Self-organization of polarized cell signaling via autocrine circuits: computational model analysis

Interlinked switch circuits of biological intelligence

Eukaryotic cells learn and adapt via unknown network architectures. Recent work demonstrated a circuit of two GTPases used by cells to overcome growth factor scarcity, encouraging our view that artificial and biological intelligence share strikingly similar design principles and that cells function as deep reinforcement learning (RL) agents in uncertain environments.

A systems perspective of heterocellular signaling

Signal exchange between different cell types is essential for development and function of multicellular organisms, and its dysregulation is causal in many diseases. Unfortunately, most cell-signaling work has employed single cell types grown under conditions unrelated to their native context. Recent technical developments have started to provide the tools needed to follow signaling between multiple cell types, but gaps in the information they provide have limited their usefulness in building realistic models of heterocellular signaling. Currently, only targeted assays have the necessary sensitivity, selectivity, and spatial resolution to usefully probe heterocellular signaling processes, but these are best used to test specific, mechanistic models. Decades of systems biology research with monocultures has provided a solid foundation for building models of heterocellular signaling, but current models lack a realistic description of regulated proteolysis and the feedback processes triggered within and between cells. Identification and understanding of key regulatory processes in the extracellular environment and of recursive signaling patterns between cells will be essential to building predictive models of heterocellular systems.

Theoretical method for evaluation of therapeutic effects and adverse effects of epidermal growth factor receptor tyrosine kinase inhibitors in clinical treatment

Epidermal growth factor receptor tyrosine kinase inhibitors (EGFR-TKIs) are used for non-small cell lung cancer patients with an EGFR gene mutation. However, skin disorders are known as adverse events. In the present study, we investigated whether EGFR-TK occupancy is useful as an index for assessing clinical efficacy and adverse events for the proper use and development of EGFR-TKIs. Average binding occupancies (Φ _ss) of EGFR-TKIs, gefitinib and erlotinib, for the EGFR-TK of cancer or skin cells were calculated. The relationships of Φ _ss with response rate (RR) or frequency of rash were analyzed using the ternary complex model. Then, the relationships between the dose of EGFR-TKIs and RR or frequency of rash were examined. Gefitinib showed a greater difference for Φ _ss value for both wild-type and mutant EGFR as compared to erlotinib at usual dose. The RR increased in a nonlinear manner rapidly rising when Φ _ss exceeded 95%. It was thought that a very high Φ _ss value might be needed to obtain the therapeutic effect of EGFR-TKIs. Meanwhile, the frequency of rash increased in a linear manner along with elevation of Φ _ss. It was shown that the K _d ratio (K _d for mutant/K _d for wild type) was less than 0.001, when the high RR and low frequency of rash were obtained simultaneously. The results showed that the therapeutic effects and skin disorder can be assessed by using Φ _ss. Furthermore, it is likely that a proper choice of drug and dose can be made by using Φ _ss in EGFR-TKI therapy.

Relaxation oscillations and hierarchy of feedbacks in MAPK signaling

We formulated a computational model for a MAPK signaling cascade downstream of the EGF receptor to investigate how interlinked positive and negative feedback loops process EGF signals into ERK pulses of constant amplitude but dose-dependent duration and frequency. A positive feedback loop involving RAS and SOS, which leads to bistability and allows for switch-like responses to inputs, is nested within a negative feedback loop that encompasses RAS and RAF, MEK, and ERK that inhibits SOS via phosphorylation. This negative feedback, operating on a longer time scale, changes switch-like behavior into oscillations having a period of 1 hour or longer. Two auxiliary negative feedback loops, from ERK to MEK and RAF, placed downstream of the positive feedback, shape the temporal ERK activity profile but are dispensable for oscillations. Thus, the positive feedback introduces a hierarchy among negative feedback loops, such that the effect of a negative feedback depends on its position with respect to the positive feedback loop. Furthermore, a combination of the fast positive feedback involving slow-diffusing membrane components with slower negative feedbacks involving faster diffusing cytoplasmic components leads to local excitation/global inhibition dynamics, which allows the MAPK cascade to transmit paracrine EGF signals into spatially non-uniform ERK activity pulses.

Mechanotransduction

Physical forces are central players in development and morphogenesis, provide an ever-present backdrop influencing physiological functions, and contribute to a variety of pathologies. Mechanotransduction encompasses the rich variety of ways in which cells and tissues convert cues from their physical environment into biochemical signals. These cues include tensile, compressive and shear stresses, and the stiffness or elastic modulus of the tissues in which cells reside. This article focuses on the proximal events that lead directly from a change in physical state to a change in cell-signaling state. A large body of evidence demonstrates a prominent role for the extracellular matrix, the intracellular cytoskeleton, and the cell matrix adhesions that link these networks in transduction of the mechanical environment. Recent work emphasizes the important role of physical unfolding or conformational changes in proteins induced by mechanical loading, with examples identified both within the focal adhesion complex at the cell-matrix interface and in extracellular matrix proteins themselves. Beyond these adhesion and matrix-based mechanisms, classical and new mechanisms of mechanotransduction reside in stretch-activated ion channels, the coupling of physical forces to interstitial autocrine and paracrine signaling, force-induced activation of extracellular proteins, and physical effects directly transmitted to the cell's nucleus. Rapid progress is leading to detailed delineation of molecular mechanisms by which the physical environment shapes cellular signaling events, opening up avenues for exploring how mechanotransduction pathways are integrated into physiological and pathophysiological cellular and tissue processes.

Mathematical modeling of eukaryotic cell migration: insights beyond experiments

A migrating cell is a molecular machine made of tens of thousands of short-lived and interacting parts. Understanding migration means understanding the self-organization of these parts into a system of functional units. This task is one of tackling complexity: First, the system integrates numerous chemical and mechanical component processes. Second, these processes are connected in feedback interactions and over a large range of spatial and temporal scales. Third, many processes are stochastic, which leads to heterogeneous migration behaviors. Early on in the research of cell migration it became evident that this complexity exceeds human intuition. Thus, the cell migration community has led the charge to build mathematical models that could integrate the diverse experimental observations and measurements in consistent frameworks, first in conceptual and more recently in molecularly explicit models. The main goal of this review is to sift through a series of important conceptual and explicit mathematical models of cell migration and to evaluate their contribution to the field in their ability to integrate critical experimental data.

An investigation of spatial signal transduction in cellular networks

Background

Spatial signal transduction plays a vital role in many intracellular processes such as eukaryotic chemotaxis, polarity generation and cell division. Furthermore it is being increasingly realized that the spatial dimension to signalling may play an important role in other apparently purely temporal signal transduction processes. It is increasingly being recognized that a conceptual basis for studying spatial signal transduction in signalling networks is necessary.

Results

In this work we examine spatial signal transduction in a series of standard motifs/networks. These networks include coherent and incoherent feedforward, positive and negative feedback, cyclic motifs, monostable switches, bistable switches and negative feedback oscillators. In all these cases, the driving signal has spatial variation. For each network we consider two cases, one where all elements are essentially non-diffusible, and the other where one of the network elements may be highly diffusible. A careful analysis of steady state signal transduction provides many insights into the behaviour of all these modules. While in the non-diffusible case for the most part, spatial signalling reflects the temporal signalling behaviour, in the diffusible cases, we see significant differences between spatial and temporal signalling characteristics. Our results demonstrate that the presence of diffusible elements in the networks provides important constraints and capabilities for signalling.

Conclusions

Our results provide a systematic basis for understanding spatial signalling in networks and the role of diffusible elements therein. This provides many insights into the signal transduction capabilities and constraints in such networks and suggests ways in which cellular signalling and information processing is organized to conform to or bypass those constraints. It also provides a framework for starting to understand the organization and regulation of spatial signal transduction in individual processes.

Lipopolysaccharide disrupts the directional persistence of alveolar myofibroblast migration through EGF receptor

Bronchopulmonary dysplasia (BPD) is characterized by alveolar simplification with decreased alveolar number and increased airspace size. Formation of alveoli involves a process known as secondary septation triggered by myofibroblasts. This study investigated the underlying mechanisms of altered lung morphogenesis in a rat model of BPD induced by intra-amniotic injection of lipopolysaccharide (LPS). Results showed that LPS disrupted alveolar morphology and led to abnormal localization of myofibroblasts in the lung of newborn rats, mostly in primary septa with few in secondary septa. To identify potential mechanisms, in vitro experiments were carried out to observe the migration behavior of myofibroblasts. The migration speed of lung myofibroblasts increased with LPS treatment, whereas the directional persistence decreased. We found that LPS induced activation of EGFR and overexpression of its ligand, TGF-α in myofibroblasts. AG1478, an EGFR inhibitor, abrogated the enhanced locomotivity of myofibroblasts by LPS and also increased the directional persistence of myofibroblast migration. Myofibroblasts showed a high asymmetry of phospho-EGFR localization, which was absent after LPS treatment. Application of rhTGF-α to myofibroblasts decreased the directional persistence. Our findings indicated that asymmetry of phospho-EGFR localization in myofibroblasts was important for cell migration and its directional persistence. We speculate that LPS exposure disrupts the asymmetric localization of phospho-EGFR, leading to decreased stability of cell polarity and final abnormal location of myofibroblasts in vivo, which is critical to secondary septation and may contribute to the arrested alveolar development in BPD.

Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization

Cell polarization, in which substances previously uniformly distributed become asymmetric due to external or/and internal stimulation, is a fundamental process underlying cell mobility, cell division, and other polarized functions. The yeast cell S. cerevisiae has been a model system to study cell polarization. During mating, yeast cells sense shallow external spatial gradients and respond by creating steeper internal gradients of protein aligned with the external cue. The complex spatial dynamics during yeast mating polarization consists of positive feedback, degradation, global negative feedback control, and cooperative effects in protein synthesis. Understanding such complex regulations and interactions is critical to studying many important characteristics in cell polarization including signal amplification, tracking dynamic signals, and potential trade-off between achieving both objectives in a robust fashion. In this paper, we study some of these questions by analyzing several models with different spatial complexity: two compartments, three compartments, and continuum in space. The step-wise approach allows detailed characterization of properties of the steady state of the system, providing more insights for biological regulations during cell polarization. For cases without membrane diffusion, our study reveals that increasing the number of spatial compartments results in an increase in the number of steady-state solutions, in particular, the number of stable steady-state solutions, with the continuum models possessing infinitely many steady-state solutions. Through both analysis and simulations, we find that stronger positive feedback, reduced diffusion, and a shallower ligand gradient all result in more steady-state solutions, although most of these are not optimally aligned with the gradient. We explore in the different settings the relationship between the number of steady-state solutions and the extent and accuracy of the polarization. Taken together these results furnish a detailed description of the factors that influence the tradeoff between a single correctly aligned but poorly polarized stable steady-state solution versus multiple more highly polarized stable steady-state solutions that may be incorrectly aligned with the external gradient.

Exact solutions to a spatially extended model of kinase-receptor interaction

B and Mast cells are activated by the aggregation of the immune receptors. Motivated by this phenomena we consider a simple spatially extended model of mutual interaction of kinases and membrane receptors. It is assumed that kinase activates membrane receptors and in turn the kinase molecules bound to the active receptors are activated by transphosphorylation. Such a type of interaction implies positive feedback and may lead to bistability. In this study we apply the Steklov eigenproblem theory to analyze the linearized model and find exact solutions in the case of non-uniformly distributed membrane receptors. This approach allows us to determine the critical value of receptor dephosphorylation rate at which cell activation (by arbitrary small perturbation of the inactive state) is possible. We found that cell sensitivity grows with decreasing kinase diffusion and increasing anisotropy of the receptor distribution. Moreover, these two effects are cooperating. We showed that the cell activity can be abruptly triggered by the formation of the receptor aggregate. Since the considered activation mechanism is not based on receptor crosslinking by polyvalent antigens, the proposed model can also explain B cell activation due to receptor aggregation following binding of monovalent antigens presented on the antigen presenting cell.

A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues

Polarization, a primary step in the response of an individual eukaryotic cell to a spatial stimulus, has attracted numerous theoretical treatments complementing experimental studies in a variety of cell types. While the phenomenon itself is universal, details differ across cell types, and across classes of models that have been proposed. Most models address how symmetry breaking leads to polarization, some in abstract settings, others based on specific biochemistry. Here, we compare polarization in response to a stimulus (e.g., a chemoattractant) in cells typically used in experiments (yeast, amoebae, leukocytes, keratocytes, fibroblasts, and neurons), and, in parallel, responses of several prototypical models to typical stimulation protocols. We find that the diversity of cell behaviors is reflected by a diversity of models, and that some, but not all models, can account for amplification of stimulus, maintenance of polarity, adaptation, sensitivity to new signals, and robustness.

Spatial aspects of intracellular information processing

The computational properties of intracellular biochemical networks, for which the cell is assumed to be a 'well-mixed' reactor, have already been widely characterized. What has so far not received systematic treatment is the important role of space in many intracellular computations. Spatial network computations can be divided into two broad categories: those required for essential spatial processes (e.g. polarization, chemotaxis, division, and development) and those for which space is simply used as an extra dimension to expand the computational power of the network. Several pertinent recent examples of each category are discussed that illustrate the often conceptually subtle role of space in the processing of intracellular information.

Modeling robustness tradeoffs in yeast cell polarization induced by spatial gradients

Cells localize (polarize) internal components to specific locations in response to external signals such as spatial gradients. For example, yeast cells form a mating projection toward the source of mating pheromone. There are specific challenges associated with cell polarization including amplification of shallow external gradients of ligand to produce steep internal gradients of protein components (e.g. localized distribution), response over a broad range of ligand concentrations, and tracking of moving signal sources. In this work, we investigated the tradeoffs among these performance objectives using a generic model that captures the basic spatial dynamics of polarization in yeast cells, which are small. We varied the positive feedback, cooperativity, and diffusion coefficients in the model to explore the nature of this tradeoff. Increasing the positive feedback gain resulted in better amplification, but also produced multiple steady-states and hysteresis that prevented the tracking of directional changes of the gradient. Feedforward/feedback coincidence detection in the positive feedback loop and multi-stage amplification both improved tracking with only a modest loss of amplification. Surprisingly, we found that introducing lateral surface diffusion increased the robustness of polarization and collapsed the multiple steady-states to a single steady-state at the cost of a reduction in polarization. Finally, in a more mechanistic model of yeast cell polarization, a surface diffusion coefficient between 0.01 and 0.001 µm(2)/s produced the best polarization performance, and this range is close to the measured value. The model also showed good gradient-sensitivity and dynamic range. This research is significant because it provides an in-depth analysis of the performance tradeoffs that confront biological systems that sense and respond to chemical spatial gradients, proposes strategies for balancing this tradeoff, highlights the critical role of lateral diffusion of proteins in the membrane on the robustness of polarization, and furnishes a framework for future spatial models of yeast cell polarization.

Dynamics of Cdc42 network embodies a Turing-type mechanism of yeast cell polarity

Complex biochemical networks can be understood by identifying their principal regulatory motifs and mode of action. We model the early phase of budding yeast cellular polarization and show that the biochemical processes in the presumptive bud site comprise a Turing-type mechanism. The roles of the prototypical activator and substrate are played by GTPase Cdc42 in its active and inactive states, respectively. We demonstrate that the nucleotide cycling of Cdc42 converts cellular energy into a stable cluster of activated Cdc42. This energy drives a continuous membrane-cytoplasmic exchange of the cluster components to counteract diffusive spread of the cluster. This exchange explains why only one bud forms per cell cycle, because the winner-takes-all competition of candidate sites inevitably selects a single site.

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.

Wave-pinning and cell polarity from a bistable reaction-diffusion system

Motile eukaryotic cells polarize in response to external signals. Numerous mechanisms have been suggested to account for this symmetry breaking and for the ensuing robust polarization. Implicated in this process are various proteins that are recruited to the plasma membrane and segregate at an emergent front or back of the polarizing cell. Among these are PI3K, PTEN, and members of the Rho family GTPases such as Cdc42, Rac, and Rho. Many such proteins, including the Rho GTPases, cycle between active membrane-bound forms and inactive cytosolic forms. In previous work, we have shown that this property, together with appropriate crosstalk, endows a biochemical circuit (Cdc42, Rac, and Rho) with the property of inherent polarizability. Here we show that this property is present in an even simpler system comprised of a single active/inactive protein pair with positive feedback to its own activation. The simplicity of this minimal system also allows us to explain the mechanism using insights from mathematical analysis. The basic idea resides in a well-known property of reaction-diffusion systems with bistable kinetics, namely, propagation of fronts. However, it crucially depends on exchange between active and inactive forms of the chemicals with unequal rates of diffusion, and overall conservation to pin the waves into a stable polar distribution. We refer to these dynamics as wave-pinning and we show that this phenomenon is distinct from Turing-instability-generated pattern formation that occurs in reaction-diffusion systems that appear to be very similar. We explain the mathematical basis of the phenomenon, relate it to spatial segregation of Rho GTPases, and show how it can account for spatial amplification and maintenance of polarity, as well as sensitivity to new stimuli typical in polarization of eukaryotic cells.

Simulating non-small cell lung cancer with a multiscale agent-based model

BACKGROUND

The epidermal growth factor receptor (EGFR) is frequently overexpressed in many cancers, including non-small cell lung cancer (NSCLC). In silico modeling is considered to be an increasingly promising tool to add useful insights into the dynamics of the EGFR signal transduction pathway. However, most of the previous modeling work focused on the molecular or the cellular level only, neglecting the crucial feedback between these scales as well as the interaction with the heterogeneous biochemical microenvironment.

RESULTS

We developed a multiscale model for investigating expansion dynamics of NSCLC within a two-dimensional in silico microenvironment. At the molecular level, a specific EGFR-ERK intracellular signal transduction pathway was implemented. Dynamical alterations of these molecules were used to trigger phenotypic changes at the cellular level. Examining the relationship between extrinsic ligand concentrations, intrinsic molecular profiles and microscopic patterns, the results confirmed that increasing the amount of available growth factor leads to a spatially more aggressive cancer system. Moreover, for the cell closest to nutrient abundance, a phase-transition emerges where a minimal increase in extrinsic ligand abolishes the proliferative phenotype altogether.

CONCLUSION

Our in silico results indicate that in NSCLC, in the presence of a strong extrinsic chemotactic stimulus (and depending on the cell's location) downstream EGFR-ERK signaling may be processed more efficiently, thereby yielding a migration-dominant cell phenotype and overall, an accelerated spatio-temporal expansion rate.

Multiple-state reactions between the epidermal growth factor receptor and Grb2 as observed by using single-molecule analysis

Phosphorylation of the cytoplasmic tyrosine residues of the epidermal growth factor receptor (EGFR) upon binding of EGF induces recognition of various intracellular signaling molecules, including Grb2. Here, the reaction kinetics between EGFR and Grb2 was analyzed by visualizing single molecules of Grb2 conjugated to the fluorophore Cy3 (Cy3-Grb2). The plasma membrane fraction was purified from human epithelial carcinoma A431 cells after stimulation with EGF and attached to coverslips. Unitary events of association and dissociation of Cy3-Grb2 on the EGFR in the membrane fraction were observed at different concentrations of Grb2 (0.1-100 nM). The dissociation kinetics could be explained by using a multiple-exponential function with a major (>90%) dissociation rate of 8 s(-1) and a few minor components, suggesting the presence of multiple bound states. In contrast, the association kinetics could be described by a stretched exponential function, suggesting the presence of multiple reaction channels from many unbound substates. Transitions between the unbound substates were also suggested. Unexpectedly, the rate of association was not proportional to the Grb2 concentration: an increase in Cy3-Grb2 concentration by a factor of 10 induced an increase in the reaction frequency approximately by a factor of three. This effect can compensate for fluctuation of the signal transduction from EGFR to Grb2 caused by variations in the expression level of Grb2 in living cells.

EGF-receptor-mediated mammary epithelial cell migration is driven by sustained ERK signaling from autocrine stimulation

EGF family ligands are synthesized as membrane-anchored precursors whose proteolytic release yields mature diffusible factors that can activate cell surface receptors in autocrine or paracrine mode. Expression of these ligands is altered in pathological states and in physiological processes, such as development and tissue regeneration. Despite the widely documented biological importance of autocrine EGF signaling, quantitative relationships between protease-mediated ligand release and consequent cell behavior have not been rigorously investigated. We thus explored the relationship between autocrine EGF release rates and cell behavioral responses along with activation of ERK, a key downstream signal, by expressing chimeric ligand precursors and modulating their proteolytic shedding using a metalloprotease inhibitor in human mammary epithelial cells. We found that ERK activation increased monotonically with increasing ligand release rate despite concomitant downregulation of EGF receptor levels. Cell migration speed was directly related to ligand release rate and proportional to steady-state phospho-ERK levels. Moreover, migration speed was significantly greater for autocrine stimulation compared with exogenous stimulation, even at comparable phospho-ERK levels. By contrast, cell proliferation rates were approximately equivalent at all ligand release rates and were similar regardless of whether the ligand was presented endogenously or exogenously. Thus, in our mammary epithelial cell system, migration and proliferation are differentially sensitive to the mode of EGF ligand presentation.

Stochastic resonant signaling in enzyme cascades

We observe the phenomenon of stochastic resonant signaling in signal amplification enzyme cascades, where certain optimal reaction rates minimize the average threshold-crossing time. We develop a new analytical technique to obtain the mean first passage time, based on a novel decomposition of the master equation. Our analytical results are in good agreement with the exact numerical simulations. We demonstrate that resonant behavior may be a ubiquitous phenomenon in stochastic threshold crossing in cell signaling. The physical principles behind this phenomenon are elucidated.

Potential for control of signaling pathways via cell size and shape

BACKGROUND

In order for signals generated at the plasma membrane to reach intracellular targets, activated messengers, such as G proteins and phosphoproteins, must diffuse through the cytoplasm. If the deactivators of these messengers, GTPase activating proteins (GAPs) and phosphatases, respectively, are sufficiently active in the cytoplasm, then the signal could in principle decay before reaching the target and a stable spatial gradient in phosphostate would be generated. Recent experiments document the existence of such gradients in living cells and suggest a role for them in mitotic spindle morphogenesis and cell migration. However, how such systems behave theoretically when embedded in a cell of varying size or shape has not been considered.

RESULTS

Here we use a simple mathematical model to explore the theoretical consequences of a plasma membrane bound activator (i.e., guanine nucleotide exchange factor, GEF, or kinase) and a cytoplasmic deactivator (i.e., GAP or phosphatase), and we find that as a model cell grows, the substrate becomes progressively dephosphorylated as a result of decreased proximity to the activator. Conversely, as a cell spreads and flattens, the substrate becomes globally phosphorylated because of increased proximity of the substrate to the activator. Similarly, in the leading edge of polarized cells and in protrusions such as lamellipodia or filopodia, the substrate is highly phosphorylated. As a specific test of the model, we found that the experimentally observed preferential activation of the G protein Cdc42 in the periphery of fibroblasts that was recently reported is consistent with model predictions.

CONCLUSIONS

We conclude that cell-signaling pathways can theoretically be turned on and off, both locally and globally, in response to alterations in cell size and shape.

Phosphoinositides and Rho proteins spatially regulate actin polymerization to initiate and maintain directed movement in a one-dimensional model of a motile cell

Gradient sensing, polarization, and chemotaxis of motile cells involve the actin cytoskeleton, and regulatory modules, including the phosphoinositides (PIs), their kinases/phosphatases, and small GTPases (Rho proteins). Here we model their individual components (PIP1, PIP2, PIP3; PTEN, PI3K, PI5K; Cdc42, Rac, Rho; Arp2/3, and actin), their interconversions, interactions, and modular functions in the context of a one-dimensional dynamic model for protrusive cell motility, with parameter values derived from in vitro and in vivo studies. In response to a spatially graded stimulus, the model produces stable amplified internal profiles of regulatory components, and initiates persistent motility (consistent with experimental observations). By connecting the modules, we find that Rho GTPases work as a spatial switch, and that the PIs filter noise, and define the front versus back. Relatively fast PI diffusion also leads to selection of a unique pattern of Rho distribution from a collection of possible patterns. We use the model to explore the importance of specific hypothesized interactions, to explore mutant phenotypes, and to study the role of actin polymerization in the maintenance of the PI asymmetry. We also suggest a mechanism to explain the spatial exclusion of Cdc42 and PTEN and the inability of cells lacking active Cdc42 to properly detect chemoattractant gradients.

An integrated agent-mathematical model of the effect of intercellular signalling via the epidermal growth factor receptor on cell proliferation

We have previously developed Epitheliome, a software agent representation of the growth and repair characteristics of epithelial cell populations, where cell behaviour is governed by a number of simple rules. In this paper, we describe how this model has been extended to incorporate an example of a molecular 'mechanism' behind a rule-in this case, how signalling by both endogenous and exogenous ligands of the epidermal growth factor receptor (EGFR) can impact on the proliferation of cell agents. We have developed a mathematical model representing release of endogenous ligand by cells, three-dimensional diffusion of the secreted molecules through a volume of cell culture medium, ligand-receptor binding, and bound receptor internalization and trafficking. Information relating to quantities of molecular species associated with each cell agent is frequently exchanged between the agent and signalling models, and the ratio of bound to free receptors determines cell cycle progression and hence the proliferative behaviour of the cell agents. We have applied this integrated model to examine the effect of plating density on tissue growth via autocrine/paracrine signalling. This predicts that cell growth is dependent on the concentration of exogenous ligand, but where this is limited, then growth becomes dependent on cell density and the availability of endogenous ligand. We have further modified the calcium concentration of the medium to modulate the formation of intercellular bonds between cells and shown that the increased propensity for cells to form colonies in physiological calcium does not result in significantly different patterns of receptor occupancy. In conclusion, our approach demonstrates that by combining agent-based and mathematical modelling paradigms, it is possible to probe the complex feedback relationship between the behaviour of individual cells and their interaction with one another and their environment.

EGF-ERBB signalling: towards the systems level

Signalling through the ERBB/HER receptors is intricately involved in human cancer and already serves as a target for several cancer drugs. Because of its inherent complexity, it is useful to envision ERBB signalling as a bow-tie-configured, evolvable network, which shares modularity, redundancy and control circuits with robust biological and engineered systems. Because network fragility is an inevitable trade-off of robustness, systems-level understanding is expected to generate therapeutic opportunities to intercept aberrant network activation.

Rules for modeling signal-transduction systems

Formalized rules for protein-protein interactions have recently been introduced to represent the binding and enzymatic activities of proteins in cellular signaling. Rules encode an understanding of how a system works in terms of the biomolecules in the system and their possible states and interactions. A set of rules can be as easy to read as a diagrammatic interaction map, but unlike most such maps, rules have precise interpretations. Rules can be processed to automatically generate a mathematical or computational model for a system, which enables explanatory and predictive insights into the system's behavior. Rules are independent units of a model specification that facilitate model revision. Instead of changing a large number of equations or lines of code, as may be required in the case of a conventional mathematical model, a protein interaction can be introduced or modified simply by adding or changing a single rule that represents the interaction of interest. Rules can be defined and visualized by using graphs, so no specialized training in mathematics or computer science is necessary to create models or to take advantage of the representational precision of rules. Rules can be encoded in a machine-readable format to enable electronic storage and exchange of models, as well as basic knowledge about protein-protein interactions. Here, we review the motivation for rule-based modeling; applications of the approach; and issues that arise in model specification, simulation, and testing. We also discuss rule visualization and exchange and the software available for rule-based modeling.

Cell-signalling dynamics in time and space

The specificity of cellular responses to receptor stimulation is encoded by the spatial and temporal dynamics of downstream signalling networks. Temporal dynamics are coupled to spatial gradients of signalling activities, which guide pivotal intracellular processes and tightly regulate signal propagation across a cell. Computational models provide insights into the complex relationships between the stimuli and the cellular responses, and reveal the mechanisms that are responsible for signal amplification, noise reduction and generation of discontinuous bistable dynamics or oscillations.

Computational modelling of the receptor-tyrosine-kinase-activated MAPK pathway

The MAPK (mitogen-activated protein kinase) pathway is one of the most important and intensively studied signalling pathways. It is at the heart of a molecular-signalling network that governs the growth, proliferation, differentiation and survival of many, if not all, cell types. It is de-regulated in various diseases, ranging from cancer to immunological, inflammatory and degenerative syndromes, and thus represents an important drug target. Over recent years, the computational or mathematical modelling of biological systems has become increasingly valuable, and there is now a wide variety of mathematical models of the MAPK pathway which have led to some novel insights and predictions as to how this system functions. In the present review we give an overview of the processes involved in modelling a biological system using the popular approach of ordinary differential equations. Focusing on the MAPK pathway, we introduce the features and functions of the pathway itself before comparing the available models and describing what new biological insights they have led to.

The effects of EGF-receptor density on multiscale tumor growth patterns

We studied the effects of epidermal growth factor receptor (EGFR) density on tumor growth dynamics, both on the sub- and the multi-cellular level using our previously developed model. This algorithm simulates the growth of a brain tumor using a multi-scale two-dimensional agent-based approach with an integrated transforming growth factor alpha (TGFalpha) induced EGFR-gene-protein interaction network. The results confirm that increasing cell receptor density correlates with an acceleration of the tumor system's spatio-temporal expansion dynamics. This multicellular behavior cannot be explained solely on the basis of spatial sub-cellular dynamics, which remain qualitatively similar amongst the three glioma cell lines investigated here in silico. Rather, we find that cells with higher EGFR density show an early increase in the phenotypic switching activity between proliferative and migratory traits, linked to a higher level of initial auto-stimulation by the PLCgamma-mediated TGFalpha-EGFR autocrine network. This indicates a more active protein level interaction in these chemotactically acting tumor systems and supports the role of post-translational regulation for the implemented EGFR pathway. Implications of these results for experimental cancer research are discussed.

Spontaneous polarization in eukaryotic gradient sensing: a mathematical model based on mutual inhibition of frontness and backness pathways

A key problem of eukaryotic cell motility is the signaling mechanism of chemoattractant gradient sensing. Recent experiments have revealed the molecular correlate of gradient sensing: Frontness molecules, such as PI3P and Rac, localize at the front end of the cell, and backness molecules, such as Rho and myosin II, accumulate at the back of the cell. Importantly, this frontness-backness polarization occurs spontaneously even if the cells are exposed to uniform chemoattractant profiles. The spontaneous polarization suggests that the gradient sensing machinery undergoes a Turing bifurcation. This has led to several classical activator-inhibitor and activator-substrate models which identify the frontness molecules with the activator. Conspicuously absent from these models is any accounting of the backness molecules. This stands in sharp contrast to experiments which show that the backness pathways inhibit the frontness pathways. Here, we formulate a model based on the mutually inhibitory interaction between the frontness and backness pathways. The model builds upon the mutual inhibition model proposed by Bourne and coworkers [Xu et al., 2003. Divergent signals and cytoskeletal assemblies regulate self-organizing polarity in neutrophils. Cell 114, 201-214.]. We show that mutual inhibition alone, without the help of any positive feedback (autocatalysis), can trigger spontaneous polarization of the frontness and backness pathways. The spatial distribution of the frontness and backness molecules in response to inhibition and activation of the frontness and backness pathways are consistent with those observed in experiments. Furthermore, depending on the parameter values, the model yields spatial distributions corresponding to chemoattraction (frontness pathways in-phase with the external gradient) and chemorepulsion (frontness pathways out-of-phase with the external gradient). Analysis of the model suggests a mechanism for the chemorepulsion-to-chemoattraction transition observed in neurons.

Current challenges in quantitative modeling of epidermal growth factor signaling

Over the last decade, epidermal growth factor (EGF) signaling has been used repeatedly as a test-bed for pioneering computational systems biology. Recent breakthroughs in our molecular understanding of EGF signaling pose new challenges for mathematical modeling strategies. Three key areas emerge as particularly relevant: the pervasive importance of compartmentalization and endosomal trafficking; the complexity of signalosome complexes; and the regulatory influence of diffusion and spatiality. Each one of them demands a drastic change in current computational approaches. We discuss recent developments in the field that address these emerging aspects in a new generation of more realistic - and potential more useful - models of EGF signaling.

Cholesterol dictates the freedom of EGF receptors and HER2 in the plane of the membrane

The flow of information through the epidermal growth factor receptor (EGFR) is shaped by molecular interactions in the plasma membrane. The EGFR is associated with lipid rafts, but their role in modulating receptor mobility and subsequent interactions is unclear. To investigate the role of nanoscale rafts in EGFR dynamics, we used single-molecule fluorescence imaging to track individual receptors and their dimerization partner, human epidermal growth factor receptor 2 (HER2), in the membrane of human mammary epithelial cells. We found that the motion of both receptors was interrupted by dwellings within nanodomains. EGFR was significantly less mobile than HER2. This difference was likely due to F-actin because its depolymerization led to similar diffusion patterns between the EGFR and HER2. Manipulations of membrane cholesterol content dramatically altered the diffusion pattern of both receptors. Cholesterol depletion led to almost complete confinement of the receptors, whereas cholesterol enrichment extended the boundaries of the restricted areas. Interestingly, F-actin depolymerization partially restored receptor mobility in cholesterol-depleted membranes. Our observations suggest that membrane cholesterol provides a dynamic environment that facilitates the free motion of EGFR and HER2, possibly by modulating the dynamic state of F-actin. The association of the receptors with lipid rafts could therefore promote their rapid interactions only upon ligand stimulation.

Simulating the impact of a molecular 'decision-process' on cellular phenotype and multicellular patterns in brain tumors

Experimental evidence indicates that human brain cancer cells proliferate or migrate, yet do not display both phenotypes at the same time. Here, we present a novel computational model simulating this cellular decision-process leading up to either phenotype based on a molecular interaction network of genes and proteins. The model's regulatory network consists of the epidermal growth factor receptor (EGFR), its ligand transforming growth factor-alpha (TGF alpha), the downstream enzyme phospholipaseC-gamma (PLC gamma) and a mitosis-associated response pathway. This network is activated by autocrine TGF alpha secretion, and the EGFR-dependent downstream signaling this step triggers, as well as modulated by an extrinsic nutritive glucose gradient. Employing a framework of mass action kinetics within a multiscale agent-based environment, we analyse both the emergent multicellular behavior of tumor growth and the single-cell molecular profiles that change over time and space. Our results show that one can indeed simulate the dichotomy between cell migration and proliferation based solely on an EGFR decision network. It turns out that these behavioral decisions on the single cell level impact the spatial dynamics of the entire cancerous system. Furthermore, the simulation results yield intriguing experimentally testable hypotheses also on the sub-cellular level such as spatial cytosolic polarization of PLC gamma towards an extrinsic chemotactic gradient. Implications of these results for future works, both on the modeling and experimental side are discussed.

Intracoronary administration of FGF-2: a computational model of myocardial deposition and retention

This study uses a computational model to characterize the myocardial deposition and retention of basic fibroblast growth factor (FGF-2) at the cellular level after intracoronary (IC) administration of exogenous FGF-2. The model is applied to the in situ conditions present within the myocardium of a dog for which the plasma pharmacokinetics resulting from IC injection of FGF-2 were recorded. Our estimates show that the processes involved in FGF-2 signaling are not diffusion limited; rather, the response time is determined by the reaction time of FGF-2 binding to cell surface receptors. Additionally, the processes of receptor secretion and internalization are found to play crucial roles in the FGF-2 dynamics; future experiments are required to quantify these processes. The model predictions obtained in this study suggest that IC administration of FGF-2 via either a single bolus or repetitive injections causes a transient increase (time scale of hours) in myocardial FGF-2 concentration if the endogenous level of free interstitial FGF-2 is low enough to allow permeation of FGF-2 molecules from the microvascular to the interstitial spaces. The model shows that the majority (64%) of the extracellular FGF-2 ligands are located within the interstitium, and similar fractions are found in the basement membrane and extracellular matrix. Among the FGF-2 molecules found within the interstitium, 2% are free and 98% are bound to interstitial heparan sulfate proteoglycans. These results support the theory of extracellular control of the bioavailability of FGF-2 via dynamic storage of FGF-2 within the basement membrane and extracellular matrix.