Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Receptor-based mechanism of relative sensing and cell memory in mammalian signaling networks

Detecting relative rather than absolute changes in extracellular signals enables cells to make decisions in constantly fluctuating environments. It is currently not well understood how mammalian signaling networks store the memories of past stimuli and subsequently use them to compute relative signals, that is perform fold change detection. Using the growth factor-activated PI3K-Akt signaling pathway, we develop here computational and analytical models, and experimentally validate a novel non-transcriptional mechanism of relative sensing in mammalian cells. This mechanism relies on a new form of cellular memory, where cells effectively encode past stimulation levels in the abundance of cognate receptors on the cell surface. The surface receptor abundance is regulated by background signal-dependent receptor endocytosis and down-regulation. We show the robustness and specificity of relative sensing for two physiologically important ligands, epidermal growth factor (EGF) and hepatocyte growth factor (HGF), and across wide ranges of background stimuli. Our results suggest that similar mechanisms of cell memory and fold change detection may be important in diverse signaling cascades and multiple biological contexts.

Ligand Binding Dynamics for Pre-dimerised G Protein-Coupled Receptor Homodimers: Linear Models and Analytical Solutions

Evidence suggests that many G protein-coupled receptors (GPCRs) are bound together forming dimers. The implications of dimerisation for cellular signalling outcomes, and ultimately drug discovery and therapeutics, remain unclear. Consideration of ligand binding and signalling via receptor dimers is therefore required as an addition to classical receptor theory, which is largely built on assumptions of monomeric receptors. A key factor in developing theoretical models of dimer signalling is cooperativity across the dimer, whereby binding of a ligand to one protomer affects the binding of a ligand to the other protomer. Here, we present and analyse linear models for one-ligand and two-ligand binding dynamics at homodimerised receptors, as an essential building block in the development of dimerised receptor theory. For systems at equilibrium, we compute analytical solutions for total bound labelled ligand and derive conditions on the cooperativity factors under which multiphasic log dose–response curves are expected. This could help explain data extracted from pharmacological experiments that do not fit to the standard Hill curves that are often used in this type of analysis. For the time-dependent problems, we also obtain analytical solutions. For the single-ligand case, the construction of the analytical solution is straightforward; it is bi-exponential in time, sharing a similar structure to the well-known monomeric competition dynamics of Motulsky–Mahan. We suggest that this model is therefore practically usable by the pharmacologist towards developing insights into the potential dynamics and consequences of dimerised receptors.

General principles of binding between cell surface receptors and multi-specific ligands: A computational study

The interactions between membrane receptors and extracellular ligands control cell-cell and cell-substrate adhesion, and environmental responsiveness by representing the initial steps of cell signaling pathways. These interactions can be spatial-temporally regulated when different extracellular ligands are tethered. The detailed mechanisms of this spatial-temporal regulation, including the competition between distinct ligands with overlapping binding sites and the conformational flexibility in multi-specific ligand assemblies have not been quantitatively evaluated. We present a new coarse-grained model to realistically simulate the binding process between multi-specific ligands and membrane receptors on cell surfaces. The model simplifies each receptor and each binding site in a multi-specific ligand as a rigid body. Different numbers or types of ligands are spatially organized together in the simulation. These designs were used to test the relation between the overall binding of a multi-specific ligand and the affinity of its cognate binding site. When a variety of ligands are exposed to cells expressing different densities of surface receptors, we demonstrated that ligands with reduced affinities have higher specificity to distinguish cells based on the relative concentrations of their receptors. Finally, modification of intramolecular flexibility was shown to play a role in optimizing the binding between receptors and ligands. In summary, our studies bring new insights to the general principles of ligand-receptor interactions. Future applications of our method will pave the way for new strategies to generate next-generation biologics.

In order to adapt to surrounding environments, multiple signaling pathways have been evolved in cells. The first step of these pathways is to detect external stimuli, which is conducted by the dynamic interactions between cell surface receptors and extracellular ligands. As a result, recognition of extracellular ligands by cell surface receptors is an indispensable component of many physiological or pathological activities. In both natural selection and drug design, the presence of multiple binding sites in extracellular ligand complexes (so-called multi-specific ligands) is a common strategy to target different receptors on surface of the same cell. Such spatial organization of ligand binding sites can elaborately modulate the downstream signaling pathways. However, our understanding to the interactions between multi-specific ligands and membrane receptors is largely limited by the fact that these interactions are difficult to quantify and they have only been successfully measured in a very small number of cases in vivo. Using a simple computational model, we can realistically simulate the binding process between specially designed multi-specific ligands and membrane receptors on cell surfaces. This study therefore provides a useful pathway to unravel basic mechanisms of ligand-receptor interactions and design principles for new drug candidates.

Effect of Spatial Inhomogeneities on the Membrane Surface on Receptor Dimerization and Signal Initiation

Important signal transduction pathways originate on the plasma membrane, where microdomains may transiently entrap diffusing receptors. This results in a non-random distribution of receptors even in the resting state, which can be visualized as “clusters” by high resolution imaging methods. Here, we explore how spatial in-homogeneities in the plasma membrane might influence the dimerization and phosphorylation status of ErbB2 and ErbB3, two receptor tyrosine kinases that preferentially heterodimerize and are often co-expressed in cancer. This theoretical study is based upon spatial stochastic simulations of the two-dimensional membrane landscape, where variables include differential distributions and overlap of transient confinement zones (“domains”) for the two receptor species. The in silico model is parameterized and validated using data from single particle tracking experiments. We report key differences in signaling output based on the degree of overlap between domains and the relative retention of receptors in such domains, expressed as escape probability. Results predict that a high overlap of domains, which favors transient co-confinement of both receptor species, will enhance the rate of hetero-interactions. Where domains do not overlap, simulations confirm expectations that homo-interactions are favored. Since ErbB3 is uniquely dependent on ErbB2 interactions for activation of its catalytic activity, variations in domain overlap or escape probability markedly alter the predicted patterns and time course of ErbB3 and ErbB2 phosphorylation. Taken together, these results implicate membrane domain organization as an important modulator of signal initiation, motivating the design of novel experimental approaches to measure these important parameters across a wider range of receptor systems.

Thermodynamic and kinetic characterization of transmembrane helix association

The transient dimerization of transmembrane proteins is an important event in several cellular processes and here we use coarse-grain and meso-scale modeling methods to quantify their underlying dynamics.

Decoding information in cell shape

Shape is an indicator of cell health. But how is the information in shape decoded? We hypothesize that decoding occurs by modulation of signaling through changes in plasma membrane curvature. Using analytical approaches and numerical simulations, we studied how elongation of cell shape affects plasma membrane signaling. Mathematical analyses reveal transient accumulation of activated receptors at regions of higher curvature with increasing cell eccentricity. This distribution of activated receptors is periodic, following the Mathieu function, and it arises from local imbalance between reaction and diffusion of soluble ligands and receptors in the plane of the membrane. Numerical simulations show that transient microdomains of activated receptors amplify signals to downstream protein kinases. For growth factor receptor pathways, increasing cell eccentricity elevates the levels of activated cytoplasmic Src and nuclear MAPK1,2. These predictions were experimentally validated by changing cellular eccentricity, showing that shape is a locus of retrievable information storage in cells.

A tunable coarse-grained model for ligand-receptor interaction

Cell-surface receptors are the most common target for therapeutic drugs. The design and optimization of next generation synthetic drugs require a detailed understanding of the interaction with their corresponding receptors. Mathematical approximations to study ligand-receptor systems based on reaction kinetics strongly simplify the spatial constraints of the interaction, while full atomistic ligand-receptor models do not allow for a statistical many-particle analysis, due to their high computational requirements. Here we present a generic coarse-grained model for ligand-receptor systems that accounts for the essential spatial characteristics of the interaction, while allowing statistical analysis. The model captures the main features of ligand-receptor kinetics, such as diffusion dependence of affinity and dissociation rates. Our model is used to characterize chimeric compounds, designed to take advantage of the receptor over-expression phenotype of certain diseases to selectively target unhealthy cells. Molecular dynamics simulations of chimeric ligands are used to study how selectivity can be optimized based on receptor abundance, ligand-receptor affinity and length of the linker between both ligand subunits. Overall, this coarse-grained model is a useful approximation in the study of systems with complex ligand-receptor interactions or spatial constraints.

The current importance of cell surface receptors as primary targets for drug treatment explains the increasing interest in a mathematical and quantitative description of the process of ligand-receptor interaction. Recently, a new generation of synthetic chimeric ligands has been developed to selectively target unhealthy cells, without harming healthy tissue. To understand these and other types of complex ligand-receptor systems, conventional chemical interaction models often rely on simplifications and assumptions about the spatial characteristics of the system, while full atomistic molecular dynamics simulations are too computationally demanding to perform many particle statistical analysis. In this paper, we describe a novel approach to model the interaction between ligands and receptors based on a coarse grained approximation that includes explicitly both spatial and kinetic details of the interaction, while allowing many-particle simulations and therefore, statistical analysis at biologically relevant time scales. The model is used to study the binding properties of generic chimeric ligands to understand how cell specificity is achieved, its dependence on receptor concentration and the influence of the distance between subunits of the chimera. Overall, this approach proves optimal to study other ligand-receptor systems with complex spatial regulation, such as receptor clustering, multimerization and multivalent asymmetric ligand binding.

Analytical solution of steady-state equations for chemical reaction networks with bilinear rate laws

True steady states are a rare occurrence in living organisms, yet their knowledge is essential for quasi-steady-state approximations, multistability analysis, and other important tools in the investigation of chemical reaction networks (CRN) used to describe molecular processes on the cellular level. Here, we present an approach that can provide closed form steady-state solutions to complex systems, resulting from CRN with binary reactions and mass-action rate laws. We map the nonlinear algebraic problem of finding steady states onto a linear problem in a higher-dimensional space. We show that the linearized version of the steady-state equations obeys the linear conservation laws of the original CRN. We identify two classes of problems for which complete, minimally parameterized solutions may be obtained using only the machinery of linear systems and a judicious choice of the variables used as free parameters. We exemplify our method, providing explicit formulae, on CRN describing signal initiation of two important types of RTK receptor-ligand systems, VEGF and EGF-ErbB1.

Mathematical simulation of membrane protein clustering for efficient signal transduction

Initiation and propagation of cell signaling depend on productive interactions among signaling proteins at the plasma membrane. These diffusion-limited interactions can be influenced by features of the membrane that introduce barriers, such as cytoskeletal corrals, or microdomains that transiently confine both transmembrane receptors and membrane-tethered peripheral proteins. Membrane topographical features can lead to clustering of receptors and other membrane components, even under very dynamic conditions. This review considers the experimental and mathematical evidence that protein clustering impacts cell signaling in complex ways. Simulation approaches used to consider these stochastic processes are discussed.

The shuttling scaffold model for prevention of yeast pheromone pathway misactivation

The molecular scaffold in the yeast pheromone pathway, Ste5, shuttles continuously between the nucleus and the cytoplasm. Ste5 undergoes oligomerization reaction in the nucleus. Upon pheromone stimulation, the Ste5 dimer is rapidly exported out of the nucleus and recruited to the plasma membrane for pathway activation. This clever device on part of the yeast cell is thought to prevent pathway misactivation at high enough levels of Ste5 in the absence of pheromone. We have built a spatiotemporal model of signaling in this pathway to describe its regulation. Our present work underscores the importance of spatial modeling of cell signaling networks to understand their control and functioning.

Membrane microdomains emergence through non-homogeneous diffusion

Background

In the classical view, cell membrane proteins undergo isotropic random motion, that is a 2D Brownian diffusion that should result in an homogeneous distribution of concentration. It is, however, far from the reality: Membrane proteins can assemble into so-called microdomains (sometimes called lipid rafts) which also display a specific lipid composition. We propose a simple mechanism that is able to explain the colocalization of protein and lipid rafts.

Results

Using very simple mathematical models and particle simulations, we show that a variation of membrane viscosity directly leads to variation of the local concentration of diffusive particles. Since specific lipid phases in the membrane can account for diffusion variation, we show that, in such a situation, the freely diffusing proteins (or any other component) still undergo a Brownian motion but concentrate in areas of lower diffusion. The amount of this so-called overconcentration at equilibrium issimply related to the ratio of diffusion coefficients between zones of high and low diffusion. Expanding the model to include particle interaction, we show that inhomogeneous diffusion can impact particles clusterization as well. The clusters of particles were more numerous and appear for a lower value of interaction strength in the zones of low diffusion compared to zones of high diffusion.

Conclusion

Provided we assume stable viscosity heterogeneity in the membrane, our model propose a simple mechanism to explain particle concentration heterogeneity. It has also a non-trivial impact on density of particles when interaction is added. This could potentially have an impact on membrane chemical reactions and oligomerization.

Unified regression model of binding equilibria in crowded environments

Molecular crowding is a critical feature distinguishing intracellular environments from idealized solution-based environments and is essential to understanding numerous biochemical reactions, from protein folding to signal transduction. Many biochemical reactions are dramatically altered by crowding, yet it is extremely difficult to predict how crowding will quantitatively affect any particular reaction systems. We previously developed a novel stochastic off-lattice model to efficiently simulate binding reactions across wide parameter ranges in various crowded conditions. We now show that a polynomial regression model can incorporate several interrelated parameters influencing chemistry under crowded conditions. The unified model of binding equilibria accurately reproduces the results of particle simulations over a broad range of variation of six physical parameters that collectively yield a complicated, non-linear crowding effect. The work represents an important step toward the long-term goal of computationally tractable predictive models of reaction chemistry in the cellular environment.

Mathematical modeling of K-Ras nanocluster formation on the plasma membrane

K-Ras functions as a critical node in the mitogen-activated protein kinase (MAPK) pathway that regulates key cellular functions including proliferation, differentiation, and apoptosis. Following growth factor receptor activation K-Ras.GTP forms nanoclusters on the plasma membrane through interaction with the scaffold protein galectin-3. The generation of nanoclusters is essential for high fidelity signal transduction via the MAPK pathway. To explore the mechanisms underlying K-Ras.GTP nanocluster formation, we developed a mathematical model of K-Ras-galectin-3 interactions. We designed a computational method to calculate protein collision rates based on experimentally determined protein diffusion rates and diffusion mechanisms and used a genetic algorithm to search the values of key model parameters. The optimal estimated model parameters were validated using experimental data. The resulting model accurately replicates critical features of K-Ras nanoclustering, including a fixed ratio of clustered K-Ras.GTP to monomeric K-Ras.GTP that is independent of the concentration of K-Ras.GTP. The model reproduces experimental results showing that the cytosolic level of galectin-3 determines the magnitude of the K-Ras.GTP clustered fraction and illustrates that nanoclustering is regulated by key nonequilibrium processes. Our kinetic model identifies a potential biophysical mechanism for K-Ras nanoclustering and suggests general principles that may be relevant for other plasma-membrane-localized proteins.

Quantitative understanding of cell signaling: the importance of membrane organization

Systems biology modeling of signal transduction pathways traditionally employs ordinary differential equations, deterministic models based on the assumptions of spatial homogeneity. However, this can be a poor approximation for certain aspects of signal transduction, especially its initial steps: the cell membrane exhibits significant spatial organization, with diffusion rates approximately two orders of magnitude slower than those in the cytosol. Thus, to unravel the complexities of signaling pathways, quantitative models must consider spatial organization as an important feature of cell signaling. Furthermore, spatial separation limits the number of molecules that can physically interact, requiring stochastic simulation methods that account for individual molecules. Herein, we discuss the need for mathematical models and experiments that appreciate the importance of spatial organization in the membrane.

Spatially distributed cell signalling

Emerging evidence indicates that complex spatial gradients and (micro)domains of signalling activities arise from distinct cellular localization of opposing enzymes, such as a kinase and phosphatase, in signal transduction cascades. Often, an interacting, active form of a target protein has a lower diffusivity than an inactive form, and this leads to spatial gradients of the protein abundance in the cytoplasm. A spatially distributed signalling cascade can create step-like activation profiles, which decay at successive distances from the cell surface, assigning digital positional information to different regions in the cell. Feedback and feedforward network motifs control activity patterns, allowing signalling networks to serve as cellular devices for spatial computations.

Lipid raft-mediated regulation of G-protein coupled receptor signaling by ligands which influence receptor dimerization: a computational study

G-protein coupled receptors (GPCRs) are the largest family of cell surface receptors; they activate heterotrimeric G-proteins in response to ligand stimulation. Although many GPCRs have been shown to form homo- and/or heterodimers on the cell membrane, the purpose of this dimerization is not known. Recent research has shown that receptor dimerization may have a role in organization of receptors on the cell surface. In addition, microdomains on the cell membrane termed lipid rafts have been shown to play a role in GPCR localization. Using a combination of stochastic (Monte Carlo) and deterministic modeling, we propose a novel mechanism for lipid raft partitioning of GPCRs based on reversible dimerization of receptors and then demonstrate that such localization can affect GPCR signaling. Modeling results are consistent with a variety of experimental data indicating that lipid rafts have a role in amplification or attenuation of G-protein signaling. Thus our work suggests a new mechanism by which dimerization-inducing or inhibiting characteristics of ligands can influence GPCR signaling by controlling receptor organization on the cell membrane.

Coupled stochastic spatial and non-spatial simulations of ErbB1 signaling pathways demonstrate the importance of spatial organization in signal transduction

Background

The ErbB family of receptors activates intracellular signaling pathways that control cellular proliferation, growth, differentiation and apoptosis. Given these central roles, it is not surprising that overexpression of the ErbB receptors is often associated with carcinogenesis. Therefore, extensive laboratory studies have been devoted to understanding the signaling events associated with ErbB activation.

Methodology/Principal Findings

Systems biology has contributed significantly to our current understanding of ErbB signaling networks. However, although computational models have grown in complexity over the years, little work has been done to consider the spatial-temporal dynamics of receptor interactions and to evaluate how spatial organization of membrane receptors influences signaling transduction. Herein, we explore the impact of spatial organization of the epidermal growth factor receptor (ErbB1/EGFR) on the initiation of downstream signaling. We describe the development of an algorithm that couples a spatial stochastic model of membrane receptors with a nonspatial stochastic model of the reactions and interactions in the cytosol. This novel algorithm provides a computationally efficient method to evaluate the effects of spatial heterogeneity on the coupling of receptors to cytosolic signaling partners.

Conclusions/Significance

Mathematical models of signal transduction rarely consider the contributions of spatial organization due to high computational costs. A hybrid stochastic approach simplifies analyses of the spatio-temporal aspects of cell signaling and, as an example, demonstrates that receptor clustering contributes significantly to the efficiency of signal propagation from ligand-engaged growth factor receptors.

Spine neck geometry determines spino-dendritic cross-talk in the presence of mobile endogenous calcium binding proteins

Dendritic spines are thought to compartmentalize second messengers like Ca2+. The notion of isolated spine signaling, however, was challenged by the recent finding that under certain conditions mobile endogenous Ca(2+)-binding proteins may break the spine limit and lead to activation of Ca(2+)-dependent dendritic signaling cascades. Since the size of spines is variable, the spine neck may be an important regulator of this spino-dendritic crosstalk. We tested this hypothesis by using an experimentally defined, kinetic computer model in which spines of Purkinje neurons were coupled to their parent dendrite by necks of variable geometry. We show that Ca2+ signaling and calmodulin activation in spines with long necks is essentially isolated from the dendrite, while stubby spines show a strong coupling with their dendrite, mediated particularly by calbindin D28k. We conclude that the spine neck geometry, in close interplay with mobile Ca(2+)-binding proteins, regulates the spino-dendritic crosstalk.

Versatile analysis of single-molecule tracking data by comprehensive testing against Monte Carlo simulations

We propose here an approach for the analysis of single-molecule trajectories which is based on a comprehensive comparison of an experimental data set with multiple Monte Carlo simulations of the diffusion process. It allows quantitative data analysis, particularly whenever analytical treatment of a model is infeasible. Simulations are performed on a discrete parameter space and compared with the experimental results by a nonparametric statistical test. The method provides a matrix of p-values that assess the probability for having observed the experimental data at each setting of the model parameters. We show the testing approach for three typical situations observed in the cellular plasma membrane: i), free Brownian motion of the tracer, ii), hop diffusion of the tracer in a periodic meshwork of squares, and iii), transient binding of the tracer to slowly diffusing structures. By plotting the p-value as a function of the model parameters, one can easily identify the most consistent parameter settings but also recover mutual dependencies and ambiguities which are difficult to determine by standard fitting routines. Finally, we used the test to reanalyze previous data obtained on the diffusion of the glycosylphosphatidylinositol-protein CD59 in the plasma membrane of the human T24 cell line.

Signal transduction at point-blank range: analysis of a spatial coupling mechanism for pathway crosstalk

The plasma membrane provides a physical platform for the orchestration of molecular interactions and biochemical conversions involved in the early stages of receptor-mediated signal transduction in living cells. In that context, we introduce here the concept of spatial coupling, wherein simultaneous recruitment of different enzymes to the same receptor scaffold facilitates crosstalk between different signaling pathways through the local release and capture of activated signaling molecules. To study the spatiotemporal dynamics of this mechanism, we have developed a Brownian dynamics modeling approach and applied it to the receptor-mediated activation of Ras and the cooperative recruitment of phosphoinositide 3-kinase (PI3K) by activated receptors and Ras. Various analyses of the model simulations show that cooperative assembly of multimolecular complexes nucleated by activated receptors is facilitated by the local release and capture of membrane-anchored signaling molecules (such as active Ras) from/by receptor-bound signaling proteins. In the case of Ras/PI3K crosstalk, the model predicts that PI3K is more likely to be recruited by activated receptors bound or recently visited by the enzyme that activates Ras. By this mechanism, receptor-bound PI3K is stabilized through short-range, diffusion-controlled capture of active Ras and Ras/PI3K complexes released from the receptor complex. We contend that this mechanism is a means by which signaling pathways are propagated and spatially coordinated for efficient crosstalk between them.

Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm

We present a general-purpose model for biomolecular simulations at the molecular level that incorporates stochasticity, spatial dependence, and volume exclusion, using diffusing and reacting particles with physical dimensions. To validate the model, we first established the formal relationship between the microscopic model parameters (timestep, move length, and reaction probabilities) and the macroscopic coefficients for diffusion and reaction rate. We then compared simulation results with Smoluchowski theory for diffusion-limited irreversible reactions and the best available approximation for diffusion-influenced reversible reactions. To simulate the volumetric effects of a crowded intracellular environment, we created a virtual cytoplasm composed of a heterogeneous population of particles diffusing at rates appropriate to their size. The particle-size distribution was estimated from the relative abundance, mass, and stoichiometries of protein complexes using an experimentally derived proteome catalog from Escherichia coli K12. Simulated diffusion constants exhibited anomalous behavior as a function of time and crowding. Although significant, the volumetric impact of crowding on diffusion cannot fully account for retarded protein mobility in vivo, suggesting that other biophysical factors are at play. The simulated effect of crowding on barnase-barstar dimerization, an experimentally characterized example of a bimolecular association reaction, reveals a biphasic time course, indicating that crowding exerts different effects over different timescales. These observations illustrate that quantitative realism in biosimulation will depend to some extent on mesoscale phenomena that are not currently well understood.

Modelling spatio-temporal interactions within the cell

Biological phenomena at the cellular level can be represented by various types of mathematical formulations. Such representations allow us to carry out numerical simulations that provide mechanistic insights into complex behaviours of biological systems and also generate hypotheses that can be experimentally tested. Currently, we are particularly interested in spatio-temporal representations of dynamic cellular phenomena and how such models can be used to understand biological specificity in functional responses. This review describes the capability and limitations of the approaches used to study spatio-temporal dynamics of cell signalling components.

Dimerization of VEGF receptors and implications for signal transduction: a computational study

Vascular endothelial growth factor (VEGF) is a potent cytokine involved in the induction of neovascularization. Secreted as a cysteine-linked dimer, it has two binding sites at opposite poles through which it may bind VEGF receptors (VEGFRs), receptor tyrosine kinases found on the surface of endothelial and other cells. The binding of a VEGF molecule to two VEGFR molecules induces transphosphorylation of the intracellular domains of the receptors, leading to signal transduction. The dominant mechanism of receptor dimerization is not clear: the receptors may be present in an inactive pre-dimerized form, VEGF binding first to one of the receptors, the second receptor then ideally located for dimerization; or VEGF may bind receptor monomers on the cell surface, which then diffuse and bind to available unligated receptor monomers to complete the activation. Both processes take place and one or other may dominate on different cell types. We demonstrate the impact of dimerization mechanism on the binding of VEGF to the cell surface and on the formation of active signaling receptor complexes. We describe two methods to determine which process dominates, based on binding and phosphorylation assays. The presence of two VEGF receptor populations, VEGFR1 and VEGFR2, can result in receptor heterodimer formation. Our simulations predict that heterodimers will comprise 10-50% of the active, signaling VEGF receptor complexes, and that heterodimers will form at the expense of homodimers of VEGFR1 when VEGFR2 populations are larger. These results have significant implications for VEGF signal transduction and interpretation of experimental studies. These results may be applicable to other ligand-receptor pairs, in particular PDGF.

Accommodating space, time and randomness in network simulation

Interest in the possibility of dynamically simulating complex cellular processes has escalated markedly in recent years. This interest has been fuelled by three factors: the generally accepted value in understanding living processes as integrated systems; the dramatic increase in computational capability; and the availability of new or improved technology for making the quantitative measurements that are needed to drive and validate cellular simulations. Between the extremes of atom-scale and organism-scale simulation is a vast middle-ground requiring simulation strategies that are capable of dealing with a range of spatial, temporal and molecular abundance scales that are crucial for a comprehensive understanding of integrative cell biology. Although at an early stage, methodological improvements and the development of computational platforms provide some hope that simulations will emerge that can bridge the gap between network models and the true operation of the cell as a complex machine.