Modeling and simulation of biological systems with stochasticity

Mathematical modeling is a powerful approach for understanding the complexity of biological systems. Recently, several successful attempts have been made for simulating complex biological processes like metabolic pathways, gene regulatory networks and cell signaling pathways. The pathway models have not only generated experimentally verifiable hypothesis but have also provided valuable insights into the behavior of complex biological systems. Many recent studies have confirmed the phenotypic variability of organisms to an inherent stochasticity that operates at a basal level of gene expression. Due to this reason, development of novel mathematical representations and simulations algorithms are critical for successful modeling efforts in biological systems. The key is to find a biologically relevant representation for each representation. Although mathematically rigorous and physically consistent, stochastic algorithms are computationally expensive, they have been successfully used to model probabilistic events in the cell. This paper offers an overview of various mathematical and computational approaches for modeling stochastic phenomena in cellular systems.     

Toward computational systems biology

The development and successful application of high-throughput technologies are transforming biological research. The large quantities of data being generated by these technologies have led to the emergence of systems biology, which emphasizes large-scale, parallel characterization of biological systems and integration of fragmentary information into a coherent whole. Complementing the reductionist approach that has dominated biology for the last century, mathematical modeling is becoming a powerful tool to achieve an integrated understanding of complex biological systems and to guide experimental efforts of engineering biological systems for practical applications. Here I give an overview of current mainstream approaches in modeling biological systems, highlight specific applications of modeling in various settings, and point out future research opportunities and challenges.     

On the limitations of standard statistical modeling in biological systems: A full Bayesian approach for biology

One of the most important scientific challenges today is the quantitative and predictive understanding of biological function. Classical mathematical and computational approaches have been enormously successful in modeling inert matter, but they may be inadequate to address inherent features of biological systems. We address the conceptual and methodological obstacles that lie in the inverse problem in biological systems modeling. We introduce a full Bayesian approach (FBA), a theoretical framework to study biological function, in which probability distributions are conditional on biophysical information that physically resides in the biological system that is studied by the scientist.     

Attraction Basins as Gauges of Robustness against Boundary Conditions in Biological Complex Systems

One fundamental concept in the context of biological systems on which researches have flourished in the past decade is that of the apparent robustness of these systems, i.e., their ability to resist to perturbations or constraints induced by external or boundary elements such as electromagnetic fields acting on neural networks, micro-RNAs acting on genetic networks and even hormone flows acting both on neural and genetic networks. Recent studies have shown the importance of addressing the question of the environmental robustness of biological networks such as neural and genetic networks. In some cases, external regulatory elements can be given a relevant formal representation by assimilating them to or modeling them by boundary conditions. This article presents a generic mathematical approach to understand the influence of boundary elements on the dynamics of regulation networks, considering their attraction basins as gauges of their robustness. The application of this method on a real genetic regulation network will point out a mathematical explanation of a biological phenomenon which has only been observed experimentally until now, namely the necessity of the presence of gibberellin for the flower of the plant Arabidopsis thaliana to develop normally.     

Systems biology in animal sciences

Systems biology is a rapidly expanding field of research and is applied in a number of biological disciplines. In animal sciences, omics approaches are increasingly used, yielding vast amounts of data, but systems biology approaches to extract understanding from these data of biological processes and animal traits are not yet frequently used. This paper aims to explain what systems biology is and which areas of animal sciences could benefit from systems biology approaches. Systems biology aims to understand whole biological systems working as a unit, rather than investigating their individual components. Therefore, systems biology can be considered a holistic approach, as opposed to reductionism. The recently developed 'omics' technologies enable biological sciences to characterize the molecular components of life with ever increasing speed, yielding vast amounts of data. However, biological functions do not follow from the simple addition of the properties of system components, but rather arise from the dynamic interactions of these components. Systems biology combines statistics, bioinformatics and mathematical modeling to integrate and analyze large amounts of data in order to extract a better understanding of the biology from these huge data sets and to predict the behavior of biological systems. A 'system' approach and mathematical modeling in biological sciences are not new in itself, as they were used in biochemistry, physiology and genetics long before the name systems biology was coined. However, the present combination of mass biological data and of computational and modeling tools is unprecedented and truly represents a major paradigm shift in biology. Significant advances have been made using systems biology approaches, especially in the field of bacterial and eukaryotic cells and in human medicine. Similarly, progress is being made with 'system approaches' in animal sciences, providing exciting opportunities to predict and modulate animal traits.     

Predictive landscapes hidden beneath biological cellular automata

To celebrate Hans Frauenfelder’s achievements, we examine energy(-like) “landscapes” for complex living systems. Energy landscapes summarize all possible dynamics of some physical systems. Energy(-like) landscapes can explain some biomolecular processes, including gene expression and, as Frauenfelder showed, protein folding. But energy-like landscapes and existing frameworks like statistical mechanics seem impractical for describing many living systems. Difficulties stem from living systems being high dimensional, nonlinear, and governed by many, tightly coupled constituents that are noisy. The predominant modeling approach is devising differential equations that are tailored to each living system. This ad hoc approach faces the notorious “parameter problem”: models have numerous nonlinear, mathematical functions with unknown parameter values, even for describing just a few intracellular processes. One cannot measure many intracellular parameters or can only measure them as snapshots in time. Another modeling approach uses cellular automata to represent living systems as discrete dynamical systems with binary variables. Quantitative (Hamiltonian-based) rules can dictate cellular automata (e.g., Cellular Potts Model). But numerous biological features, in current practice, are qualitatively described rather than quantitatively (e.g., gene is (highly) expressed or not (highly) expressed). Cellular automata governed by verbal rules are useful representations for living systems and can mitigate the parameter problem. However, they can yield complex dynamics that are difficult to understand because the automata-governing rules are not quantitative and much of the existing mathematical tools and theorems apply to continuous but not discrete dynamical systems. Recent studies found ways to overcome this challenge. These studies either discovered or suggest an existence of predictive “landscapes” whose shapes are described by Lyapunov functions and yield “equations of motion” for a “pseudo-particle.” The pseudo-particle represents the entire cellular lattice and moves on the landscape, thereby giving a low-dimensional representation of the cellular automata dynamics. We outline this promising modeling strategy.

     

Mathematical Models Light Up Plant Signaling

Plants respond to changes in the environment by triggering a suite of regulatory networks that control and synchronize molecular signaling in different tissues, organs, and the whole plant. Molecular studies through genetic and environmental perturbations, particularly in the model plant Arabidopsis thaliana, have revealed many of the mechanisms by which these responses are actuated. In recent years, mathematical modeling has become a complementary tool to the experimental approach that has furthered our understanding of biological mechanisms. In this review, we present modeling examples encompassing a range of different biological processes, in particular those regulated by light. Current issues and future directions in the modeling of plant systems are discussed.     

Plant Metabolic Modeling: Achieving New Insight into Metabolism and Metabolic Engineering

Models are used to represent aspects of the real world for specific purposes, and mathematical models have opened up new approaches in studying the behavior and complexity of biological systems. However, modeling is often time-consuming and requires significant computational resources for data development, data analysis, and simulation. Computational modeling has been successfully applied as an aid for metabolic engineering in microorganisms. But such model-based approaches have only recently been extended to plant metabolic engineering, mainly due to greater pathway complexity in plants and their highly compartmentalized cellular structure. Recent progress in plant systems biology and bioinformatics has begun to disentangle this complexity and facilitate the creation of efficient plant metabolic models. This review highlights several aspects of plant metabolic modeling in the context of understanding, predicting and modifying complex plant metabolism. We discuss opportunities for engineering photosynthetic carbon metabolism, sucrose synthesis, and the tricarboxylic acid cycle in leaves and oil synthesis in seeds and the application of metabolic modeling to the study of plant acclimation to the environment. The aim of the review is to offer a current perspective for plant biologists without requiring specialized knowledge of bioinformatics or systems biology.     

Modeling cancer: integration of "omics" information in dynamic systems

The last 10 years have seen the rise of many technologies that produce an unprecedented amount of genome-scale data from many organisms. Although the research community has been successful in exploring these data, many challenges still persist. One of them is the effective integration of such data sets directly into approaches based on mathematical modeling of biological systems. Applications in cancer are a good example. The bridge between information and modeling in cancer can be achieved by two major types of complementary strategies. First, there is a bottom-up approach, in which data generates information about structure and relationship between components of a given system. In addition, there is a top-down approach, where cybernetic and systems-theoretical knowledge are used to create models that describe mechanisms and dynamics of the system. These approaches can also be linked to yield multi-scale models combining detailed mechanism and wide biological scope. Here we give an overall picture of this field and discuss possible strategies to approach the major challenges ahead.     

The systems biology of signaling pathways

Signaling networks play the central role in the regulation of processes in a single cell and in the entire body. A recent breakthrough in technologies for systems biology, which combine experimental and mathematical methods, permits scientists to model signaling pathways in an individual cell and in cell populations. This approach provides new information on mechanisms that regulate a variety of biological processes. Here we discuss the mathematical formalisms that are applied to signaling pathway modeling and relevant experimental methods.     

A top-down approach to mechanistic biological modeling: application to the single-chain antibody folding pathway

A top-down approach to mechanistic modeling of biological systems is presented and exemplified with the development of a hypothesis-driven mathematical model for single-chain antibody fragment (scFv) folding in Saccharomyces cerevisiae by mediators BiP and PDI. In this approach, model development starts with construction of the most basic mathematical model—typically consisting of predetermined or newly-elucidated biological behavior motifs—capable of reproducing desired biological behaviors. From this point, mechanistic detail is added incrementally and systematically, and the effects of each addition are evaluated. This approach follows the typical progression of experimental data availability in that higher-order, lumped measurements are often more prevalent initially than specific, mechanistic ones. It also necessarily provides the modeler with insight into the structural requirements and performance capabilities of the resulting detailed mechanistic model, which facilitates further analysis. The top-down approach to mechanistic modeling identified three such requirements and a branched dependency-degradation competition motif critical for the scFv folding model to reproduce experimentally observed scFv folding dependencies on BiP and PDI and increased production when both species are overexpressed and promoted straightforward prediction of parameter dependencies. It also prescribed modification of the guiding hypothesis to capture BiP and PDI synergy.     

Metabolic ensemble modeling for strain engineers

Previous mathematical modeling efforts have made significant contributions to the development of systems biology for predicting biological behavior quantitatively. However, dynamic metabolic model construction remains challenging due to uncertainties in mechanistic structures and parameters. In addition, parameter estimation and model validation often require designated experiments conducted only for purpose of modeling. Such difficulties have hampered the progress of modeling in biology and biotechnology. To circumvent these problems, ensemble approaches have been used to account for uncertainties in model structure and parameters. Specifically, this review focuses on approaches that utilize readily available fermentation data for parameter screening and model validation. Time course data for metabolite measurements, if available, can further calibrate the model. The basis for this approach is explained in non-mathematical terms accessible to experimentalists. Information gained from such an approach has been shown to be useful in designing Escherichia coli strains for metabolic engineering and synthetic biology.     

计算系统生物学:理论、方法及在药物研发中的应用

计算系统生物学是一个多学科交叉的新兴领域,旨在通过整合海量数据建立其生物系统相互作用的复杂网络。数据的整合和模型的建立需要发展合适的数学方法和软件工具,这也是计算系统生物学的主要任务。生物系统模型有助于从整体上理解生物体的内在功能和特性。同时,生物网络模型在药物研发中的应用也越来越受到制药企业以及新药研发机构的重视,如用于特异性药物作用靶点的预测和药物毒性评估等。该文简要介绍计算系统生物学的常见网络和计算模型,以及建立模型所用的研究方法,并阐述其在建模和分析中的作用及面临的问题和挑战。     

Space as the final frontier in stochastic simulations of biological systems

Recent technological and theoretical advances are only now allowing the simulation of detailed kinetic models of biological systems that reflect the stochastic movement and reactivity of individual molecules within cellular compartments. The behavior of many systems could not be properly understood without this level of resolution, opening up new perspectives of using computer simulations to accelerate biological research. We review the modeling methodology applied to stochastic spatial models, also to the attention of non-expert potential users. Modeling choices, current limitations and perspectives of improvement of current general-purpose modeling/simulation platforms for biological systems are discussed.     

Simple network motifs can capture key characteristics of the floral transition in Arabidopsis

The floral transition is a key decision during plant development. While different species have evolved diverse pathways to respond to different environmental cues to flower in the correct season, key properties such as irreversibility and robustness to fluctuating signals appear to be conserved. We have used mathematical modeling to demonstrate how minimal regulatory networks of core components are sufficient to capture these behaviors. Simplified models inevitably miss finer details of the biological system, yet they provide a tractable route to understanding the overall system behavior. We combined models with experimental data to qualitatively reproduce characteristics of the floral transition and to quantitatively scale the network to fit with available leaf numbers. Our study highlights the value of pursuing an iterative approach combining modeling with experimental work to capture key features of complex systems.