IOP PUBLISHING
PHYSICAL BIOLOGY
doi:10.1088/1478-3975/4/1/N01
Phys. Biol. 4 (2007) 60–63
COMMENT
Comment on elastic network models and proteins M F Thorpe Center for Biological Physics, Bateman Physical Sciences, Arizona State University, Tempe, AZ 85287-1504, USA E-mail:
[email protected]
Received 23 June 2006 Accepted for publication 23 November 2006 Published 3 April 2007 Online at stacks.iop.org/PhysBio/4/60 Abstract Elastic network models have been used to study the properties of coarse grained models of proteins and larger biomolecular complexes. In this comment, we point out that it is important to build rotational symmetry, as well as translational symmetry, into these models that are designed to describe the rigidity, and the associated low-frequency deformations. This leads to strong restrictions on what form of interactions can be used. In particular, the only allowed two-center harmonic interactions are those corresponding to Hooke springs. Additional complexity can be introduced if required by using three-center harmonic interactions.
Background In the last decade, various simple network models have been used to describe the large-scale motions of proteins and protein complexes where more detailed classical phenomenological potentials [1, 2] involving all atoms cannot be used because of the restrictions on the amount of time that can be covered in computer simulations. The reader is referred to a recent review [3] for a discussion of present applications. The present resurgence of interest in such models can be traced to Tirion [4], whose work put earlier work [5–7] in an easy-to-use form. Network models fall into two broad classes, those with Hooke springs which describe the rigidity of the protein [4], and those using a connectivity or Kirchoff matrix, which describe the connectivity of the protein [8]. Such network models provide important insights into the nature of macromolecular motions, and the relationships to biological functions, that would be hard to obtain using any other approach. For this reason it seems this is an appropriate time to pause and examine the underpinnings of these approaches. Atomic force constant models have been used for nearly a century to describe the vibrations in molecules [9, 10] and solids [11, 12] with considerable success. Only in more recent times has it been possible to derive the force constant parameters used in these models from first principle 1478-3975/07/010060+04$30.00
quantum mechanical calculations [13, 14] but this is not yet feasible in proteins because of the large number of atoms involved. Therefore phenomenological atomic models continue to provide an attractive approach in proteins and protein complexes as they can capture the main features, with the force constants being fitted to experimental observables a posteriori. An additional and novel feature of elastic network models used to describe proteins is that in most cases only some degrees of freedom are included, usually those associated with the Cα atoms on the backbone [15], both because this reduces the number of degrees of freedom that need to be considered, and because the Cα atoms provide equivalent markers across the spatial extent of the protein, making it not unreasonable for the interactions to be set equal. This can be thought of as roughly, but not exactly, replacing the protein by a uniform elastic continuum (the difference being in the way the surface of the protein is treated), coupled with a finite element approach, where the Cα atoms form the grid points (finite element methods are used in biology to study the elastic response of bones for example [16]). The idea behind using only the Cα atoms in the basis set is that the other degrees of freedom are renormalized out. While in general this would lead to frequency-dependent force constants, this frequency dependence can be reasonably neglected in the low-frequency
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domain as being small, so that the force constants are taken to be constant and frequency independent and then subsequently fitted to experimental variables, often the mean B values (Debye–Waller factors) as determined from x-ray diffraction [8]. As there are many more B values than force constants, network models lead to significant predictions, which are in good agreement with experiment in many cases.
Symmetry Symmetry places severe restrictions on the form of the interactions that can be used in elastic network models. The harmonic potential energy function [12] is written as αβ β 1 α β ui − uαj Kij ui − uj , (1) V = 2 i,j,α,β where the angular brackets indicate that each pair of atoms i, j is included only once in the summation, and the factor 1/2 is conventionally used in the definition of the force αβ constants Kij . The displacements uαi , with α = (x, y, z) being the Cartesian components, are the small movements of atoms i away from assumed equilibrium positions. The symmetric tensor K contains six constants for each pair of atoms. This potential (1) is quadratic in the displacements and two-center (involving two sites). By including only differences in the displacements uαi − uαj in the potential (1), translational invariance is properly built in from the outset. This means that if the displacements are given by a rigid body translation (2) u i = d, where the vector d is independent of the site i, then the set (2) leads to V = 0, and hence no energy being associated with the macroscopic rigid body translations of the whole macromolecule. This also leads to a proper description of the low-lying ‘acoustic’ modes where neighboring atoms move approximately in the same direction, but with a small phase shifts [17]). In this comment, we point out that it is equally important to build in rotational invariance. A general discussion of the restrictions imposed on molecular potentials is given by Herzberg [9] in chapter 2. A small rigid body rotation by an angle of θ for all sites about an (arbitrary) axis of rotation z is described by i ⊗ zˆ ), (3) u i = θ (R i is a vector from any point on the axis of rotation to where R the site i, and zˆ is a unit vector along the axis of rotation. The only invariant form of the potential (1) consistent with (3) is αβ β given by Kij = Kij riα rj leading to 1 V2 = Kij [( ui − u j ).r ij ]2 , (4) 2 i,j y where r i = rix , ri , riz is the unit vector connecting sites i, j. This simple form, involving only a dot product, is equivalent to joining sites i and j by an unstretched Hooke spring of strength Kij and linearizing. Equation (4) is the form used by Tirion [4] (see equation (4)) and in the subsequent work of Hinsen and
i
j
i
j
Figure 1. The left panel shows the displacements associated with central forces between atoms i and j, which correspond to a Hooke spring connecting the two sites and leads to both translational and rotational invariance as described by equation (4). The right panel shows the transverse displacements, which lead to translational but not rotational invariance and therefore should not have any associated restoring force.
collaborators [18–20]. The potential (4) is the only two-center harmonic potential that is consistent with both translational and rotational invariance which of course is very important to build into a model, in order to get the highly correlated directionality and phasing between local displacements in the low-frequency modes. This is the form used in what has become known as the elastic network model (ENM) and is sometimes referred to as the anisotropic network model [21]. Another example of the potential (1) that has been widely used and applied to biomolecules is the Gaussian network model (GNM) [8, 22–24] where αβ
Kij = Kijα δαβ leads to a Born model [11, 25, 26], with 1 α ui − uαj Kijα uαi − uαj = Vx + Vy + Vz , V = 2 i,j α
(5)
(6)
which decouples into three independent pieces Vx , Vy , Vz that are similar in form. It is tempting to use this form involving Vx as this involves a scalar variable at each site rather than a vector variable and so reduces the dimensionality of the problem from 3N down to N, where N is the number of nodes, often taken to be the Cα atoms. If we insert the rigid rotation (3) into the Born model (6) we see that instead of getting zero, we obtain an energy associated with the rotation that increases as O(N) and so scales with the volume of the protein, leading to ever larger energies associated with rigid body rotations by a fixed angle as the size of the protein becomes larger. The Hessians formed from the potentials in the GNM of proteins, each containing a single zero frequency, gives a total of three zero-frequency modes. In a properly rotationally invariant theory, there are a total of six zerofrequency modes associated with any free-standing body. The lack of rotational invariance in the general case (1) is most easily seen in the Born potential (6) and is illustrated in figure 1.
Discussion The GNM has superficial similarities [8] to the work of Flory [27] who considers the statistics associated with the end-to-end distance of a finite polymer chain and the partition function. This model was set up by Flory to study the statistics associated with the random walk of the freely jointed chain and not the vibrational states of a polymer chain which is quite distinct, and requires force constant matrices with spring constants. 61
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That is the Flory Gaussian distributions are associated entirely with entropy involving the random walk statistics of polymers, whereas the theory of lattice vibrations has both enthalpic and entropic contributions associated with the (small amplitude) vibrations about equilibrium positions, which are controlled by much shorter range coupling constants. The Flory theory involves matrix elements between sites i and j that are given by −1 (7) γij ∼ rij2 , 2 where the mean square distance rij ∼ n, where n is the number of links separating sites i and j in the polymer chain. This comes from the random walk statistics obeyed by polymer chains at large n, and hence leads to a slow fall-off in γij ∼ 1/n, which are therefore quite long ranged. This result would not be true or appropriate in the densely cross-linked protein structure and correlated structure of a polypeptide chain in a protein. The Born model was first used in lattice dynamics almost a century ago by Max Born [11, 25, 26], before the detailed atomic nature of matter was properly understood. The Born model was the subject of controversy [28] until its lack of rotational invariance was fully understood [29]. More complex potentials than (4) can be constructed, but as pointed out by Keating [29] these must involve the dot products of differences in the displacements. The only two-center invariant is given by (4). The next harmonic term is given by the three-center contribution to the potential 1 Kij k [( ui − u j ).r j k + ( uk − u j ).r j i ]2 , (8) V3 = 2 i,j,k where the two pairs of atoms i, j and k, j share a common vertex j. Such terms can be used to introduce additional parameters that can also then be subsequently fitted to experimentally measured quantities. Both the two-center and three-center harmonic terms (4) and (7) can be derived from writing down invariant dot products of vectors involving the displaced nodes and linearizing as advocated by Keating [29] to give (8). It is particularly important to build in rotational invariance in proteins at the potential level in order to describe various hinge motions, etc involving protruding groups of atoms moving toward each other as the hinge closes and binds a ligand as in for example adenylate kinase [30]. Subsequently, further parameters could be introduced via the three-center Keating-type forces (8). These are known to be important in small molecules, diamond, silica, polymers, etc, and indeed anywhere where covalent bonds are present. Their magnitude is typically ∼20% of the central forces, and they are probably important for hinge and similar motions. GNMs are quite distinct from ENMs as they involve a scalar variable at each site (rather than a vector variable) and so have no directional properties. Whereas ENMs couple vector displacements via Hooke springs and so probe elasticity, GNMs couple scalar quantities and hence probe connectivity. GNMs do not involve a 3N × 3N dynamical matrix but rather an N × N connectivity matrix. The GNM has been used successfully in describing B values because the B factor is largely determined by how close to the surface you are, and the connectivity matrix is a convenient phenomenological way 62
of computing this. Note that the agreement Halle [31] gets ˚ around each in figure 3 by placing spheres of radius ∼7 A non-hydrogen atom, and counting the number of atoms inside this sphere which is inversely proportional to the B value. The ˚ is close to the cutoff used in the GNM and so value of 7 A appears to set the relevant length scale. Network models have also been extended to couple together blocks in proteins [32, 33], where physical insight is used in choosing the blocks which are assumed to be more rigid than the protein as a whole. These blocks have six degrees of freedom rather than the three used above with atoms considered as point objects. Here it is also important to write down potential functions that ensure both translational and rotational invariance between the blocks and this is most easily accomplished by joining the blocks by a sufficient number of Hooke springs, located at some convenient anchor points within the blocks.
Summary The ENM, which uses Hooke springs between pairs of atoms, is firmly based in fundamentals, obeys all the translational and rotational requirements, uses standard dynamical matrices, and so is justified a priori when studying displacements, elasticity, etc. The GNM couples scalar rather than vector variables at each site and so describes connectivity rather than elasticity. The potentials for the GNM and ENM are given as equations (1) and (2) in the commentary by Rader and Bahar [34], but they do not make any comment on the large conceptual differences between these two potentials, in that they relate to connectivity and rigidity respectively. The GNM cannot be regarded as a theory of harmonic vibrations, but rather as a description of the protein connectivity. Therefore, the good agreement between the GNM with observed B values (Debye–Waller factors) in proteins places it in the same class as the work of Halle [31], who emphasizes the distance from the surface being associated with small B values, etc, and hence use of the GNM must be justified a posteriori.
Acknowledgment We acknowledge support from NSF under grant no DMR0078361 and NIH under grant no GM067249.
References [1] Brooks B R et al 1983 CHARMM: a program for macromolecular energy minimization, and dynamics calculations J. Comput. Chem. 4 187–217 [2] Pearlman D A et al 1995 AMBER, a computer program for applying molecular mechanics, normal mode analysis, molecular dynamics and free energy calculations to elucidate the structures and energies of molecules Comput. Phys. Commun. 91 1–41 [3] Tama F and Brooks C L 2006 Symmetry, form, and shape: guiding principles for robustness in macromolecular machines Annu. Rev. Biophys. Biomol. Struct. 35 115–33 [4] Tirion M M 1996 Large amplitude elastic motions in proteins from single-parameter atomic analysis Phys. Rev. Lett. 77 1905–8
Comment
[5] Brooks B and Karplus M 1983 Harmonic dynamics of proteins: normal modes and fluctuations in bovine pancreatic trypsin inhibitor Proc. Natl Acad. Sci. USA 80 6571–5 [6] Go N, Noguti T and Nishikawa T 1983 Dynamics of a small globular protein in terms of low-frequency vibrational modes Proc. Natl Acad. Sci. USA 80 3696–700 [7] Levitt M, Sander C and Stern P S 1985 Protein normal-mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme J. Mol. Biol. 181 423–47 [8] Bahar I, Atilgan A R and Erman B 1997 Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential Fold. Des. 2 173–81 [9] Herzberg G 1945 Infrared and Raman Spectra of Polyatomic Molecules (New York: Van Nostrand) [10] Wilson E B J, Decius J C and Cross P C 1955 Molecular Vibrations: The theory of Infrared and Raman Vibrational Spectra (New York: Dover) [11] Born M and Huang K 1954 Dynamical Theory of Crystal Lattices Chapter V (New York: Oxford University Press) [12] Maradudin A A, Montroll E W and Weiss G H 1963 Theory of Lattice Dynamics in the Harmonic Approximation (New York: Academic) [13] Car R and Parrinello M 1985 Unified approach for molecular-dynamics and density-functional theory Phys. Rev. Lett. 55 2471–4 [14] Adams G B et al 1991 1st-principles quantum-moleculardynamics study of the vibrations of icosahedral C60 Phys. Rev. B 44 4052–5 [15] Tama F and Sanejouand Y H 2001 Conformational change of proteins arising from normal mode calculations Protein Eng. 14 1–6 [16] Lewis G and Duggineni R 2006 Finite element analysis of a three-dimensional model of a proximal femur-cemented femoral THJR component construct: influence of assigned interface conditions on strain energy density Biomed. Mater. Eng. 16 319–27 [17] Ashcroft N W and Mermin N D 1976 Solid State Physics vol 21 (New York: Holt, Rinehart and Winston) 826 pp [18] Hinsen K 1998 Analysis of domain motions by approximate normal mode calculations Proteins 33 3–417
[19] Hinsen K, Thomas A and Field M J 1999 Analysis of domain motions in large proteins Proteins 34 3–369 [20] Hinsen K and Kneller G R 1999 A simplified force field for describing vibrational protein dynamoics over the whole frequency range J. Chem. Phys. 24 10766–9 [21] Atilgan A R et al 2001 Anisotropy of fluctuation dynamics of proteins with an elastic network model Biophys. J. 80 1–505 [22] Rader A J et al 2006 The Gaussian network model: theory and applications Normal Mode Analysis. Theory and Applications to Biological and Chemical Systems ed Q Cui and I Bahar (London: Chapman and Hall) pp 41–64 [23] Bahar I et al 1998 Vibrational dynamics of folded proteins: significance of slow and fast motions in relation to function and stability Phys. Rev. Lett. 80 2733–6 [24] Haliloglu T, Bahar I and Erman B 1997 Gaussian dynamics of folded proteins Phys. Rev. Lett. 79 3090–3 [25] Born M and von Karman T 1912 On fluctuations in spatial grids Physik. Z. 13 297–309 [26] Born M 1914 The crystal lattice theory of diamonds Ann. Physik. 44 605–42 [27] Flory P J 1976 Statistical mechanics of random networks Proc. R. Soc. A 351 351–80 [28] Lax M 1965 Lattice Dynamics (New York: Pergamon) p 583 [29] Keating P N 1966 Effect of invariance requirements on elastic strain energy of crystals with application to diamond structure Phys. Rev. 145 637 [30] Creighton T E 1996 Proteins: Structures and Molecular Properties 2nd edn (New York: Freeman) [31] Halle B 2002 Flexibility and packing in proteins Proc. Natl Acad. Sci. USA 99 3–1274 [32] Durand P, Trinquier G and Sanejouand Y H 1994 New approach for determining low-frequency normal-modes in macromolecules Biopolymers 34 6–759 [33] Tama F et al 2000 Building-block approach for determining low-frequency normal modes of macromolecules Proteins 41 1–1 [34] Bahar I and Rader A J 2005 Coarse-grained normal mode analysis in structural biology Curr. Opin. Struct. Biol. 15 586–92
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PHYSICAL BIOLOGY
doi:10.1088/1478-3975/4/1/N02
Phys. Biol. 4 (2007) 64–65
REPLY
Reply to ‘Comment on elastic network models and proteins’ Ivet Bahar1, Chakra Chennubhotla1 and Burak Erman2 1 Department of Computational Biology, School of Medicine, University of Pittsburgh, Pittsburgh, PA 15213, USA 2 Department of Chemical and Biological Engineering, Koc University, Sariyer 34450, Istanbul, Turkey
Received 7 September 2006 Accepted for publication 28 November 2006 Published 3 April 2007 Online at stacks.iop.org/PhysBio/4/64 Abstract The Gaussian network model (GNM) is based on the underlying assumption of isotropic and Gaussian fluctuations of residues controlled by inter-residue contact topology, similar to the statistical theory of junction fluctuations developed by Flory and coworkers for polymer networks. We clarify here these underlying assumptions and emphasize the fact that the GNM predictions (e.g. mean-square fluctuations, cross-correlations and global mode shapes) are rotationally invariant and more accurate than their counterparts predicted by elastic network normal mode analysis.
Below are three major points that we would like to clarify in response to Thorpe’s comment [1]: 1. Two different and opposing views of equilibrium fluctuations in proteins are in question here. In the first picture, the residues may be assumed to be rigidly fixed at lattice-like points, and exhibit small amplitude fluctuations, similar to those in crystals. In the second view, the fluctuations are larger similar to those of the junctions in a polymer network. The preceding comment is based on the first view. The Gaussian network model (GNM) assumes the second view. The statement ‘The GNM has superficial similarities to the work of Flory [ . . . ]’ is misleading. The foundations of the model can be traced back to two seminal papers from Flory and Pearson [2, 3] on the statistical thermodynamics and fluctuation dynamics of polymer networks, which, in turn, are based on the original work of James and Guth in the early 1940s [4]. The theory set forth by these pioneering scientists has since been extensively used in rubber elasticity and polymer physics [5–8]. The GNM of protein dynamics [9, 10] has utilized the key physical concepts, mathematical formulations and even the notation, adopted in these studies. 2. The GNM is based on two assumptions: (i) the extent of the fluctuations of a given residue decreases as the number 1478-3975/07/010064+02$30.00
of other residues sharing its fluctuation volume increases, and (ii) the fluctuations are isotropic and Gaussian. There are no other assumptions in the model. These are also the two basic assumptions of the theory of random amorphous elastic networks [2, 5, 6]. That each residue has a welldefined mean position is implicit in these assumptions. The only information needed to build the model is the ‘identity’ (index) of amino acids (network nodes) that interact with each other (that are connected by an elastic spring), and in the GNM this information maps to the Kirchhoff matrix of inter-residue contacts. The concern about the lack of rotational invariance raised in the preceding comment is therefore irrelevant. How can a model that does not use as input any directional or vectorial quantities, and not even residue coordinates (but contact topology, exclusively), be frame dependent or yield any results that are rotationally variant? 3. In the literature there has been a tendency to view the anisotropic network model (ANM) [11], or the normal mode analyses (NMA) with uniform harmonic potentials (originally introduced by Tirion [12] and Hinsen [13] at the atomic and residue levels, respectively) also referred to as elastic network model (ENM) NMA [14, 15], as an improved version of the GNM, in which the isotropy assumption of the GNM has been removed. The same mistake is seen in the preceding comment [1]. To avoid
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further confusion, we would like to emphasize again that; • The ANM (or ENM NMA) is not an improved version of the GNM. In contrast, it is based on a harmonic potential that is physically less realistic than that implicitly underlying the GNM. • The information on the three-dimensional deformation vectors in the ANM—as opposed to deformation magnitudes only in the GNM, is obtained at the expense of adopting this less realistic potential. The reader is referred to two recent reviews for explicit expressions of the underlying potentials and their implications [16, 17]. • Not surprisingly, the mean-square fluctuations computed by the GNM have been shown in many applications to agree better with experimental data than those predicted by the ANM. See for example the work of Phillips and coworkers [18]. • We have extensively analyzed the predictions of the GNM and ANM with two servers [19, 20] that we have developed for the respective GNM and ANM analyses of PDB structures. Our recommendation is to use the GNM (rather than ANM) for evaluating mean-square fluctuations of residues and their crosscorrelations, as well as the profiles of residue displacement sizes in collective modes, and to resort to the ANM only for exploring properties that cannot be computed using the GNM, such as deformation vectors or molecular motions/movies. In conclusion, the two assumptions inherent to the GNM lead to a simple Hamiltonian that defines the equilibrium dynamics of the network. There is no allusion to the random walk of polymer chains in the model, and in this respect the preceding comment misses the entire point. The efforts therein to remove the isotropy assumption of the GNM by invoking the analogy to Born potential may lead to another model of different complexity. However, the predictive value of such a model above that of the GNM cannot be assessed a priori. In the absence of quantitative formulation and comparison with existing evidence, the utility and significance of the preceding comment remains questionable. Finally, it should be acknowledged that the network model and methodology used in the GNM has proven useful not only in rubber elasticity and protein dynamics, but in other disciplines (e.g. computer science and information theory) that utilize the concepts of graph theory and the Laplacian (or Kirchhoff) matrix [21].
References [1] Thorpe M 2007 Comment on elastic network models and proteins Phys. Biol. 4 60–3 [2] Flory P J 1976 Statistical thermodynamics of random networks Proc. R. Soc. A 351 351–80 [3] Pearson D S 1977 Scattered intensity from a chain in a rubber network Macromolecules 10 696–701 [4] James H M and Guth E 1943 Theory of the elastic properties of rubber J. Chem. Phys. 11 455–81 [5] Flory P J and Erman B 1982 Theory of elasticity of polymer networks 3 Macromolecules 15 800–5 [6] Erman B and Flory P J 1982 Relationship between stress strain and molecular constitution of polymer networks comparison of theory with experiments Macromolecules 15 806–11 [7] Mark J E and Erman B 1988 Rubberlike Elasticity A Molecular Primer (New York: Wiley) [8] Kloczkowski A, Mark J E and Erman B 1989 Chain dimensions and fluctuations in random elastomeric networks I phantom gaussian networks in the undeformed state Macromolecules 22 1423–32 [9] Bahar I, Atilgan A R and Erman B 1997 Direct evaluation of thermal fluctuations in protein using a single parameter harmonic potential Folding Des. 2 173–81 [10] Haliloglu T, Bahar I and Erman B 1997 Gaussian dynamics of folded proteins Phys. Rev. Lett. 79 3090–3 [11] Atilgan A R, Durell S R, Jernigan R L, Demirel M C, Keskin O and Bahar I 2001 Anisotropy of fluctuation dynamics of proteins with an elastic network model Biophys. J. 80 505–15 [12] Tirion M M 1996 Large amplitude elastic motions in proteins from a single-parameter atomic analysis Phys. Rev. Lett. 77 1905–8 [13] Hinsen K 1998 Analysis of domain motions by approximate normal mode calculations Proteins 33 417–29 [14] Tama F and Sanejouand Y H 2001 Conformational change of proteins arising from normal mode calculations Protein Eng. 14 1–6 [15] Ma J 2005 Usefulness and limitations of normal mode analysis modeling dynamics of biomolecular complexes Structure 13 373–80 [16] Chennubhotla C, Rader A J, Yang L-W and Bahar I 2005 Elastic network models for understanding biomolecular machinery: from enzymes to supramolecular assemblies Phys. Biol. 2 S173–S180 [17] Bahar I and Rader A J 2005 Coarse-grained normal modes in structural biology Curr. Opin. Struct. Biol. 15 586–92 [18] Kundu S, Melton J S, Sorensen D C and Phillipd G N Jr 2002 Dynamics of proteins in crystals: comparison of experiment with simple models Biophys. J. 83 723–32 [19] Yang L-W, Rader A J, Li X, Jursa C J, Chen S C, Karimi H A and Bahar I 2006 oGNM: online computation of structural dynamics using the Gaussian network model Nucl. Acids Res. 34 W24–31 [20] Eran E, Yang L-W and Bahar I 2006 Anisotropic network model: systematic evaluation and a new web interface Bioinformatics 22 2619–27 [21] Chung F R K 1997 Spectral Graph Theory (CBMS Lecture) (Providence, RI: American Mathematical Society)
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