Computational approaches for modeling regulatory cellular networks

Cellular components interact with each other to form networks that process information and evoke biological responses. A deep understanding of the behavior of these networks requires the development and analysis of mathematical models. In this article, different types of mathematical representations for modeling signaling networks are described, and the advantages and disadvantages of each type are discussed. Two experimentally well-studied signaling networks are then used as examples to illustrate the insight that could be gained through modeling. Finally, the modeling approach is expanded to describe how signaling networks might regulate cellular machines and evoke phenotypic behaviors.     

Mathematical modeling as a tool for investigating cell cycle control networks

Although not a traditional experimental "method," mathematical modeling can provide a powerful approach for investigating complex cell signaling networks, such as those that regulate the eukaryotic cell division cycle. We describe here one modeling approach based on expressing the rates of biochemical reactions in terms of nonlinear ordinary differential equations. We discuss the steps and challenges in assigning numerical values to model parameters and the importance of experimental testing of a mathematical model. We illustrate this approach throughout with the simple and well-characterized example of mitotic cell cycles in frog egg extracts. To facilitate new modeling efforts, we describe several publicly available modeling environments, each with a collection of integrated programs for mathematical modeling. This review is intended to justify the place of mathematical modeling as a standard method for studying molecular regulatory networks and to guide the non-expert to initiate modeling projects in order to gain a systems-level perspective for complex control systems.     

The transition from differential equations to Boolean networks: a case study in simplifying a regulatory network model

Methods for modeling cellular regulatory networks as diverse as differential equations and Boolean networks co-exist, however, without much closer correspondence to each other. With the example system of the fission yeast cell cycle control network, we here discuss these two approaches with respect to each other. We find that a Boolean network model can be formulated as a specific coarse-grained limit of the more detailed differential equations model for this system. This demonstrates the mathematical foundation on which Boolean networks can be applied to biological regulatory networks in a controlled way.     

Short-term information processing, long-term responses: Insights by mathematical modeling of signal transduction. Early activation dynamics of key signaling mediators can be predictive for cell fate decisions

How do cells interpret information from their environment and translate it into specific cell fate decisions? We propose that cell fate is already encoded in early signaling events and thus can be predicted from defined signal properties. Specifically, we hypothesize that the time integral of activated key signaling molecules can be correlated to cellular behavior such as proliferation or differentiation. The identification of these decisive key signal mediators and their connection to cell fate is facilitated by mathematical modeling. A possible mechanistic linkage between signaling dynamics and cellular function is the directed control of gene regulatory networks by defined signals. Targeted experiments in combination with mathematical modeling can increase our understanding of how cells process information and realize distinct cell fates.     

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

ABSTRACT: 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.     

Metabolic modeling of microbial strains in silico

The large volume of genome-scale data that is being produced and made available in databases on the World Wide Web is demanding the development of integrated mathematical models of cellular processes. The analysis of reconstructed metabolic networks as systems leads to the development of an in silico or computer representation of collections of cellular metabolic constituents, their interactions and their integrated function as a whole. The use of quantitative analysis methods to generate testable hypotheses and drive experimentation at a whole-genome level signals the advent of a systemic modeling approach to cellular and molecular biology.     

A survey on methods for modeling and analyzing integrated biological networks

Understanding how cellular systems build up integrated responses to their dynamically changing environment is one of the open questions in Systems Biology. Despite their intertwinement, signaling networks, gene regulation and metabolism have been frequently modeled independently in the context of well-defined subsystems. For this purpose, several mathematical formalisms have been developed according to the features of each particular network under study. Nonetheless, a deeper understanding of cellular behavior requires the integration of these various systems into a model capable of capturing how they operate as an ensemble. With the recent advances in the "omics" technologies, more data is becoming available and, thus, recent efforts have been driven toward this integrated modeling approach. We herein review and discuss methodological frameworks currently available for modeling and analyzing integrated biological networks, in particular metabolic, gene regulatory and signaling networks. These include network-based methods and Chemical Organization Theory, Flux-Balance Analysis and its extensions, logical discrete modeling, Petri Nets, traditional kinetic modeling, Hybrid Systems and stochastic models. Comparisons are also established regarding data requirements, scalability with network size and computational burden. The methods are illustrated with successful case studies in large-scale genome models and in particular subsystems of various organisms.     

Hybrid framework for feasible modeling of an edge illumination X-ray phase-contrast imaging system at a human scale

Here, we present a hybrid approach for simulating an edge illumination X-ray phase-contrast imaging (EIXPCi) set-up using graphics processor units (GPU) with a high degree of accuracy. In this study, the applicability of pixel, mesh and non-uniform rational B-splines (NURBS) objects to carry out realistic maps of X-ray phase-contrast distribution at a human scale is accounted for by using numerical anthropomorphic phantoms and a very fast and robust simulation framework which integrates total interaction probabilities along selected X-ray paths. We exploit the mathematical and algorithmic properties of NURBS and describe how to represent human scale phantoms in an edge illumination X-ray phase-contrast model. The presented implementation allows the modeling of a variety of physical interactions of x-rays with different mathematically described objects and the recording of quantities, e.g. path integrals, interaction sites and deposited energies. Furthermore, our efficient, scalable and optimized hybrid Monte Carlo and ray-tracing projector can be used in iterative reconstruction algorithms on multi GPU heterogeneous systems. The preliminary results of our innovative approach show the fine performance of an edge illumination X-ray phase-contrast medical imaging system on various human-like soft tissues with noticeably reduced computation time. Our approach to the EIXPCi modeling confirms that building a true imaging system at a human scale should be possible and the simulations presented here aim at its future development.     

Understanding biological functions through molecular networks

The completion of genome sequences and subsequent high-throughput mapping of molecular networks have allowed us to study biology from the network perspective. Experimental, statistical and mathematical modeling approaches have been employed to study the structure, function and dynamics of molecular networks, and begin to reveal important links of various network properties to the functions of the biological systems. In agreement with these functional links, evolutionary selection of a network is apparently based on the function, rather than directly on the structure of the network. Dynamic modularity is one of the prominent features of molecular networks. Taking advantage of such a feature may simplify network-based biological studies through construction of process-specific modular networks and provide functional and mechanistic insights linking genotypic variations to complex traits or diseases, which is likely to be a key approach in the next wave of understanding complex human diseases. With the development of ready-to-use network analysis and modeling tools the networks approaches will be infused into everyday biological research in the near future.     

时延细胞神经网络的全局稳定性分析

本文利用 Ｌｙａｐｕｎｏｖ泛函方法和一些分析技巧得到了一类时延细胞神经网络ＤＣＮＮ全局渐进稳定性的若干新的判据；这些判据可用于设计出全局稳定的各种动态网络，且在信号处理，特别是动态图象处理中具有重要的意义．     

Exploring overlapping functional units with various structure in protein interaction networks

Revealing functional units in protein-protein interaction (PPI) networks are important for understanding cellular functional organization. Current algorithms for identifying functional units mainly focus on cohesive protein complexes which have more internal interactions than external interactions. Most of these approaches do not handle overlaps among complexes since they usually allow a protein to belong to only one complex. Moreover, recent studies have shown that other non-cohesive structural functional units beyond complexes also exist in PPI networks. Thus previous algorithms that just focus on non-overlapping cohesive complexes are not able to present the biological reality fully. Here, we develop a new regularized sparse random graph model (RSRGM) to explore overlapping and various structural functional units in PPI networks. RSRGM is principally dominated by two model parameters. One is used to define the functional units as groups of proteins that have similar patterns of connections to others, which allows RSRGM to detect non-cohesive structural functional units. The other one is used to represent the degree of proteins belonging to the units, which supports a protein belonging to more than one revealed unit. We also propose a regularizer to control the smoothness between the estimators of these two parameters. Experimental results on four S. cerevisiae PPI networks show that the performance of RSRGM on detecting cohesive complexes and overlapping complexes is superior to that of previous competing algorithms. Moreover, RSRGM has the ability to discover biological significant functional units besides complexes.     

Critical conditions for pattern formation and in vitro tubulogenesis driven by cellular traction fields

In vitro angiogenesis assays have shown that tubulogenesis of endothelial cells within biogels, like collagen or fibrin gels, only appears for a critical range of experimental parameter values. These experiments have enabled us to develop and validate a theoretical model in which mechanical interactions of endothelial cells with extracellular matrix influence both active cell migration--haptotaxis--and cellular traction forces. Depending on the number of cells, cell motility and biogel rheological properties, various 2D endothelial patterns can be generated, from non-connected stripe patterns to fully connected networks, which mimic the spatial organization of capillary structures. The model quantitatively and qualitatively reproduces the range of critical values of cell densities and fibrin concentrations for which these cell networks are experimentally observed. We illustrate how cell motility is associated to the self-enhancement of the local traction fields exerted within the biogel in order to produce a pre-patterning of this matrix and subsequent formation of tubular structures, above critical thresholds corresponding to bifurcation points of the mathematical model. The dynamics of this morphogenetic process is discussed in the light of videomicroscopy time lapse sequences of endothelial cells (EAhy926 line) in fibrin gels. Our modeling approach also explains how the progressive appearance and morphology of the cellular networks are modified by gradients of extracellular matrix thickness.     

The statistical mechanics of complex signaling networks: nerve growth factor signaling

The inherent complexity of cellular signaling networks and their importance to a wide range of cellular functions necessitates the development of modeling methods that can be applied toward making predictions and highlighting the appropriate experiments to test our understanding of how these systems are designed and function. We use methods of statistical mechanics to extract useful predictions for complex cellular signaling networks. A key difficulty with signaling models is that, while significant effort is being made to experimentally measure the rate constants for individual steps in these networks, many of the parameters required to describe their behavior remain unknown or at best represent estimates. To establish the usefulness of our approach, we have applied our methods toward modeling the nerve growth factor (NGF)-induced differentiation of neuronal cells. In particular, we study the actions of NGF and mitogenic epidermal growth factor (EGF) in rat pheochromocytoma (PC12) cells. Through a network of intermediate signaling proteins, each of these growth factors stimulates extracellular regulated kinase (Erk) phosphorylation with distinct dynamical profiles. Using our modeling approach, we are able to predict the influence of specific signaling modules in determining the integrated cellular response to the two growth factors. Our methods also raise some interesting insights into the design and possible evolution of cellular systems, highlighting an inherent property of these systems that we call 'sloppiness.'