CAN COMPLEX CELLULAR PROCESSES BE ... - Semantic Scholar

January 30, 2009 19:16 WSPC/185-JBCB

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Journal of Bioinformatics and Computational Biology Vol. 7, No. 1 (2009) 243–268 c Imperial College Press

CAN COMPLEX CELLULAR PROCESSES BE GOVERNED BY SIMPLE LINEAR RULES?

KUMAR SELVARAJOO∗ , MASARU TOMITA and MASA TSUCHIYA† Institute for Advanced Biosciences Keio University, Baba-Cho, 14-1, Tsuruoka Yamagata, 997-0035, Japan ∗[email protected] †[email protected] Received 12 May 2008 Revised 22 October 2008 Accepted 22 October 2008

Complex living systems have shown remarkably well-orchestrated, self-organized, robust, and stable behavior under a wide range of perturbations. However, despite the recent generation of high-throughput experimental datasets, basic cellular processes such as division, differentiation, and apoptosis still remain elusive. One of the key reasons is the lack of understanding of the governing principles of complex living systems. Here, we have reviewed the success of perturbation–response approaches, where without the requirement of detailed in vivo physiological parameters, the analysis of temporal concentration or activation response unravels biological network features such as causal relationships of reactant species, regulatory motifs, etc. Our review shows that simple linear rules govern the response behavior of biological networks in an ensemble of cells. It is daunting to know why such simplicity could hold in a complex heterogeneous environment. Provided physical reasons can be explained for these phenomena, major advancement in the understanding of basic cellular processes could be achieved. Keywords: Complex processes; simple governing rules; perturbation–response approaches; systems biology; mass-action response.

1. Introduction Cellular systems are characterized by the complex interplay of DNA, RNA, proteins, and metabolites to achieve specific goals: cell division, differentiation, apoptosis, etc. To unravel the cellular complexity, high-throughput experimental technologies have been recently developed to generate large datasets at the level of genomics, proteomics, and metabolomics.1–3 Although this is valuable advancement in modern biology, cellular properties such as growth, ageing, morphology, and immune response still remain largely elusive. To understand such complex and dynamic behavior of living systems, which may be governed by key regulatory principles, 243


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the development of systems biology approaches which integrates theoretical concepts with experimental methodologies is required. In recent years, various studies have demonstrated that biological networks are robust to random errors.4,5 Typically, random deletions, mutations, or duplications of genes have been shown not to affect the overall network behavior or phenotypic outcome of living systems, revealing the persistence of stable and robust behavior under diverse perturbations.6 Ishii et al.,7 for instance, demonstrated robustness of biological systems by showing that metabolic gene disruption did not affect the levels of most metabolites. This was achieved through the activation of other genes and proteins which resulted in rechannelling of metabolic fluxes. On the other hand, the stability of biological network was demonstrated by Leibler and colleagues8,9 who created a very simple two-state kinetic-model of E. coli chemotactic network and showed that the variation of biochemical parameters does not significantly affect bacterial chemotaxis in silico and in vivo. They demonstrated that bacterial adaptation property is a consequence of network’s connectivity and does not require the precision of parameter values. Despite being robust and stable, biological systems have been shown to be fragile on a number of occasions and are vulnerable to attacks.10–13 The removal of a relatively few highly-linked molecules (molecules that interact with many other molecules) can lead to system failure, a property found in scale-free networks.14 This postulation suggests that biological networks are not connected randomly, but centers around a small proportion of “hub” and “connector” elements.5,14–16 For example, the activation of tumor necrosis factor (TNF)-receptor–associated factor 6 (TRAF6) occurs in several distinct signaling cascades. Targeted disruption of this molecule in mice has led to systemic failure and premature death,17,18 suggesting that TRAF6 is a “hub/connector” element. Similarly, “hub/connector” molecules such as cycA or cdk2 together with p53 are also identified in p53 network, a well-known tumor suppressing pathway that controls apoptosis, cell cycle, etc.19 Therefore, catastrophic failure can occur due to the lost of function of such crucial family of “hub/connector” molecules. Although this important information (robust to perturbation, fragile to targeted attacks) about the organization of biological networks has been uncovered, the comprehensive understanding of dynamic cellular behavior and its control mechanisms remains poorly understood. 2. Modular Biological Networks The enormous complexity of biological networks could be simplified according to the modular nature of cellular functions.5,20–22 For instance, the understanding of extracellular signal propagation into the nucleus is obtained by the modularization of signal transduction cascades.23 Similarly, we can think of transcription modules generated by common transcription factors24 with co-regulating microRNA.25 In biochemical regulation, distinct modules such as glycolysis and Krebs Cycle also exist.


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To interpret the dynamics and regulatory mechanisms, the analysis of molecular interactions within a network module is required. Usually, there are a large number of molecules present within a module that can potentially react with one another through probabilistic or stochastic processes. However, well-defined signal transduction module in living systems cannot result through random collisions or interactions. The innate immunity, through the Toll-like receptors (TLRs), exhibits highly complex but coordinated modular networks of the mammalian defense system.26,27 Upon the recognition of pathogens-associated molecular patterns (ligand) by immune cells such as macrophages and dendritic cells, the TLRs recruit adaptor molecule myeloid differentiation primary-response protein 88 (MyD88) to its intracellular Toll/IL-1 receptor (TIR) domain, resulting in the phosphorylation of Iκb kinases (IKKs) and MAP kinases (MAPKs) that activate key transcription factors (TFs), nuclear factor–κB (NF-κB) and activator protein (AP)-1, respectively.23,28 Subsequently, these activated TFs control the rate of RNA polymerase activities of proinflammatory cytokine genes in the nucleus by binding to the relevant DNA promoter regions (binding sites). After the process of transcription and the subsequent export of the mRNA out of the nucleus, proinflammatory proteins are translated and further complex adaptive immune regulation is established by the immune cells.26,27 Although transcriptional and translational processes have been shown to be stochastic in single cells,29,30 in an ensemble of cells, deterministic (averaging) collective phenotypic behavior emerges.16,31–33 In addition, the activation of specific kinases and phosphatases acting on specific substrates to conduct directed reactions suggests that the occurrence of highly sophisticated and orchestrated signal transduction cannot be possible by random or stochastic interactions alone. Thus, there must be organizational principles to explain the highly coordinated “average” behavior of biological network modules.34,35

3. Perturbation-Response Approach to Analyze Modular Biological Networks In the quest to uncover “rules” to understand dynamic behavior, control mechanisms, and emergent properties in biology, deciphering governing principles of network interactions within network modules is the key first step. The main difficulty in doing this has been the determination of detailed molecular reaction principles and associated parameter values that govern each interaction (“bottom-up” approach). Mostly, the reaction parameters are primarily determined in vitro in which the species in question has been deliberately purified from its physiologic neighbors. Whereas in vivo, reaction parameters can vary quite significantly due to the presence of other interactions apart from the simple isolated in vitro reaction, such as feedback loops (FBL), feedforward loops (FFL), crosstalk mechanisms, branching reactions, etc. Thus, bottom-up methodologies relying deeply on reaction parameters suffer from huge setbacks.36,37


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To shed light on this difficulty, the utilization of “top-down” dynamical approaches that can analyze the response behavior of biological networks without being highly sensitive to the accuracy of parameter values is necessary. Recently, several methodologies that do not rely on kinetic parameters for the determination of network structure within functional modules have been developed. These can be broadly classified into quantitative and qualitative approaches. Examples of quantitative approaches include the perturbation–response experiments, which infer network connectives using perturbation and activation dynamics of reaction species,38−40 and sequential dynamical approaches, which decipher causal networks by mapping input and output data.41,42 The qualitative approaches consists of stoichiometric methods, which deal with reaction stoichiometry to identify reaction mechanisms,43,44 singular-value decomposition methods, statistical clustering for identifying coordinated behavior from highthroughput data,45,46 and Bayesian network approaches for constructing network connectivity using probabilistic graphs.47,48 Here, we review the quantitative perturbation–response approaches focusing on how simple rules can be found, and how these rules lead to uncovering important network features such as signaling flux propagation, network topology including feedback loops and missing molecules. The basic principle is to induce a controlled perturbation of a certain (input) reaction species of a system kept at steady-state and monitor the response of the concentration/activation levels of other species (output) of the system. These studies show evidences, as described below, that first-order mass-action response equations lead to the uncovering of simple linear rules that govern the dynamic behavior of biological network modules. 4. Examples of Perturbation-Response Approaches 4.1. Detecting feedback control of IKK activity using computational model based on mass-action response equations Werner et al.49 developed a computational model based on simple mass-action response equations with IKK perturbation to study the functional pleiotropism (immune, inflammation, survival, apoptosis, etc.) of signal-responsive transcription factor NF-κB. Using the model, they investigated the capability of the same IKK– IκB–NF-κB signaling module to elucidate different gene expression programs for different stimuli in murine embryonic fibroblasts. Since the temporal control of NF-κB activity can lead to selective gene expression,50 stimulus-specific temporal control of IKK activity might allow for distinct biological responses if signal processing within the IKK–IκB–NF-κB signaling module results in distinct NF-κB activity profiles. Compared with experimental data for both the tumor-necrosis factor (TNF) and lipopolysaccharide (LPS) stimulation, their results suggested that the mechanism for the rapid termination of IKK activity in TNF stimulation is due to negative


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feedback control by post induction of A20, while the prolonged activation of IKK observed in LPS stimulation is due to positive feedback of autocrine TNF signaling. Therefore, IKK activity can be switched by regulating certain key molecules such as A20 for distinct cellular response. Although these data show that perturbation– response model, utilizing mass-action response equations, is useful to infer key biology network features, we show below that even first-order mass-action response model can decipher such biological features. 4.2. Discovering switching behavior of MAPK signaling between epidermal growth factor (EGF) and neuronal growth factor (NGF) stimuli Modular Response Analysis (MRA) developed by Kholodenko et al.37,51,52 is an example of first-order response approach derived from Jacobian matrix (see Sec. 4.4) where experimentally measured steady-state levels at different nodes (species) in a network are compared after successive small perturbations are given to all nodes. A perturbation, pj , given to a steady-state, xi , propagates through the local network and relaxes to a new steady-state, which can be measured experimentally to determine the global response coefficient Rij (= ∂lnxi /∂pj ). The local response coefficients rij for the network or module are computed from global response coefficient such that j rij Rjk = 0 to infer connectivity of all the nodes within the network. A direct effect of node j on node i is quantified by the local response coefficient rij expressed as the fractional change ∆xi /xi brought about a small fractional change in xj : ∆xi /xi ∂ ln xi rij = lim = . ∆xj →0 ∆xj /xj ∂ ln xj Thus, rij > 0 indicates that node j activates node i, rij < 0 indicates that node j inhibits node i, and rij = 0 indicates that node j and node i are not connected (simple rules). Therefore, to determine the connectivities between species, (1) all species in a network should be known and (2) perturbation of all species within the network is required. Santos et al.52 used MRA and revealed that MAPK signaling, conserved in numerous cellular processes and governing cell fate, results in stimulus specific differential activation of cellular processes. Their study (1) determined the wiring of the MAPK network; (2) found that EGF, which stimulates EGF receptor (EGFR), elicits negative feedback from ERK, whereas NGF, which activates TrkA receptor, induces positive feedback; (3) found that the positive feedback not only sustained ERK activity for long periods, but also induced switch-like, bistable dynamics of the NGF-stimulated ERK cascade; and (4) showed rewiring of the EGF-stimulated MAPK network by concomitant activation of protein kinase C (PKC). Thus, analysis of the sign and absolute values of local and global response coefficients based on Jacobian matrix (first-order) was able to extract the connectivity and regulatory motifs of MAPK network.


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4.3. Detecting connectivities of reaction molecules by first-order mass-action response equations Ross and colleagues53,54 applied pulse perturbation to deterministic kinetic evolution equation and constructed rules for detecting causal connectivities of species in a reaction network. By studying the effects of giving small perturbation to the concentration of one or more species and using first-order mass-action response equations to monitor the response profiles of other species within a mass-conserved system, they proposed the analysis of time to reach peak, and that the actual peak levels for all species can be used to determine the causal relationships or connectivities between them. The basic principle suggests that simple linear rules might exist for the propagation of small perturbation (response wave) within a biochemical system. That is, the response rate of species in a mass-conserved system at initial steady-state can be approximated by the first order mass-action response equation, given small perturbation to one or more species. To illustrate, let us consider a linear-chain of reactions (X1 → X2 → X3 → · · ·) at steady-state condition.53 If the concentration of X1 is pulse perturbed, concentrations of X2 , X3 , X4 , etc. will increase, go through a maximum, and then decrease back to their steady-state values in sequential order (stable perturbation-response phenomenon) (Fig. 1). Thus, the temporal order of the responses can yield the causal (direct) connectivities of the species in the reaction mechanism. The experiments preserve the total input and output fluxes. Under such condition, a linear superposition of propagation response waves connects the species between input and output fluxes. Thus, simple linear rules can be derived for the

Fig. 1. Temporal response profiles of concentration in linear-chain of the first-order mass-action reaction network for pulse perturbation of species X1 (see main text). The units are arbitrary, scaled by rate coefficients. Figure adapted from Vance et al.53


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system, notably, (1) the time to reach peak values (u1 , u2 , etc. in Fig. 1) increases and its amplitude decreases as we move down the reaction network, unless there are other features such as feedback reactions; (2) the initial response gradient can be used to determine the location of a reactant species in a network, i.e. the steepest gradient is the closest to the perturbed reactant species and the lowest gradient is the farthest; (3) reactant species that are not connected to the system do not show any response profile; (4) since law of mass conservation is used for pulse propagation, the sum of all species’ deviations from steady-state (weighted by stoichiometric coefficients) is constant. Therefore, it can help to determine correct stoichiometric coefficients. Using the simple linear rules, Torralba et al.55 investigated the glycolysis pathway, a well-appreciated network module involving energy metabolism. They confirmed that simple linear rules can be used to reinfer the causal connectivity of glycolysis pathway. In other words, perturbation–response study clearly provides evidence that complex biological networks could be approximated by first-order mass-action response equations resulting in the linear superposition of propagation response waves.

4.4. Uncovering innate immune signaling features using the first-order mass-action response equations In our own work, we utilized the perturbation–response approach and investigated the well-characterized innate immune response to invading pathogen based on the TLR4 signaling pathways. Constructing the first-order mass-action response equations, we predicted and validated the presence of novel (1) signaling intermediates,56 (2) crosstalk mechanisms,57,58 and (3) signaling flux redistribution (SFR)59 of the TLR4 signaling. To illustrate our approach, we perturb a stable biological network consisting of n species from reference steady-states. In general, the resultant changes in concentration of the species are governed by the kinetic evolution equation: ∂Xi = Fi (X1 , X2 , . . . , Xn ), ∂t

i = 1, . . . , n,

(1)

where the corresponding vector form of Eq. (1) is ∂X ∂t = F (X). F is a vector of any nonlinear functions including diffusion and reactions of the species vector X = (X1 , X2 , . . . , Xn ), which represents activated concentration levels of signaling molecules, for example, through phosphorylation, or binding concentration of transcription factors to promoter regions. The response to perturbation can be written by X = X 0 + δX, where X 0 is the reference steady-state vector and δX is the relative response from steady-states (δX t=0 = 0). We consider small perturbation around steady-state in which higher-order terms become negligible in Eq. (1), resulting in approximation of the first-order term.


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∼ ∂F (X) In vector form dδX dt = ∂X |X=X 0 δX (note the change from partial derivative to total derivative of time), where the zeroth order term F (X 0 ) = 0 at the steady(X) |X=X 0 . The state X 0 and the Jacobian matrix or linear stability matrix, J = ∂F∂X elements of J are chosen by fitting δX with corresponding experimental profiles and knowing the activation network topology. Note that Jacobian matrix elements (response coefficients) can include not only reaction information, but also spatial information such as diffusion and transport mechanisms. Hence, the amount of response (flux propagated) along a signaling pathway can be determined using first-order mass-action response equations, dδX dt = J δX. 4.4.1. Strategy to analyze the topology and dynamics of biological pathways using the first-order mass-action response equations We developed a strategy to analyze the topology and dynamics of biological pathways based on perturbation–response experiments. As an illustration, we investigated the TLR4 signaling. Initially, we obtained the biological network from wellknown pathway databases such as the Kyoto Encyclopedia of Genes and Genomes (KEGG) database60 and BioCarta (http://biocarta.com/) (step 1 of Fig. 2). Using this topology, we developed a computational model representing each reaction by first-order mass-action response equation with pulse perturbation given to the receptor (step 2). The parameters of the model (elements of J or response coefficients) were estimated by fitting the simulation profiles of downstream molecules, such as NF-κB and MAP kinases, with corresponding activation profiles of wildtype cells for a period up to 1 hr representing primary signaling process (step 3). If the model successfully fits the wildtype profiles of all tested molecules, we accept the model (step 4, Reference Model). Otherwise, we adjust the model topology or parameter values (response coefficients) until the model is able to fit the experimental profiles reasonably (step 5). There can be several Js that could fit wildtype activation profiles of NF-κB and MAP kinases. To reduce the parameter space for J, we utilized the temporal experimental profiles of other conditions such as MyD88 knock-out (KO) and TRAF6 KO. Each in silico KO was performed by setting the response coefficients involving the KO molecule null. If our KO simulation does not compare well with experiments, we retune the response coefficients till the model simulations fit the activation profiles of both wildtype and KO conditions (step 6). If unsuccessful, we modify the topology so that a better fit can be obtained, for example, adding novel signaling intermediates to obtain the delayed activation56 or crosstalk mechanism to provide an alternative source of activation in the KO condition57,58 (step 5 and go to step 3 to repeat process). We repeat the procedure of modifying parameter values and topology until we obtain reasonable predictions for all conditions. At this point, we accept the model and call it the Robust Model (step 7). Once this is reached, the comparison of our initial model topology with the final Robust Model will result in the discovery of novel network features.


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Fig. 2. Strategy to analyze the topology and dynamics of biological pathways using the first-order mass-action response equations; see main text for details.

As future work, we could consider secondary signaling (at later time points) such as autocrine or paracrine events49,61 by creating a separate secondary signaling module, and applying the above strategy to connect it with the primary signaling module.


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(a)

(b)

Fig. 3. Schematic representation of TLR4 MyD88- and TRIF-dependent signaling pathways. (a) Pathway obtained directly from KEGG database,60 (b) Predicted novel signaling intermediates56 and crosstalk mechanisms.57,58 Note that the crosstalk mechanisms between TRIF to TRAF6 and TRIF to RIP1 were absent from the KEGG database prior to the publication of Refs. 57 and 58.

4.4.2. Predicting missing molecules/processes in TLR4 signaling In TLR4 signaling, upon LPS recognition, MyD88 and TIR-domain–containing adaptor inducing IFN-β (TRIF) bind to TLR4 and trigger the MyD88-dependent and TRIF-dependent pathways, respectively [Fig. 3(a)]. Based on known signaling topology, we created a wildtype model, as described above (steps 1 to 4) and performed in silico MyD88 KO (shutting down the MyD88-dependent pathway, step 6). We realized that our initial model simulation could not reproduce the delayed experimental time–series activation profile of downstream NF-κB. Time delay processes can occur due to several reasons such as directed transport machinery, complex formation, and missing reactions. The Jacobian matrix J can represent such information through additional terms. For example, to simulate the desired 20-minutes delayed experimental activation of NF-κB in MyD88 KO compared with wildtype, we added five novel terms (intermediates) along the TRIF-dependent pathway [Fig. 3(b), steps 5 and 6] to reproduce the delayed activation. It is important to note that to maintain peak activation level, reducing response coefficients alone cannot provide time-delay activation profile. Although the five additional hypothetical terms may not exactly represent five actual intermediate molecules, these terms broadly represent missing biological molecules/processes. Our predicted novel terms56 were subsequently confirmed experimentally to be: (1) phosphorylation of TRAM by PKCε upstream of TRIF,62


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(2) the binding of TRAF3 to TRIF,63 and (3) the upstream sequential events (endocytosis of TLR4) leading to TRAM activation.64 Applying our approach further on the TLR4 signaling and analyzing both NFκB and MAP kinases in wildtype, MyD88 KO, TRAF6 KO, TRIF KO, TAK1 KO, and MyD88/TRAF6 double KO, we were able to propose two additional crosstalk mechanisms between TRIF and TRAF6, and between TRIF and RIP1, which are necessary to explain the induction of MyD88-dependent genes Tnf and Ilb in MyD88 KO.58 4.4.3. Discovering Signaling Flux Redistribution (SFR) at pathway junctions More recently, we demonstrated that the removal and addition of one molecule at a signaling pathway junction, enhances and impairs, respectively, the entire alternative pathway through Signaling Flux Redistribution (SFR).59 Using our TLR4 model, we investigated the TRIF (TRAM)-dependent pathway in wildtype and MyD88 KO condition and found that in silico removal of MyD88 resulted in the increased activation of TRAM-dependent pathway (Fig. 4). The removal of MyD88 solely abolishes the propagation of signal transduction from TLR4 to MyD88. As a result, interaction between TLR4 and TRAM increases due to the law of mass conservation derived from first-order response: d(T LR4) d(M yD88pathway) d(T RAM pathway) + + = 0, (2) − dt dt dt LR4) d(MyD88pathway) is rate of TLR4 activation by LPS perturbation, where d(Tdt dt d(T RAMpathway) and are signaling flux of MyD88 and TRAM pathways, respecdt d(MyD88pathway) d(T RAMpathway) tively. In in silico MyD88 KO, = 0 and dt dt increases, resulting in enhanced activation of the entire TRAM-dependent pathway, i.e. SFR. Furthermore, we examined the signaling outcome of overexpressing MyD88 on the TRAM-dependent pathway. Our simulations resulted in reduced activation of TRAM-dependent molecules. That is, when molecules at pathway junctions are reduced and enhanced, the activation of alternative pathways is enhanced and reduced, respectively. We describe this rechannelling of signal transduction, which was validated experimentally at two pathway junctions of TLR4, as SFR. Although we demonstrated SFR for molecules with common binding domain, from the law of mass flow conservation, SFR might also occur between molecules with different binding domains at pathway junctions (Fig. 5). Thus, SFR, a simple linear rule, illustrates a novel mechanism for enhanced activation of alternative pathways when molecules at pathway junctions are removed. 4.5. Reverse engineering to decipher local biological network connectivities: the Non-Integral Connectivity Method (NICM) So far, we have shown that complex cellular processes can be understood by simple linear rules derived from the first-order response equations. Using this approach


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Fig. 4. Enhancement of TRAM-dependent pathway in MyD88 KO due to SFR. Simulation profiles (arbitrary units) of (a) TRAM activation, (b) IRF3 activation, and (c) Cxcl10 induction in the wildtype (thin lines), MyD88 KO (dotted lines), and two-fold overexpression (thick lines) of MyD88. (d) Schematic drawing of SFR. Top: Wildtype. Fluxes propagate through both the MyD88-dependent and TRAM-dependent pathways. Middle: MyD88 KO. More fluxes propagate or overflow through the TRAM-dependent pathway resulting in increased Cxcl10 induction. Bottom: MyD88 overexpression by two-fold. Figure adapted from Selvarajoo et al.59

to analyze network modules is forward engineering, where the initial network modules are first obtained from the current knowledge and subsequently modified to fit experimental outcome. In this section, we show a reversed engineering approach, where a priori knowledge of a network module (all species within the module) is not required to analyze its biological response. We review the Non-Integral Connectivity Method (NICM)65 which uses the first-order response


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(a)

(b)

(c) Fig. 5. Possible mechanisms of action for SFR (but not limited). (a) Competition: molecules X and Y compete to bind with molecule Z. X and Y share binding sites at Z. (b) Steric hindrance: when X binds to Z, the complex prevents the binding of Y to another binding site at Z. (c) Conformational change: when X binds to Z, structural changes to Z lowers the affinity of Y binding to Z. Figure adapted from Selvarajoo et al.59

solution of δX(sum-of-exponentials) and delineates them into formation and depletion propagation response waves. Such delineation of response waves can be used for the construction of local biological network connectivities. For pulse perturbation, we can linearize Eq. (1) to solve δX by linear trans−1 JP δY . The formation, δX = P δY and diagonalizing J, such that dδY dt = P


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diagonal elements of P −1 JP are eigenvalues of the Jacobian matrix, when matrix P = (ˆ e1 , eˆ2 , . . . , eˆn ) is given by column unit eigenvector eî of J (i.e., J eî = λi eî with eigenvalue λi , i = 1, 2, . . . , n). Then, the solution of δX = (δX1 , δX2 , . . . , δXn ) becomes the summation of exponential terms, n ai e−λi t , (3) δX = i=1

where ai is perturbation coefficient vector, (ai1 , ai2 , . . . , ain ). Theoretical analysis shows that the eigenvalues λi are nonpositive, because otherwise the output signal will be unbounded. The eigenvalues are usually simple (distinct) and real; multiple or complex eigenvalues might exist. If complex values exist, then they occur in conjugated pairs which produce trigonometric functions of time in Eq. (3), modulated by exponentially decaying amplitudes. In some cases, such linearization, however, can lead to error compensation or amplification.66 To illustrate NICM, let us consider an example of pulse perturbation, (α, 0) at t = 0, given to a simple two-intermediate first-order mass-action model, X = k1 k2 (X1 , X2 ) : X1 −→ X2 −→. The perturbation wave δX = (δX 1 δX2 ), applied to the system with rate constants k1 and k2 for X1 and X2 , where X1 has k1 as depletion, and X2 has k1 as formation and k2 as depletion, can be represented by: dδX −k1 0 = δX. (4) k1 −k2 dt With initial conditions X 0 , the temporal profiles of intermediates are X 0 + δX and yield sum-of-exponentials solution: δX1 = αe−k1 t , and αk1 (e−k1 t − e−k2 t ). δX2 = k2 − k1

(5) (6)

Factorize Eq. (6) with respect to e−k1t if k2 > k1 (or e−k2t if k1 > k2 ) and obtain: αk1 (1 − e−(k2 −k1 )t )e−k1 t . (7) k2 − k1 In NICM, we use Eq. (7) as foundation to set up a generalized expression directly for the concentration or activated concentration levels of an intermediate X from steady-state, acting along a pathway, to consist of both the formation and depletion wave terms: δX2 =

δ X = α × (1 − e − p t ) × e− p t 1

Perturbation

Formation wave

2

,

(8)

Depletion wave

where α represents the amount of perturbation, p1 and p2 represent the measure of formation and depletion response propagation waves, respectively. In general, p1 and p2 are not equal to k1 and k2 , respectively and are determined by fitting with experimental data.60


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Symbolically, the formation and depletion wave terms are defined by: p1

Formation wave term: δX = (1 − e−p1 t ), Depletion wave term: δX = e−p2 t ,

X

p2 X

Combining the two terms with unit perturbation (i.e. α = 1), we obtain Eq. (8): X = (1 − e−p1 t )e−p2 t p1

p2 X

When a perturbation of α (where α can be a constant or function of time) is given to X1 , it induces changes to X1 (δX1 ) which subsequently induces changes to X2 (δX2 ) and so on, until the perturbation α is propagated throughout the network. α (t = 0) p2

p1 X1

X2

Mathematically: δX1 = αe−p1 t , δX2 = α(1 − e

(9) −p1 t

)e

−p2 t

.

(10)

For mass-conservation, the depletion coefficient of X1 , p1 , is equal to the formation coefficient of X2 . Basically, the time-events of the formation and depletion wave terms for each reactant species obey the law of superposition of propagation waves, which is the essence of NICM. This can be used to construct theoretical (NICM) motifs of commonly found subnetwork structures such as linear-chain–, merging–, branching–, reversible–, FBL–, and FFL–motifs (Table 1). By fitting experimental concentration profiles with commonly found network motifs constructed using NICM, we are able to detect the local connectivity of reactant species. In time-dependent perturbation or regulation such as FBL and FFL, p1 and p2 may not be constant values. For illustration, the construction of FBL motif is an extension of linear chain motif with nonconstant power of exponential: p1 A

p2 B

C

δA = αe−p1 (C)t , δB = α(1 − e

−p1 (C)t

δC = α(1 − e

−p1 (C)t

(11) )e

−p2 t

,

)(1 − e

(12) −p2 t

),

(13)

where p1 (C) is a function of species C concentration/activation level. For positive FBL, p1 is a function such that it increases in line with C. For negative FBL, p1 is


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Table 1. NICM to detect commonly-found network motifs. NICM expressions based on response propagation rules

Motif

δA = αe−p1 t

Linear-Chain Motif α (t = 0)

δB = α(1 − e−p1 t )e−p2 t

p2

p1 A

B

C

δC = α(1 − e−p1 t )(1 − e−p2 t ) δA = αe−p1 t

Merging Motif α (t = 0)

δB = βe−p2 t

A

p1

δC = [α(1 − e−p1 t ) + β(1 − e−p2 t )]e−p3 t

D

C

β (t = 0)

p3 p2

B

δD = [α(1 − e−p1 t ) + β(1 − e−p2 t )](1 − e−p3 t ) δA = αe−p1 t

Diverging Motif β , p2

α (t=0) p1

A

C

B

δB = α(1 − e−p1 t )e−p2 t δC = β(1 − e−p1 t )(1 − e−p2 t )

δ, p2

D

δD = δ(1 − e−p1 t )(1 − e−p2 t ) Note: β + δ = 1 for mass-conserved system (i) Linear-Chain Motif with Reversible Step

δA(i) = αe−p1 t δA(ii) = B ∗ (1−e−p3 t ) = β(1−e−p1 t )(1−e−p3 t )

α (t=0) p1 A

δA = δA(i) + δA(ii)

B p3

δA = αe−p1 t + β(1 − e−p1 t )(1 − e−p3 t ) For A: α (t=0) p3

p1 A

B

δB(i) = B ∗ e−p3 t = β(1 − e−p1 t )e−p3 t

A

A*

B*

(i)

(ii)

δB(ii) = A∗ = (α − β)(1 − e−p1 t )

For B:

δB = δB(i) + δB(ii)

p1

p3 B

δB = β(1 − e−p1 t )e−p3 t + (α − β)(1 − e−p1 t )

p1 A

B A*

B* (i)

(ii)


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Table 1. (Continued ) NICM expressions based on response propagation rules

Motif (ii) Linear-Chain Motif with Reversible Step and additional reactant α (t = 0)

δA = αe−p1 t + β(1 − e−p1 t )(1 − e−p3 t )e−p2 t " δB =

p1

β(1 − e−p1 t )e−p3 t

p2 B

A

+(α − β)(1 − e−p1 t )

# e−p2 t

C

δC = α(1 − e−p1 t )(1 − e−p2 t )

p3

For A: p1

p3

A

B

A p2

C

For B:

p1

p3

p1 A

B

B

B*

A* p2 C

δA = αe−p1 (C)t

Feedback Motif α (t = 0)

δB = α(1 − e−p1 (C)t )e−p2 t p2

p1 A

B

C

δA = αe−p1 t

Feedforward Motif

δB = α(1 − e−p1 t )e−p2 t

α (t = 0) p1

A

δC = α(1 − e−p1 (C)t )(1 − e−p2 t )

p2

B

p3

p3

C

_

D

δC = α(1 − e−p1 t )(1 − e−p2 t )e− B

t p3

δD = α(1 − e−p1 t )(1 − e−p2 t )(1 − e− B t ) for B > 0

259


260

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a function that decreases as C increases. Similar approach can be taken for FFL. In order to determine the exact function of formation or depletion terms in FBL and FFL, we search for best fitting experimental data. To show how NICM works and to prevent false positive connectivities, we created a three-chain linear motif using first-order mass-action kinetics with predefined rate constants and pulse perturbation. By unbiasely using various NICM network motifs (Table 1), we analyzed the time-series plots of the three concentration profiles and found that only the correct motif yielded the best fitness score (Fig. 6 shows two motifs only). When investigating larger-scale networks, as future work, we plan to investigate motif-to-motif connection to form a bigger module. For networks where temporal profiles of only some species in a network are available, NICM motif may not be successfully applied. However, it can still produce useful information as demonstrated below. Using NICM, we tested the temporal glycolysis data of glucose pulse perturbation in Saccharomyces cerevisiae.67 Although the dataset is not sufficient for the reconstruction of the entire glycolysis connectivity using NICM motifs (Table 1), nevertheless, NICM deciphers some novel features of the glycolytic regulation. Some of the glycolytic metabolite profiles of Saccharomyces cerevisiae are shown in Fig. 7. We analyzed the first three metabolites: Glucose-6-phosphate (G6P), Fructose-6-phosphate (F6P), and Fructose-1,6-bisphosphate (FBP) only. As we do not have the metabolite profiles of other intercepting pathways with glycolysis, e.g. the pentose phosphate pathways (PPP), we started our analysis by assuming the local network is of the linear-chain motif (Table 1), that is, → G6P → F6P → FBP → [see Eq. (8)]. We next performed the fitting process and obtained the following expressions with the best fitness score: G6P = 4.80(1 − e−0.230t )e−0.016t + 0.90 F6P = 2.01(1 − e−0.130t )e−0.037t + 0.17 FBP = 2.68(1 − e

−0.258t

)e

−0.0003t

(14)

+ 0.11.

We observed that for G6P, the depletion coefficient is much smaller than the formation coefficient of F6P [p1 of Eqs. (8) and (14)]. If we assume that F6P is solely produced by G6P, then by conservation of mass, the formation coefficient value of F6P should be similar to that of the depletion coefficient of G6P [p2 of Eqs. (8) and (14)]. This could imply that the faster formation of F6P could be due to other intercepting pathways that we have not factored in our initial motif assumption. This result is justified by the fact that F6P is also known to be produced by PPP, which was deliberately excluded in our motif selection. Next, looking at FBP, the depletion coefficient is very small or insignificant. This indicates that FBP has reached saturation due to downstream rate-limiting mechanism. This observation is strengthened by the lower-than-expected yield of downstream metabolites like glycerol and G3P.67 Thus, the superposition of formation


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261

Fig. 6. Generation of time-series plots for linear-chain motif with reversible step using the firstorder mass-action response equations with a pulse perturbation and comparison with best fitted NICM motifs using genetic algorithm.65 Compared with (a) three linear-chain (X1 , X2 , X3 ) NICM motif and (b) three linear-chain NICM motif with reversible step (Table 1). The maximum error for (a) is 42% and for (b) is 4% (details in Selvarajoo and Tsuchiya65 ). The x-axis represents time and y-axis represents concentration, both in arbitrary units. For all subsequent plots we adopt the same representation of the axes (x: time; y: concentration of reactant). Solid lines indicate the first-order mass-action kinetics with pulse perturbation and dotted lines indicate the result of NICM motifs.


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K. Selvarajoo, M. Tomita & M. Tsuchiya GLU

extracellular cytoplasm

GLU

ATP ADP

G6P F6P PPP

ATP ADP

FBP

DHAP

GLY

ATP ADP

G3P

NADPH NADH

BPG

ATP ADP

3-PG

PEP

ATP ADP

PYR

mitochondria TCA Cycle

(a)

(b)

(c)

(d)

Fig. 7. Glycolytic pathway and temporal profiles of glycolytic metabolite concentrations in Saccharomyces cerevisiae. (a) Glycolytic pathway. The experimental concentration profiles (exp) and NICM fitting (sim) of (b) G6P, F6P, (c) FBP, 3PG, (d) G3P, PEP, and (e) PYR. The experimental data were obtained from Theobald et al.67 The NICM expression with coefficient values for each metabolite is shown in Eq. (10). Figure adapted from Selvarajoo and Tsuchiya.65 Abbreviations: G6P, glucose-6-phosphate; F6P, fructose-6-phosphate; FBP, fructose 1,6-bisphosphate; G3P, glyceraldehyde-3-phosphate; GLY, glycerol; BPG, 1,3-bisphosphoglycerate, DHAP, dihydroxyacetone phosphate; 3PG, 3-phosphoglycerate; PEP, phosphoenol-pyruvate; PYR, pyruvate; PPP, pentose-phosphate pathway; TCA, tricarboxylic acid; ATP, Adenosine-5 -triphosphate; ADP, Adenosine diphosphate; NADH, Nicotinamide adenine dinucleotide; NADPH, Nicotinamide adenine dinucleotide phosphate.


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263

(e) Fig. 7. (Continued )

and depletion wave terms can be used in perturbation–response experiments to detect novel regulatory or local connectivities of biological networks. Interestingly, further experimental evidence of the superposition of formation and depletion propagation waves, which reveals the first-order mass-action response, can be visibly seen in the temporal activation profiles of phosphoproteomics data generated for epidermal growth factor signaling [Figs. 3(a) and (b) of Ref. 68]. 5. Future Work Our review indicates that approximation of the kinetic evolution equation to the first-order mass-action response equation is valid for understanding complex biological networks (e.g. signaling pathways) given small perturbation. This result shows that simple linear rules govern the intricate molecular interactions of biological network module. It is daunting to know why such simplicity could hold in a heterogeneous environment where diffusion processes and molecular crowding are expected to make the response highly complex and nonlinear.69,70 One of the reasons could be that deterministic approaches are possible for representing ensemble of cells, such as cell-culture, which show collective averaging behavior.16,31–33 This, however, is in contrast to single-cell behavior where stochastic molecular phenotypes are often observed.29,30 Hence, the future work could focus on how and why complex biological-system behavior of ensemble of cells can be explained by linear response equations. 6. Conclusion The research on biological network analysis is rapidly evolving. To determine the complex, dynamic interactions among species in a large-scale network, recent high-throughput time-series generation of gene expressions, phosphoproteins, and metabolite concentrations is indispensable. Without the discovery of organizing


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principles that govern the behavior of complex biological networks, it will be difficult for biologists to analyze and interpret data generated by such large-scale experiments. Several studies have indicated that ensemble of cells display collective behavior which is deterministic (averaging), robust, highly predictable, and stable under drastic environment perturbations. Under these circumstances, we have reviewed that simple linear rules derived from the first-order mass-action response equations can be used to determine the causal relationships between biological networks. This simplicity surprisingly holds in a highly anticipated complex heterogeneous environment.

Acknowledgments We are grateful to A Guiliani (L’Istituto Superiore di Sanit` a, Italy) and T Nishioka, (Keio University, Japan) for discussions. We thank M Helmy and V Piras (Keio University) for technical support. This work was supported by Japan Science and Technology Agency/Core Research for Evolutional Science and Technology (JST CREST), Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT), and research fund by Yamagata Prefecture and Tsuruoka City, Japan.

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Kumar Selvarajoo obtained his Master of Engineering degree in Aeronautics from the Imperial College of Science, Technology and Medicine, London, in 1997. Thereafter, he worked in defense-related industries for a few years, working on computational simulations on heat transfer problems using finite elements and computational fluid dynamics before embarking on a joint Systems Biology Ph.D. project with the Nanyang Technological University and the National Cancer Centre, Singapore. During his Ph.D. candidature, he was appointed as Director of Technology


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of Systome Therapeutics, a spin-off company of Biotech Research Ventures and the National Cancer Centre. After completing his Ph.D. in 2004, he joined the Bioinformatics Institute (BII), Singapore as a project leader. In 2006, he was appointed as Assistant Professor of the Institute for Advanced Biosciences, Keio University where he continues his pursuit to better understand the fundamental properties of biological systems by focusing on integrated quantitative systems biology approaches. Masaru Tomita is Director of the Institute for Advanced Biosciences, Keio University. He received his B.S. (1981) in Mathematics from Keio University, M.S. (1983) and Ph.D. (1985) in Computer Science from Carnegie Mellon University, and another Ph.D. (1998) in Molecular Biology from Keio University. Dr. Tomita is a recipient of the prestigious Presidential Young Investigators Award from the National Science Foundation of the USA (1988). From October 2005 to September 2007, he was Dean of Faculty of Environment and Information Studies, Keio University. His research areas include systems biology, metabolomics, bioinformatics, and biological simulation. Masa Tsuchiya has conducted a broad area of research from physics, mathematics, chemistry, and biology. He graduated from the Osaka University (biophysical engineering), and conducted particle physics research (theoretical and experimental) at graduate school of Osaka University and National Laboratory for High Energy Physics (KEK) in Japan. He moved to the USA to continue theoretical physics (Ph.D. from the University of West Virginia) and carried out post-doctorial research (chaos on molecular dynamics) at Cornell University. Due to the death of his mother to cancer, he changed his focus to molecular biology and genetics. In 1999, he joined the Department of Chemistry at Stanford University as a member of the academic staff and moved to Stanford Medical School for human genomic research. In 2004, he joined Bioinformatics Institute, Singapore as a Senior Scientist (cancer biology) and since 2006, he has been an Associate Professor (systems immunology) at Institute for Advanced Biosciences (IAB), Keio University, Japan.