A network model of early events in epidermal growth factor receptor signaling that accounts for combinatorial complexity

Catalyst: Fast and flexible modeling of reaction networks

We introduce Catalyst.jl, a flexible and feature-filled Julia library for modeling and high-performance simulation of chemical reaction networks (CRNs). Catalyst supports simulating stochastic chemical kinetics (jump process), chemical Langevin equation (stochastic differential equation), and reaction rate equation (ordinary differential equation) representations for CRNs. Through comprehensive benchmarks, we demonstrate that Catalyst simulation runtimes are often one to two orders of magnitude faster than other popular tools. More broadly, Catalyst acts as both a domain-specific language and an intermediate representation for symbolically encoding CRN models as Julia-native objects. This enables a pipeline of symbolically specifying, analyzing, and modifying CRNs; converting Catalyst models to symbolic representations of concrete mathematical models; and generating compiled code for numerical solvers. Leveraging ModelingToolkit.jl and Symbolics.jl, Catalyst models can be analyzed, simplified, and compiled into optimized representations for use in numerical solvers. Finally, we demonstrate Catalyst's broad extensibility and composability by highlighting how it can compose with a variety of Julia libraries, and how existing open-source biological modeling projects have extended its intermediate representation.

Mechanistic model of MAPK signaling reveals how allostery and rewiring contribute to drug resistance

BRAF is prototypical of oncogenes that can be targeted therapeutically and the treatment of BRAF^V600E melanomas with RAF and MEK inhibitors results in rapid tumor regression. However, drug-induced rewiring generates a drug adapted state thought to be involved in acquired resistance and disease recurrence. In this article, we study mechanisms of adaptive rewiring in BRAF^V600E melanoma cells using an energy-based implementation of ordinary differential equation (ODE) modeling in combination with proteomic, transcriptomic and imaging data. We develop a method for causal tracing of ODE models and identify two parallel MAPK reaction channels that are differentially sensitive to RAF and MEK inhibitors due to differences in protein oligomerization and drug binding. We describe how these channels, and timescale separation between immediate-early signaling and transcriptional feedback, create a state in which the RAS-regulated MAPK channel can be activated by growth factors under conditions in which the BRAF^V600E -driven channel is fully inhibited. Further development of the approaches in this article is expected to yield a unified model of adaptive drug resistance in melanoma.

Comparison of rule- and ordinary differential equation-based dynamic model of DARPP-32 signalling network

Dynamic modelling has considerably improved our understanding of complex molecular mechanisms. Ordinary differential equations (ODEs) are the most detailed and popular approach to modelling the dynamics of molecular systems. However, their application in signalling networks, characterised by multi-state molecular complexes, can be prohibitive. Contemporary modelling methods, such as rule- based (RB) modelling, have addressed these issues. The advantages of RB modelling over ODEs have been presented and discussed in numerous reviews. In this study, we conduct a direct comparison of the time courses of a molecular system founded on the same reaction network but encoded in the two frameworks. To make such a comparison, a set of reactions that underlie an ODE model was manually encoded in the Kappa language, one of the RB implementations. A comparison of the models was performed at the level of model specification and dynamics, acquired through model simulations. In line with previous reports, we confirm that the Kappa model recapitulates the general dynamics of its ODE counterpart with minor differences. These occur when molecules have multiple sites binding the same interactor. Furthermore, activation of these molecules in the RB model is slower than in the ODE one. As reported for other molecular systems, we find that, also for the DARPP-32 reaction network, the RB representation offers a more expressive and flexible syntax that facilitates access to fine details of the model, easing model reuse. In parallel with these analyses, we report a refactored model of the DARPP-32 interaction network that can serve as a canvas for the development of more complex dynamic models to study this important molecular system.

Atlas: automatic modeling of regulation of bacterial gene expression and metabolism using rule-based languages

MOTIVATION

Cells are complex systems composed of hundreds of genes whose products interact to produce elaborated behaviors. To control such behaviors, cells rely on transcription factors to regulate gene expression, and gene regulatory networks (GRNs) are employed to describe and understand such behavior. However, GRNs are static models, and dynamic models are difficult to obtain due to their size, complexity, stochastic dynamics and interactions with other cell processes.

RESULTS

We developed Atlas, a Python software that converts genome graphs and gene regulatory, interaction and metabolic networks into dynamic models. The software employs these biological networks to write rule-based models for the PySB framework. The underlying method is a divide-and-conquer strategy to obtain sub-models and combine them later into an ensemble model. To exemplify the utility of Atlas, we used networks of varying size and complexity of Escherichia coli and evaluated in silico modifications, such as gene knockouts and the insertion of promoters and terminators. Moreover, the methodology could be applied to the dynamic modeling of natural and synthetic networks of any bacteria.

AVAILABILITY AND IMPLEMENTATION

Code, models and tutorials are available online (https://github.com/networkbiolab/atlas).

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Applications of personalised signalling network models in precision oncology

As our ability to provide in-depth, patient-specific characterisation of the molecular alterations within tumours rapidly improves, it is becoming apparent that new approaches will be required to leverage the power of this data and derive the full benefit for each individual patient. Systems biology approaches are beginning to emerge within this field as a potential method of incorporating large volumes of network level data and distilling a coherent, clinically-relevant prediction of drug response. However, the initial promise of this developing field is yet to be realised. Here we argue that in order to develop these precise models of individual drug response and tailor treatment accordingly, we will need to develop mathematical models capable of capturing both the dynamic nature of drug-response signalling networks and key patient-specific information such as mutation status or expression profiles. We also review the modelling approaches commonly utilised within this field, and outline recent examples of their use in furthering the application of systems biology for a precision medicine approach to cancer treatment.

Parallel Tempering with Lasso for model reduction in systems biology

Systems Biology models reveal relationships between signaling inputs and observable molecular or cellular behaviors. The complexity of these models, however, often obscures key elements that regulate emergent properties. We use a Bayesian model reduction approach that combines Parallel Tempering with Lasso regularization to identify minimal subsets of reactions in a signaling network that are sufficient to reproduce experimentally observed data. The Bayesian approach finds distinct reduced models that fit data equivalently. A variant of this approach that uses Lasso to perform selection at the level of reaction modules is applied to the NF-κB signaling network to test the necessity of feedback loops for responses to pulsatile and continuous pathway stimulation. Taken together, our results demonstrate that Bayesian parameter estimation combined with regularization can isolate and reveal core motifs sufficient to explain data from complex signaling systems.

Cells respond to diverse environmental cues using complex networks of interacting proteins and other biomolecules. Mathematical and computational models have become invaluable tools to understand these networks and make informed predictions to rationally perturb cell behavior. However, the complexity of detailed models that try to capture all known biochemical elements of signaling networks often makes it difficult to determine the key regulatory elements that are responsible for specific cell behaviors. Here, we present a Bayesian computational approach, PTLasso, to automatically extract minimal subsets of detailed models that are sufficient to explain experimental data. The method simultaneously calibrates and reduces models, and the Bayesian approach samples globally, allowing us to find alternate mechanistic explanations for the data if present. We demonstrate the method on both synthetic and real biological data and show that PTLasso is an effective method to isolate distinct parts of a larger signaling model that are sufficient for specific data.

Universal Properties of Concentration Sensing in Large Ligand-Receptor Networks

Cells estimate concentrations of chemical ligands in their environment using a limited set of receptors. Recent work has shown that the temporal sequence of binding and unbinding events on just a single receptor can be used to estimate the concentrations of multiple ligands. Here, for a network of many ligands and many receptors, we show that such temporal sequences can be used to estimate the concentration of a few times as many ligand species as there are receptors. Crucially, we show that the spectrum of the inverse covariance matrix of these estimates has several universal properties, which we trace to properties of Vandermonde matrices. We argue that this can be used by cells in realistic biochemical decoding networks.

Interactive Multiresolution Visualization of Cellular Network Processes

Visualization plays a central role in the analysis of biochemical network models to identify patterns that arise from reaction dynamics and perform model exploratory analysis. To facilitate these analyses, we developed PyViPR, a visualization tool that generates static and dynamic representations of biochemical network processes within a Python-based environment. PyViPR embeds network visualizations within Jupyter notebooks, thus enabling integration with modeling, simulation, and analysis workflows. To present the capabilities of PyViPR, we explore execution mechanisms of extrinsic apoptosis in HeLa cells. We show that community-detection algorithms identify groups of molecular species that capture key biological functions and ease exploration of the apoptosis network. We then show how different kinetic parameter sets that fit the experimental data equally well exhibit significantly different signal-execution dynamics as the system progresses toward mitochondrial outer-membrane permeabilization. Therefore, PyViPR aids the conceptual understanding of dynamic network processes and accelerates hypothesis generation for further testing and validation.

Pleione: A tool for statistical and multi-objective calibration of Rule-based models

Mathematical models based on Ordinary Differential Equations (ODEs) are frequently used to describe and simulate biological systems. Nevertheless, such models are often difficult to understand. Unlike ODE models, Rule-Based Models (RBMs) utilise formal language to describe reactions as a cumulative number of statements that are easier to understand and correct. They are also gaining popularity because of their conciseness and simulation flexibility. However, RBMs generally lack tools to perform further analysis that requires simulation. This situation arises because exact and approximate simulations are computationally intensive. Translating RBMs into ODEs is commonly used to reduce simulation time, but this technique may be prohibitive due to combinatorial explosion. Here, we present the software called Pleione to calibrate RBMs. Parameter calibration is essential given the incomplete experimental determination of reaction rates and the goal of using models to reproduce experimental data. The software distributes stochastic simulations and calculations and incorporates equivalence tests to determine the fitness of RBMs compared with data. The primary features of Pleione were thoroughly tested on a model of gene regulation in Escherichia coli. Pleione yielded satisfactory results regarding calculation time and error reduction for multiple simulators, models, parameter search strategies, and computing infrastructures.

PyBioNetFit and the Biological Property Specification Language

In systems biology modeling, important steps include model parameterization, uncertainty quantification, and evaluation of agreement with experimental observations. To help modelers perform these steps, we developed the software PyBioNetFit, which in addition supports checking models against known system properties and solving design problems. PyBioNetFit introduces Biological Property Specification Language (BPSL) for the formal declaration of system properties. BPSL allows qualitative data to be used alone or in combination with quantitative data. PyBioNetFit performs parameterization with parallelized metaheuristic optimization algorithms that work directly with existing model definition standards: BioNetGen Language (BNGL) and Systems Biology Markup Language (SBML). We demonstrate PyBioNetFit's capabilities by solving various example problems, including the challenging problem of parameterizing a 153-parameter model of cell cycle control in yeast based on both quantitative and qualitative data. We demonstrate the model checking and design applications of PyBioNetFit and BPSL by analyzing a model of targeted drug interventions in autophagy signaling.

Markov Chain Aggregation and Its Application to Rule-Based Modelling

Rule-based modelling allows to represent molecular interactions in a compact and natural way. The underlying molecular dynamics, by the laws of stochastic chemical kinetics, behaves as a continuous-time Markov chain. However, this Markov chain enumerates all possible reaction mixtures, rendering the analysis of the chain computationally demanding and often prohibitive in practice. We here describe how it is possible to efficiently find a smaller, aggregate chain, which preserves certain properties of the original one. Formal methods and lumpability notions are used to define algorithms for automated and efficient construction of such smaller chains (without ever constructing the original ones). We here illustrate the method on an example and we discuss the applicability of the method in the context of modelling large signaling pathways.

Intrinsic limits of information transmission in biochemical signalling motifs

All living things have evolved to sense changes in their environment in order to respond in adaptive ways. At the cellular level, these sensing systems generally involve receptor molecules at the cell surface, which detect changes outside the cell and relay those changes to the appropriate response elements downstream. With the advent of experimental technologies that can track signalling at the single-cell level, it has become clear that many signalling systems exhibit significant levels of 'noise,' manifesting as differential responses of otherwise identical cells to the same environment. This noise has a large impact on the capacity of cell signalling networks to transmit information from the environment. Application of information theory to experimental data has found that all systems studied to date encode less than 2.5 bits of information, with the majority transmitting significantly less than 1 bit. Given the growing interest in applying information theory to biological data, it is crucial to understand whether the low values observed to date represent some sort of intrinsic limit on information flow given the inherently stochastic nature of biochemical signalling events. In this work, we used a series of computational models to explore how much information a variety of common 'signalling motifs' can encode. We found that the majority of these motifs, which serve as the basic building blocks of cell signalling networks, can encode far more information (4-6 bits) than has ever been observed experimentally. In addition to providing a consistent framework for estimating information-theoretic quantities from experimental data, our findings suggest that the low levels of information flow observed so far in living system are not necessarily due to intrinsic limitations. Further experimental work will be needed to understand whether certain cell signalling systems actually can approach the intrinsic limits described here, and to understand the sources and purpose of the variation that reduces information flow in living cells.

Using Equation-Free Computation to Accelerate Network-Free Stochastic Simulation of Chemical Kinetics

The chemical kinetics of many complex systems can be concisely represented by reaction rules, which can be used to generate reaction events via a kinetic Monte Carlo method that has been termed network-free simulation. Here, we demonstrate accelerated network-free simulation through a novel approach to equation-free computation. In this process, variables are introduced that approximately capture system state. Derivatives of these variables are estimated using short bursts of exact stochastic simulation and finite differencing. The variables are then projected forward in time via a numerical integration scheme, after which a new exact stochastic simulation is initialized and the whole process repeats. The projection step increases efficiency by bypassing the firing of numerous individual reaction events. As we show, the projected variables may be defined as populations of building blocks of chemical species. The maximal number of connected molecules included in these building blocks determines the degree of approximation. Equation-free acceleration of network-free simulation is found to be both accurate and efficient.

Generalizing Gillespie's Direct Method to Enable Network-Free Simulations

Gillespie's direct method for stochastic simulation of chemical kinetics is a staple of computational systems biology research. However, the algorithm requires explicit enumeration of all reactions and all chemical species that may arise in the system. In many cases, this is not feasible due to the combinatorial explosion of reactions and species in biological networks. Rule-based modeling frameworks provide a way to exactly represent networks containing such combinatorial complexity, and generalizations of Gillespie's direct method have been developed as simulation engines for rule-based modeling languages. Here, we provide both a high-level description of the algorithms underlying the simulation engines, termed network-free simulation algorithms, and how they have been applied in systems biology research. We also define a generic rule-based modeling framework and describe a number of technical details required for adapting Gillespie's direct method for network-free simulation. Finally, we briefly discuss potential avenues for advancing network-free simulation and the role they continue to play in modeling dynamical systems in biology.

An Overview of Network-Based and -Free Approaches for Stochastic Simulation of Biochemical Systems

Stochastic simulation has been widely used to model the dynamics of biochemical reaction networks. Several algorithms have been proposed that are exact solutions of the chemical master equation, following the work of Gillespie. These stochastic simulation approaches can be broadly classified into two categories: network-based and -free simulation. The network-based approach requires that the full network of reactions be established at the start, while the network-free approach is based on reaction rules that encode classes of reactions, and by applying rule transformations, it generates reaction events as they are needed without ever having to derive the entire network. In this study, we compare the efficiency and limitations of several available implementations of these two approaches. The results allow for an informed selection of the implementation and methodology for specific biochemical modeling applications.

Automated visualization of rule-based models

Frameworks such as BioNetGen, Kappa and Simmune use "reaction rules" to specify biochemical interactions compactly, where each rule specifies a mechanism such as binding or phosphorylation and its structural requirements. Current rule-based models of signaling pathways have tens to hundreds of rules, and these numbers are expected to increase as more molecule types and pathways are added. Visual representations are critical for conveying rule-based models, but current approaches to show rules and interactions between rules scale poorly with model size. Also, inferring design motifs that emerge from biochemical interactions is an open problem, so current approaches to visualize model architecture rely on manual interpretation of the model. Here, we present three new visualization tools that constitute an automated visualization framework for rule-based models: (i) a compact rule visualization that efficiently displays each rule, (ii) the atom-rule graph that conveys regulatory interactions in the model as a bipartite network, and (iii) a tunable compression pipeline that incorporates expert knowledge and produces compact diagrams of model architecture when applied to the atom-rule graph. The compressed graphs convey network motifs and architectural features useful for understanding both small and large rule-based models, as we show by application to specific examples. Our tools also produce more readable diagrams than current approaches, as we show by comparing visualizations of 27 published models using standard graph metrics. We provide an implementation in the open source and freely available BioNetGen framework, but the underlying methods are general and can be applied to rule-based models from the Kappa and Simmune frameworks also. We expect that these tools will promote communication and analysis of rule-based models and their eventual integration into comprehensive whole-cell models.

Robustness analysis of the detailed kinetic model of an ErbB signaling network by using dynamic sensitivity

The ErbB receptor signaling pathway plays an important role in the regulation of cellular proliferation, survival and differentiation, and dysregulation of the pathway is linked to various types of human cancer. Mathematical models have been developed as a practical complementary approach to deciphering the complexity of ErbB receptor signaling and elucidating how the pathways discriminate between ligands to induce different cell fates. In this study, we developed a simulator to accurately calculate the dynamic sensitivity of extracellular-signal-regulated kinase (ERK) activity (ERK*) and Akt activity (Akt*), downstream of the ErbB receptors stimulated with epidermal growth factor (EGF) and heregulin (HRG). To demonstrate the feasibility of this simulator, we estimated how the reactions critically responsible for ERK* and Akt* change with time and in response to different doses of EGF and HRG, and predicted that only a small number of reactions determine ERK* and Akt*. ERK* increased steeply with increasing HRG dose until saturation, while showing a gently rising response to EGF. Akt* had a gradual wide-range response to HRG and a blunt response to EGF. Akt* was sensitive to perturbations of intracellular kinetics, while ERK* was more robust due to multiple, negative feedback loops. Overall, the simulator predicted reactions that were critically responsible for ERK* and Akt* in response to the dose of EGF and HRG, illustrated the response characteristics of ERK* and Akt*, and estimated mechanisms for generating robustness in the ErbB signaling network.

Quantitative computational models of molecular self-assembly in systems biology

Molecular self-assembly is the dominant form of chemical reaction in living systems, yet efforts at systems biology modeling are only beginning to appreciate the need for and challenges to accurate quantitative modeling of self-assembly. Self-assembly reactions are essential to nearly every important process in cell and molecular biology and handling them is thus a necessary step in building comprehensive models of complex cellular systems. They present exceptional challenges, however, to standard methods for simulating complex systems. While the general systems biology world is just beginning to deal with these challenges, there is an extensive literature dealing with them for more specialized self-assembly modeling. This review will examine the challenges of self-assembly modeling, nascent efforts to deal with these challenges in the systems modeling community, and some of the solutions offered in prior work on self-assembly specifically. The review concludes with some consideration of the likely role of self-assembly in the future of complex biological system models more generally.

LASSIE: simulating large-scale models of biochemical systems on GPUs

BACKGROUND

Mathematical modeling and in silico analysis are widely acknowledged as complementary tools to biological laboratory methods, to achieve a thorough understanding of emergent behaviors of cellular processes in both physiological and perturbed conditions. Though, the simulation of large-scale models-consisting in hundreds or thousands of reactions and molecular species-can rapidly overtake the capabilities of Central Processing Units (CPUs). The purpose of this work is to exploit alternative high-performance computing solutions, such as Graphics Processing Units (GPUs), to allow the investigation of these models at reduced computational costs.

RESULTS

LASSIE is a "black-box" GPU-accelerated deterministic simulator, specifically designed for large-scale models and not requiring any expertise in mathematical modeling, simulation algorithms or GPU programming. Given a reaction-based model of a cellular process, LASSIE automatically generates the corresponding system of Ordinary Differential Equations (ODEs), assuming mass-action kinetics. The numerical solution of the ODEs is obtained by automatically switching between the Runge-Kutta-Fehlberg method in the absence of stiffness, and the Backward Differentiation Formulae of first order in presence of stiffness. The computational performance of LASSIE are assessed using a set of randomly generated synthetic reaction-based models of increasing size, ranging from 64 to 8192 reactions and species, and compared to a CPU-implementation of the LSODA numerical integration algorithm.

CONCLUSIONS

LASSIE adopts a novel fine-grained parallelization strategy to distribute on the GPU cores all the calculations required to solve the system of ODEs. By virtue of this implementation, LASSIE achieves up to 92× speed-up with respect to LSODA, therefore reducing the running time from approximately 1 month down to 8 h to simulate models consisting in, for instance, four thousands of reactions and species. Notably, thanks to its smaller memory footprint, LASSIE is able to perform fast simulations of even larger models, whereby the tested CPU-implementation of LSODA failed to reach termination. LASSIE is therefore expected to make an important breakthrough in Systems Biology applications, for the execution of faster and in-depth computational analyses of large-scale models of complex biological systems.

Computational investigation of sphingosine kinase 1 (SphK1) and calcium dependent ERK1/2 activation downstream of VEGFR2 in endothelial cells

Vascular endothelial growth factor (VEGF) is a powerful regulator of neovascularization. VEGF binding to its cognate receptor, VEGFR2, activates a number of signaling pathways including ERK1/2. Activation of ERK1/2 is experimentally shown to involve sphingosine kinase 1 (SphK1) activation and its calcium-dependent translocation downstream of ERK1/2. Here we construct a rule-based computational model of signaling downstream of VEGFR2, by including SphK1 and calcium positive feedback mechanisms, and investigate their consequences on ERK1/2 activation. The model predicts the existence of VEGF threshold in ERK1/2 activation that can be continuously tuned by cellular concentrations of SphK1 and sphingosine 1 phosphate (S1P). The computer model also predicts powerful effects of perturbations in plasma and ER calcium pump rates and the current through the CRAC channels on ERK1/2 activation dynamics, highlighting the critical role of intracellular calcium in shaping the pERK1/2 signal. The model is then utilized to simulate anti-angiogenic therapeutic interventions targeting VEGFR2-ERK1/2 axis. Simulations indicate that monotherapies that exclusively target VEGFR2 phosphorylation, VEGF, or VEGFR2 are ineffective in shutting down signaling to ERK1/2. By simulating therapeutic strategies that target multiple nodes of the pathway such as Raf and SphK1, we conclude that combination therapy should be much more effective in blocking VEGF signaling to EKR1/2. The model has important implications for interventions that target signaling pathways in angiogenesis relevant to cancer, vascular diseases, and wound healing.

Vascular endothelial growth factor (VEGF) signaling is a potent regulator of angiogenesis, the growth and development of new vessels out of a preexisting vascular network. Angiogenesis requires enhanced survival, proliferation, and motility of the vascular endothelial cells. Crucial signaling endpoints in VEGF-mediated angiogenic response include elevation in intracellular calcium and the activation of the proteins ERK1 and 2 (ERK1/2). In this study, we have developed a novel computer model for the activation of ERK1/2 and calcium downstream of VEGF receptor type 2 (VEGFR2). Our model is the first of its kind to incorporate and investigate the consequences of calcium elevation and the role of a cellular lipid modifier known as sphingosine kinase 1 (SphK1). We also utilize the model to simulate therapeutic strategies targeting VEGF signaling to ERK1/2 indicating inefficiency of single therapies known as tyrosine kinase inhibitors (TKI) that target receptor phosphorylation. Computer simulations indicate that combination therapy is essential for effective blockade of this important pathway. Our results have important implications for human diseases such as cancer where plethora of anti-VEGF therapies are currently employed. Overall, our computer model sheds new light on a complex feedback involving SphK1 and calcium that radically alters the response of cells to VEGF.

Modeling of Receptor Tyrosine Kinase Signaling: Computational and Experimental Protocols

The advent of systems biology has convincingly demonstrated that the integration of experiments and dynamic modelling is a powerful approach to understand the cellular network biology. Here we present experimental and computational protocols that are necessary for applying this integrative approach to the quantitative studies of receptor tyrosine kinase (RTK) signaling networks. Signaling by RTKs controls multiple cellular processes, including the regulation of cell survival, motility, proliferation, differentiation, glucose metabolism, and apoptosis. We describe methods of model building and training on experimentally obtained quantitative datasets, as well as experimental methods of obtaining quantitative dose-response and temporal dependencies of protein phosphorylation and activities. The presented methods make possible (1) both the fine-grained modeling of complex signaling dynamics and identification of salient, course-grained network structures (such as feedback loops) that bring about intricate dynamics, and (2) experimental validation of dynamic models.

Extending rule-based methods to model molecular geometry and 3D model resolution

BACKGROUND

Computational modeling is an important tool for the study of complex biochemical processes associated with cell signaling networks. However, it is challenging to simulate processes that involve hundreds of large molecules due to the high computational cost of such simulations. Rule-based modeling is a method that can be used to simulate these processes with reasonably low computational cost, but traditional rule-based modeling approaches do not include details of molecular geometry. The incorporation of geometry into biochemical models can more accurately capture details of these processes, and may lead to insights into how geometry affects the products that form. Furthermore, geometric rule-based modeling can be used to complement other computational methods that explicitly represent molecular geometry in order to quantify binding site accessibility and steric effects.

RESULTS

We propose a novel implementation of rule-based modeling that encodes details of molecular geometry into the rules and binding rates. We demonstrate how rules are constructed according to the molecular curvature. We then perform a study of antigen-antibody aggregation using our proposed method. We simulate the binding of antibody complexes to binding regions of the shrimp allergen Pen a 1 using a previously developed 3D rigid-body Monte Carlo simulation, and we analyze the aggregate sizes. Then, using our novel approach, we optimize a rule-based model according to the geometry of the Pen a 1 molecule and the data from the Monte Carlo simulation. We use the distances between the binding regions of Pen a 1 to optimize the rules and binding rates. We perform this procedure for multiple conformations of Pen a 1 and analyze the impact of conformation and resolution on the optimal rule-based model.

CONCLUSIONS

We find that the optimized rule-based models provide information about the average steric hindrance between binding regions and the probability that antibodies will bind to these regions. These optimized models quantify the variation in aggregate size that results from differences in molecular geometry and from model resolution.

A rule-based model of insulin signalling pathway

BACKGROUND

The insulin signalling pathway (ISP) is an important biochemical pathway, which regulates some fundamental biological functions such as glucose and lipid metabolism, protein synthesis, cell proliferation, cell differentiation and apoptosis. In the last years, different mathematical models based on ordinary differential equations have been proposed in the literature to describe specific features of the ISP, thus providing a description of the behaviour of the system and its emerging properties. However, protein-protein interactions potentially generate a multiplicity of distinct chemical species, an issue referred to as "combinatorial complexity", which results in defining a high number of state variables equal to the number of possible protein modifications. This often leads to complex, error prone and difficult to handle model definitions.

RESULTS

In this work, we present a comprehensive model of the ISP, which integrates three models previously available in the literature by using the rule-based modelling (RBM) approach. RBM allows for a simple description of a number of signalling pathway characteristics, such as the phosphorylation of signalling proteins at multiple sites with different effects, the simultaneous interaction of many molecules of the signalling pathways with several binding partners, and the information about subcellular localization where reactions take place. Thanks to its modularity, it also allows an easy integration of different pathways. After RBM specification, we simulated the dynamic behaviour of the ISP model and validated it using experimental data. We the examined the predicted profiles of all the active species and clustered them in four clusters according to their dynamic behaviour. Finally, we used parametric sensitivity analysis to show the role of negative feedback loops in controlling the robustness of the system.

CONCLUSIONS

The presented ISP model is a powerful tool for data simulation and can be used in combination with experimental approaches to guide the experimental design. The model is available at http://sysbiobig.dei.unipd.it/ was submitted to Biomodels Database ( https://www.ebi.ac.uk/biomodels-main/ # MODEL 1604100005).

Systems Medicine: The Future of Medical Genomics, Healthcare, and Wellness

Recent advances in genomics have led to the rapid and relatively inexpensive collection of patient molecular data including multiple types of omics data. The integration of these data with clinical measurements has the potential to impact on our understanding of the molecular basis of disease and on disease management. Systems medicine is an approach to understanding disease through an integration of large patient datasets. It offers the possibility for personalized strategies for healthcare through the development of a new taxonomy of disease. Advanced computing will be an important component in effectively implementing systems medicine. In this chapter we describe three computational challenges associated with systems medicine: disease subtype discovery using integrated datasets, obtaining a mechanistic understanding of disease, and the development of an informatics platform for the mining, analysis, and visualization of data emerging from translational medicine studies.

Use of mechanistic models to integrate and analyze multiple proteomic datasets

Proteins in cell signaling networks tend to interact promiscuously through low-affinity interactions. Consequently, evaluating the physiological importance of mapped interactions can be difficult. Attempts to do so have tended to focus on single, measurable physicochemical factors, such as affinity or abundance. For example, interaction importance has been assessed on the basis of the relative affinities of binding partners for a protein of interest, such as a receptor. However, multiple factors can be expected to simultaneously influence the recruitment of proteins to a receptor (and the potential of these proteins to contribute to receptor signaling), including affinity, abundance, and competition, which is a network property. Here, we demonstrate that measurements of protein copy numbers and binding affinities can be integrated within the framework of a mechanistic, computational model that accounts for mass action and competition. We use cell line-specific models to rank the relative importance of protein-protein interactions in the epidermal growth factor receptor (EGFR) signaling network for 11 different cell lines. Each model accounts for experimentally characterized interactions of six autophosphorylation sites in EGFR with proteins containing a Src homology 2 and/or phosphotyrosine-binding domain. We measure importance as the predicted maximal extent of recruitment of a protein to EGFR following ligand-stimulated activation of EGFR signaling. We find that interactions ranked highly by this metric include experimentally detected interactions. Proteins with high importance rank in multiple cell lines include proteins with recognized, well-characterized roles in EGFR signaling, such as GRB2 and SHC1, as well as a protein with a less well-defined role, YES1. Our results reveal potential cell line-specific differences in recruitment.

Orchestration of ErbB3 signaling through heterointeractions and homointeractions

Members of the ErbB family of receptor tyrosine kinases are capable of both homointeractions and heterointeractions. Because each receptor has a unique set of binding sites for downstream signaling partners and differential catalytic activity, subtle shifts in their combinatorial interplay may have a large effect on signaling outcomes. The overexpression and mutation of ErbB family members are common in numerous human cancers and shift the balance of activation within the signaling network. Here we report the development of a spatial stochastic model that addresses the dynamics of ErbB3 homodimerization and heterodimerization with ErbB2. The model is based on experimental measures for diffusion, dimer off-rates, kinase activity, and dephosphorylation. We also report computational analysis of ErbB3 mutations, generating the prediction that activating mutations in the intracellular and extracellular domains may be subdivided into classes with distinct underlying mechanisms. We show experimental evidence for an ErbB3 gain-of-function point mutation located in the C-lobe asymmetric dimerization interface, which shows enhanced phosphorylation at low ligand dose associated with increased kinase activity.

Receptor Level Mechanisms Are Required for Epidermal Growth Factor (EGF)-stimulated Extracellular Signal-regulated Kinase (ERK) Activity Pulses

In both physiological and cell culture systems, EGF-stimulated ERK activity occurs in discrete pulses within individual cells. Many feedback loops are present in the EGF receptor (EGFR)-ERK network, but the mechanisms driving pulsatile ERK kinetics are unknown. Here, we find that in cells that respond to EGF with frequency-modulated pulsatile ERK activity, stimulation through a heterologous TrkA receptor system results in non-pulsatile, amplitude-modulated activation of ERK. We further dissect the kinetics of pulse activity using a combination of FRET- and translocation-based reporters and find that EGFR activity is required to maintain ERK activity throughout the 10-20-minute lifetime of pulses. Together, these data indicate that feedbacks operating within the core Ras-Raf-MEK-ERK cascade are insufficient to drive discrete pulses of ERK activity and instead implicate mechanisms acting at the level of EGFR.

Analytical reduction of combinatorial complexity arising from multiple protein modification sites

Combinatorial complexity is a major obstacle to ordinary differential equation (ODE) modelling of biochemical networks. For example, a protein with 10 sites that can each be unphosphorylated, phosphorylated or bound to adaptor protein requires 3(10) ODEs. This problem is often dealt with by making ad hoc assumptions which have unclear validity and disallow modelling of site-specific dynamics. Such site-specific dynamics, however, are important in many biological systems. We show here that for a common biological situation where adaptors bind modified sites, binding is slow relative to modification/demodification, and binding to one modified site hinders binding to other sites, for a protein with n modification sites and m adaptor proteins the number of ODEs needed to simulate the site-specific dynamics of biologically relevant, lumped bound adaptor states is independent of the number of modification sites and equal to m + 1, giving a significant reduction in system size. These considerations can be relaxed considerably while retaining reasonably accurate descriptions of the true system dynamics. We apply the theory to model, using only 11 ODEs, the dynamics of ligand-induced phosphorylation of nine tyrosines on epidermal growth factor receptor (EGFR) and primary recruitment of six signalling proteins (Grb2, PI3K, PLCγ1, SHP2, RasA1 and Shc1). The model quantitatively accounts for experimentally determined site-specific phosphorylation and dephosphorylation rates, differential affinities of binding proteins for the phosphorylated sites and binding protein expression levels. Analysis suggests that local concentration of site-specific phosphatases such as SHP2 in membrane subdomains by a factor of approximately 10(7) is critical for effective site-specific regulation. We further show how our framework can be extended with minimal effort to consider binding cooperativity between Grb2 and c-Cbl, which is important for receptor trafficking. Our theory has potentially broad application to reduce combinatorial complexity and allow practical simulation of a variety ODE models relevant to systems biology and pharmacology applications to allow exploration of key aspects of complexity that control signal flux.

Quantitative analysis reveals how EGFR activation and downregulation are coupled in normal but not in cancer cells

Ubiquitination of the epidermal growth factor receptor (EGFR) that occurs when Cbl and Grb2 bind to three phosphotyrosine residues (pY1045, pY1068 and pY1086) on the receptor displays a sharp threshold effect as a function of EGF concentration. Here we use a simple modelling approach together with experiments to show that the establishment of the threshold requires both the multiplicity of binding sites and cooperative binding of Cbl and Grb2 to the EGFR. While the threshold is remarkably robust, a more sophisticated model predicted that it could be modulated as a function of EGFR levels on the cell surface. We confirmed experimentally that the system has evolved to perform optimally at physiological levels of EGFR. As a consequence, this system displays an intrinsic weakness that causes--at the supraphysiological levels of receptor and/or ligand associated with cancer--uncoupling of the mechanisms leading to signalling through phosphorylation and attenuation through ubiquitination.

Modeling for (physical) biologists: an introduction to the rule-based approach

Models that capture the chemical kinetics of cellular regulatory networks can be specified in terms of rules for biomolecular interactions. A rule defines a generalized reaction, meaning a reaction that permits multiple reactants, each capable of participating in a characteristic transformation and each possessing certain, specified properties, which may be local, such as the state of a particular site or domain of a protein. In other words, a rule defines a transformation and the properties that reactants must possess to participate in the transformation. A rule also provides a rate law. A rule-based approach to modeling enables consideration of mechanistic details at the level of functional sites of biomolecules and provides a facile and visual means for constructing computational models, which can be analyzed to study how system-level behaviors emerge from component interactions.

Functional aspects of the EGF-induced MAP kinase cascade: a complex self-organizing system approach

The EGF-induced MAP kinase cascade is one of the most important and best characterized networks in intracellular signalling. It has a vital role in the development and maturation of living organisms. However, when deregulated, it is involved in the onset of a number of diseases. Based on a computational model describing a "surface" and an "internalized" parallel route, we use systems biology techniques to characterize aspects of the network's functional organization. We examine the re-organization of protein groups from low to high external stimulation, define functional groups of proteins within the network, determine the parameter best encoding for input intensity and predict the effect of protein removal to the system's output response. Extensive functional re-organization of proteins is observed in the lower end of stimulus concentrations. As we move to higher concentrations the variability is less pronounced. 6 functional groups have emerged from a consensus clustering approach, reflecting different dynamical aspects of the network. Mutual information investigation revealed that the maximum activation rate of the two output proteins best encodes for stimulus intensity. Removal of each protein of the network resulted in a range of graded effects, from complete silencing to intense activation. Our results provide a new "vista" of the EGF-induced MAP kinase cascade, from the perspective of complex self-organizing systems. Functional grouping of the proteins reveals an organizational scheme contrasting the current understanding of modular topology. The six identified groups may provide the means to experimentally follow the dynamics of this complex network. Also, the vulnerability analysis approach may be used for the development of novel therapeutic targets in the context of personalized medicine.

Dynamic transition states of ErbB1 phosphorylation predicted by spatial stochastic modeling

ErbB1 overexpression is strongly linked to carcinogenesis, motivating better understanding of erbB1 dimerization and activation. Recent single-particle-tracking data have provided improved measures of dimer lifetimes and strong evidence that transient receptor coconfinement promotes repeated interactions between erbB1 monomers. Here, spatial stochastic simulations explore the potential impact of these parameters on erbB1 phosphorylation kinetics. This rule-based mathematical model incorporates structural evidence for conformational flux of the erbB1 extracellular domains, as well as asymmetrical orientation of erbB1 cytoplasmic kinase domains during dimerization. The asymmetric dimer model considers the theoretical consequences of restricted transactivation of erbB1 receptors within a dimer, where the N-lobe of one monomer docks with the C-lobe of the second monomer and triggers its catalytic activity. The dynamic nature of the erbB1 phosphorylation state is shown by monitoring activation states of individual monomers as they diffuse, bind, and rebind after ligand addition. The model reveals the complex interplay between interacting liganded and nonliganded species and the influence of their distribution and abundance within features of the membrane landscape.

Exact hybrid particle/population simulation of rule-based models of biochemical systems

Detailed modeling and simulation of biochemical systems is complicated by the problem of combinatorial complexity, an explosion in the number of species and reactions due to myriad protein-protein interactions and post-translational modifications. Rule-based modeling overcomes this problem by representing molecules as structured objects and encoding their interactions as pattern-based rules. This greatly simplifies the process of model specification, avoiding the tedious and error prone task of manually enumerating all species and reactions that can potentially exist in a system. From a simulation perspective, rule-based models can be expanded algorithmically into fully-enumerated reaction networks and simulated using a variety of network-based simulation methods, such as ordinary differential equations or Gillespie's algorithm, provided that the network is not exceedingly large. Alternatively, rule-based models can be simulated directly using particle-based kinetic Monte Carlo methods. This “network-free” approach produces exact stochastic trajectories with a computational cost that is independent of network size. However, memory and run time costs increase with the number of particles, limiting the size of system that can be feasibly simulated. Here, we present a hybrid particle/population simulation method that combines the best attributes of both the network-based and network-free approaches. The method takes as input a rule-based model and a user-specified subset of species to treat as population variables rather than as particles. The model is then transformed by a process of “partial network expansion” into a dynamically equivalent form that can be simulated using a population-adapted network-free simulator. The transformation method has been implemented within the open-source rule-based modeling platform BioNetGen, and resulting hybrid models can be simulated using the particle-based simulator NFsim. Performance tests show that significant memory savings can be achieved using the new approach and a monetary cost analysis provides a practical measure of its utility.

Rule-based modeling is a modeling paradigm that addresses the problem of combinatorial complexity in biochemical systems. The key idea is to specify only those components of a biological macromolecule that are directly involved in a biochemical transformation. Until recently, this “pattern-based” approach greatly simplified the process of model building but did nothing to improve the performance of model simulation. This changed with the introduction of “network-free” simulation methods, which operate directly on the compressed rule set of a rule-based model rather than on a fully-enumerated set of reactions and species. However, these methods represent every molecule in a system as a particle, limiting their use to systems containing less than a few million molecules. Here, we describe an extension to the network-free approach that treats rare, complex species as particles and plentiful, simple species as population variables, while retaining the exact dynamics of the model system. By making more efficient use of computational resources for species that do not require the level of detail of a particle representation, this hybrid particle/population approach can simulate systems much larger than is possible using network-free methods and is an important step towards realizing the practical simulation of detailed, mechanistic models of whole cells.

Rule-based modeling: a computational approach for studying biomolecular site dynamics in cell signaling systems

Rule-based modeling was developed to address the limitations of traditional approaches for modeling chemical kinetics in cell signaling systems. These systems consist of multiple interacting biomolecules (e.g., proteins), which themselves consist of multiple parts (e.g., domains, linear motifs, and sites of phosphorylation). Consequently, biomolecules that mediate information processing generally have the potential to interact in multiple ways, with the number of possible complexes and posttranslational modification states tending to grow exponentially with the number of binary interactions considered. As a result, only large reaction networks capture all possible consequences of the molecular interactions that occur in a cell signaling system, which is problematic because traditional modeling approaches for chemical kinetics (e.g., ordinary differential equations) require explicit network specification. This problem is circumvented through representation of interactions in terms of local rules. With this approach, network specification is implicit and model specification is concise. Concise representation results in a coarse graining of chemical kinetics, which is introduced because all reactions implied by a rule inherit the rate law associated with that rule. Coarse graining can be appropriate if interactions are modular, and the coarseness of a model can be adjusted as needed. Rules can be specified using specialized model-specification languages, and recently developed tools designed for specification of rule-based models allow one to leverage powerful software engineering capabilities. A rule-based model comprises a set of rules, which can be processed by general-purpose simulation and analysis tools to achieve different objectives (e.g., to perform either a deterministic or stochastic simulation).

Integrative modelling of the influence of MAPK network on cancer cell fate decision

The Mitogen-Activated Protein Kinase (MAPK) network consists of tightly interconnected signalling pathways involved in diverse cellular processes, such as cell cycle, survival, apoptosis and differentiation. Although several studies reported the involvement of these signalling cascades in cancer deregulations, the precise mechanisms underlying their influence on the balance between cell proliferation and cell death (cell fate decision) in pathological circumstances remain elusive. Based on an extensive analysis of published data, we have built a comprehensive and generic reaction map for the MAPK signalling network, using CellDesigner software. In order to explore the MAPK responses to different stimuli and better understand their contributions to cell fate decision, we have considered the most crucial components and interactions and encoded them into a logical model, using the software GINsim. Our logical model analysis particularly focuses on urinary bladder cancer, where MAPK network deregulations have often been associated with specific phenotypes. To cope with the combinatorial explosion of the number of states, we have applied novel algorithms for model reduction and for the compression of state transition graphs, both implemented into the software GINsim. The results of systematic simulations for different signal combinations and network perturbations were found globally coherent with published data. In silico experiments further enabled us to delineate the roles of specific components, cross-talks and regulatory feedbacks in cell fate decision. Finally, tentative proliferative or anti-proliferative mechanisms can be connected with established bladder cancer deregulations, namely Epidermal Growth Factor Receptor (EGFR) over-expression and Fibroblast Growth Factor Receptor 3 (FGFR3) activating mutations.

Depending on environmental conditions, strongly intertwined cellular signalling pathways are activated, involving activation/inactivation of proteins and genes in response to external and/or internal stimuli. Alterations of some components of these pathways can lead to wrong cell behaviours. For instance, cancer-related deregulations lead to high proliferation of malignant cells enabling sustained tumour growth. Understanding the precise mechanisms underlying these pathways is necessary to delineate efficient therapeutical approaches for each specific tumour type. We particularly focused on the Mitogen-Activated Protein Kinase (MAPK) signalling network, whose involvement in cancer is well established, although the precise conditions leading to its positive or negative influence on cell proliferation are still poorly understood. We tackled this problem by first collecting sparse published biological information into a comprehensive map describing the MAPK network in terms of stylised chemical reactions. This information source was then used to build a dynamical Boolean model recapitulating network responses to characteristic stimuli observed in selected bladder cancers. Systematic model simulations further allowed us to link specific network components and interactions with proliferative/anti-proliferative cell responses.

Machines vs. ensembles: effective MAPK signaling through heterogeneous sets of protein complexes

Despite the importance of intracellular signaling networks, there is currently no consensus regarding the fundamental nature of the protein complexes such networks employ. One prominent view involves stable signaling machines with well-defined quaternary structures. The combinatorial complexity of signaling networks has led to an opposing perspective, namely that signaling proceeds via heterogeneous pleiomorphic ensembles of transient complexes. Since many hypotheses regarding network function rely on how we conceptualize signaling complexes, resolving this issue is a central problem in systems biology. Unfortunately, direct experimental characterization of these complexes has proven technologically difficult, while combinatorial complexity has prevented traditional modeling methods from approaching this question. Here we employ rule-based modeling, a technique that overcomes these limitations, to construct a model of the yeast pheromone signaling network. We found that this model exhibits significant ensemble character while generating reliable responses that match experimental observations. To contrast the ensemble behavior, we constructed a model that employs hierarchical assembly pathways to produce scaffold-based signaling machines. We found that this machine model could not replicate the experimentally observed combinatorial inhibition that arises when the scaffold is overexpressed. This finding provides evidence against the hierarchical assembly of machines in the pheromone signaling network and suggests that machines and ensembles may serve distinct purposes in vivo. In some cases, e.g. core enzymatic activities like protein synthesis and degradation, machines assembled via hierarchical energy landscapes may provide functional stability for the cell. In other cases, such as signaling, ensembles may represent a form of weak linkage, facilitating variation and plasticity in network evolution. The capacity of ensembles to signal effectively will ultimately shape how we conceptualize the function, evolution and engineering of signaling networks.

Intracellular signaling networks are central to a cell's ability to adapt to its environment. Developing the capacity to effectively manipulate such networks would have a wide range of applications, from cancer therapy to synthetic biology. This requires a thorough understanding of the mechanisms of signal transduction, particularly the kinds of protein complexes that are formed during transmission of extracellular information to the nucleus. Traditionally, signaling complexes have been largely perceived (albeit often implicitly) as machine-like structures. However, the number of molecular complexes that could theoretically be formed by complex signaling networks is astronomically large. This has led to the pleiomorphic ensemble hypothesis, which posits that diverse and rapidly changing sets of transient protein complexes can transmit and process information. Our goal was to use computational approaches, specifically rule-based modeling, to test these hypotheses. We constructed a model of the prototypical yeast mating pathway and found significant ensemble-like behavior. Our results thus demonstrated that ensembles can in fact transmit extracellular signals with minimal noise. Additionally, a comparison of this model with one tailored to generate machine-like complexes displayed notable phenotypic differences, revealing potential advantages for ensemble-like signaling. Our demonstration that ensembles can function effectively will have a significant impact on how we conceptualize signaling and other processes inside cells.

In silico modeling: methods and applications to trauma and sepsis

OBJECTIVES

To familiarize clinicians with advances in computational disease modeling applied to trauma and sepsis.

DATA SOURCES

PubMed search and review of relevant medical literature.

SUMMARY

Definitions, key methods, and applications of computational modeling to trauma and sepsis are reviewed.

CONCLUSIONS

Computational modeling of inflammation and organ dysfunction at the cellular, organ, whole-organism, and population levels has suggested a positive feedback cycle of inflammation → damage → inflammation that manifests via organ-specific inflammatory switching networks. This structure may manifest as multicompartment "tipping points" that drive multiple organ dysfunction. This process may be amenable to rational inflammation reprogramming.

Programming biological models in Python using PySB

PySB is a framework for creating biological models as Python programs using a high-level, action-oriented vocabulary that promotes transparency, extensibility and reusability. PySB interoperates with many existing modeling tools and supports distributed model development.

PySB models are programs and leverage existing programming tools for documentation, testing, and collaborative development.

Reusable functions can encode common low-level biochemical processes as well as high-level modules, making models transparent and concise.

Modeling workflow is accelerated through close integration with Python numerical tools and interoperability with existing modeling software.

We demonstrate the use of PySB to encode 15 alternative hypotheses for the mitochondrial regulation of apoptosis, including a new ‘Embedded Together' model based on recent biochemical findings.

Mathematical equations are fundamental to modeling biological networks, but as networks get large and revisions frequent, it becomes difficult to manage equations directly or to combine previously developed models. Multiple simultaneous efforts to create graphical standards, rule-based languages, and integrated software workbenches aim to simplify biological modeling but none fully meets the need for transparent, extensible, and reusable models. In this paper we describe PySB, an approach in which models are not only created using programs, they are programs. PySB draws on programmatic modeling concepts from little b and ProMot, the rule-based languages BioNetGen and Kappa and the growing library of Python numerical tools. Central to PySB is a library of macros encoding familiar biochemical actions such as binding, catalysis, and polymerization, making it possible to use a high-level, action-oriented vocabulary to construct detailed models. As Python programs, PySB models leverage tools and practices from the open-source software community, substantially advancing our ability to distribute and manage the work of testing biochemical hypotheses. We illustrate these ideas using new and previously published models of apoptosis.

Leveraging modeling approaches: reaction networks and rules

We have witnessed an explosive growth in research involving mathematical models and computer simulations of intracellular molecular interactions, ranging from metabolic pathways to signaling and gene regulatory networks. Many software tools have been developed to aid in the study of such biological systems, some of which have a wealth of features for model building and visualization, and powerful capabilities for simulation and data analysis. Novel high-resolution and/or high-throughput experimental techniques have led to an abundance of qualitative and quantitative data related to the spatiotemporal distribution of molecules and complexes, their interactions kinetics, and functional modifications. Based on this information, computational biology researchers are attempting to build larger and more detailed models. However, this has proved to be a major challenge. Traditionally, modeling tools require the explicit specification of all molecular species and interactions in a model, which can quickly become a major limitation in the case of complex networks - the number of ways biomolecules can combine to form multimolecular complexes can be combinatorially large. Recently, a new breed of software tools has been created to address the problems faced when building models marked by combinatorial complexity. These have a different approach for model specification, using reaction rules and species patterns. Here we compare the traditional modeling approach with the new rule-based methods. We make a case for combining the capabilities of conventional simulation software with the unique features and flexibility of a rule-based approach in a single software platform for building models of molecular interaction networks.

Specification, annotation, visualization and simulation of a large rule-based model for ERBB receptor signaling

BACKGROUND

Mathematical/computational models are needed to understand cell signaling networks, which are complex. Signaling proteins contain multiple functional components and multiple sites of post-translational modification. The multiplicity of components and sites of modification ensures that interactions among signaling proteins have the potential to generate myriad protein complexes and post-translational modification states. As a result, the number of chemical species that can be populated in a cell signaling network, and hence the number of equations in an ordinary differential equation model required to capture the dynamics of these species, is prohibitively large. To overcome this problem, the rule-based modeling approach has been developed for representing interactions within signaling networks efficiently and compactly through coarse-graining of the chemical kinetics of molecular interactions.

RESULTS

Here, we provide a demonstration that the rule-based modeling approach can be used to specify and simulate a large model for ERBB receptor signaling that accounts for site-specific details of protein-protein interactions. The model is considered large because it corresponds to a reaction network containing more reactions than can be practically enumerated. The model encompasses activation of ERK and Akt, and it can be simulated using a network-free simulator, such as NFsim, to generate time courses of phosphorylation for 55 individual serine, threonine, and tyrosine residues. The model is annotated and visualized in the form of an extended contact map.

CONCLUSIONS

With the development of software that implements novel computational methods for calculating the dynamics of large-scale rule-based representations of cellular signaling networks, it is now possible to build and analyze models that include a significant fraction of the protein interactions that comprise a signaling network, with incorporation of the site-specific details of the interactions. Modeling at this level of detail is important for understanding cellular signaling.

Dramatic reduction of dimensionality in large biochemical networks owing to strong pair correlations

Large multi-dimensionality of high-throughput datasets pertaining to cell signalling and gene regulation renders it difficult to extract mechanisms underlying the complex kinetics involving various biochemical compounds (e.g. proteins and lipids). Data-driven models often circumvent this difficulty by using pair correlations of the protein expression levels to produce a small number (fewer than 10) of principal components, each a linear combination of the concentrations, to successfully model how cells respond to different stimuli. However, it is not understood if this reduction is specific to a particular biological system or to nature of the stimuli used in these experiments. We study temporal changes in pair correlations, described by the covariance matrix, between concentrations of different molecular species that evolve following deterministic mass-action kinetics in large biologically relevant reaction networks and show that this dramatic reduction of dimensions (from hundreds to less than five) arises from the strong correlations between different species at any time and is insensitive to the form of the nonlinear interactions, network architecture, and to a wide range of values of rate constants and concentrations. We relate temporal changes in the eigenvalue spectrum of the covariance matrix to low-dimensional, local changes in directions of the system trajectory embedded in much larger dimensions using elementary differential geometry. We illustrate how to extract biologically relevant insights such as identifying significant timescales and groups of correlated chemical species from our analysis. Our work provides for the first time, to our knowledge, a theoretical underpinning for the successful experimental analysis and points to a way to extract mechanisms from large-scale high-throughput datasets.

Rapid phospho-turnover by receptor tyrosine kinases impacts downstream signaling and drug binding

Epidermal growth factor receptors (ErbB1-4) are oncogenic receptor tyrosine kinases (RTKs) that regulate diverse cellular processes. In this study, we combine measurement and mathematical modeling to quantify phospho-turnover at ErbB receptors in human cells and to determine the consequences for signaling and drug binding. We find that phosphotyrosine residues on ErbB1 have half-lives of a few seconds and therefore turn over 100-1000 times in the course of a typical immediate-early response to ligand. Rapid phospho-turnover is also observed for EGF-activated ErbB2 and ErbB3, unrelated RTKs, and multiple intracellular adaptor proteins and signaling kinases. Thus, the complexes formed on the cytoplasmic tail of active receptors and the downstream signaling kinases they control are highly dynamic and antagonized by potent phosphatases. We develop a kinetic scheme for binding of anti-ErbB1 drugs to receptors and show that rapid phospho-turnover significantly impacts their mechanisms of action.

Modeling formalisms in Systems Biology

Systems Biology has taken advantage of computational tools and high-throughput experimental data to model several biological processes. These include signaling, gene regulatory, and metabolic networks. However, most of these models are specific to each kind of network. Their interconnection demands a whole-cell modeling framework for a complete understanding of cellular systems. We describe the features required by an integrated framework for modeling, analyzing and simulating biological processes, and review several modeling formalisms that have been used in Systems Biology including Boolean networks, Bayesian networks, Petri nets, process algebras, constraint-based models, differential equations, rule-based models, interacting state machines, cellular automata, and agent-based models. We compare the features provided by different formalisms, and discuss recent approaches in the integration of these formalisms, as well as possible directions for the future.

Guidelines for visualizing and annotating rule-based models

Rule-based modeling provides a means to represent cell signaling systems in a way that captures site-specific details of molecular interactions. For rule-based models to be more widely understood and (re)used, conventions for model visualization and annotation are needed. We have developed the concepts of an extended contact map and a model guide for illustrating and annotating rule-based models. An extended contact map represents the scope of a model by providing an illustration of each molecule, molecular component, direct physical interaction, post-translational modification, and enzyme-substrate relationship considered in a model. A map can also illustrate allosteric effects, structural relationships among molecular components, and compartmental locations of molecules. A model guide associates elements of a contact map with annotation and elements of an underlying model, which may be fully or partially specified. A guide can also serve to document the biological knowledge upon which a model is based. We provide examples of a map and guide for a published rule-based model that characterizes early events in IgE receptor (FcεRI) signaling. We also provide examples of how to visualize a variety of processes that are common in cell signaling systems but not considered in the example model, such as ubiquitination. An extended contact map and an associated guide can document knowledge of a cell signaling system in a form that is visual as well as executable. As a tool for model annotation, a map and guide can communicate the content of a model clearly and with precision, even for large models.

Timescale analysis of rule-based biochemical reaction networks

The flow of information within a cell is governed by a series of protein-protein interactions that can be described as a reaction network. Mathematical models of biochemical reaction networks can be constructed by repetitively applying specific rules that define how reactants interact and what new species are formed on reaction. To aid in understanding the underlying biochemistry, timescale analysis is one method developed to prune the size of the reaction network. In this work, we extend the methods associated with timescale analysis to reaction rules instead of the species contained within the network. To illustrate this approach, we applied timescale analysis to a simple receptor-ligand binding model and a rule-based model of interleukin-12 (IL-12) signaling in naïve CD4+ T cells. The IL-12 signaling pathway includes multiple protein-protein interactions that collectively transmit information; however, the level of mechanistic detail sufficient to capture the observed dynamics has not been justified based on the available data. The analysis correctly predicted that reactions associated with Janus Kinase 2 and Tyrosine Kinase 2 binding to their corresponding receptor exist at a pseudo-equilibrium. By contrast, reactions associated with ligand binding and receptor turnover regulate cellular response to IL-12. An empirical Bayesian approach was used to estimate the uncertainty in the timescales. This approach complements existing rank- and flux-based methods that can be used to interrogate complex reaction networks. Ultimately, timescale analysis of rule-based models is a computational tool that can be used to reveal the biochemical steps that regulate signaling dynamics.

Regulation of hSos1 activity is a system-level property generated by its multi-domain structure

The multi-domain protein hSos1 plays a major role in cell growth and differentiation through its Ras-specific guanine nucleotide exchange domain whose complex regulation involves intra-molecular, inter-domain rearrangements. We present a stochastic mathematical model describing intra-molecular regulation of hSos1 activity. The population macroscopic effect is reproduced through a Monte-Carlo approach. Key model parameters have been experimentally determined by BIAcore analysis. Complementation experiments of a Saccharomyces cerevisiae cdc25(ts) strain with Sos deletion mutants provided a comprehensive data set for estimation of unknown parameters and model validation. The model is robust against parameter alteration and describes both the behavior of Sos deletion mutants and modulation of activity of the full length molecule under physiological conditions. By incorporating the calculated effect of amino acid changes at an inter-domain interface, the behavior of a mutant correlating with a developmental syndrome could be simulated, further validating the model. The activation state of Ras-specific guanine nucleotide exchange domain of hSos1 arises as an "emergent property" of its multi-domain structure that allows multi-level integration of a complex network of intra- and inter-molecular signals.

A Boolean probabilistic model of metabolic adaptation to oxygen in relation to iron homeostasis and oxidative stress

Background

In aerobically grown cells, iron homeostasis and oxidative stress are tightly linked processes implicated in a growing number of diseases. The deregulation of iron homeostasis due to gene defects or environmental stresses leads to a wide range of diseases with consequences for cellular metabolism that remain poorly understood. The modelling of iron homeostasis in relation to the main features of metabolism, energy production and oxidative stress may provide new clues to the ways in which changes in biological processes in a normal cell lead to disease.

Results

Using a methodology based on probabilistic Boolean modelling, we constructed the first model of yeast iron homeostasis including oxygen-related reactions in the frame of central metabolism. The resulting model of 642 elements and 1007 reactions was validated by comparing simulations with a large body of experimental results (147 phenotypes and 11 metabolic flux experiments). We removed every gene, thus generating in silico mutants. The simulations of the different mutants gave rise to a remarkably accurate qualitative description of most of the experimental phenotype (overall consistency > 91.5%). A second validation involved analysing the anaerobiosis to aerobiosis transition. Therefore, we compared the simulations of our model with different levels of oxygen to experimental metabolic flux data. The simulations reproducted accurately ten out of the eleven metabolic fluxes. We show here that our probabilistic Boolean modelling strategy provides a useful description of the dynamics of a complex biological system. A clustering analysis of the simulations of all in silico mutations led to the identification of clear phenotypic profiles, thus providing new insights into some metabolic response to stress conditions. Finally, the model was also used to explore several new hypothesis in order to better understand some unexpected phenotypes in given mutants.

Conclusions

All these results show that this model, and the underlying modelling strategy, are powerful tools for improving our understanding of complex biological problems.