∫∫ dxdy
( x , y )∈ plane z
(2.8)
∫∫ dx' dy'
( x ', y ' )∈ plane z '
in which, the potential VAB was adopted a Lennard-Jones type, e.g., 6 12 r 0, AB r 0, AB ρ ρ 0 + V AB (r ; r ') = V AB (r ) = V AB − 2 r r
( r = rρ − rρ' )
(2.9)
One thing that needs mentioning here about the two-layer interaction strength is that so long as the formulas and parameters of the van der Waals potentials between the pair of molecules in concern are given, these χ's can be calculated analytically and thought of as constant matrices as inputs into the next step (solving of Eqs. (2.14)-(2.17) and Eq. 2.3) calculation. Hence the free energy is as follows, which is a functional of the space-dependent solvent volume fraction and the pdfs:
33
A[ϕ S ( z ), P Aα , P Bβ , PCγ ] = k B T N A ∑ P Aα log P Aα + r B N A ∑ P Bβ log P Bβ + r C N A ∑ PCγ log PCγ {α } {β } {γ } + k B T Ar v−1 ∫ dz ϕ S ( z ) log ϕ S ( z )
+ k B T [N A log(s A σ ) + r B N A log(r B s B σ ) + r C N A log(r C sC σ )]
(2.10)
+ r B N 2A ∑∑ ∫∫ dzdz ' χ 'AB ( z − z ' ) P Aα P Bβ n Aα ( z ) n Bβ ( z ' ) {α } {β }
+ r B r C N 2A ∑∑ ∫∫ dzdz ' χ BC ( z − z ' ) P Bβ PCγ n Bβ ( z ) nCγ ( z ' ) {β } {γ }
+ rC N
∑∑ ∫∫ dzdz ' χ ( z − z' ) P γ P α n γ (z ) n α ( z' ) {γ } {α }
2 A
C
CA
A
C
A
As a natural result, we define an intensive quantity of the free energy of the system, which is the area density of the dimensionless free energy: a[ϕ S ( z ), P Aα , P Bβ , PCγ ] :=
A k B T Ar
= σ ∑ P Aα log P Aα + r B σ ∑ P Bβ log P Bβ + r C σ ∑ PCγ log PCγ {α }
{β }
{γ }
+ v −1 ∫ dz ϕ S ( z ) log ϕ S ( z )
+ σ log(s A σ ) + r B σ log(r B s B σ ) + r C σ log(r C sC σ )
(2.11)
+ r B σ 2 ∑∑ ∫∫ dzdz ' χ AB ( z − z ' ) P Aα P Bβ n Aα ( z ) n Bβ ( z ' ) {α } {β }
+ r B r C σ 2 ∑∑ ∫∫ dzdz ' χ BC ( z − z ' ) P Bβ PCγ n Bβ ( z ) nCγ ( z ' ) {β } {γ }
+ r C σ 2 ∑∑ ∫∫ dzdz ' χ CA ( z − z ' ) PCγ P Aα nCγ (z ) n Aα ( z ' ) {γ } {α }
where χ ≡ χ ' Ar / k B T for each χ' for convenience. Next we rewrite the constraint equation and introduce the Lagrangian multiplier function π(z):
34
g[ϕ S ( z ), P Aα , P Bβ , PCγ ] = ∫ dzπ ( z )vσ ∑ P Aα n Aα ( z ) + r B ∑ P Bβ n Bβ ( z ) + r C ∑ PCγ nCγ (z ) + ϕ S ( z ) − 1 = 0 {α } {β } {γ } (2.12) The Lagrangian multiplier function π(z) has relation to the solvent osmotic pressure, which is defined as –(∂a/∂Ar){NR, NL, NS, T} (Szleifer and Carignano 1996).
D) Minimization of Free Energy
We execute the functional derivatives (or equivalent variations) of quantities a and g in equations (2.11) and (2.12) with respect to each of the variables xi
(xi ∈ {ϕ S (z ), P Aα , P Bβ , PCγ }) , applying the following relation: δa δg + =0 δ xi δ xi
(2.13)
This gives us a set of coupled nonlinear equations for ϕ S ( z ) , P Aα , P Bβ , P Cγ and
π (z ) . Together with the original constraint equation (2.3), they form a closed set of simultaneous equations:
ϕ S ( z ) = e − vπ ( z )
(2.14)
P Aα =
AC AB exp− r B ∑ M αβ P Bβ − r C ∑ M αγ PCγ {β } {γ } g A E Aα
(2.15)
P Bβ =
BA BC exp− ∑ M βα PCγ P Aα − r C ∑ M βγ {γ } g B E Bβ {α }
(2.16)
PCγ =
CB CA exp− ∑ M γα − r B ∑ M γβ P Bβ P Aα {β } g C E Cγ {α }
(2.17)
1
1
1
35
σ ∑ P Aα n Aα (z )v + r B σ ∑ P Bβ n Bβ ( z )v + r C σ ∑ PCγ nCγ ( z )v + ϕ S ( z ) = 1 {α }
{β }
(2.3)
{γ }
where gX (X=A, B or C) is a normalization factor, for example, gA makes PAα satisfy P α = 1 , so, ∑ {α } A
gA = ∑ {α }
AC AB exp− r B ∑ M αβ P Bβ − r C ∑ M αγ PCγ . {β } {γ } E Aα 1
The E terms in each of the denominators of Eq. (2.15)-(2.17), which is related to the spatial distribution of matter of conformation, is defined: E Xξ ≡ exp[v ∫ dzπ ( z ) n Xξ ( z )]
( X ∈ {A, B, C}, ξ ∈ {α , β , γ })
(2.18)
The matrices M's, the “interaction energy matrices”, containing the inter-species interactions (between AB, BC and CA) such that the value of M goes to zero when there is no interaction between the species, are defined as: XY M ξτ ≡ σ ∫∫ dzdz' n Xξ ( z )χ XY ( z − z ' ) nYτ ( z ')
( X , Y ∈ {A, B, C}, ξ ,τ ∈ {α , β , γ }) (2.19)
Formally, the three equations, i.e. Eq. (2.15)-(2.17) can be expressed by one universal equation: Pi,α i =
E i,α i 2 exp − r i + j ∑ M αi ,ii,+αji+ j P i ∈ {A, B, C } ∏ i + j ,α i + j {α i+ j } g i j =1
(2.20)
In Eq. (2.20), {αi} denotes the space of conformations belonging to polymer species i (i=A, B or C). Besides rB and rC, we need to introduce an rA, which is the ratio of the number of species A to itself, of course ought to be 1. Also we invent a cyclic convention, e.g., A+1=B, B+1=C and C+1=A and A+2=(A+1)+1. Again gi (i=A, B or C) is the normalization factor to assure the probability interpretation, e.g.
P α = 1 , thus ∑ { } i,
αi
36
i
2 g i = ∑ E i ,α i ∏ exp − r i + j ∑ M αi ,ii,+αji+ j P i + j ,α i+ j {α i+ j } {α i } j =1
(2.21)
The contribution due to the spatially distribution of matter is defined as: E i ,α i := exp[− v ∫ dzπ ( z ) ni ,α i ( z )]
i ∈ {A, B, C}
(2.22)
The interaction matrices, M's, have the following structure: i, j M α i ,α j := σ ∫∫ dzdz' ni ,α i (z )χ ij ( z − z ' ) n j ,α j ( z ') i, j ∈ {A, B, C}
(2.23)
In the most explicit form, equations (2.15)-(2.17), (or Eq. 2.20) are expressed as:
P A,α
− v ∫ dzπ ( z ) n A,α ( z ) __________ ______ 1 exp− r B σ ∑ ∫∫ dzdz ' χ AB ( z − z ' ) n A,α ( z ) n B , β ( z ') P B , β = {β } gA __________ ______ − σ dzdz ' χ CA ( z − z ' ) nC ,γ ( z ) n A,α ( z ') PC ,γ ∫∫ r C {∑ γ}
(2.24)
P B,β
− v ∫ dzπ (z ) n B ,β ( z ) __________ ______ 1 exp− r B r C σ ∑ ∫∫ dzdz ' χ BC ( z − z ' ) n B , β (z ) nC ,γ ( z ') PC ,γ = {γ } gB __________ ______ − σ dzdz ' χ AB ( z − z ' ) n A,α ( z ) n B , β ( z ') P A,α ∫∫ r B {∑ α}
(2.25)
PC ,γ
− v ∫ dzπ ( z ) nC ,γ ( z ) __________ ______ 1 exp− r C σ ∑ ∫∫ dzdz ' χ CA ( z − z ' ) nC ,γ ( z ) n A,α ( z ') P R ,α = {α } gC __________ ______ − dzdz ' χ BC ( z − z ' ) n B , β ( z ) nC ,γ ( z ') P B , β ∫∫ r B r C σ {∑ β}
(2.26)
37
E) Numerical Solution
Equations (2.14)-(2.17) and (2.3) provide us with a closed set. We solve ϕ S ( z ) , P Aα , P Bβ , PCγ and π ( z ) numerically as functions of a set of key regulating parameters {σ, D, rB, rC}. Known quantities as input include real constants, e.g. v, the volume of solvent, which does not change throughout our calculation; and pseudo-constants, including: 1) spatial distribution of matter {nAα(z), nBβ(z), nCγ(z)} derived from molecular sampling; 2) Two-layer interaction strength matrices χ XY ( z − z ' ) (X,Y=A,B or C) being calculated beforehand. The reason they are called pseudo-constants is that they are firststep preparations into the numerical solution of Eqs. (2.14)-(2.17) and Eq. (2.3), so in the latter step, they are fixed. However, during the preparation step, 1) can be changed by constraint added to torsion angles; 2) can be changed by varying parameters in the Lennard-Jones potentials (see Eq. 2.9). The simultaneous equations (2.14)-(2.17) and (2.3) were solved numerically by firstly discretizing the space along z direction into layers with finite thickness δ. We use index i to refer to a specific layer. Through discretization of space, the constraint equation can be discretized into a sequence of equations, each referring to the incompressibility in a layer accordingly. The constraint equation in layer i is as follows.
σv −vπ ( i ) ) ) ) =1 ∑ P Aα n Aα (i) + r B ∑ P Bβ n Bβ (i) + r C ∑ PCγ n Cγ (i) + e δ {α } {β } {γ }
(2.27)
in which π(i) is the value of π(z) in the ith layer, n) Aα (i ) , n) Bβ (i) and n) Cγ (i) are numbers of segments in the ith layer for molecules A, B and C, respectively. For simplicity if a 38
quantity has index i should be thought of as the value evaluated at iδ, for example,
π (i ) ≡ π (iδ ) . The last term on the left hand side of Eq. (2.27) comes directly from Eq. (2.14). We used the iteration method for the 4 equations below to seek the solution for the unknowns P Aα , P Bβ , PCγ and π ( z ) : n P Aα =
1 AB n −1 AC n −1 exp − v ∑ π n−1 (i) n) Aα (i ) − r B ∑ M αβ P Bβ − r C ∑ M αγ PCγ n {β } {γ } gA i
(2.28)
n P Bβ =
1 BC n −1 BA n −1 exp− v ∑ π n −1 (i ) n) Bβ (i) − r C ∑ M βγ PCγ − ∑ M βα P Aα n {γ } {α } gB i
(2.29)
n P Cγ =
1 CA n −1 CB n −1 exp− v ∑ π n −1 (i) n) Cγ (i) − ∑ M γα P Aα − r B ∑ M γβ P Bβ n {α } {β } gC i
(2.30)
σv v π n (i ) = − log 1 − ∑ P nAα n) Aα (i ) + r B ∑ P nBβ n) Bβ (i ) + r C ∑ PCn γ n) Cγ (i ) (2.31) {β } {γ } δ {α }
The superscripts n and n-1 refer to the orders of iteration. The interaction energy matrices M ξτXY present in Eq. (2.15)-(2.17) are calculated as the following taking that between species A and B as an example: AB M αβ ≡ σ ∫∫ dzdz ' χ AB ( z − z ' ) n Aα ( z ) n Bβ ( z ') ≈ σ ∑ n) Aα (i ) ∑ χ AB (i, j ) n) Bβ ( j ) i
(2.32)
j = i −1,i ,i +1
The van der Waals potential decreases sharply with distance and the van der Waals energy between a layer and its second nearest neighbor layer is about two orders of magnitude lower than that within the same layer, therefore we only consider interactions between the same layer and the nearest neighbor for our calculation. The
39
two-layer interaction strength matrix χ 'AB (i, j ) in the equation above, with a minor difference compared to its continuous formula in Eq. (2.8) is evaluated as:
χ 'AB (i, j ) =
∫∫) dxdydz (
( x , y , z ∈ layer i
ρ ρ ∫∫) dx' dy' dz 'V (r ; r ') AB
x ', y ', z ' ∈layer j
∫∫ dxdydz
( x , y , z )∈ layer i
∫∫ dx' dy' dz '
( x ', y ', z ' )∈layer
(2.33)
j
ρ ρ The Lennard-Jones potential V AB (r ; r ') is the same as that defined in Eq. (2.9).
One thing that needs attention here is that the χ ' calculated from Eq. (2.33) is proportional to A−r 1 , the area of membrane of concern, since χ ≡ χ ' Ar / k B T containing an Ar, χ is a quantity of finite value which is independent of Ar. And in reality, neither Ar nor the total number of receptor molecules NA is an explicit parameter in our calculation, instead, only σ, which is their quotient, is needed. As there are no precise data for the parameters in Lennard-Jones potential for each conformation of receptor and ligand, we assume literature values to give an orderof-magnitude study. And for r0's, which are the average equilibrium distances in Lennard-Jones potential, we assigned a unique value in our simulation, 3.0 Å, which is close to the equilibrium locations for the van der Waals interactions between many molecules or atomic groups (Parsegian 2005). And V 0 is the approximated interaction energy in the range between 10-20 kcal/mol (~15-30 kBT, which is the typical range for receptor-ligand bonds). The volume of one segment of receptor is taken to be 90 Å3, an average of the van der Waals volume of amino acids (Creighton 1992). The thickness of a layer δ is chosen to be 6.0 Å. The coarse-grained molecular simulations were carried out 40
using an in-house program written in C programming language and the data was analyzed using Matlab.
2.3 RESULTS
After the equations (2.14)-(2.17) and (2.3) have been solved, they are substituted back into Eq. (2.11) to quantify the area density of the dimensionless free energy, i.e. in unit of kBT, the free energy per unit area. Receptor-mediated adhesion to ligand coated surfaces was analyzed for three cases, namely surfaces coated with synthetic polymer (PEG) chains, surfaces coated with protein ligands and surfaces coated with both polymers and proteins above. The focus of this work is to develop a theoretical framework to understand the interactions and the free energy costs associated with cellmatrix interactions and while we only considered three sample cases, our method allows us to capture the free energy of systems involving multiple proteins and polymers. The major effort of this work is to study the response of the free energy to the variation of a set of key parameters, i.e. the surface coverage (area density) of receptor, interaction distance between membrane and substrate, the ratio(s) of ligand(s) to receptor, equilibrium energy in the Lennard-Jones potential, confinement ligand conformations, addition of ligand species, etc.
A) Interaction with Polymer Coated Surface
We first consider a simple case of receptor-polymer adhesion in which protein ligand (molecule C) is absent. The free energy term “a” defined in Eq. (2.11) is plotted versus polymer surface coverage in Figure 2.4a. The free energy “a” given in all of our 41
a)
35 30 25
b)
d = 20 d = 22 d = 24 d = 26 d = 28
20
σB= 0.5
15
σB= 0.2
10
a
20
5
a
15
0
10
-5
5
-10
0
-15
-5 0.0
-20 0.0 0.2 0.4 0.6 0.8 1.0 Receptor Surface Coverage ( σA)
0.1 0.2 0.3 0.4 0.5 Polymer Surface Coverage (σB)
Surface Coverage (σB)
c)
0.1
-2.000
0.2
15.10
0.3
32.20
0.4 0.5 20
22
24
26
28
Distance
Figure 2.4 Free Energy of Receptor-Polymer System
Free energy a (in unit of av/δ) plotted as a function of surface coverage for a number of separation distances, D (in unit of layers with each layer equal to 6.0 Å). σB is defined as σB = rBσ, the area density of B, i.e. PEG ligand. The free energy increases with increase in surface coverage. b) Free energy a plotted as a function of variations in receptor density. At low surface polymer coverage, free energy is lower at higher receptor expression and at high surface coverage the free energy is lower at low receptor expression. The results shown are for D = 20. c) A free energy contour plot showing the regions of highest and lowest free energy as a function of surface coverage and separation distance. figures is a dimensionless quantity: av/δ. As the surface coverage is increased, the free energy also increases that is consistent with previous studies of polymer-gel interactions 42
(Huang, Szleifer et al. 2001) for the range of density. As a result of van der Waals interactions, increase in surface coverage results in a marginal increase in overall interaction, but the entropic penalty due to increase in ligand concentration dominates the free energy term and results in increase in free energy. We also note distance dependence in free energy, where increase in free energy is associated with increase in the separation between the cell and the ligand coated surface. The big “fork” observed between the short distance 20 (or 22) and the others is probably due to the typical lengths of receptors and ligands being simulated. Less than d=22, these adhesion molecules have good contact and therefore have sufficient interaction. However, beyond the well-contacted separation, the interaction is weak. The enhancement of density further increases this difference, leading to the “fork” as we observe. To capture the features of numerous experiments, where the integrin expression is modulated by either genetic manipulation or integrin blocking antibodies, we study the role of receptor density on overall free energy a. Figure 2.4b captures the scenario when the integrin density is regulated while the polymer coverage is kept constant. The most interesting feature is the compensation between the favorable internal energy and the unfavorable entropic term. At low integrin expression level, the interaction energy term dominates and the overall free energy is lower for the higher polymer coverage, on the other hand, at higher values of integrin activation, the entropic factor dominates and the system with a lower polymer surface coverage shows lower free energy. This balance provides an interesting perspective on an optimal surface design strategy for maximal adhesion.
43
The free energy contour plot shown in Figure 2.4c captures the subtle aspects of variation of the free energy surfaces as a function of either the separation or the surface coverage. The simulations show that minimum free energy occurs at low separation distances and low surface coverage, while high surface coverage and greater distances lead to higher free energy values.
B) Interaction with Protein Coated Surface
Cell adhesion is often studied on surfaces coated with cell adhesion proteins, such as collagen, laminin, fibronectin etc (Benoit and Gaub 2002; Zhu 2002; Chang 2006). By coating the substrate with 20 amino acid long peptides, we have applied our method to study such systems, in which molecule protein ligand is present with synthetic polymer being blocked. Figure 2.5a shows that for a system comprising of adhesion receptor interacting with a surface coated with proteins, free energy costs are higher for systems with greater distances. This plot is in qualitative agreement with that of the receptorpolymer system shown in Figure 2.4. To probe the role of protein conformations on cell adhesion, we simulated the 20 amino acid long surface peptide to occupy only a single basin (in this case, β basin) in the phi/psi conformational space (see Figure 2.5). The motivation to study this interaction was to probe the role of protein conformations on overall free energy. In addition, cell adhesion proteins, such as collagen are conformationally rigid and have a preference for left-handed alpha helix or PP-II basin (Bella, Eaton et al. 1994; Kramer, Bella et al. 1999; Persikov, Ramshaw et al. 2000). By forcing our peptide to occupy only one basin, we are 44
able to study both qualitative and quantitative changes in free energy as a function of conformational sampling. The free energy of a conformationally restricted surface peptide is shown in Figures 2.5c and 2.5d. We note that there are both qualitative and quantitative differences between the free energy of free peptide and that of
a
20 15 10 5 0 -5 0.0
c) 30 25
a
20
0.1
D = 20 D = 22 D = 24 D = 26 D = 28
10 5 0 0.2 0.4 0.6 0.8 Protein Surface Coverage ( σC)
-0.7000 2.550 5.800
0.2
9.050 12.30 15.55
0.3
18.80 22.05 25.30
0.4 0.5 0.6 0.7 20
0.2 0.4 0.6 0.8 Protein Surface Coverage ( σC)
15
-5 0.0
b) Surface Coverage ( σ C)
25
D = 20 D = 22 D = 24 D = 26 D = 28
22
24
26
28
Distance
d)
0.1
Surface Coverage (σ C)
a)30
-0.7000
0.2 9.850
0.3 20.40
0.4 0.5 0.6 0.7 20
22
24
26
28
Distance
Figure 2.5 Free Energy of Receptor-Peptide System
a) Free energy a plotted as a function of surface coverage for a number of separation distances, D. σC is defined as σC = rCσ, the area density of C, i.e. peptide ligand. The free energy increases with increase in surface coverage. b) A free energy contour plot showing the regions of highest and lowest free energy as a function of surface coverage and separation distance. c) Free energy of receptor-peptide system with the peptide restricted to occupy only a single (β) basin in the phi/psi space. The figure as well as the free energy contour plot (d) shows qualitative and quantitative differences in the free energy upon conformational confinement. 45
conformationally restricted one. The major difference is that the restricted free energy is typically smaller. Because we randomly selected torsion angles from these basins, this indicates the β basin has lower free energy than other basins. Similar scenarios can be expected for situations where amino acids are forced to occupy other specific basins.
C) Interaction with Mixed Surface
The next system of our study was a more complex mixed system where the substrate was coated with both the polymer and the peptide. The length of the surface polymer was 20 monomers and that of the protein was 20 amino acids, too. Mixed systems, such as these are becoming common in biotech applications where PEG gels are fortified with amino acids to study adhesion, proteolysis and structural changes upon cell binding and migration (Lutolf, Lauer-Fields et al. 2003; Raeber, Lutolf et al. 2005). Figure 2.6a shows the free energy for a system where the surface was covered with equal amounts of PEG and protein molecules. In these systems, not only the interaction between receptor and surface ligands was considered but we also accounted for the interaction between the surface protein and the surface polymers. This multi-species system shows an interesting bimodal behavior in free energy. For short distances, we noticed a decrease in free energy upon increase in surface coverage. This is due to receptor-ligand and ligand-ligand interaction that affects the internal energy part of the free energy. On the other hand, at greater distances we noticed an increase in free energy upon increase in surface coverage, primarily due to weaker receptor-ligand interactions. Thus for mixed species interactions with adhesion receptor we note that both entropic and 46
40 30
a
20
D = 20 D = 22 D = 24 D = 26 D = 28
b) Surface Coverage (σ B+ σ c)
a)
10 0 -10
c)
40 30
a
20
0.1 0.2 0.3 0.4 0.5 Surface Coverage ( σB+ σc)
D = 20 D = 22 D = 24 D = 26 D = 28
10 0 -10 -20 -30 0.0
e)
40
a
30
0.1 0.2 0.3 0.4 0.5 Surface Coverage ( σB+ σc)
0.2
7.000
0.3
30.20
0.4 0.5 20
22
24
26
28
d) 0.1
-19.70
0.2
4.950
0.3
29.60
0.4 0.5 20
22
24
26
28
Distance
Figure 2.6 Receptor-Mixed Ligands System D = 20 D = 22 D = 24 D = 26 D = 28
20 10 0 -10 0.0
-16.20
Distance
Surface Coverage (σ B+ σ c)
-20 0.0
0.1
0.1 0.2 0.3 0.4 0.5 Surface Coverage ( 0.75 σB+ 0.25 σc)
a) Free energy “a” plotted as a function of surface coverage for a number of separation distances, D. The free energy shows a bimodal behavior showing an increase upon increasing in surface coverage for higher separation distances and showing a decrease upon increasing surface coverage for lower separation distances. The system plotted is with the surface coverage ratio, σ=σB=σC. Contour plot showing free energy surface for a receptormixed species system. c) Upon confinement of the protein
ligand to a single basin, the bimodal trend increases and shows a decrease in free energy for D = 22, which previously showed only a monotonic increase with increase in surface coverage. d) Contour plot for a mixed system with the peptide ligand confined to a single basin in the conformational space. e) By changing the ratio of surface coverage of polymer to protein, we can quantitatively change the overall free energy. The system plotted has the surface coverage ratio σB=1.5σ and σC=0.5σ.
47
internal energy factors compete in determining the overall magnitude of free energy. This is distinct from our previous observations of single species systems where entropic factors nearly always determined the nature of the free energy landscape. Our simulations also probed the effect of conformational restrictions in surface proteins. When we forced the protein molecules to only occupy β conformations (Figure 2.6c), the free energy at lower separation showed a similar trend as that in Figure 2.6a. However, unlike the constrain-free conformations, we noticed that at higher surface coverages the free energy decreased at even greater separations. For example, at D =22 in Figure 2.6a, we saw a monotonic increase in free energy, however, in a mixed system, the same separation shows a decrease in free energy at greater surface coverage. The probability distribution function depends upon the conformations of the protein and the overall conformational difference affects the cell binding at lower distances while at greater distances the effect of conformational sampling is relatively small. The quantitative differences are also obvious from the contour plot in Figure 2.6d. Finally, we probe the interaction of the adhesion receptor in a multi-species matrix by altering the interaction energy V0 in Eq. (2.9). Figure 2.7 shows the effect from changes in van der Waals interaction on overall adhesion free energy. The figure shows the free energy profile of a system, where the equilibrium interaction energy V0 is 2-fold greater than that in Figure 2.6. While we do not provide a mathematical basis for changing the interaction term by 2-fold, there can be a number of possibilities to describe such a scenario. Our model currently does not take into account electrostatic effects which may increase the interaction, additionally factors such as ion presence and pH 48
D = 20 D = 22 D = 24 D = 26 D = 28
40 20
b)
a
0 -20 -40
c)
40
a
20
D= D= D= D= D=
20 22 24 26 28
-50.20
0.2
-10.00
30.20
0.3 0.4
22
24
26
28
Distance
d)
0 -20 -40 -60 0.0
0.1
0.5 20
0.1 0.2 0.3 0.4 0.5 Surface Coverage ( σB+ σc)
Surface Coverage (σ B+ σ c)
-60 0.0
Surface Coverage (σB+ σ c)
a)
0.1 0.2 0.3 0.4 0.5 Surface Coverage ( σB+ σc)
0.1
-52.50
0.2
-11.45
0.3
29.60
0.4 0.5 20
22
24
26
28
Distance
Figure 2.7 Effect of Changes in Receptor-Ligand Interaction
The figures are similar to Figure 6a-d except the interaction term between the protein and the receptor is 2x of what was used in Figure 6a-d. The proteinpolymer and the polymer-receptor interactions remain as in the previous cases. The overall free energy of the system shows higher sensitivity on conformational confinement (7c and 7d). changes may also increase attractive interaction. Similar to the trend observed in Figure 2.6, we also note a strong dependence of free energy on conformational preferences suggesting a design strategy where a set of mutations may allow us to fine tune cellsurface adhesion. Another interesting feature is the balance between energetic term and 49
the entropic term. So far we have only seen that the interaction terms are dominated by entropic factors, but we note that by modulating receptor-ligand interaction, we are able to shift the free energy balance such that maximum adhesion occurs at higher surface coverage.
2.4 CONCLUSION
In this chapter, we introduced a statistical thermodynamic model of receptormediated cell adhesion to polymer or protein or both coated surfaces, which is a foundation for our next models. Our model incorporates the conformations of both the adhesion receptors as well as the surface ligands. The free energy surfaces, plotted simultaneously as a function of surface coverage and separation distance provide a useful way of visualizing subtle changes in the free energy landscape due to changes in interaction energy or conformational sampling. Our model can be used to predict the behavior of materials designed to modulate natural and artificial cell adhesion, as both increasing cell adhesion (such as wound healing) and decreasing integrin mediated adhesion (such as tumor) is highly desirable for a number of therapeutic applications. The competition between the two factors in free energy, namely favorable internal energy and unfavorable entropic contribution is clearly visible in our systems. The interaction of receptor-mediated adhesion with a mixed polymer/protein ligand surface also provides an interesting theoretical perspective on a growing number of biomaterials where polymers and proteins are in contact to achieve the desired result. 50
Unlike previous computational models, where adhesion was only studied at a continuum level, our results probe receptor mediated cell adhesion at the molecular level. A molecular level treatment can be easily extended to study adhesion as a function of amino acid composition of the proteins. In addition, designing novel substrates based on amino acids with known conformational propensities can be achieved by extending our framework to include amino acid propensities. Even though our model takes into account a number of realistic scenarios encountered in therapeutic, biochemical and biotechnological environments, there are a number of limitations in this model. First of all, our model does not include electrostatic and hydrophobic effects between receptors and the surface molecules. Such interactions will give a more accurate depiction of the overall free energy landscape. The receptor is modeled after the extra-cellular I-domain of integrins, but does not include the tertiary or the secondary structure information. Also, we do not explicitly include protein side chains, and assume an average hard sphere to represent a generic side chain. Finally, the effects of the solvent such as ionic strength or the presence of counter-ions is not taken into account. In spite of these limitations, our model provides a novel perspective on cellmatrix adhesion at the molecular level and makes several predictions that can be tested experimentally using cell force apparatus, optical traps or magnetic tweezers. We hope that recent advances in instrumentation and cell adhesion experiments will rigorously test our predictions and will provide new information that will help us in validating and improving our models. 51
Chapter 3 Model II: Cell Adhesion to Motile Homogeneous Nano Ligands: Effects of Ligand Size and Concentration 3.1 INTRODUCTION
Cells interact with both tethered and motile ligands in their extra-cellular environment to initiate and regulate signaling, adhesion and migration. For example, integrin-mediated cell adhesion and soluble integrin ligands contribute to maintaining COX-2 steady-state levels in endothelial cells by the combined prevention of lysosomaldependent degradation and the stimulation of mRNA synthesis involving multiple signaling pathways (Zaric and Ruegg 2005). Fas-Fas ligand is one of the bestcharacterized cell surface systems that regulate apoptosis. The concentration of the soluble ligand for the Fas receptor in serum of women with uterine tumors was examined by Anasz et al. (Kondera-Anasz, Mielczarek-Palacz et al. 2005). It was found that apoptosis that appears to occur in the cancerous cells of malignant uterine tumors is asso-
Figure 3.1 Cartoon Depicting Motile Ligand System
The system contains receptor, ligand and solvent. The solvent molecules occupy the entire volume that is not occupied by ligands or receptors. The receptors can be in bound (with orange chain in the Figure) or unbound/free (red chain) state and the unbound nano-ligand molecules (blue spheres) are free to move. 52
ciated with high level of soluble Fas ligand in serum. Soluble forms of the Notch ligands Jagged1 and Delta1 induced fibroblast growth factor receptor-dependent cell transformation in NIH3T3 fibroblasts. These soluble ligands were found to promote in vivo Tumorigenicity in NIH3T3 Fibroblasts with Distinct Phenotypes (Urs, Roudabush et al. 2008). Background soluble CD40 ligand is expressed on platelets and released from them on activation. It was investigated the predictive value of soluble CD40 ligand as a marker for clinical outcome and the therapeutic effect of glycoprotein IIb/IIIa receptor inhibition in patients with acute coronary syndromes. And it was found that elevation of soluble CD40 ligand levels indicated an increased risk of cardiovascular events (Heeschen, Dimmeler et al. 2003). Interactions between cell surface receptors and freely moving ligands in solution activate various signaling pathways and are essential for the normal functions of cells. In the previous chapter the treatment of receptor-ligand interactions in cell adhesion has focused mainly on tethered ligands ignoring the motile ligands in solution. A comprehensive thermodynamic treatment of interaction of cell surface receptors with mobile ligands is not only essential for fundamental understanding of cell-matrix interactions but may also provide valuable input in cancer research, drug design and a number of other biotech applications. In this chapter, we extend a previously developed and validated mean field approach (Carignano and Szleifer 1993; Huang, Szleifer et al. 2001; Longo and Szleifer 2005; Yang and Zaman 2007) to study the thermodynamics of interaction between membrane receptors and homogeneous mobile ligands in solution and focus on ligand concentration, size and interaction energy.
53
Our studies show that the free energy of interaction between the receptors and the nano-sized ligands depends strongly on the ligand size and the results at low and high concentrations show completely different trends that cannot be explained by simple scaling laws. In addition we also observe various regimes of strong and weak adhesion as a function of ligand size and concentrations. Our calculations provide insights into understanding cell-matrix interactions at a fundamental level as well as to identify potential avenues for fabrication of nano-ligands for therapeutic and bio-technological purposes. Our results provide insights into how the size of the nano-ligands regulates adhesion and suggest avenues for designing efficient ligands for biotech applications.
3.2 MODEL DEVELOPMENT
We will first introduce the theory and then discuss the simulations of molecules and choices of parameters in the later part of this chapter. Our system of interest contains three species, namely receptors, ligands and solvent as illustrated in Figure 3.1. The receptors are tethered on the cell’s membrane and occur in either free or bound (to the ligand) state. The rest of the space between the cell membrane and the substrate is filled with solvent molecules or motile ligands. In our system, the temperature, volume and particle number of each species are considered constant, therefore satisfying the requirements for a canonical ensemble, where the Helmholtz Free Energy is minimized at equilibrium. We will first describe each term in the formulation of Helmholtz free energy and then minimize it to derive a
54
series of relations for the thermodynamic quantities concerning receptors, extra-cellular ligands and the solvent. The free energy of the system of concern can be separated into two categories, the entropy-related terms and the interaction terms. We first examine and describe the entropic contribution and then turn our attention to the interaction terms. The entropy of our system can be separated into the following five components: 1) the translational (or mixing) entropy of solvent SS (the subscript S denotes “solvent”); 2) the mixing entropy of free nano-ligands SL, (the subscript L denotes “ligand”); 3) the translational entropy of receptors (including both bound and unbound receptors) on cell’s membrane SR, (the subscript R denotes “receptor”); 4) the mixing entropy of free and bound receptors on cell’s membrane SF-B (the subscript F-B denotes “free and bound”); 5) the conformational entropy of both free and bound receptors SC (the subscript C denotes “conformational”, and the necessity to get conformational entropy involved has been previously presented in 2.2 A) of model I). Next, we will quantify each of the five entropies, ignoring the Boltzmann constant kB. 1) The mixing entropy of solvent can be defined as: S S = − Ar ∫ dz ρ S ( z ) log[ρ S (z )V S ] =−
Ar ∫ dz ϕ S (z )log ϕ S (z ) VS
(3.1)
where Ar is the cross sectional area, or equivalently the area of the side of membrane against the substrate or the area of substrate of concern, ρS(z) is the volume density of solvent, i.e. the number of solvent molecules per unit area at height z, VS is the van der 55
Waals volume of solvent. In the 2nd equality, φS(z) is the volume fraction of solvent at z. There is a simple relation between the volume fraction and volume density: φS(z)=ρS (z)*VS. As shown in Figure 3.1, z is the spatial coordinate normal to the representative plane of the membrane or that of the substrate. 2) The second component of entropy, namely the mixing entropy of the unbound/free nano-ligand is defined by in the following:
S L = − Ar ∫ dz ρ L ( z ) log[ρ L ( z )V L ]
(3.2)
where VL is the volume of an extracellular nano-ligand and ρL(z) is the volume density for free (non-bound) nano ligands, i.e. the number of free ligands per unit volume at z. 3) The third type of mixing entropy, i.e. mixing entropy of all the receptors is defined as:
S R = − N R log ϕ R
(3.3)
NR is the total number of receptors and φR is the area fraction i.e. the fraction of the area covered by receptors tethered on the membrane. 4) Next we define the mixing entropy of the free receptor and the bound receptorligand complex in bulk mixture:
S F − B = − N R [ f F log f F + f B log f B ]
(3.4)
Here fF is the fraction of free receptors, so NR fF is the total number of free receptors. And fB is the fraction of bound receptors. Therefore, an instant relation between fF and fB is that: fF + fB =1 because there is no third condition of existence for receptor molecules.
56
5) The conformational entropies of free and bound receptors added up are given by:
S C = − N R f F ∑ P α log P α − N R f B ∑ P β log P β {α }
(3.5)
{β }
where Pα and Pβ are probability density functions (pdf, introduced in detail in section 2.2 A of model I), i.e., Pα is the probability of finding the α conformation of a free receptor. Pβ is the probability of finding the β conformation of a bound receptor. The conformations of free and bound receptors are derived from simulation and will be revisited in the end of the model development section. We now focus our attention on interactions between the components of the system. Firstly, we look at the attractive interactions. Our approach in this study is different from the model in chapter 2, where the receptor-ligand binding is treated as van der Waals interactions. In this model, they are separated into explicit and implicit components. The explicit interaction IE (the subscript E denotes "explicit") refers to binding energy with respect to the receptor-ligand pair (the strength ranges from in the range of to bigger than hydrogen bond (Helm, Knoll et al. 1991)), which is defined as:
IE = − NR f Bε0
(3.6)
ε0 is the magnitude of binding energy of one receptor-ligand bond. The implicit
interactions II (the subscript I denotes "implicit") are defined as:
57
2
Ar d z1 d z 2 χ '11 ( z1 − z 2 )[1 − ϕ S ( z1)][1 − ϕ S ( z 2 )] II = 2V 2S ∫∫ 2
+ A2r ∫∫ d z1 d z 2 χ '12 ( z1 − z 2 )[1 − ϕ S ( z1)]ϕ S ( z 2 ) VS
(3.7)
2
+ Ar 2 ∫∫ d z1 d z 2 χ '22 ( z1 − z 2 )ϕ S ( z1)ϕ S ( z 2 ) 2V S
In equation (3.7), the three terms are implicit interactions between a) polymers, b) polymer and solvent, and c) solvent and solvent, separately, in the mean-field framework (see section 2.2 C of model I for details). χ ij ' ( z1 − z 2 ) 's are the “two-layer interaction strength matrices” similar to those introduced in section 2.2 C) of chapter 2. Here we use “polymer” to denote both receptor and ligand nano-particle (either free or bound) and they are assumed to be indistinguishable in all types of implicit interactions for simplicity. With the help of the above equations (equations 3.1-3.7), we can construct a transformed free energy (i.e. the area density of dimensionless free energy) of the system a[ Pα ;P β ;ϕ S ( z );ρ E (z ); f F ] := βA / Ar =
1 VS
∫ dz ϕ (z )log ϕ (z ) + ∫ dz ρ (z )log[ρ (z )V ] S
S
L
L
L
+ σ R log(s σ R ) + σ R [ f F log f F + f B log f B ]
as follows:
+ σ R f F ∑ Pα log Pα + f B ∑ P β log P β {β } {α } −σ R f Bε 0 d z1 d z 2 χ 11 ( z1 − z 2 )[1 − ϕ S ( z1)][1 − ϕ S (z 2 )] 2V 2S ∫∫ 1 + 2 ∫∫ d z1 d z 2 χ 12 ( z1 − z 2 )[1 − ϕ S ( z1)]ϕ S ( z 2 ) VS 1 + d z1 d z 2 χ 22 ( z1 − z 2 )ϕ S ( z1)ϕ S ( z 2 ) 2V 2S ∫∫ +
1
58
(3.8)
In the above equation, A is the free energy of the system, β = 1/kBT and σR is the surface coverage (i.e. area density) of receptors defined as σR:=NR/Ar. “s” is the area covered by the tethering segment of a receptor molecule on the membrane (see Eq. 3.3 φR = s* σR). Similar to that being defined in section 2.2 C) of model I: χ ij := β Ar χ 'ij . Again as has been introduced in section 2.2 B) of model I, in order to take account of short distance interaction, i.e. steric effect, we introduce the packing constraint, or volume filling equation (Huang, Szleifer et al. 2001; Yang and Zaman 2007) which assumes that the space between the membrane and the substrate is filled up leaving no vacuum, and at the same time, no overlapping of molecules in space.
ϕ S (z ) + ∫ dz ' ρ L (z ')V L n(L ) (z; z ') + σ R V S f F ∑ Pα n Fα (z ) + f B ∑ P β n Bβ (z ) = 1 2
{α }
{β }
(3.9) where dz ' n(L2 ) ( z; z ') is the fraction of volume of a ligand particle within (z, z+dz) which is centered at z’. nFα(z) and nBβ(z) are segment linear number densities (similar to nAα(z) etc. in section 2.2 A of model I) at z of free and bound receptors in conformations α and β, respectively, which are obtained from the simulation of receptors. We minimize the free energy in Eq. (3.8) with respect to the pdf's Pα and Pβ, ρL(z), φS(z) and fF taking account the constraint Eq.(3.9) and the normalization equation fF + fB
=1 and obtain the following relations: Pα =
[
]
1 exp − ∫ dzπ ( z ) n Fα ( z ) gF
(3.10)
59
Pβ =
[
]
1 exp − ∫ dzπ ( z ) n Bβ ( z ) gB
ρ L ( z )V L =
1 exp− V L ∫ dz ' π ( z ') n(L2 ) ( z ' ; z ) gL VS
π ( z ) = − log ϕ S ( z ) + +
1 VS
(3.11)
1 VS
(3.12)
∫ dz' χ ( z − z ' )[1 − ϕ (z ')] 11
S
∫ dz ' χ ( z − z' )[2 ϕ (z ') − 1] − V ∫ dz ' χ ( z − z' )ϕ (z') 1
12
S
22
(3.13)
S
S
fF g = F exp(−β ε 0) 1− f F g B
(3.14)
where gF, gB and gL are normalization factors (for the former two, are also conformational partition functions of correspondent molecules, (Huang, Szleifer et al. 2001; Yang and Zaman 2007)) for free, bound receptors and nano-ligands, respectively, in which Pα and Pβ
are
both
normalized
to
1.
Therefore,
{α '}
[
[
]
g F = ∑ exp − ∫ dzπ ( z ) n Fα ' ( z )
and
]
g B = ∑ exp − ∫ dzπ ( z ) n Bβ ' ( z ) . ρL(z) also needs normalization, which will be discussed {β '}
after the discretization of the above equations. π(z) is the Lagrange multiplier, which is the “modified” osmotic pressure. The Boltzmann-type distribution between the 2-energy state of receptor is a natural result of a canonical ({Ni}, V, T) ensemble, which also indicates that the detailed balance is retained. Equations (3.9)-(3.14) form a closed set of equations supplied with fF + fB =1. In order to solve the 6 simultaneous equations, we discretize these equations (Szleifer and Carignano 1996), by separating the space between the two parallel plates in Figure 3.1 into a series of layers by virtual planes perpendicular to z-axis with equal separation δ 60
(similar to what's been done in section 2.2 E) of model I). The discretization of space gives:
π (i ) = − log ϕ S (i ) +
δ
∑ χ 11 (i, j )[1 − ϕ S ( j )] + ∑ χ 12 (i, j )[2 ϕ S ( j ) − 1] − ∑ χ 22 (i, j )ϕ S ( j ) VS j j j
(3.15)
Pα =
1 exp− ∑ π ( j ) n) Fα ( j ) gF j
(3.16)
Pβ =
1 exp− ∑ π ( j ) n) Bβ ( j ) gB j
(3.17)
ρ L (i )V L =
fF=
V 1 exp− L ∑ π ( j ) n) (L2 ) ( j; i ) gL VS j
gF g F + g B eβ ε 0
(3.18)
(3.19)
ϕ S (i ) = 1 − ∑ ϕ L ( j ) nˆ (L2 ) (i; j ) − σ R V S f F ∑ Pα n) Fα (i ) + (1 − f F )∑ P β n) Bβ (i ) (3.20) j
{α }
{β }
in which, i, j are indices for layers. Again, for simplicity we adopted this notation. In reality the value of any quantity attached with an index i should be understood as its continuous representation function evaluated at iδ, e.g. φS(i):= φS(iδ). The two-point volume contribution function nˆ (L2 ) (i; j ) , which we add a hat to make difference between its continuous identity n(L2 ) ( z; z ') by that the former is no longer a linear density, but fraction, therefore, a dimensionless quantity. Its definition is: nˆ (L2 ) (i; j ) := δ ∗ n(L2 ) (iδ ; jδ ) . 61
Similarly, the spatial distributions of segments, e.g. n) Fα ( j ) = δ ∗ nFα ( j ) , are changed from densities to counts by adding a superscript (i.e. hat). Now we take care of the normalization factor for extra-cellular nano ligand gL. Since: ∑ ∑ ρ ( j )V
( 2) ( 2) n L (i; j )δ Ar =∑ ρ L ( j )V L ∑ n L (i; j )δ Ar = V L ∑ [ρ L ( j )δ Ar ] = V L (r L − f B ) N R j i j j i , gL is normalized such that ∑ ρ L (i ) = σ R (r L − f B ) . In the formulas above, r L := N L , NL δ NR i L
L
is the total number of extra-cellular ligands including both free and bound. Similar as that presented in section 2.2 C) of model I, the mean-field average “bar” on top of the interaction χ’s was calculated as follows: Ar
χ MN (i, j ) =
∫∫∫
ρ r 1 ∈ layer i
ρ d r1
ρ
∫∫∫ d r
ρ r 2 ∈ layer j
ρ ∫∫∫ d r 1
ρ r 1 ∈ layer i
2
β V MN (rρ1 , rρ2 ) , M , N ∈ {1,2}
ρ ∫∫∫ d r 2
(3.21)
ρ r 2 ∈ layer j
The pair-wise implicit interaction energy is based upon the Lennard-Jones potential:
( )( )
r0 ρ ρ MN V MN (r 1 , r 2 ) = V 0 − 2 ρ ρ r1 − r 2
6
r0 + ρ ρ r1 − r 2
12
(3.22)
The simultaneous equations (3.15) - (3.20) can be solved by standard numerical methods. For our calculation, we have adopted Broyden's method (Press, Teukolsky et al. 2007) for numerical calculation of the simultaneous equations. After being solved, these equations can be substituted back into Eq. (3.8) to calculate the free energy a. A second
62
natural result of our calculation is the chemical equilibrium constant Keq defined in the following (Pauling 1970): K eq =
[RL] = [R][L]
fB
f F (r L − f B )([R ] + [RL])
=
f BD f F (r L − f B )σ R
(3.23)
D is the separation between the membrane and the substrate. In the bracket, [RL], [R] and [L] are concentrations for bound receptor-ligand, free receptor and free ligand, respectively. The binding free energy G (Pauling 1970) for the reversible reaction R + L ⇔ RL is defined as:
G = − RT log K eq
(3.24)
R is the gas constant. The unit of Keq is the same as the volume and only the difference in binding free energy is meaningful. We simulated 106 conformations of free receptors each containing 200 amino acids (which is on the order of the length of extra-cellular domain of integrins). Similar studies on tethered polymer systems have shown that 106 conformations provide an adequate picture of the sample space of conformational space (Szleifer and Carignano 1996; Yang and Zaman 2007). Further sampling of the conformational space does not lead to any significant improvement of results. Free receptors were simulated by a coarsegrained method described in section 2.2 A) of model I. The method incorporates Cα, carboxyl group and amino group in amino acid backbone and uses hard-spheres for side chains. The dihedral angles were chosen randomly (with equal weight in the allowed region of Ramachandran plot) from allowed basins in the Phi/Psi map of amino acids. The segments were checked for self-avoidance and uniqueness. The ligands are modeled
63
as spherical nano-size particles. We adopt a similar method for the generation of bound receptors to Longo et al. (Longo and Szleifer 2005). The nano sized extracellular sphere is attached to the free end of a conformation of free receptor. However, one free receptor has the potential to allow for a number of bound conformations dependent on how the spherical ligand is oriented with respect to the receptor. For each conformation of free receptor, we sampled the orientation angles, which were divided into 12 equal intervals from 0 to 2π. Self-avoidance is also checked for the spherical ligand. In our calculations, the nano-ligands simulated ranged from 1.0 to 2.0 nm in radius. Finally, the number of bound receptor conformations was between 4x106–8x106 conformations. The simulations were carried out using in-house programs written in C and analyzed using Matlab. The following is a summary of all the quantities introduced above, which are divided into three categories: 1) Constants, including D (the distance between two boundaries), β (the Boltzmann factor that carries the same information as temperature T of the system, is assumed to be the room temperature), rL (ligand receptor ratio), Vs (the volume of one solvent molecule), s (the area on the membrane covered by the tethering segment), and V0's and r0’s in Lennard-Jones potential; 2) Inputs that are obtained as results of our simulation, which are also taken as constants throughout the calculation, including nFα(z) (conformation-dependent free receptor segment number density), nBβ(z) (conformation-dependent bound receptor segment number density), and n(L2 ) ( z; z ') (free ligand volume correlation); 3) Parameters selected to control features of the system, including VL, (the volume of one free ligand), or equivalently, RL, (the radius of ligand due to its spherical geometry), σR (the surface coverage of receptor), and ε0 (the explicit 64
binding energy of one receptor-ligand pair); 4) Unknowns to be solved, including φS(z) (the spatial dependent solvent volume fraction), ρL(z) (the spatial dependent number density of ligand), fF (the free receptor fraction among all receptors), fB (the bound receptor fraction among all receptors), Pα (the free receptor pdf), Pβ (the bound receptor pdf), and a (the area density of dimensionless free energy); 5) Quantities without explicit values, which are introduced for clearness of presentation, including Ar (the area of cell membrane or substrate in equivalence), NR (the total number of receptor), and NL (the total number of ligand). The constants and parameters used in this work are listed as follows. The following order of magnitude estimates were used in our numerical calculations. The volume of solvent molecule VS is assumed to be identical to that of one segment of receptor and chosen as 90A3 same as in model I. Receptor ligand binding energy ε0 falls in the range between 5kBT and 30kBT. The layer width δ is chosen as 0.6nm and D=53δ. the area on membrane covered by the tethering segment s=30A2. As in a number of in vitro and in vivo environments, the ligands are in excess to receptors, we fix the ratio rL=1.5 for the purposes of our calculations. We adopted V 110 = 0.6 k B T , 12 22 V 0 = 0.3 k B T , V 0 = 0.15 k B T , and r0=0.3nm.
3.3 RESULTS
To study the role of the size of ligand nano-particle, concentration and interaction between cell adhesion receptors and mobile nano-sized ligands, we focus on two key thermodynamic parameters, namely chemical equilibrium constant Keq, which is related to the binding free energy of interaction between the receptors and ligands defined in 65
equation (3.24), and the Helmholtz free energy of the system, given by equation (3.8). The Helmholtz free energy provides the free energy of the entire system including the effects of the solvent and entropic contributions due to mixing of species and attractive/repulsive interactions. However, the binding free energy quantifies the difficulty of the binding of receptor and ligand, which, of course is coupled to other effects, e.g. steric, van der Waals interactions, conformations, etc. For clarity, we divide our results into three main categories: a) Effects of nanoligand size, concentration and receptor-ligand interaction energy at low concentrations; b) These effects for concentrated system and c) Comparison of probability density functions (pdf) for different nano ligand sizes and regions of concentrations.
A) Free Energy and Binding Status for Dilute Systems
We first pay attention to the effects of nano-particle size at low concentration. In our calculations, we have kept the ligand-to-receptor ratio constant, with the ligand always 1.5 times that of the number of available receptors. This situation mimics numerous in vivo and experimental scenarios where ligands are often in excess. Figure 3.2 A) illustrates the effect of ligand size (at low concentrations) on the chemical equilibrium binding constant. We observe two fundamental trends. First of all, the –log Keq term (which is a measure of binding free energy) is lower for bigger particles and becomes less negative as the particle radius goes down from 2.0 nm to 1.0 nm. The second key feature is that increase in concentration has a stronger effect on larger nanoparticles than the smaller ones. This is evident by the slopes of 1.0 and 2.0 nm particles in 66
1.0 nm 1.5 nm 2.0 nm
-25
B)
- log Keq
-50
0
30
40
-15
-75
-20
-100 -125
-25
-150 -175 -200
1.0 nm 1.5 nm 2.0 nm
-10
Helmholtz Free Energy (βA)(Vs/δS)
A)
-30
0
10
20
30
40
2
10
20
30
40
2
Surface Coverage "σ"(x 1000 /µm )
Surface Coverage "σ" (x 1000 /µm )
ε0= 5kT
C)
ε0= 10 kT ε0=15 kT
- log Keq
-70
ε0=30k T
-80 -90
-100 -110
0
10
20
2
Surface Coverage "σ" (x 1000 /µm )
Figure 3.2 Free Energies in Dilute Systems
The thermodynamic behavior of the system at low concentration region is studied. The ligand concentration is fixed at 1.5x the receptor concentration. A) –log Keq (which is a measure of the binding free energy and only the difference between binding free energies is meaningful) is plotted against concentration for nano-ligands of various radii. The binding free energy increases with the higher slopes for particles of the larger radii and remains flatter for smaller sized ligands. B) The Helmholtz free energy, which incorporates the entropic effects of mixing of the solvent and the ligands, also shows higher sensitivity of larger sized ligands to variations in concentration at low concentrations. C) The binding free energy scales linearly with variations in interaction energy. Figure 3.2 A). Both of these features suggest that at lower concentrations of the ligands, larger particles are more sensitive to fluctuations in concentration than smaller ones. In 67
this region, for all the ligand sizes, the receptor molecules are almost all in the bound state, with fF extremely close to zero. Therefore, the rather low binding free energy might literarily be considered as minus infinity, indicating that at this region of concentration, and at the binding energy, the reaction in reality is not reversible, but leads only to the bound species. Although the number of binding free energy at this region might lack precise meaning, the trends of changes with ligand size and concentration are reasonable. The results depicted in Figure 3.2 A) are at 15 kBT binding energy. Results at other values of interaction energies, ε0, are qualitatively similar (data not shown here). Somewhat similar trend is observed in the Helmholtz free energy of the system. In this case, we observe once again that bigger nano-particles are more sensitive to concentration effects while smaller nano-particles show a much more insensitive regime upon changes in ligand concentrations. Next, we want to study the extent of interactions between nano-particles and adhesion receptors as a function of interaction energy ε0. In order to develop this understanding, we studied the effects of various interaction energies ranging from 5 kBT (i.e. a weakly interacting system) to 30 kBT (i.e. in the range of typical biological interactions). Figure 3.2 C) shows the effect for 1.0 nm ligand particles. We note that as the interaction energy is increased, system remains qualitatively unperturbed, with only quantitative scaling proportional to the interaction energy. This is consistent with our previous models and studies (Yang and Zaman 2007), where interaction energies scale the effects on overall free energies for most distances and concentrations. A more interesting effect is observed when the results of 1.0 nm particles are 68
1.0 nm 30
10 20
-60
30
Interaction Energy ε0 (kT)
-50
- log Keq
Int era cti on (kT Ene rgy ε ) 0
-40
-52.59
30
20
-48.81
25
-45.02 -41.24 -37.46
20
-33.68 -29.90
15 10 5
40
-60.15 -56.37
-30
10
5 10 15 20 25 30 35 40 2
Surface Coverage σ (x 1000 /µm )
2
Surface Coverage σ (x 1000 /µm )
1.5 nm 30
-60
-107.0 -102.1
20
-100
30
40
30
20
10
-110
-92.40
25
-87.53 -82.67 -77.80
20
-72.94 -68.07
(kT)
- log Keq
-90
10
(kT)
Inter actio n
Ene rgy ε
0
-80
Interaction Energy ε0
-97.27
-70
15 10 5
2
Surface Coverage σ (x 1000 /µm )
5 10 15 20 25 30 35 40 2 Surface Coverage σ (x 1000 /µm )
2.0 nm
Inter actio n En ergy ε (kT) 0
-140 -160 10
- log Keq
-120
-180
20 30
-200 40 30 20 10 2 Surface Coverage σ (x 1000 /µm )
Figure 3.3 Landscapes and Contours of Binding Free Energy
–log Keq versus concentration and interaction energies for three sizes of ligand shows tilting of the landscape with increase in size. The landscape is linear with concentration at large particle sizes whereas at lower sizes it shows saturation at higher concentrations. compared to 1.5 and 2.0 nm particles in Figure 3.3. As we move from lower to higher concentrations, we note that the landscape tilts upward or “away” from lower 69
concentrations. In other words, as the nano-particle size is increased, the landscape becomes linear and flat. This is interesting, as these results suggest that at smaller nanoparticle size, the system begins to saturate and becomes independent of concentration. On the other hand, for larger nano-particles, increase in concentration results in a linear increase in the binding free energy. This result is consistent with results of Figure 3.2 where we observe a similar linear dependence on concentration at higher nano-particle sizes, while for smaller nano-ligands the higher concentrations show only a marginal change in the binding free energy.
B) Free Energy and Binding Status for Concentrated Systems
In order to develop a comprehensive understanding of receptor-ligand interactions, it is essential that we study the system at both low and high ligand concentrations. To study the effect at higher concentrations, we calculate thermodynamic parameters at concentrations ranging from 10 kilo to 650 kilo receptors per squared microns. Similar to their lower concentration counterparts, we focus on both the nanoparticle size and interaction energies. As we move to higher concentrations, a very different picture of interaction emerges between the receptors and the ligands. These results at higher concentrations suggest a non-linear dependence on concentration, an effect that cannot be explained by simple scaling laws. First of all, we note that what seemed like a plateau or saturation in “–log Keq” for 1.0 nm ligands is in fact only an intermediate regime in the equilibrium constant as at very high concentrations the “– log Keq” term increases sharply (Figure 3.4A). This is 70
40
B)
Helmholtz Free Energy (βA)(Vs/δS)
- log Keq
A)
20 0 -20 5 kT 10 kT 15 kT 30 kT
-40 -60 0
100 200 300 400 500 600 700
-10 -20 -30 -40 -50 -60 -70
2
1.0nm 1.5 nm 2.0 nm
-150 -200
Helmholtz Free Energy (βA)(Vs/δS)
- log Keq
-50
-100
Surface Coverage σ (x 1000 /µm )
D)
0
0 100 200 300 400 500 600 700 800 2
Surface Coverage σ (x 1000 /µm )
C) 50
5 kT 10 kT 15 kT 30 kT
-10 -20 -30 -40
1.0nm 1.5 nm 2.0 nm
-50 -60 -70
0 100 200 300 400 500 600 700
0 100 200 300 400 500 600 700 2
2
Surface Coverage σ (x 1000 /µm )
Surface Coverage σ (x 1000 /µm )
Figure 3.4 Free Energies at High Concentration Region
A) –log Keq versus concentration for 1.0 nm ligands. The binding free energy scales linearly with binding energy but displays non-linearity in response to concentration of solutes. B) The Helmholtz free energy of the system shows non-monotonic with concentration but invariance to interaction energy. The upward slope at very high concentrations points to systems where the concentration of particles is very high and entropic effect dominates. C) The effect of nano-ligand size on –log Keq (binding free energy) is shown for three systems of 1.0, 1.5 and 2.0 nm ligands. At low concentrations the binding free energy of larger particles is lower but at higher concentrations this trend reverses completely. The results of 1.0 are computed at the highest concentrations due to packing and fixed total volume constraints. Similarly higher concentrations of 1.5 nm and 2.0 nm ligands can be studied as opposed to 1.0 nm. The “crossovers” among the curves are observed D) The Helmholtz free energy of the system also suggests that as the concentration of the particles increases, the free energy of 2.0 and 1.5 nm ligands begins to saturate while 1.0 nm ligands continue to decrease the overall free energy of the system, except at very high concentrations where the solvent entropic effects are potentially negligible and the Helmholtz free energy begins to increase slightly. 71
valid for all interaction energies, with similar scaling as observed at lower concentrations. The Helmholtz free energy shows a similar trend where the effect of interaction energy is negligible. However, an interesting feature appears at very high concentrations for 1.0 nm ligands, where the Helmholtz free energy starts to increase in a sharp trend contrast to lower concentrations where it continues to decrease monotonically (Figure 3.4B). Comparison of receptor-ligand interaction at high concentrations as a function of ligand size shows a sharp departure from interactions at lower concentrations (Figure 3.4C). At lower concentrations, the effects are mainly linear, with the biggest size having the most sensitivity to concentration and having the lowest binding free energy. Contrary to these results, we observe a different scenario at high concentrations. The –log Keq term first increases for all three particle sizes and while it reaches a maximum for 1.5 and 2.0 nm and then begins to saturate, a similar trend never occurs for 1.0 nm particles (Figure 3.4C). The second aspect is that at lower concentrations, 1.5 and 2.0 nm particles have a lower binding free energy ( -RT log Keq) than 1.0 nm but at higher concentrations this trend changes, resulting in lower values for 1.0 nm particles (Figure 3.4C). While the binding free energy for 1.0 nm particles continues to increase for even higher concentrations (beyond 400), effects at such concentrations cannot be observed for 1.5 and 2.0 nm due to volume constraints with larger particles. Nonetheless, these results point to several interesting features. Firstly, we note that the observations made at lower concentrations regarding the sensitivity of particle size to concentration no longer remain true at higher concentrations. As a matter of fact, this behavior reverses completely (comparison of Figure 3.2 and Figure 3.4) and it is now the larger nano-particles that 72
show signs of saturation of binding Free energy, whereas lower concentration particles systems continue to have higher free energies. A deeper physical insight into this behavior requires an understanding of the equilibrium constant. The equilibrium constant defined in Eq. (3.23) allows us to deduce that -log Keq can be approximated as log (fF/fB) + log (rL- fB)+ log (σR). The second term of the equation can be ignored since rL = 1.5 and fB is between 0 and 1 and thus log (rL- fB) is quite close to 1. The last term, i.e. log (σR) is also negligible for surface coverages that do not differ by an order of magnitude or more. Thus, the most important is the first term (many orders of magnitudes variation) and hence the binding free energy is determined mostly by the ratio of the fraction of the free ligands with respect to that of the bound. At low surface coverages, i.e. dilute solution, most receptors are in bound state, therefore, fF is close to 0 and fB is close to 1. Since the ratio of the two is a very small positive number, the logarithm of the ratio is negative. We also note that all the three curves show that with the increase of receptors coverage, the binding free energy increases monotonically. This is primarily a consequence of the monotonic increase of fF, (or monotonic decrease of fB). On the other hand, at high surface coverages, the behavior of size 1.0 nm and those of 1.5 and 2.0 nm are disparate, due to the binding status of receptors, or equivalently, the binding free energy. For the 1.0 nm case, the binding free energy shows a change in sign. The fraction of bound receptors, fB, decreases from 0.99999984 at 400 x103 receptors/µm2 to 0.9934417 at 500 x103 receptors/µm2, but jumps down to 1.18 * 10-5 at 600 x 103 receptors/µm2. The decrease in the fraction of bound receptors for 1.0 nm particles shows that with the increase of surface coverage, the receptors undergo a transition from a bound dominant to an 73
unbound dominant regime. This transition takes place where the solvent is no long the dominant species in the system, the total volume it occupies is less than that by solutes. On the other hand, for larger sizes, 1.5 nm and 2.0 nm, there's no similar transition in appearance and the bound state always remains dominant. This observation can be related to the modified osmotic pressure π ( z ) .
1.0 nm 1.5 nm 2.0 nm
Osmotic Pressure π (z )
5
4
3
2
1
0
-1 0
10
20
30
40
50
60
z (layer Count)
Figure 3.5 Spatial Distribution of Modified Osmotic Pressure π(z) π(z) for systems containing nano particles of three sizes, i.e. 1.0nm, 1.5nm and 2.0nm. Receptor-ligand binding energy ε0 is chosen as 15 kBT uniformly. The curves for the two larger sizes are close to each other while the 1.0nm system is well above the others and has larger fluctuation at both ends.
According to Eq. (3.14),
[
]
f F gF = exp(− β ε 0 ) where gF and gB, are defined as f B gB
[
]
g F = ∑ exp − ∫ dzπ (z) n Fα (z) , g B = ∑ exp − ∫ dzπ (z) n Bβ (z) , respectively. There are {α }
{β }
three possible reasons that can be attributed to the observed behavior of free energy due 74
to size, namely 1) the modified osmotic pressure function π ( z ) ; 2) the conformationdependent segment number density n Bβ ( z ) for bound receptors and 3) the number of bound receptors. While we believe that the difference observed as a function of the ligand size is due to synergy of the 3 factors listed above, the dominant factor is the modified osmotic pressure function. As is shown in Figure 3.5, at most layers (i.e. z values), the 1.0-nm-size values for π ( z ) are bigger than 1.5 and 2.0 nm systems by almost a factor of 3. This is because the term
∫ dz n α (z ) in the equation above is a constant which is the F
number of monomers for free receptor. Similarly
∫ dz n β (z ) B
is another constant and
related to the equivalent number of segments for a bound receptor (which includes a receptor and a nano-ligand) and thus is always bigger than
∫ dz n α (z ) by a constant. So F
imagine we translate π ( z ) to π ( z ) + constant (here the constant is ~3),
fF will be fB
significantly larger due to the exponential operation. And as we explained, each exponent in the numerator and denominator depends not on the conformation, but the species. To be precise, the quotient is related only to the difference of the numbers of segments between a free and a bound receptor. The oscillations found close to the substrate may indicate an alternation of the increase and decrease of motile ligands’ and solvent’s concentrations in the region where receptors can barely reach. Typically a jump in the behavior of a system with the variation of a parameter is associated with a characteristic quantity of the same dimension. In our system, somewhere between the radii 1.0 and 1.5nm, there should be a critical radius that 75
separates the likeness of the behavior of the system. We note that this may be connected to the size of the receptor we simulated and the characteristic quantity in our system may be the volume of the receptor. The receptors in our system are identically 200mers with the volume of each monomer 90Å3. Therefore, the total volume of one receptor is 18nm3. The volumes of the three ligands 1.0nm, 1.5nm and 2.0nm in our simulation are 4.19nm3, 14.14nm3 and 33.51nm3, respectively. If the ligand’s volume is close to or larger than that of the receptor, the system at high concentration will behave similar to the 2.0nm one. On the other hand, if the volume of the ligand is much smaller than that of the receptor, it will appear akin to the 1.0nm system. Although ligand is a geometric sphere—the most concentrated volume in 3D space, the overall structure of the receptor molecules may play a role in determine where precisely in the vicinity of the receptor’s volume the “critical” volume for ligand is located. Another interesting result of our calculation is the crossing over of free energies (see Figure 3.4C). This phenomenon occurs first for 2.0 and 1.5 nm, then for 2.0 and 1.0 nm and finally for 1.0 and 1.5 nm particles. Before these regions, the larger sizes have a lower – log Keq value and after these crossovers they have a higher – log Keq value. These cross-over regions represent the various regimes of the system as a function of concentration. The Helmholtz free energy of the entire system also shows an interesting nonlinear trend at higher concentrations (Figure 3.4D). Unlike the – log Keq, the binding free energy of interaction, there are no cross-over concentrations observed in the Helmholtz free energy. However, we observe saturation in 2.0 nm and 1.5 nm systems at 76
concentrations that correspond to saturation in the –log Keq plot. Interestingly, at no concentration do we observe the Helmholtz Free Energy of the overall system at 1.0 nm become lower than 1.5 or 2.0 nm systems. The 2.0 nm system is always at lower free energy, followed by 1.5 and then 1.0 nm. While this is consistent with observations at lower concentration, the saturation effects are only observed at higher concentrations. In summary, results at lower and higher concentrations of nano-particles paint a strikingly different picture about the sensitivity of thermodynamic interactions on particle size and concentration. In order to get physical insights into the overall Helmholtz Free energy at various ligand sizes, we study the contribution of each individual components (equations 3.1-3.8) contributing to the free energy. The results are tabulated in Tables 3-1 and 3-2. Table 3-1 represents contributions of various components of the Helmholtz free energy for 1.0 nm ligand size while Table 3-2 shows the results for 2.0 nm. In both tables, each component of contribution is tabulated with reference to the starting surface coverage of receptors, hence the first column, being the reference is uniformly zero. For both Table 3-1 and Table 3-2, at low surface coverages, the quantity that varies most significantly is the van der Waals interaction between polymers (here polymers include both receptors and ligands). The vdw interaction term changes more significantly in the case of 1.0 nm than 2.0 nm system (second columns of Tables 3-1 and 3-2). The initial steep region of Figure 3.4D for 2.0 nm region corresponds to a decrease in van der Waals interaction by 1.35 while the saturation region corresponds to only a marginal (0.11) change in van der Waals interaction term. Thus we note that for 2.0 nm ligands, the shift from a rapidly 77
Table 3-1 Contribution to Helmholtz Free Energy for 1.0 nm Ligand System
Surface Coverage (103/µm) vdw int. of p-p vdw int. of p-s vdw int. of s-s Solvent entropy Free receptor conformational entropy Bound receptor conformational entropy Receptor-ligand binding Free ligand entropy Mixing of receptor Mixing of free & bound receptor
400 0 0 0 0
500 -7.51 1.06 1.35 -0.81
600 -16.68 3.79 2.27 0.99
650 -21.89 5.78 2.58 3.31
700 -27.53 8.20 2.78 7.14
750 -33.59 11.03 2.88 13.89
0
-0.01
-1.12
-1.18
-1.23
-1.23
0 0 0 0 0
-0.19 -0.24 0.01 -0.02 -3.29E-03
0.86 1.00 0.05 -0.03 -1.44E-05
0.86 1.00 0.09 -0.04 1.73E-07
0.86 1.00 0.14 -0.04 1.73E-07
0.86 1.00 0.20 -0.05 1.73E-07
Table 3-2 Contribution to Helmholtz Free Energy for 2.0 nm Ligand System
Surface Coverage (103/µm) vdw int. of p-p vdw int. of p-s vdw int. of s-s Solvent entropy Free receptor conformation Bound receptor conformation Receptor-ligand binding Free ligand entropy
80 90
100
110
120
130
140
150
160
169
0 0 0 0 0 0
-1.35 -0.02 0.35 -0.46 -3.86E08 -0.03
-1.46 -0.02 0.37 -0.49 -1.37E07 -0.05
-1.52 -0.02 0.39 -0.51 -2.90E07 -0.08
-1.56 -0.02 0.40 -0.52 -5.23E07 -0.10
-1.60 -0.01 0.41 -0.54 -8.78E07 -0.12
-1.63 -0.01 0.42 -0.55 -1.42E06 -0.15
-1.67 -0.01 0.42 -0.56 -2.26E06 -0.17
-1.70 -0.01 0.43 -0.56 -3.59E06 -0.19
-1.73 -0.01 0.44 -0.57 -5.46E06 -0.21
0
-0.03
-0.05
-0.08
-0.10
-0.13
-0.15
-0.18
-0.20
-0.22
0
-2.08E03
0.01
0.02
0.04
0.06
0.08
0.11
0.14
0.16
Mixing of receptor
0
-4.58E03
-0.01
-0.01
-0.02
-0.02
-0.03
-0.03
-0.03
-0.04
Mixing of free & bound receptor
0
-4.67E08
-1.54E07
-3.10E07
-5.40E07
-8.76E07
-1.38E06
-2.12E06
-3.27E06
-4.82E06
78
decreasing Helmholtz free energy to a more saturated region is mainly due to change in polymer-polymer interactions. Results in Table 3-1 also explain the minimum observed in Figure 3.4D for the case of 1.0 nm systems at high coverage. We note that at lower coverages the total of all three vdw interactions dominates the solvent entropic contribution, but at extremely high coverages, e.g. from 700 to 750, the former is dominated by the entropic penalty of solvent. The tables show consistency with the binding free energy behavior through the fF and fB related terms. For example, in Table 31, the receptor-ligand binding component undergoes a significant change from 500 to 600, where fB reduces from 0.993 to 1.2*10-5. Beyond 600, fB is nearly zero and the receptor-ligand binding has no contribution. Our observations about the binding Free energy are supported by the data in Tables 3-1 and Table 3-2. For systems of all three sizes, the fraction of receptors bound to the ligand increase as the surface coverage increases. However, at high coverage of receptors (or equivalently at high concentration of ligand) the 1.0 nm system changes from a region where the bound receptors rule to a region where the free receptors dominate. On the other hand, 1.5 nm and 2.0 nm systems are always in the regimes where bound receptors dominate as is seen in Figure 3.4 for binding free energy. Due to the similarity between 1.5 nm and 2.0 nm systems and the disparity from 1.0 nm, we can conclude from all these that the particle size plays an important role in the features of the system and there's a transition somewhere between 1.0 nm and 1.5 nm for high concentration behavior.
79
C) Comparison of Probability Density Functions
Figure 3.6 shows the big picture emerging from our calculations by depicting the variation in the conformation of free ligands at low and high concentrations for three sizes in histograms. Our calculation suggests that as the particle size goes up, the peak is “squeezed”. In other words, more conformations occupy probabilities next to the peak and the variance becomes smaller. This is analogous to having ground and excited states, where the region of lowest probability is the “ground state” and those of higher probability are the “excited states”. The most obvious change observed with the increase of particle size is that more particles shift from the ground state to the 1st excited state. In case of small sized nano-particles (1.0 nm), a few “special” conformations that have higher probabilities can be picked out to minimize the free energy. However, in larger systems, such as 2.0 nm, one ligand occupies a larger region of continuous space, and fewer “special” conformations (in free or bound space) are observed. As a result, the probability distribution is smoothed out and we observe a tighter distribution of conformations. We also note that the graphs for 1.5 nm, especially at high concentrations, are much closer to the 2.0 nm than to 1.0 nm. This is consistent with the behavior of other quantities, such as fF, and other parameters listed in Tables 3-1 and 3-2. Next, we study the effect of concentration. Compared to high concentration situations, the behavior of the system at low concentration is quite similar to each other. This is because in dilute solutions or mixtures, the receptors have less steric constraint than in the concentrated environments and therefore their probability distribution is more dependent on the spatial
80
distribution of the segments of each conformation, which is unchanged after these conformations are generated in our simulation.
Figure 3.6 Probability Density Functions at Low and High Concentration
The histograms of probability of occurrence for free receptors at low (black) and high (red) concentrations for 1.0, 1.5 and 2.0 nm ligands are shown. The vertical dimension is the number of conformations. The horizontal is the probability. Each bin indicates the number of conformations falling into the corresponding range of probability. The low concentration corresponds to 10 x 1000 receptors while the high values correspond to the maximum for each size in Figures 3.4C and 3.4D. The results suggest the presence of various regimes in free energy as a function of size. For smaller ligands more receptor conformations (at both low and high concentrations) are in located in the lowest probability region. At high concentration, most of the receptor conformations in 1.0nm ligand systems are in the smallest probability bin. 81
3.4 CONCLUSION
The results of our calculation point to a number of interesting features that suggest presence of sensitive dependence of thermodynamic properties on nano-ligand size and concentration. Interestingly, the effects of interaction energies between particles and their receptors are relatively mild. At low and high concentrations of ligands, increase in interaction energy results only in scaling of binding and Helmholtz free energies. On the other hand, the ligand-solvent-receptor system shifts from regimes of lower binding free energy with bigger particles at low concentration to lower free energy at high concentration with smaller nano-ligands. Of particular interest is at high concentration regime, small ligand system undergoes a transition from bound to free dominance. However, this transition does not happen to those containing larger ligands. This suggests that the efficiency of nano-ligands for adhesion depends strongly on size and concentration of ligand. Especially at all concentrations allowed the adhesive interactions favor larger particles. These results have strong implications for biotechnology applications of nano particles, particularly for nano-targeting of receptors for drug delivery, anti-body design and cell adhesion studies. Our results suggest that any design of nano-particles for biotech applications should consider not only the size and concentration but simultaneous effects of size and concentration on overall adhesion. Larger particles may be favored at 82
lower concentrations but as the concentration increases, smaller particles may be able to lead to stronger adhesion. There is an ongoing experimental collaboration (Zaman and Cummings ongoing) to study the effect of size of spherical nano-ligands on cell adhesion. Collaboration with the Nanomaterials Institute at Oak Ridge is focused on functionalizing various sized nano-particles, homogeneously coated with fibronectin molecules (a class of proteins that interacts with integrins and binds to integrins), and observing their intake into the cells. In spite of useful insights into cell adhesion and receptor-ligand interactions, our model has a number of limitations. First of all, the model does not account for more types of interactions, e.g. electrostatic interaction, hydrophobic effects, etc. The mechanical properties of the membrane are also beyond the scope of the model. Finally, while we address ligand diffusion in the solvent and mixing of receptors, we do not incorporate receptor clustering on the cell surface. However, in spite of these limitations, our model presents a quantitative approach to study and characterize cell matrix interactions between receptors and mobile ligands. Previous studies of cell matrix interactions have focused primarily on tethered ligands and have ignored the physical and chemical effects due to ligand motility. Our model fills this gap and provides a quantitative framework that will aid researchers in designing more efficient ligands for a variety of biotech applications in tissue engineering and drug design.
83
Chapter 4 Model III: Comparison of Clustered and Unclustered System: Effect of Receptor Aggregation 4.1 INTRODUCTION
The thermodynamics of cell adhesion is regulated by the conformation, energetics and dynamics of adhesion receptors. In addition, clustering of these receptors regulates adhesion as well as downstream signaling. In this chapter, we present an integrated model incorporating Monte Carlo simulations and a mean field approach to study the effect of receptor clustering on free energy of cell adhesion. The Monte Carlo simulation incorporates receptor diffusion and dimerization and provides input for the mean field computation of free energy of adhesion in a receptor-ligand-solvent system. Our results show that cell adhesion is regulated by co-operativity between receptor density and interaction energy. Receptor-ligand binding is an important mechanism in the maintenance of cellular homeostasis. The binding of extra-cellular matrix ligands to trans-membrane integrins triggers many cellular processes, e.g. signal transduction, cell proliferation, differentiation, migration, etc (Hynes 1987; Lauffenburger and Horwitz 1996; Critchley 2000; Hynes and Zhao 2000; Petit and Thiery 2000; Hynes 2002; Ridley, Schwartz et al. 2003; Camacho-Leal, Zhai et al. 2007). A key feature of receptor activity is the formation of aggregates or clusters (Irvine, Hue et al. 2002; Brinkerhoff, Woolf et al. 2004; 84
Brinkerhoff and Linderman 2005; Subramanian, Tsai et al. 2006; Camacho-Leal, Zhai et al. 2007). These clusters initiate focal adhesions and trigger numerous signaling pathways by providing a platform for protein interactions leading to protein phosphorylation and cytoskeletal attachment. Integrin clusters continuously form and disperse in migrating cells, and numerous kinases, phosphatases, and adaptor molecules dynamically regulate focal adhesion signaling (Lauffenburger and Horwitz 1996). A quantitative description of cell adhesion therefore requires understanding the effect of clustering on cell-matrix interactions. Receptor-ligand interactions in cellular systems have been studied through a variety of computational and experimental methods (Hammer and Lauffenburger 1987; Dembo, Torney et al. 1988; DiMilla, Barbee et al. 1991; Hammer and Apte 1992; Ward and Hammer 1992; Dickinson and Tranquillo 1993; Hammer, Tempelman et al. 1993; Kuo and Lauffenburger 1993; Ward, Dembo et al. 1995; Palecek, Loftus et al. 1997; Palecek, Horwitz et al. 1999; Paku, Tovari et al. 2003). However, the thermodynamic aspects of cell adhesion, mediated by receptor clustering, have been elusive. It is not clear under what regimes clustering leads to decrease in free energy of interaction and under what circumstances receptor clustering has negligible or even a negative effect on binding. In order to answer this question, we have developed a framework integrating a lattice free Monte Carlo model with a previously described mean field theory for cell adhesion in the above two models (Yang and Zaman 2007; Yang and Zaman 2008). Application of a lattice-free model allows us to simulate receptor diffusion on cell membranes of any curvature and hence has applications to cells embedded in matrices. 85
The Monte Carlo simulation provides equilibrium distribution of receptors that serves as an input in the mean field cell-matrix interaction model. The mean field model incorporates receptor and ligand conformations, entropic repulsion effects, as well as long- and short-range interactions of receptors and ECM molecules. The outcome of our model is a direct comparison of the free energy of adhesion in clustered and unclustered systems under varying receptor concentrations and cell-substrate interaction energies. The results of the model provide a quantitative description of the regimes where clustering provides a clear advantage in cell adhesion and regimes where cell adhesion does not depend upon receptor clustering.
Figure 4.1 Interactions between Cluster Receptors and Ligands
The cartoon depicts the system of interest. Receptors are tethered on the top plate and are 200 monomers long. The 20mer RGD ligands can bind to multiple receptors in the cluster.
86
4.2 MODEL DEVELOPMENT
Our system of interest is illustrated in Figure 4.1. The two plates are parallel to each other, with receptors tethered from the top one, ligands and solvent molecules embedded in between. Both plates are assumed to be planar and impenetrable. We apply a 2-step method to calculate the free energies of adhesion. First, we use a Monte Carlo routine to calculate the equilibrium distribution of receptor cluster size. Second, we input the results derived from the first step and calculate the free energy for clustered and unclustered systems.
A) Monte Carlo Simulation to Find Distribution of Receptor Clusters
Cell adhesion receptors are simulated by a coarse-grained method incorporating amino groups, Cα's and carboxyl groups in amino acid backbone and uses hard-spheres for side chains (Yang and Zaman 2007). The dihedral angles are chosen randomly from allowed basins in the Phi/Psi map of amino acids. The binding and interaction region of the chain is assumed to be the last 20 amino acids in the 200 amino acid receptors. The receptors can only bind to each other when the binding regions of them are within a certain interaction distance (6nm, the maximum distance at which integrins can dimerize). This coarse-grained assumption is based on numerous studies of proteinprotein interaction and binding that suggest that binding and aggregation occurring only when two protein molecules are in the correct 3D conformation. 106 unique conformations of the receptor molecules are generated using our coarse-grained simulations. 87
Once we generate unique conformations of receptors, we allow the receptors to undergo random walk and dimerize/monomerize until the equilibrium state is reached, upon which cluster sizes are counted. While previous studies have shown that binding itself is initiated by atypical conformations that arise due to fluctuations rather than equilibrated average conformations (Wong, Kuhl et al. 1997) , the purpose of our study in this work is not to model the dynamics of how cell adhesion is initiated, but to study at equilibrium, the free energy features of the receptor-ligand-solvent system. In this regard, previous computational and experimental studies (Brinkerhoff and Linderman 2005) have shown that only under circumstances where ligands are clustered or are islanded, the binding of receptors to ligands has significant effect on receptor clustering. On the other hand, it has been observed that when the ligands are homogeneously distributed, as is the case with the current study (the homogeneity along horizontal directions), the effect of ligand-receptor binding on overall clustering is minimal. Our lattice free model to calculate cluster sizes is based upon previous lattice based models (Brinkerhoff and Linderman 2005) with a number of critical improvements. Firstly, our receptors are chosen from the conformational space and hence are not identical. Secondly, we have used a lattice free model that does not suffer from unrealistic constraints of a lattice model. After the generation of unique conformations, we allow the receptors to undergo random walk with the fixation of their geometric configurations. We sample a large number of conformations (on the order of 106) to represent the set of possible conformations of receptors. Since during binding the receptors do not undergo significant 88
conformational changes, but rather experience small perturbations in conformations, our approach to simulate a large number of receptor configurations and then freezing them for free energy calculations allows us to capture the main experimental scenarios. Two receptors can bind if their binding sites are within the interaction distance. The excluded volume constraint is executed by checking monomer/segment pairwise distances to ensure the steric constraints are satisfied. The two monomers can belong to a same integrin or two different ones. At each time step, each receptor chooses from only one of the following 3 options: dimerization with a neighboring receptor, monomerization, or diffusion one step of length with respective probabilities Pdimer, Pmono, and Pmove. If more than one receptor is within the interaction distance of a monomer, the partner is chosen randomly. A reverse process is carried out for monomerization of the receptor from its partners. Lastly if a receptor is in a monomeric state and chooses to diffuse, it picks up a random direction and diffuses one step of length l (l=1nm here). If a receptor is a dimer, it can either approach or move away from its partner by length l or move together with its partner by a distance l/2. The estimate of the probabilities Pdimer, Pmono, and Pmove is based on assumption of Poisson processes with small time step ∆t, e.g.
P mono = exp(− k mono
(k ∆t )∑ ∞
n =1
mono ∆t ) = 1 − exp(− k mono ∆t ) ≈ k mono ∆t n!
n
(4.1)
Similarly, Pmove ≈ 4∆tDt/l2, in which “4” indicates 2D diffusion, and Pdimer ≈ kdimer∆t. Dt is the translational diffusion coefficient of receptor. The value estimated for integrin is between 10-11 and 10-10 cm2/s (Yauch, Felsenfeld et al. 1997; Buensuceso, de Virgilio et al. 2003). We adopt 10-10 cm2/s. The discretized time step ∆t is set to 2.5x10-6 89
s. Therefore, during ∆t, the probability for the receptor to move is Pmove=0.1. Next we will estimate the two parameters: kdimer and kmono. We need to keep in mind that here the kinetic rate for dimerization is not the forward kinetic rate for the binding reaction, in which the reactants have to move to neighboring locations in advance. As has been introduced in the last paragraph, when a monomer receptor chooses to dimerize, it searches within the interaction distance for a partner. Therefore, the binding possibility for these two spatially close receptors has no relation with diffusion and concerns only the pure dimerization rate. So this depends on the chance for the pair of receptors to form correct orientation. The rotational diffusion coefficient for GPCR (G protein-coupled receptor) was reported previous experiments (Saffman and Delbruck 1975), the number is 2.7x105. Using this rotational diffusion coefficient, Woolf and Lindermann (Woolf and Linderman 2003) simulated the Brownian rotation of two adjacent receptors and determined the mean time required for two adjacent receptors to enter the region for the proper alignment configurations. Under the assumption that receptors will bind once they are properly aligned, the upper bound of the dimerization rate is estimated to be 105 s-1. In our simulation, we go with kdimer=4x104 s-1. Therefore during ∆t=2.5x10-6 s, Pdimer=0.1. The monomerization kmono was estimated by Brinkerhoff and Linderman (Brinkerhoff and Linderman 2005) to range from 10 to 103 s-1 for integrins. We take kmono=400 s-1. Therefore, Pmono=10-3 in ∆t period. To study the effect of receptor density, we go with four different receptor concentrations, 1000, 2000, 4000 and 8000 /µm2. In real simulation, we found that due to the values of three probabilities are all small compared to 1, at many time steps, the receptor does not undertake any of these changes, neither 90
dimerization/monomerization nor movement. Therefore we enforce these probabilities to be normalized to 1, which means typically one of these processes happens for any receptor. This has no effect on the final result but reduces the total number of simulation steps, which was chosen as 104. We used the following two-fold equilibrium criteria: 1) the cluster-size distribution is unchanged within tolerance; 2) the number of dimers is unchanged within tolerance. The cluster size of a receptor and receptor distribution is determined by counting all receptors within the interaction distance of receptor. Simulations are carried out multiple times to ensure that our results are not affected by initial positions of the receptors.
B) Calculation of Free Energy
In order to calculate the free energy, we define an interaction region to be 30 nm away from the cell’s membrane (the top plane in Figure 4.1). The choice of 30 nm long interaction region is based on the fact that nearly all of the receptors active sites are located within 30 nm from the cell’s membrane. This means if a ligand is beyond the interaction region, it is unable to bind. The ligands for receptors are 20 repeating units of RGD (sequence of “Arginine-Glycine-Aspartic”) peptide. The conformations of these ligands are also computed by coarse-grained dynamics, where the phi/psi angle distributions are derived from allowed phi/psi values for the three amino acids. Since each RGD monomer can potentially bind to a receptor, presence of up to 20 units allow for multiple binding of receptors to a single ligand. The number of bonds is determined by the cluster size of the receptors. The larger the cluster size, the more receptor-ligand 91
bonds form because more binding sites of ligands have partners. So here we consider a picture where receptor is more than ligand. Not all receptors can find each a partner, but when they cluster, which means they are spatially close, they can share a ligand which provides multiple binding sites. Hence we adopt an average method which takes the cluster size distribution derived from the Monte Carlo simulations as an input. For example, if the percentage of appearance of cluster sizes is (Honore, Pichard et al.), where i denotes the cluster size, and the number of bonds formed with receptors are {bi} (bi is linear to i), the average receptor-ligand bond number is given by: b = ∑ pi bi
(4.2)
i
We utilize the similar method presented in the former two models to minimize the free energy of the system to calculate the probability distribution function (pdf) of receptors and ligands as well as the spatial dependent volume fraction of solvent. The free energy of the system of interest can be described by the following formula: A = k B T N R ∑ P Rα log P Rα + k B T N L ∑ P Lβ log P Lβ {α }
{β }
T + k B Ar ∫ dz ϕ s ( z ) log ϕ s ( z ) v − e r − l b N L ∑ P L β δ (β )
(4.3)
{β }
− er −r f r − r N R where kB is the Boltzmann constant, T is the temperature of the system, considered as room temperature. NR, NL is the total number of receptor and ligand molecules (RGD proteins), respectively. PRα, PLβ is the probability of finding receptors or ligands in α or β conformation, separately, Ar is the area of any of the two boundaries and v is the van der 92
Waals volume of solvent. Our model assumes that the volume of one segment of receptor or RGD protein is equal to v. φs(z) is the volume fraction of solvent in the layer (z, z+dz). er-l and er-r are the binding energies of one receptor-ligand pair and receptor-receptor pair, respectively. Both of them are typically in the range of or bigger than hydrogen bonding. For simplicity, we assume er-l= er-r= e. δ(β) is a two-value function, which is 1 if the active sites of the RGD falls into the interaction region to receptor, and 0 otherwise. δ(β) is derived from our simulation of RGD conformations. In this model, we ignore van der Waals interactions to emphasize on explicit binding. fr-r is the number of receptorreceptor bonds divided by NR, which is derived from the Monte Carlo simulation. Dividing Eq. 4.3) by kB TAr yields: a=
A k B T Ar
= σ ∑ P Rα log P Rα + r L σ ∑ P Lβ log P Lβ {α }
{β }
∫ dz ϕ ( z ) logϕ ( z ) − r σb e' ∑ P β δ (β ) {β } +v
−1
s
L
(4.4)
s
L
− e' f r − r σ where σ := N R is the surface coverage of receptor. Ar receptor. e' =
r
L
:= N L is the ratio of RGD ligand to NR
e is dimensionless binding energy. Short-term repulsion (see section 2.2 kBT
B of model I) among segments of chain molecules and solvents is accounted by solvent incompressibility condition. In the layer (z, z+dz), it is of the form:
ϕ R (z ) + ϕ L (z ) + ϕ S (z ) = 1 93
(4.5)
where φ’s are volume fractions, e.g., N n R ( z ) dzv = σv ∑ P Rα n Rα (z ) ϕ R (z ) = R
Ar dz
(4.6)
{α }
is the conformational average number of segments of receptor in the layer (z, z+dz). nRα(z) is the linear number density of segments of the α conformation of receptor, which we derive from the sampling of conformations. See section 2.2 A) in model I for details. Now the constraint equation can be written as:
σv∑ P Rα nRα ( z ) + r σv ∑ P Lβ nLβ ( z ) + ϕ s ( z ) = 1 L
{α }
(4.7)
{β }
We minimize Eq. (4.4) with respect to two pdf’s PRα and PPβ and solvent fraction
ϕ s (r ) . Taking into consideration the packing constraint, Eq. (4.5), we introduce a Lagrangian multiplier x(z), which is related to the dimensionless osmotic pressure (also x(z)= vπ(z), see Eq. (2.14) of model I for π(z)). The minimization results in:
[
]
P Rα =
1 exp − ∫ dzx( z ) n Rα ( z ) gR
P Lβ =
1 exp − ∫ dzx( z ) n Lβ ( z ) + b e' δ (β ) gL
[
(4.8)
]
ϕ s ( z ) = exp[− x z ]
(4.10)
σv∑ P Rα n Rα (z ) + r σv∑ P Lβ n Lβ ( z ) + exp[− x z ] = 1 L
{α }
(4.9)
(4.11)
{β }
[
]
in which, g R := ∑ exp − ∫ dzx( z ) n Rα ( z ) is the normalization factor (partition function of {α }
conformations) for receptor, and similarly, gL. Taking surface coverage of receptors σ, volume v, average bond number b , binding energy e’, the ratio of protein to receptor rP, 94
spatial distribution of segments nRα(z) and nLβ(z), and the function to judge whether a conformation of ligand is able to bind, δ(β) as inputs, equations (4.8), (4.9) and (4.11) form a closed set. To solve the above equations, we follow the method described in section 2.2 E) of model I, in which the space between the 2 surfaces is divided into a series of layers. We then iterate the following equations to arrive at a final solution. In the following equations, subscript i is used to denote discrete layers at different heights. The equations now have the following form:
P Rα =
1 ) exp − ∑ x(i ) n Rα (i ) gR i
(4.12)
P Pβ =
1 ) exp b e' δ (β ) − ∑ x( j ) n Pβ ( j ) gP j
(4.13)
)
)
φ ∑ P Rα n Rα (i ) + r φ ∑ P Pβ n Pβ (i ) + exp(− x i ) = 1 P
{α }
where φ :=
(4.14)
{β }
N R is dimensionless surface coverage of receptor, δ is the width of one Ar δ / v
layer. Again, for simplicity if a quantity has index i should be thought of as the value evaluated at iδ, for example, x(i) := x(iδ). And the hats on top of n’s change them from linear densities to numbers. In this model, we study the effect of concentration of receptors as well as interaction energy between receptors and ligands on adhesion in clustered and unclustered systems. By comparing the overall free energy with these two variables, our
95
model can quantify the regimes where receptor clustering dominates adhesion and regimes where clustering provides no benefit in overall adhesion free energy.
4.3 RESULTS AND CONCLUSION
% of Cluster Size
100
2
1000 /µm 2 2000 /µm 2 4000 /µm 2 8000 /µm
80 60 40 20 0 0
2
4
6
8
10
12
14
Cluster Size Figure 4.2 Receptor Cluster Size Distribution
The cluster size distribution for receptor concentration 1000/µm2 2000/µm2 , 4000/µm2 and 8000/µm2. It provides the information for a certain size of cluster, how much the percentage of its occurrence is. We first study the distribution of cluster sizes (the “size” shown here in fact is the number of companions in a cluster, the real size should add 1) as a function of receptor concentration. Figure 4.2 shows the distributions of four different receptor 96
concentrations, 1000/µm2, 2000/µm2 4000/µm2 and 8000/µm2. The average cluster size increases with increase in initial concentration of the receptors, and shows that at the receptor concentration 1000/µm2, most receptors exist in the form of monomer. As the receptor concentration increases, so does the cluster size. The trend of the change of the curves with different concentrations is like that of a Poisson distribution when the λfactor increases.
Figure 4.3 Free Energy Difference for Clustered and Unclustered Systems
The free energy difference for clustered and unclustered systems as a function of receptor concentration is shown. ∆a:=acluster-auncluster is the free energy difference, in unit (A/ kB T)(v/Arδ). The results indicate that the stability of the system is a non-linear function of receptor concentration and interaction energy. At higher interaction energies, the system is highly non-linear as opposed to weakly interacting systems, which show negligible dependence on clustering.
Figure 4.3 shows the free energy advantage for receptor clustering as a function of receptor concentration and interaction energy. ∆a := acluster - auncluster is the free energy 97
difference between clustered and unclustered systems. We note that the change in free energy between clustered and unclustered systems varies nonlinearly with increase in receptor concentration. For low interaction energies, the free energy change is mainly linear, but becomes highly non-linear at higher receptor concentrations. Our results indicate that for weakly interacting systems, such as interaction of receptors with matrices or ligands with poor binding, increase in clustering provides no significant advantage in the overall stability of the system. On the other hand, for highly interacting systems, a smaller increase in receptor concentration can lead to greater stability of the system.
Figure 4.4 Landscape of Free Energy Difference
The change in free energy between clustered and unclustered systems is shown as a function of receptor concentration and interaction energies. Only energies greater than 10 kBT and receptor concentrations higher than 4000/µm2 result in decrease in the ∆a. 98
The feature of Figure 4.3 allows us to answer the following interesting and useful questions. Is clustering beneficial for adhesion? If so, in what regimes is clustering most efficient? The answer to this question is depicted in Figure 4.4. The figure shows 3D plot of the difference in free energy between the clustered and unclustered systems for various receptor concentrations and interaction energies. The figure suggests that at low interaction energies, up to 5 kBT, clustering has no effect, irrespective of the cluster size or the initial concentration. The figure also suggests that at receptor concentrations less than 4000/µm2, clustering provides no advantage over unclustered systems, regardless of interaction energy. Only at concentration of 4000/µm2 or above, and interaction energy greater than 10 kBT, can aggregation of receptors result in increase in adhesion. While interaction energies over 10 kBT are not unusual in biological systems, receptor concentration is often modulated by external or internal signaling and inhibitors. Our results show that either interaction energy or receptor concentration alone is not sufficient in yielding a significant advantage. Instead, both of these conditions are necessary, they are not sufficient by themselves. We note that by keeping energy greater than 10 kBT, but decreasing receptor density by a factor of 2 (from 8000/µm2 to 4000/µm2) would result in an unstable regime of the free energy plot, where clustering provides no benefit. Similarly, large clusters by themselves are not particularly useful at interaction energies of 10 kBT or less, as they provide no significant benefit in overall adhesion. Receptors mediate signaling through a variety of signal transduction pathways. Monovalent ligand interactions with dispersed receptors generally induce limited signaling, whereas clustering of cell-surface integrins greatly augments signal 99
transduction. We developed an integrated model of receptor clustering and binding with ligands to computationally investigate the role of adhesion in cell adhesion. Our method combines Monte Carlo studies of receptor diffusion and dimerization as well as mean field calculation of adhesion free energy. Our results show the synergy between interaction energy and cluster size in determining the free energy landscape of adhesion and provide useful information about the complex receptor-ECM interactions.
100
Chapter 5 Model IV: Regulation of Cell Adhesion Free Energy and Force by External Sliding Velocity 5.1 INTRODUCTION
Cells experience a variety of forces and external velocities in vivo. While a number of continuum based models have addressed cell substrate interactions in these dynamic environments, a molecular picture of adhesion under external sliding forces and velocities is lacking. Using a molecular thermodynamic model, that incorporates entropic and steric repulsions, molecular conformations and constraints, penetrable substrates and explicit binding interactions, we study the effect of external sliding velocities (typically shear force generated by the relative motion of a cell with respect to the fluid, for example, the translational motion of a red blood cell in blood) on cell adhesion. Our model addresses binding and unbinding interactions in the presence of external velocities and quantifies adhesion force as a function of the magnitude of external velocity, surface coverage of adhesion receptors and the strength of binding interaction. We believe that our calculations of adhesion force in realistic dynamic environments will provide a molecular level understanding of adhesion in vivo and in vitro. In addition, models such as ours will provide design strategies for next generation of biomaterials that are optimized for specific pharmaceutical and biotechnological applications.
101
Figure 5.1 Cartoon Depicting the System under Sliding Force
A cartoon depicting the system studied in our calculations. The external force acts on the surface (membrane) decorated with adhesion receptors. The receptors interact with a penetrable adhesive substrate of finite thickness.
5.2 MODEL DEVELOPMENT
A schematic of the cell-substrate system used in this study is shown in Figure 5.1. The system consists of a top plate representing the cell membrane, and a bottom “box” represents a penetrable adhesive substrate. The reason to use a “penetrable” adhesive box instead of a single plate means that the ligands are not restricted to be coated on the surface of a substrate. Instead, they are in a substrate that has thickness. This is to capture realistic scenarios in numerous experimental studies. A large number of cell adhesion studies use collagen or Matrigel (Kleinman and Martin 2005) gels as substrates. These
102
substrates are of finite thickness and are penetrable by solvent molecules and cell adhesion receptors. The top plate, i.e. the cell membrane is decorated with receptors with a uniform surface coverage σ. The conformations of the receptors are generated by simulating the allowed phi/psi angles of the integrin amino acids. The receptors are 150 amino acids long and the side chains are simulated through a hard-sphere model as described elsewhere in chapter 2. Our simulations are superior to simple hard sphere simulations that do not account for the backbone torsion angles or the side chains. For simplicity, we assume that only the terminal region (last 20 amino acids) of the receptor interacts with the ligands and the other regions of the receptors that are chemically inert. This is a reasonable approximation given that integrins bind to ligands at specific locations on the molecular surface. The penetrable adhesive substrate is assumed to contain receptor binding ligands that interact with the receptors with a binding energy “e”. To capture the effect of in vivo scenarios, where a cell is subjected to external velocities and forces, we introduce a constant sliding velocity acting on the cell membrane. The receptors are distributed in the z direction within 10 nm from the cell membrane. The thickness of the gel substrate is assumed to have a fixed height of 8 nm along the z direction. If the binding region (i.e. last 20 amino acids) of the receptor do not fall into the substrate, there is no receptor-substrate binding. Receptors that go inside the box are regarded as potentially binding receptors. This means that not all potentially binding receptors form bonds with ligands. The possibility for a bond between a receptor and a ligand is related to the sliding velocity and bond energy. This issue is discussed in 103
detail in the later part of the model development section. First, we describe our method to compute the free energy of the system of interest. The free energy can be written as: A = k B TN ∑ Pα log Pα + {α }
k B T Ar Pα δ (α ) (5.1) ∫ dz ϕ s (z ) log ϕ s (z ) − eP(V , e)N ∑ vs {α }
in which, A is the free energy of the system. The first term in Eq. (5.1) comes from the entropy of receptors where kB is the Boltzmann constant, T is the temperature of the system and N is the total number of receptors. “α” denotes the conformation of receptors and Pα is the probability of finding the conformation α of all receptors. The second term in Eq. (5.1) is rooted in the translational entropy of solvent molecules, in which Ar is the total area of the membrane, v s is the van der Waals volume of solvent molecule and
ϕ s ( z ) is the volume fraction of solvent in the layer (z, z+dz). The last term refers to the specific binding energy of receptors and ligands, in which “e” is the bond energy. The term N ∑ Pα δ (α ) is the number of potentially binding receptors with δ (α ) equal to 1 if {α }
the α th conformation has overlap with the substrate and 0 otherwise. P(V, e) is the conditional probability for a potentially binding receptor to make a bond which depends on sliding velocity (V) and bond energy (e). Dividing Eq. (5.1) by kBTAr, we arrive at: a :=
A k B T Ar
= σ ∑ Pα log Pα + {α }
1 vs
Pα δ (α ) ∫ dz ϕ (z )log ϕ (z ) − σe' P(V , e)∑ {α } s
s
104
(5.2)
where σ :=
N
is the surface coverage of receptors. e':=
Ar
e is dimensionless binding kBT
energy. The short-term strong repulsion between the monomers and the solvent is accounted for through packing constraints. In the layer (z, z+dz), the packing constraint is formulated as:
ϕ R ( z ) + ϕ s ( z ) = 1 − f L (z )
(5.3)
in which, denotes ensemble average over all conformations. Hence ϕ R ( z ) is the volume fraction of receptors in (z, z+dz), which can be expressed as:
ϕ R (z ) =
N n R ( z ) dzv = σv ∑ Pα n Rα ( z ) Ar dz {α }
(5.4)
where nR ( z ) dz is the ensemble averaged number of segments of receptor in (z, z+dz), v is the Van der Waals volume of the monomer. In the last formula of Eq. (5.4), nRα (z ) is the linear number density of segments of α conformation of receptor, which is derived from the sampling of receptors. f L ( z ) is the volume fraction of ligands in the box at z, which is either zero in the case the box is beyond z or assumed to be a constant number otherwise to approximate spatially homogeneous ligands. For our purposes, the constant value is assumed to be as 0.05 (in this piece of work we consider only dilute solution— typical for most bio-systems in which solvent is dominant and solutes occupies only a small volume fraction). We define f ( z ) := 1 − f L ( z ) . The constraint equation now becomes: 105
σv ∑ Pα n Rα ( z ) + ϕ s ( z ) = f ( z )
(5.5)
{α }
In order to calculate the free energy, we minimize the quantity “a” in Eq. (5.2) with respect to the pdf Pα and the solvent term ϕ s ( z ) . We introduce a Lagrangian multiplier x(z) in the constraint equation. The physical interpretation of x(z) is dimensionless osmotic pressure. The simultaneous equations can now be written as: Pα =
[
]
1 exp − ∫ dzx( z ) n Rα ( z ) + e' P (V , e )δ (α ) gR
(5.6)
ϕ s ( z ) = exp[− x z ]
(5.7)
σv ∑ Pα n Rα ( z ) + exp[− x z ] = f ( z )
(5.8)
{α }
[
]
In Eq. (5.6), g R ≡ ∑ exp − ∫ dzx( z ) nRα ( z ) is the partition function to normalize the pdf {α }
Pα . Equations (5.6), (5.7) and (5.8) are a closed set. Taking segment number density
nRα ( z ) , the fraction of solvent and receptors f (z ) and binding potentiality δ (α ) as input, surface coverage σ, binding energy e’ and sliding velocity V as parameters, we can calculate the free energies at a series of separations between the membrane and the far edge of the substrate. The solution of the simultaneous equations above is similar to that in the previous chapters. See, for example, section 2.2 E) for details. Finally, we address the core of this work--binding probability as a function of relative velocity and binding energy P(V, e’). Assuming a mechanical force balance in the translational plane, the average force exerted on a monomer is f, the average distance 106
between neighboring monomers or that between monomer and solvent molecules is λ. Eyring (Eyring 1935; Eyring 1938) showed that if k0 is the kinetic rate of the unbinding process in the absence of an external force, kf is the rate along the direction of the microscopic force f or equivalently, the direction of the sliding velocity. And similarly, kb is the rate of backward reaction. In the presence of an external microscopic force, these rates are given by: fλ / 2 k f = k 0 exp kBT
(5.11)
fλ / 2 k b = k 0 exp − kBT
(5.12)
We can relate the microscopic force f with the macroscopic velocity V by
fλ fλ V = λ k 0 exp − exp − 2 k B T 2kBT
(5.13)
The assumption made here is: A bound receptor chooses one of the three options at a time period ∆t: 1) hop a step (of a bond length) forward; 2) hop a step backward; 3) stay where it was. And the corresponding possibilities are: kf∆t, kb∆t, and k0∆t. The average move of the molecule is the λ(kf-kb)∆t. And because the binding rate is orders of magnitude higher than the unbinding rate, we ignore the time binding takes. After ∆t the molecules recovers the original state and the only change is that it shifts a displacement V∆t in the direction of the constant velocity. We equal the displacement of the receptor after ∆t calculated from two considerations: one is that considered under microscopic picture in which it unbinds followed by rebinds and the other under macroscopic picture in which it has a 107
constant velocity relative to the substrate. As a consequence Eq. (5.13) is obtained. Also the binding and unbinding of receptors and ligands is a kinetic process of a two-energy system. At equilibrium, the number of potentially binding receptors (those the binding sites of which fall into the substrate) is given by N’, among which, Nb are receptors in the binding state, Nf are in the free state. Due to the presence of an external force f, the energy difference between the two states is lowered and Nb/Nf=exp[(e-fλ/2)/kBT]. Therefore,
fλ fλ P( f , e') = exp e'− / 1 + exp e'− 2kBT 2 k B T
(5.14)
From Eqs. (5.13) and (5.14), we get P(V, e’), the probability of a receptor being bound under the condition it has the potential to get bound. λ is chosen to be 0.3 nm. k0 has a broad range (two orders of magnitude in α1β2-R207C and GABA binding) due to different receptor species and is affected by many factors, e.g. integrin mutations, structural change of ion channels, etc (Wagner, Czajkowski et al. 2004). And the qualitative property is sensitive to k0. We go with two numbers, first k0=10/s, which is a high value, and a low rate 0.01/s. The latter is also typical for α2β1 integrin—collagen unbinding after integrin activation (Masson-Gadais B, Pierres A et al. 1999).
5.3 RESULTS
We first check the high unbinding rate k0=10/s. In order to quantify the effect of external sliding force on cell adhesion, we first study the effect of external velocity on binding and unbinding. Figure 5.2 shows the effect of the magnitude of external velocity 108
on percentage of bonds remaining as a function of binding energy. The percentage is calculated by comparing number of available bonds to the number of bonds formed in the absence of any external sliding velocity. Our results show that at high binding interactions (such as e=10 kBT), even very high velocities, such as 1µm/s do not lead to any significant bond breakage and the cell remains strongly adhered. On the other hand, as the adhesion energy decreases, increasing sliding velocities leads to only a small fraction of bonds forming. The percentage of available bonds allows us to map the free energy landscape of
% receptor- surface bonds
cell adhesion in the presence of external sliding velocity. Figure 5.3 shows the effects of
100
10 0.01 kT 1kT 5 kT 10 kT
1 1E-4
1E-3
0.01
0.1
1
Sliding Velocity (µm/s) Figure 5.2 Binding Probability Regulated by Velocity and Binding Energy
The percentage of bonds between the cell and the substrate vary with the magnitude of external velocity and the strength of the binding interactions. At lower speeds and high interaction energies, most bonds remain intact. However, at high speeds and lower binding energies, the number of bonds formed is a small fraction of the total number of bonds formed in the absence of any external forces. 109
A)
C)
B)
D)
Figure 5.3 Free Energy of Different Binding Energy and Sliding Velocity
Free energy of cell substrate adhesion at four different binding energies is plotted. At 0.1 kBT, the free energy of the system at high velocities is fairly high and the system remains unstable (Figure 5.3A). The gap between free energies at various velocities starts to decrease as the binding interaction increases to 1 kBT (Figure 5.3B). At even higher binding interactions (Figure 5.3C; e = 5 kBT) the gap between free energies at lower and higher velocities is very small. At e = 10 kBT (Figure 5.3D), the free energy landscape is in variant to external velocities. This is because even high sliding velocities are unable to cause any significant loss in binding.
adhesion binding energy and sliding velocity on overall free energy of adhesion. The 110
results are shown for a given receptor surface coverage (0.01), results for other surface coverages are qualitatively similar and differ only in magnitude. Our results show a number of key features that have not been quantified previously in continuum level models. First of all, we note that at lowest interaction energy (Figure 5.3A), cell adhesion is most stable at lowest speeds and short distances between the substrate and the cell. As the sliding velocity increases, there are only a few bonds formed between the surface and the cell, as a result increasing free energy does not change the free energy landscape significantly. In other words, for a weakly interacting system, high velocities produce no significant change in the free energy landscape. On the other hand, for these weakly interacting systems, lower velocities have smaller effect at shorter distances, but as the separation increases, the free energy starts to increase. This is because fewer bonds are formed at greater distances and even smaller velocities are able to disrupt these bonds. Thus for a weakly interacting system, higher speeds show smaller changes in the free energy landscape as a function of distance, while lower speeds show a stable system at shorter distance but approach instabilities caused by higher velocities at greater distances. As the interaction energy is increased by an order of magnitude, we note that the gap between lower and higher velocities starts to decrease. This pattern is observed as we increase the cell substrate binding further, first to 5 kBT and 10 kBT. With increasing binding energy, the free energy landscape becomes less and less responsive to changes in velocity. At 10 kBT, the velocity has virtually no effect at all on the overall free energy landscape. This result may seem counterintuitive, but in fact can be explained by the trends observed in Figure 5.2. As the interaction energy increases, the effect of external 111
velocity reduces. As shown in Figure 5.2, at 10 kBT, even 1 µm/s does not lead to any significant unbinding. As a result, the system shows invariance to external sliding. This invariance results in the robustness of the free energy landscape, and the free energy profile of the system remains the same regardless of external velocities. The dynamics of the free energy landscape are captured in Figure 5.4 that shows the variations in the landscape as a function of binding interactions. The 3D plots demonstrate the free energy variations as a function of distance and sliding velocities. As suggested by Figure 5.4, the landscape becomes increasingly smooth as the interaction energy increases. Landscape plots such as the ones depicted in Figure 5.4 provide useful insights into the robustness and resistance of a system to external forces.
Figure 5.4 Free Energy Landscape for Binding Energy and Velocity
The free energy landscapes at various binding interactions are shown. As the binding energy increases, the free energy landscape becomes smooth and resistant to external sliding velocities. 112
A)
B) e= 5 kBT e=15 kBT e=30 kBT
% of Bonds
0.8
0.6
0.00
Free Energy (avs/δ)
1.0
0.4
0.2
0.0
V=0.1µm/s V=1µm/s V=10µm/s V=50µm/s
-0.01
-0.02
-0.03
ε = 5kBT -0.04
-0.05 0
0
10
20
30
40
50
1
2
3
4
5
6
7
8
9
Receptor Coverage (x1000/µm2)
Sliding Velocity (µm/s) C)
D) 0.00
V=0.1µm/s V=1.0µm/s V=10 µm/s V=50 µm/s
-0.01 -0.02
Free Energy (avs/δ)
Free Energy (avs/δ)
0.00
-0.03 -0.04
ε = 15kBT
-0.05 -0.06 -0.07 0
1
2
3
4
5
6
7
8
9
Receptor Coverage (x1000/µm2)
V=0.1µm/s V=1.0µm/s V=10 µm/s V=50 µm/s
-0.02
-0.04
-0.06
ε = 30kBT
-0.08
-0.10
0
1
2
3
4
5
6
7
8
9
Receptor Coverage (x1000/µm2)
Figure 5.5 Results for Low Unbinding Rate
A) The ratios of binding for receptors able to bind being regulated by the binding free energy and sliding velocity in the typical range of cellular motion. B), C) and D) Free energy curves at various binding interactions are shown. Also being studied is the effects of sliding velocity and receptor density. B) C) and D) are consistent with A)
113
Next, we change the rate constant to k0=10/s. Figure 5.5 A) displays that the probability of binding for a potentially bound receptor is highly sensitive to the receptorligand binding energy. When the binding energy is small, for example, 5 kBT, the probability is almost identically 0 in the study domain of the sliding velocity for typical cellular motion. On the other hand, when the binding energy is as big as 30 kBT, almost all the receptors are in the bound state no matter how much the velocity varies. Only for intermediate binding energy, e.g. 15 kBT, the probability shows sensitivity to the velocity, varying from 1 for low velocity to close to 0 for higher velocity. Note that 15 kBT is a normal strength for receptor-ligand bonding. Figures 5.5 B)-D) display a similar feature as Figure 5.5 A), i.e. free energy is also inert to the lowest and highest binding energy.
5.4 DISCUSSION AND CONCLUSION
Cell-substrate interactions have tremendous significance in tissue engineering, biomaterial design and pharmaceutical industries. A detailed description of these interactions will allow us to design and optimize next generation of smart biomaterials. Using a molecular thermodynamic model, we have quantified the effect of external sliding velocity on the magnitude of free energy and adhesion force in cell substrate interactions. Our model incorporates steric entropic effects of the receptor chains, solvent incompressibility and predicts adhesion forces as a function of the strength of binding interactions. Our model also maps the dynamic landscape of adhesion free energy in the presence of external sliding force and quantifies regimes for maximum and minimum adhesion. Our model is a first of its kind of study to quantify the effects of sliding 114
velocities on cell adhesion while incorporating molecular thermodynamic aspects of cell adhesion. Our current framework lays the foundation for future models, which will incorporate biochemical, biophysical and biomechanical features of cell adhesion receptors and their ligands in quantifying the molecular and cellular response to forces experienced by cells. Such models will aid researchers, both in experimental and computational bioengineering to further understand the molecular basis of cell adhesion and to quantify cell adhesion in a variety of in vivo settings.
115
Chapter 6 Model V: Estimation of Cellular Adhesion Forces Using Mean Field Theory 6.1 INTRODUCTION
A complete understanding of the interaction between the cell and the surrounding substrate requires a quantitative understanding of the force under which they adhere to each other. Using mean field theory, we provide a new and robust method to calculate this force of cellular adhesion to a ligand coated substrate in a system that contains receptors, ligands and solvent. Our calculations incorporate molecular conformations, long and short range interactions and solvent entropic effects that regulate adhesion in native environments. The results of our calculations provide a set of quantitative predictions that have been tested previously and will guide future experimental endeavors. Currently, most models to quantify forces at cellular interface to date have focused at the continuum-cell level and typically ignore molecular level events, such as conformations and mutations, thus changes in receptor structure, solvent conditions and ligand conformations are beyond the scope of these models. We apply the mean-field approach presented in the previous chapters that incorporates structural, conformational and entropic contributions due to the receptors ligands and the solvent, which allows us to calculate adhesion force explicitly. At the same time, our model is able to predict the bulk behavior that is outside the regime of single molecule simulations. 116
Figure 6.1 Tethered Receptor and Ligand System to Calculate Adhesion Force
Both conformations of the receptors and ligands are considered. The ligands are coated on a floatable surface at a distance d from the cell membrane (the top boundary). The thick black line represents the other boundary of the system, which has a constant separation D from the membrane. The solvent can be anywhere between the two boundaries. However, solutes are all between the membrane and the substrate. The results from our model are in line with previous study (Brakebusch and Fassler 2005) which points out that the membrane-substrate separations range from 1nm for the strongest, to about 50 nm for the weakest adhesions. Our model makes a series of testable predictions that will hopefully guide AFM and other force measuring experiments to quantify adhesion forces, for example at the leading edge of a migrating cell.
117
6.2 MODEL DEVELOPMENT
Our model is rooted in a mean field approach similar to the previous explained models that can be seen in the prior four chapters. The adhesion force is calculated from the spatially dependent free energy. Figure 6.1 provides a schematic of our model, where the receptors are attached to the membrane at distance D from the lower end of the system. A movable substrate is located at a distance d (< D) from the cell, from which ligands are attached. Solvent molecules are imbued throughout the system. Adhesion force is calculated from the d-dependent free energy. Briefly speaking, we minimize the free energy of the receptor-ligand-solvent system in a canonical ensemble, which is an approximation mimicking a steady-state situation in which the temperature and the volume as well as the particle numbers of the system of interest are constant. The free energy A of this receptor-ligand-solvent three-species system can be written as:
A = k B T N R ∑ Pα log Pα + k B T N L ∑ P β log P β + {α }
2 r 2 s
+ A 2v 2
+ A2r vs
{β }
∫∫ d z d z 1
∫∫ d z d z 1
2
2
k B T Ar ∫ dz ϕ s ( z ) log ϕ s ( z ) vs
' ( z1 − z 2 ) 1 − ϕ s ( z1) 1 − ϕ s ( z 2 ) χ 11
χ
' 12
( z1 − z 2 ) 1 − ϕ s ( z1) ϕ s ( z 2 )
(6.1)
2
+ Ar2 ∫∫ d z1 d z 2 χ '22 ( z1 − z 2 )ϕ s ( z1)ϕ s ( z 2 ) 2v s In the above equation, the first two terms are due to the conformational entropy of receptor and ligand. NR or NL is the number of receptor or ligand molecules, respectively. Pα or Pβ is the probability density distribution (pdf) of finding a receptor or a ligand in α 118
or β conformation, respectively. The third term is the translational entropy of solvent. Ar is the cross section area, equivalent to the area of either (top or bottom) boundary of our system. φS(z) is the volume fraction of solvent at z. vs is the van der Waals volume of a solvent molecule, which is also assumed to be equal to that of one segment of either receptor or ligand for simplicity. The last three terms are van der Waals contributions to polymer-polymer (“polymer” here includes both receptor and ligand), polymer-solvent and solvent-solvent pairwise interactions. Interaction strengths for polymers are assumed to be identical in an average sense. This assumption is not critical and can be relaxed. ' is between χ 11' is the averaged interaction strength between polymer-polymer, χ 12
polymer-solvent and χ '22 is between solvent-solvent. We define the surface coverage of rceptor σ:=NR/Ar, the ligand receptor ratio rL:=NL/NR, and a transformed interaction strength χmn:= Arβχ'mn (m,n ∈ {1,2}, β=1/kBT, T is the temperature). The area density of dimensionless free energy yields: a := β A / Ar = σ ∑ Pα log Pα + σ r L ∑ P β log P β + {α }
{β }
∫ dz ϕ ( z ) log ϕ ( z ) s
s
vs
1 d z1 d z 2 χ 11 ( z1 − z 2 )1 − ϕ s ( z1) 1 − ϕ s ( z 2 ) 2 ∫∫ 2 vs 1 + 2 ∫∫ d z1 d z 2 χ 12 ( z1 − z 2 )1 − ϕ s ( z1) ϕ s ( z 2 ) vs 1 + 2 ∫∫ d z1 d z 2 χ 22 ( z1 − z 2 )ϕ s ( z1)ϕ s ( z 2 ) 2v s +
(6.2)
To account for the strong short-term repulsions, we introduce packing constraint, which in equation can be written as: 119
ϕ s ( z ) + σ v s ∑ Pα n Rα ( z ) + r L σ v s ∑ P β n Lβ ( z ) = 1 {α }
(6.3)
{β }
The three terms on the left hand side are volume fractions of solvent, receptor and ligand, respectively. nRα(z) or nLβ(z) is the segment number density at height z of α conformation of receptor or β conformation of ligand, separately. These segment densities are counted from the conformations which are generated from molecular simulation described in detail in the previous models. For the purposes of this study, the sizes of the conformations’ spaces of receptors and ligands are chosen to be 107 and 5x106, respectively. Further enlargement of the conformational space has an ignorable effect on the results. When the cell and the substrate approach each other closer, some conformations are not allowed due to the spatial constraint and the space of conformations becomes smaller. Applying variations with respect to pdfs Pα and Pβ, volume fraction φS(z) in Eq. (6.2) and Eq. (6.3), we obtain the expressions for the following quantities which minimize the free energy in observance of the packing constraint: π (z ) = − log ϕ s ( z ) +
1 vs
1 1 ∫ dz ' χ ( z '− z )1 − ϕ ( z ') + v ∫ dz ' χ ( z '− z ) 2 ϕ ( z ') − 1 − v ∫ dz ' χ ( z '− z )ϕ ( z ') 11
s
12
s
s
22
s
(6.4)
[
]
(6.5)
[
]
(6.6)
Pα =
1 exp − ∫ dzπ ( z ) n Rα ( z ) gR
Pβ =
1 exp − ∫ dzπ ( z ) n Lβ ( z ) gL
120
s
ϕ s ( z ) = 1 − σ v s ∑ Pα n Rα ( z ) + r L ∑ P β n Lβ ( z ) {α }
(6.7)
{β }
In Eq. (6.5) and Eq. (6.6), gR and gL are normalization factors that ensure these pdfs to each add up to be 1 due to the probability interpretation. Equations (6.4)-(6.7) form a closed set and therefore are solvable. After they are solved and substituted back into Eq. (6.2), the free energy of the system can be calculated. To solve the simultaneous Eqs. (6.4)-(6.7), we apply the well-studied method presented in previous four chapters, which discretizes the space along z direction into a series of layers labeled by i. After the discretization, these equations turn out to be:
π (i ) = − log ϕ s (i ) +
δ vs
δ δ ∑ χ (i, j ) 1 − ϕ ( j ) + v ∑ χ (i, j ) 2 ϕ ( j ) − 1 − v ∑ χ (i, j )ϕ ( j ) 11
s
j
12
s
j
s
22
s
s
j
(6.8)
Pα =
1 exp− ∑ π ( j ) n) Rα ( j ) gR j
(6.9)
Pβ =
1 exp− ∑ π ( j ) n) Lβ ( j ) gL j
(6.10)
ϕ s (i ) = 1 − σ vs ∑ Pα n) Rα (i ) + r L ∑ P β n) Lβ (i ) {β } {α }
(6.11)
in which δ is the width of each layer, which we take as 0.6 nm uniformly. The hat changes those spatial distributions of segments from a linear density to a number. Furthermore, any quantity with an index "i" in the bracket should be understood as the
121
value at iδ. χ mn (i, j ) (m,n ∈ {1,2}) is the modified averaged van der Waals interaction between layers i and j, which is defined as: Ar
χ mn (i, j ) =
∫∫∫
ρ r 1 ∈ layer i
ρ d r1
ρ
∫∫∫ d r
ρ r 2 ∈ layer j
ρ ∫∫∫ d r 1
ρ r 1 ∈ layer i
2
β V mn (rρ1 , rρ2 )
(m, n ∈ {1,2})
ρ ∫∫∫ d r 2
(6.12)
ρ r 2 ∈ layer j
in which,
( )( )
r0 ρ ρ mn V mn (r 1 , r 2 ) = V 0 − 2 ρ ρ r1 − r 2
6
r0 + ρ ρ r1 − r 2
12
(6.13)
We choose r0 to be 0.3nm, V011=10kBT, V012=0.3kBT and V022=0.15kBT. Now we apply the simple thermodynamic relation to derive the adhesion force from free energy. In canonical ensemble we have: dA = − SdT − fdz + ∑ µ i d N i
(6.14)
i
The temperature of our system is assumed constant. Also the number of each of the species (receptor, ligand and solvent) remains unchanged. Therefore, we have: ∂A f = − ∂z T ,{N i}
(6.15)
We calculate a series of spatial dependent free energies and determine the force using Eq. (6.15).
6.3 RESULTS AND CONCLUSION
To test our model, we calculate adhesion forces under three receptor surface coverages: 103 /µm2, 104 /µm2 and 2*104 /µm2. For simplicity, the ligand receptor ratio rL 122
is taken as 1.0. We first calculate the free energy at varying distances “d” and use Eq. (6.15) to calculate the adhesion force. Figure 6.2 shows the force versus distance for the three surface coverages. 3
0.0
-0.2
0.005
-0.4
0.000 Force (pN/nm2)
Froce per unit area (pN/nm2)
2
10 /micron 4 2 10 /micron 4 2 2 x 10 /micron
0.2
-0.6
-0.8
-0.005 -0.010 -0.015 -0.020 12 13 14 15 16 17 18 19 Distance (nm)
-1.0 0.0
5.0
10.0
15.0
20.0
25.0
30.0
Distance (nm)
Figure 6.2 Adhesion Force versus Distance
The force density, calculated by our mean-field approach, shows a strong dependence on the adhesion receptor coverage at shorter distances but at greater separations is independent of the receptor concentration. Positive or negative force means repulsion or adhesion, respectively. The inset shows a zoomed-in view of the force density as a function of distance. Table 6-1 Components to Free Energy in the Transition Region
distance “d” (nm) 11 12 13 14
entropic contribution 0 -0.000235022 -0.000289545 -0.000387615
p-p interactions 0 0.000487725 0.000833839 0.001065458
p-s interactions 0 -0.000588931 -0.000907669 -0.001183285
123
p-p + p-s 0 -0.00010121 -7.383E-05 -0.00011783
s-s interactions 0 0.000172534 0.000245375 0.000325278
total 0 -0.0001637 -0.000118 -0.0001802
The force calculated here is in fact a force density with unit pN/nm2. The plus or minus sign means repulsion or attraction, respectively. Our results show that at large distances between the cell and the substrate, the forces are uniformly close to zero, which is a natural result considering the limited length of receptor/ligand. When these molecules are far apart only very few are able to reach each other to interact, therefore in accord with a non-interactive picture. At distances shorter than 6nm, forces are significantly higher in magnitude. At lower receptor/ligand coverages, we see a competition of entropic repulsions and energetic interactions. For example, at 103 /µm2, the cell and the substrate have repulsive forces around 14 nm. For 104 /µm2 coverage, the repulsive forces shift to the right, at 15 nm. We also note that the repulsion for higher surface coverage, e.g. 104 /µm2, is smaller than that for a lower, e.g. 103 /µm2. And repulsion does not appear in the highest density. At extremely short distance, large repulsive forces are expected because of the dominance of the strong steric repulsion. However, this behavior is not displayed in the domain studied here. In order to develop a fundamental understanding of the origin of these forces, we further investigate the role of polymer-polymer (p-p), polymer-solvent (p-s) and solventsolvent (s-s) interactions in regimes where forces go from favorable to unfavorable. Data in Table 6-1 shows the contributions of various components to free energy at multiple distances for receptor and ligand coverages both at 104 /µm2. Each value in the table is for receptor density at 104 /µm2, and other values are qualitatively similar. What vary most in magnitude are p-p interactions and p-s interactions. An interesting fact is that p-p and p-s interactions have opposite signs, and are very close in their absolute value, thus they 124
almost cancel each other. As a consequence the competition between entropic repulsion and solvent-solvent interactions become dominant. This demonstrates the necessity of incorporating conformational information and solvation in the model. With the enlargement of separation, the entropic contribution decreases, indicating an increase of entropy. The s-s VDW interaction is relatively small and goes weaker with d, thus this interaction negates some of the effects of entropic contributions. As such our model is not only able to quantify the forces, but is also able to provide a fundamental basis into the origin of the various contributors to the adhesion force between the receptors on the membrane and the ligands on the substrate. In summary, we present a novel scalable approach to investigate the distance dependent force area density between the cell and the substrate interface. Our model is consistent with other observations on how surface coverage of receptors and ligands regulates the strength of favorable adhesive interactions and is able to provide a fundamental understanding on the various contributors to adhesive forces between the membrane and the substrate ligands. We note that at high surface coverages, energetic interactions dominate entropic repulsions which results in favorable adhesive interactions for the entire domain of distances studied. At lower surface coverage of receptors and ligands (e.g. the 103 µm-2 and 104 µm-2), we observe repulsive forces between the cell and substrate at intermediate distances. While our results here are for receptors that contain 150 amino acids, changes in receptor length, composition, ligand structure and identity and coverage can be easily implemented. This makes our model applicable to a variety of biological scenarios, currently beyond the scope of traditional continuum models. Our 125
method is scalable to include additional molecular details and hence can provide researchers with a more accurate multi-scale tool to accurately quantify adhesive forces in migration and cell-matrix interactions.
126
Chapter 7 Comparison with Experiments In this chapter our results are presented with special emphasis on comparison with experimental data about cell adhesion. Two important quantities calculated with our models can be compared directly with experimental results: the binding free energy (or chemical equilibrium constant) from the motile ligands model described in chapter 3, and the force between cell and substrate depicted in chapter 6. The binding status is especially sensitive to the chemical, conformational and geometric characteristics of the receptors and ligands, and the concentrations of smaller ions. To date, there are no experimental data for the system we proposed with regards to the motile ligands. However, there is an ongoing experimental collaboration to study the effect of the sizes of spherical nanoligands on cell adhesion (Zaman and Cummings ongoing). This collaboration is focusing on functionalizing various sized nano-particles, which are homogeneously coated with fibronectin molecules to observe their transfer into the cells. For the force calculation, we can compare both the strength of adhesion and the force interaction distance profile. In the regard to the existing experimental data, Goldsmith and colleagues (Tha, Shuster et al. 1986) measured the force to remove glycophorin molecules (a membranespanning protein that carries sugar molecules for blood cell) from the membrane by measuring the shear rate necessary to dissociate red blood cells bound by antibodies and glycophorin. The strength of an individual receptor-ligand bond is observed to be between 10 and 20 pN. Evans and collaborators (Evans, Berk et al. 1991) used micropipette aspiration to measure the force necessary to dislodge a point contact between a deformable and rigidified red blood cell. The mean fracture force of a single bond is roughly 20 pN. Liang, Whitesides et al. (Liang, Smith et al. 2000) measured the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose127
presenting surfaces and found bacteria that attached to self-assembled monolayers (SAMs) required forces ranging from 3.5 to 18 pN for detachment. Chang et al. (Chang 2006) measured the adhesive force between a single Klebsiella pneumoniae type 3 fimbria and collagen IV using optical tweezers and found that the strength of the bonds is between 2 and 4 pN. Simpson measured the adhesive forces between individual
Adhesion Force (pN/bond)
Staphylococcus aureus MSCRAMMs and Protein-Coated surfaces, S. epidermidis and 25 20 15
Receptor Density: 1000/µm2
10 5 0 -5
0
50
100 150 200 250 300 Distance (A)
Figure 7.1 Comparison of Force-Distance Curve with Experiment
The left figure is excerpted from “Direct Measurements of Heterotypic Adhesion between the Cell Surface Proteins CD2 and CD48” Zhu et al. Biochemistry 2002. Force versus distance profiles between CD2 and CD48 monolayers on lipid bilayers containing NTA-DLGE and DTPC were measured using surface force apparatus. Circles indicate measurements done with membranes containing 25 mol % NTA-DLGE, and squares indicate the measurements done with 75 mol % NTA-DLGE membranes. The curves show the adhesion measured after the protein layers were left in contact for 10 min. The outward directed arrows indicate the position at which the surfaces jumped out of adhesive contact. The right plot is from our mean-field calculation.
128
fibronectin-coated surfaces, as well as Staphylococcus aureus fibronectin binding protein A mutants using optical tweezers (Simpson 2002; Simpson 2003; Simpson 2004). The reported forces range from 16-25 pN. Although the magnitude of the binding forces varies because of the dependence on the type of adhesion molecule pairs, our results from the mean-field calculation, are 1 to 3 pN per bond at 100 Å and for all distances from -2 to 35 pN per bond, which is consistent with experimentally determined values within 3σ. In regard to the comparison of our results with data for the distance profile of the forces calculated from mean-field theory, the “Direct Measurements of Heterotypic Adhesion between the Cell Surface Proteins CD2 and CD48” (Zhu 2002) will be used. Figure 7.1 displays the adhesion force measured with surface force apparatus (SFA) between CD2 and CD48 on fluid membranes. The force profiles between CD2 and CD48 monolayers bound to fluid NTA-DLGE/DTPC membranes are given. Data were obtained with 25 and 75 mol % NTA-DLGE membranes. In order to apply the JKR theory to calculate adhesion energy density, the author used “F/R” as the vertical axis, which is the force divided by the contact radius. However, we can do a conversion to the forces to see the hidden strength of the bond in this work, e.g. all the “F/R” at 50 Å measured at different approaches are around “4”, which corresponds to a strength of 8.7 pN/bond. The right figure in Figure 7.1 shows the result from our simulations, which is in essence the same as the black curve (103 µm-2) in Figure 6.2. In order to make the comparison more transparent, we made two changes: 1) To be consistent with the experimental figure, we flipped the signs of the attraction and repulsion to define the former to be plus; 2) The unit in the distance axis was changed from nm to Å. The onset of steric repulsion between these protein monolayers occurs at D ~120 Å. Upon separation (increasing D) there is a true hysteresis between the forces measured during approach (advancing) and those measured during separation (receding). The attraction force almost disappears 129
beyond 100 Å in the experimental data. This value shifts to the left at ~70 Å in our curve. This can be the result from the length of receptors we simulated, which is only 150 residues, shorter than the extracellular domain of a typical membrane receptor (~250 residues). Another interesting finding is that in the receding curves in Zhu’s, in a small region around 150 Å, repulsion dominates, which expedites the separation. According to the author, it is due to the steric effect. As a comparison, in our plot, around 130 Å, we also find repulsion. It’s difficult to ascribe this behavior to a single factor. In reality it is the combined effect of conformational entropy and osmotic pressure, which dominates the attractive interaction energy. Again, in our plot, the effect of repulsion shifts to the left. If the effect is based on steric effect, the hysteresis phenomenon cannot be produced with our model due to our incapability of simulating dynamic behaviors.
130
Chapter 8 Concluding Remarks Cell-matrix interactions regulate many events and processes in the functioning and evolution of cells. With implications from immunology to biotechnology, tissue engineering to cancer research, understanding cell matrix interactions at the fundamental thermodynamic level is essential. Our goal has been to tackle the complex cellular interactions from this aspect and we have focused on a fundamental thermodynamic quantity: the free energy. It is the energy a system of interest can gain from or lose to its environment. Compared to continuum models, our models include more molecular details. Incorporating the molecular modeling of receptor molecules (a coarse-grain simulation), our models are capable of taking into account conformations of biopolymers (e.g. receptors, ligands), besides solvation, steric, entropic, implicit and explicit, longand short-range interactions of receptors and extra-cellular matrix (ECM) molecules. Our results show how the free energy of the studied system, the binding-unbinding chemical equilibrium constant (or binding free energy) for receptor-ligand and the adhesive force are influenced by the spatial and conformational distributions of polymers, the surface coverage of receptor, the interaction distance between cell and substrate, the explicit binding energy, the implicit interaction strength, the constraints enforced to ligand’s conformations, the size of motile homogeneous nano-ligand and the clustering effect of receptors. We find that in highly concentrated systems, both Helmholtz free energy and binding free energy of receptor and ligand display a distinctive behavior, even opposite to the trends found in dilute solution. This can have significant implications in polymer 131
science (e.g. polymer brushes) and biotechnology (e.g. the devise of artificial adhesive surface in tissue engineering). The aggregation of receptors was found in numerous experiments carried out to study cellular adhesion. Incorporating the receptor clustering in our framework we found that the receptor concentration and binding energy are two key factors to the regulation of aggregation and on the free energy. We studied the influence of the free energy the non-rotational motion between a cell and substrate and find that the binding status and free energy are more sensitive to the kinetic unbinding rate and the receptor-ligand binding energy in comparison to the velocity of relative motion between a cell and the substrate for typical cell migrations. Experiments have demonstrated that the unbinding rate for different receptor-ligand pairs can have variations spanning several orders of magnitudes. This reveals the complexity and diversity of cell adhesion. The force calculated using our theory is in line with experimental reports, in both order of magnitude and distance profile. Despite reasonable agreements of our results with experiments, numerous improvements on the models can be made in future. For example, we tried to address the specificity and conformational essentiality of receptor-ligand binding. But those conformations generated from our simulation are more applicable to random coil proteins than genuine receptors. This is partially due to the lack of experimental data of receptor structures. However, as we know the more confinements adopted to simulate receptor conformations, the closer they approach to the structures in reality. We may make improvement in the modeling of receptors’ conformations by the partial embodiment of the side chains, which could allow us to potentially consider more complex interactions, 132
e.g. hydrophobic effect, hydration force. Mechanical properties, e.g. the bending and the stretching of the viscoelastic membrane, which are currently neglected, can be incorporated in the future models. Finally, our models are blind to nonequilibrium behaviors at cell-matrix interface. Despite the existence of these limitations, our models provide an alternative other than intensely studied kinetic-mechanical models to study cell adhesion. We believe with further development, more predictions will emerge from our models and hope that additional experimental verifications will highlight the wide applicability of our studies.
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Vita
Tianyi Yang was born in Rugao, Jiangsu, China, on Feb. 15, 1978, the son of Songquan Yang and Gumei Zhou. After completing his work at Rugao high school in Rugao, Jiangsu, China, he entered University of Science and Technology of China, Hefei, China in 1995. He got the Bachelor of Science degree in the department of physics in May 1999. In August 2004, he entered the department of physics, University of Texas at Austin as a graduate student. He was awarded Ph.D degree in Aug. 2009.
Permanent address: Room 301-206 Xing Fu Xin Cun, Rugao, Jiangsu, 226500, China This dissertation was typed by Tianyi Yang.
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