SmartCell, a framework to simulate cellular processes that combines stochastic approximation with diffusion and localisation: analysis of simple networks

Rule-based spatial modeling with diffusing, geometrically constrained molecules

Background

We suggest a new type of modeling approach for the coarse grained, particle-based spatial simulation of combinatorially complex chemical reaction systems. In our approach molecules possess a location in the reactor as well as an orientation and geometry, while the reactions are carried out according to a list of implicitly specified reaction rules. Because the reaction rules can contain patterns for molecules, a combinatorially complex or even infinitely sized reaction network can be defined.

For our implementation (based on LAMMPS), we have chosen an already existing formalism (BioNetGen) for the implicit specification of the reaction network. This compatibility allows to import existing models easily, i.e., only additional geometry data files have to be provided.

Results

Our simulations show that the obtained dynamics can be fundamentally different from those simulations that use classical reaction-diffusion approaches like Partial Differential Equations or Gillespie-type spatial stochastic simulation. We show, for example, that the combination of combinatorial complexity and geometric effects leads to the emergence of complex self-assemblies and transportation phenomena happening faster than diffusion (using a model of molecular walkers on microtubules). When the mentioned classical simulation approaches are applied, these aspects of modeled systems cannot be observed without very special treatment. Further more, we show that the geometric information can even change the organizational structure of the reaction system. That is, a set of chemical species that can in principle form a stationary state in a Differential Equation formalism, is potentially unstable when geometry is considered, and vice versa.

Conclusions

We conclude that our approach provides a new general framework filling a gap in between approaches with no or rigid spatial representation like Partial Differential Equations and specialized coarse-grained spatial simulation systems like those for DNA or virus capsid self-assembly.

BiGG: a Biochemical Genetic and Genomic knowledgebase of large scale metabolic reconstructions

BACKGROUND

Genome-scale metabolic reconstructions under the Constraint Based Reconstruction and Analysis (COBRA) framework are valuable tools for analyzing the metabolic capabilities of organisms and interpreting experimental data. As the number of such reconstructions and analysis methods increases, there is a greater need for data uniformity and ease of distribution and use.

DESCRIPTION

We describe BiGG, a knowledgebase of Biochemically, Genetically and Genomically structured genome-scale metabolic network reconstructions. BiGG integrates several published genome-scale metabolic networks into one resource with standard nomenclature which allows components to be compared across different organisms. BiGG can be used to browse model content, visualize metabolic pathway maps, and export SBML files of the models for further analysis by external software packages. Users may follow links from BiGG to several external databases to obtain additional information on genes, proteins, reactions, metabolites and citations of interest.

CONCLUSIONS

BiGG addresses a need in the systems biology community to have access to high quality curated metabolic models and reconstructions. It is freely available for academic use at http://bigg.ucsd.edu.

Detailed simulations of cell biology with Smoldyn 2.1

Most cellular processes depend on intracellular locations and random collisions of individual protein molecules. To model these processes, we developed algorithms to simulate the diffusion, membrane interactions, and reactions of individual molecules, and implemented these in the Smoldyn program. Compared to the popular MCell and ChemCell simulators, we found that Smoldyn was in many cases more accurate, more computationally efficient, and easier to use. Using Smoldyn, we modeled pheromone response system signaling among yeast cells of opposite mating type. This model showed that secreted Bar1 protease might help a cell identify the fittest mating partner by sharpening the pheromone concentration gradient. This model involved about 200,000 protein molecules, about 7000 cubic microns of volume, and about 75 minutes of simulated time; it took about 10 hours to run. Over the next several years, as faster computers become available, Smoldyn will allow researchers to model and explore systems the size of entire bacterial and smaller eukaryotic cells.

Probability distributed time delays: integrating spatial effects into temporal models

Background

In order to provide insights into the complex biochemical processes inside a cell, modelling approaches must find a balance between achieving an adequate representation of the physical phenomena and keeping the associated computational cost within reasonable limits. This issue is particularly stressed when spatial inhomogeneities have a significant effect on system's behaviour. In such cases, a spatially-resolved stochastic method can better portray the biological reality, but the corresponding computer simulations can in turn be prohibitively expensive.

Results

We present a method that incorporates spatial information by means of tailored, probability distributed time-delays. These distributions can be directly obtained by single in silico or a suitable set of in vitro experiments and are subsequently fed into a delay stochastic simulation algorithm (DSSA), achieving a good compromise between computational costs and a much more accurate representation of spatial processes such as molecular diffusion and translocation between cell compartments. Additionally, we present a novel alternative approach based on delay differential equations (DDE) that can be used in scenarios of high molecular concentrations and low noise propagation.

Conclusions

Our proposed methodologies accurately capture and incorporate certain spatial processes into temporal stochastic and deterministic simulations, increasing their accuracy at low computational costs. This is of particular importance given that time spans of cellular processes are generally larger (possibly by several orders of magnitude) than those achievable by current spatially-resolved stochastic simulators. Hence, our methodology allows users to explore cellular scenarios under the effects of diffusion and stochasticity in time spans that were, until now, simply unfeasible. Our methodologies are supported by theoretical considerations on the different modelling regimes, i.e. spatial vs. delay-temporal, as indicated by the corresponding Master Equations and presented elsewhere.

Multi-scale spatio-temporal modeling: lifelines of microorganisms in bioreactors and tracking molecules in cells

Agent-based models are rigorous tools for simulating the interactions of individual entities, such as organisms or molecules within cells and assessing their effects on the dynamic behavior of the system as a whole. In context with bioprocess and biosystems engineering there are several interesting and important applications. This contribution aims at introducing this strategy with the aid of two examples characterized by striking distinctions in the scale of the individual entities and the mode of their interactions. In the first example a structured-segregated model is applied to travel along the lifelines of single cells in the environment of a three-dimensional turbulent field of a stirred bioreactor. The modeling approach is based on an Euler-Lagrange formulation of the system. The strategy permits one to account for the heterogeneity present in real reactors in both the fluid and cellular phases, respectively. The individual response of the cells to local variations in the extracellular concentrations is pictured by a dynamically structured model of the key reactions of the central metabolism. The approach permits analysis of the lifelines of individual cells in space and time.The second application of the individual modeling approach deals with dynamic modeling of signal transduction pathways in individual cells. Usually signal transduction networks are portrayed as being wired together in a spatially defined manner. Living circuitry, however, is placed in highly malleable internal architecture. Creating a homogenous bag of molecules, a well-mixed system, the dynamic behavior of which is modeled with a set of ordinary differential equations is normally not valid. The dynamics of the MAP kinase and a steroid hormone pathway serve as examples to illustrate how single molecule tracking can be linked with the stochasticity of biochemical reactions, where diffusion and reaction occur in a probabilistic manner. The problem of hindered diffusion caused by macromolecular crowding is also taken into account.

A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of E. coli MinE to E-ring formation

Many important cellular processes are regulated by reaction-diffusion (RD) of molecules that takes place both in the cytoplasm and on the membrane. To model and analyze such multicompartmental processes, we developed a lattice-based Monte Carlo method, Spatiocyte that supports RD in volume and surface compartments at single molecule resolution. Stochasticity in RD and the excluded volume effect brought by intracellular molecular crowding, both of which can significantly affect RD and thus, cellular processes, are also supported. We verified the method by comparing simulation results of diffusion, irreversible and reversible reactions with the predicted analytical and best available numerical solutions. Moreover, to directly compare the localization patterns of molecules in fluorescence microscopy images with simulation, we devised a visualization method that mimics the microphotography process by showing the trajectory of simulated molecules averaged according to the camera exposure time. In the rod-shaped bacterium Escherichia coli, the division site is suppressed at the cell poles by periodic pole-to-pole oscillations of the Min proteins (MinC, MinD and MinE) arising from carefully orchestrated RD in both cytoplasm and membrane compartments. Using Spatiocyte we could model and reproduce the in vivo MinDE localization dynamics by accounting for the previously reported properties of MinE. Our results suggest that the MinE ring, which is essential in preventing polar septation, is largely composed of MinE that is transiently attached to the membrane independently after recruited by MinD. Overall, Spatiocyte allows simulation and visualization of complex spatial and reaction-diffusion mediated cellular processes in volumes and surfaces. As we showed, it can potentially provide mechanistic insights otherwise difficult to obtain experimentally.

Stochastic simulation of signal transduction: impact of the cellular architecture on diffusion

The transduction of signals depends on the translocation of signaling molecules to specific targets. Undirected diffusion processes play a key role in the bridging of spaces between different cellular compartments. The diffusion of the molecules is, in turn, governed by the intracellular architecture. Molecular crowding and the cytoskeleton decrease macroscopic diffusion. This article shows the use of a stochastic simulation method to study the effects of the cytoskeleton structure on the mobility of macromolecules. Brownian dynamics and single particle tracking were used to simulate the diffusion process of individual molecules through a model cytoskeleton. The resulting average effective diffusion is in line with data obtained in the in vitro and in vivo experiments. It shows that the cytoskeleton structure strongly influences the diffusion of macromolecules. The simulation method used also allows the inclusion of reactions in order to model complete signaling pathways in their spatio-temporal dynamics, taking into account the effects of the cellular architecture.

Cell type-specific importance of ras-c-raf complex association rate constants for MAPK signaling

We generated 17 c-Raf (RAF proto-oncogene serine-threonine protein kinase) mutants with altered Ras-Raf association and dissociation rates to investigate the role of electrostatically driven Ras-Raf association rates on epidermal growth factor (EGF)-activated mitogen-activated protein kinase (MAPK) signal transduction. Some of these mutants had compensating changes in association and dissociation rates, enabling the effects of changes in association rate to be distinguished from those of changes in affinity. In rabbit kidney (RK13) cells, these mutants affected downstream signaling, with changes in Ras-c-Raf association rates having a greater effect on MAPK signaling than did similar changes in dissociation rates. Mutants with compensating decreases in both association and dissociation rates stimulated less extracellular signal-regulated kinase (ERK)-dependent reporter activity than did wild-type c-Raf, whereas the converse was true for mutants with increased association and dissociation rates. In marked contrast, the mutants had little or no effect on signaling in human embryonic kidney (HEK) 293 cells. These two cell lines also showed distinct patterns of EGF-dependent ERK phosphorylation and signaling: ERK activation and signaling were transient in HEK293 cells and sustained in RK13 cells, with the difference resulting from the lack of negative feedback from ERK to Sos (Son of Sevenless) in the latter. Computer simulation revealed that, in the presence of negative feedback, changes in the rate of Ras-c-Raf binding have little effect on ERK activation. Thus, EGF-MAPK activation kinetics and feedback regulation is cell type specific and depends on the network topology.

Biochemical simulations: stochastic, approximate stochastic and hybrid approaches

Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems. In particular, stochastic simulation methods have attracted increasing interest recently. In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers. Numerous stochastic, approximate stochastic and hybrid simulation methods have been proposed in the literature. In this article, they are systematically reviewed in order to guide the researcher and help her find the appropriate method for a specific problem.

Kinetic modeling of biological systems

The dynamics of how the constituent components of a natural system interact defines the spatio-temporal response of the system to stimuli. Modeling the kinetics of the processes that represent a biophysical system has long been pursued with the aim of improving our understanding of the studied system. Due to the unique properties of biological systems, in addition to the usual difficulties faced in modeling the dynamics of physical or chemical systems, biological simulations encounter difficulties that result from intrinsic multi-scale and stochastic nature of the biological processes.This chapter discusses the implications for simulation of models involving interacting species with very low copy numbers, which often occur in biological systems and give rise to significant relative fluctuations. The conditions necessitating the use of stochastic kinetic simulation methods and the mathematical foundations of the stochastic simulation algorithms are presented. How the well-organized structural hierarchies often seen in biological systems can lead to multi-scale problems and the possible ways to address the encountered computational difficulties are discussed. We present the details of the existing kinetic simulation methods and discuss their strengths and shortcomings. A list of the publicly available kinetic simulation tools and our reflections for future prospects are also provided.

Association rate constants of ras-effector interactions are evolutionarily conserved

Evolutionary conservation of protein interaction properties has been shown to be a valuable indication for functional importance. Here we use homology interface modeling of 10 Ras-effector complexes by selecting ortholog proteins from 12 organisms representing the major eukaryotic branches, except plants. We find that with increasing divergence time the sequence similarity decreases with respect to the human protein, but the affinities and association rate constants are conserved as predicted by the protein design algorithm, FoldX. In parallel we have done computer simulations on a minimal network based on Ras-effector interactions, and our results indicate that in the absence of negative feedback, changes in kinetics that result in similar binding constants have strong consequences on network behavior. This, together with the previous results, suggests an important biological role, not only for equilibrium binding constants but also for kinetics in signaling processes involving Ras-effector interactions. Our findings are important to take into consideration in system biology approaches and simulations of biological networks.

Cellular signal transductions processes are based on protein interactions. Proteins can either associate transiently with each other or form stable complexes, and the strength of the interaction is described by the affinity (the affinity is the ratio between the rate of dissociation and association). Protein complexes with similar affinities can bind and dissociate with different rates, and these rates describe the kinetic properties of protein binding. These kinetic rates are important for signaling; however, to what extent individual changes in such rate constants are biologically important or whether the affinity is more crucial might be different in different signaling processes. In this study we analyze whether association rates are conserved during evolution, because evolutionary conservation of protein biochemical properties is usually a valuable indication of its importance. We analyzed the binding of Ras proteins to effector domains, which are central proteins in many signal transduction pathways, in different organisms. On the basis of homology modeling and energy calculations we find that association rates are conserved, although the sequence similarity decreases compared to the human protein. Our finding should encourage further analysis of the importance of kinetics for cellular signal transduction.

Reaction Brownian dynamics and the effect of spatial fluctuations on the gain of a push-pull network

Brownian Dynamics algorithms have been widely used for simulating systems in soft-condensed matter physics. In recent times, their application has been extended to the simulation of coarse-grained models of biochemical networks. In these models, components move by diffusion and interact with one another upon contact. However, when reactions are incorporated into a Brownian dynamics algorithm, care must be taken to avoid violations of the detailed-balance rule, which would introduce systematic errors in the simulation. We present a Brownian dynamics algorithm for simulating reaction-diffusion systems that rigorously obeys detailed balance for equilibrium reactions. By comparing the simulation results to exact analytical results for a bimolecular reaction, we show that the algorithm correctly reproduces both equilibrium and dynamical quantities. We apply our scheme to a "push-pull" network in which two antagonistic enzymes covalently modify a substrate. Our results highlight that spatial fluctuations of the network components can strongly reduce the gain of the response of a biochemical network.

A hybrid agent-based approach for modeling microbiological systems

Models for systems biology commonly adopt Differential Equations or Agent-Based modeling approaches for simulating the processes as a whole. Models based on differential equations presuppose phenomenological intracellular behavioral mechanisms, while models based on Multi-Agent approach often use directly translated, and quantitatively less precise if-then logical rule constructs. We propose an extendible systems model based on a hybrid agent-based approach where biological cells are modeled as individuals (agents) while molecules are represented by quantities. This hybridization in entity representation entails a combined modeling strategy with agent-based behavioral rules and differential equations, thereby balancing the requirements of extendible model granularity with computational tractability. We demonstrate the efficacy of this approach with models of chemotaxis involving an assay of 10(3) cells and 1.2x10(6) molecules. The model produces cell migration patterns that are comparable to laboratory observations.

GridCell: a stochastic particle-based biological system simulator

Background

Realistic biochemical simulators aim to improve our understanding of many biological processes that would be otherwise very difficult to monitor in experimental studies. Increasingly accurate simulators may provide insights into the regulation of biological processes due to stochastic or spatial effects.

Results

We have developed GridCell as a three-dimensional simulation environment for investigating the behaviour of biochemical networks under a variety of spatial influences including crowding, recruitment and localization. GridCell enables the tracking and characterization of individual particles, leading to insights on the behaviour of low copy number molecules participating in signaling networks. The simulation space is divided into a discrete 3D grid that provides ideal support for particle collisions without distance calculation and particle search. SBML support enables existing networks to be simulated and visualized. The user interface provides intuitive navigation that facilitates insights into species behaviour across spatial and temporal dimensions. We demonstrate the effect of crowing on a Michaelis-Menten system.

Conclusion

GridCell is an effective stochastic particle simulator designed to track the progress of individual particles in a three-dimensional space in which spatial influences such as crowding, co-localization and recruitment may be investigated.

Simulated de novo assembly of golgi compartments by selective cargo capture during vesicle budding and targeted vesicle fusion

The Golgi apparatus is comprised of stacked cisternal membranes forming subcompartments specialized for posttranslational processing of newly synthesized secretory cargo. Recent experimental evidence indicates that the Golgi apparatus can undergo de novo biogenesis from the endoplasmic reticulum, but the mechanism by which the membranes self assemble into compartmentalized structures remains unknown. We developed a discrete-event computer simulation model to test whether two fundamental mechanisms-vesicle-coat-mediated selective concentration of soluble N-ethylmaleimide-sensitive factor attachment protein receptor (SNARE) proteins during vesicle formation, and SNARE-mediated selective fusion of vesicles-suffice to generate and maintain compartments. Simulations verified that this minimal model is adequate for homeostasis of preestablished compartments, even in response to small perturbations, and for de novo formation of stable compartments. The model led to a novel prediction that Golgi size is in part dependent on target SNARE expression level. This prediction was supported by a demonstration that exogenous expression of the Golgi target SNARE syntaxin-5 alters Golgi size in living cells.

Operating regimes of signaling cycles: statics, dynamics, and noise filtering

A ubiquitous building block of signaling pathways is a cycle of covalent modification (e.g., phosphorylation and dephosphorylation in MAPK cascades). Our paper explores the kind of information processing and filtering that can be accomplished by this simple biochemical circuit. Signaling cycles are particularly known for exhibiting a highly sigmoidal (ultrasensitive) input–output characteristic in a certain steady-state regime. Here, we systematically study the cycle's steady-state behavior and its response to time-varying stimuli. We demonstrate that the cycle can actually operate in four different regimes, each with its specific input–output characteristics. These results are obtained using the total quasi–steady-state approximation, which is more generally valid than the typically used Michaelis-Menten approximation for enzymatic reactions. We invoke experimental data that suggest the possibility of signaling cycles operating in one of the new regimes. We then consider the cycle's dynamic behavior, which has so far been relatively neglected. We demonstrate that the intrinsic architecture of the cycles makes them act—in all four regimes—as tunable low-pass filters, filtering out high-frequency fluctuations or noise in signals and environmental cues. Moreover, the cutoff frequency can be adjusted by the cell. Numerical simulations show that our analytical results hold well even for noise of large amplitude. We suggest that noise filtering and tunability make signaling cycles versatile components of more elaborate cell-signaling pathways.

A cell is subjected to constantly changing environments and time-varying stimuli. Signals sensed at the cell surface are transmitted inside the cell by signaling pathways. Such pathways can transform signals in diverse ways and perform some preliminary information processing. A ubiquitous building block of signaling pathways is a simple biochemical cycle involving covalent modification of an enzyme–substrate pair. Our paper is devoted to fully characterizing the static and dynamic behavior of this simple cycle, an essential first step in understanding the behavior of interconnections of such cycles. It is known that a signaling cycle can function as a static switch, with the steady-state output being an “ultrasensitive” function of the input, i.e., changing from a low to high value for only a small change in the input. We show that there are in fact precisely four major regimes of static and dynamic operation (with ultrasensitive being one of the static regimes). Each regime has its own input–output characteristics. Despite the distinctive features of these four regimes, they all respond to time-varying stimuli by filtering out high-frequency fluctuations or noise in their inputs, while passing through the lower-frequency information-bearing variations. A cell can select the regime and tune the noise-filtering characteristics of the individual cycles in a specific signaling pathway. This tunability makes signaling cycles versatile components of elaborate cell-signaling pathways.

Discrete stochastic simulation of cell signaling: comparison of computational tools

Several stochastic simulation tools have been developed recently for cell signaling. A comparative evaluation of the stochastic simulation tools is needed to highlight the current state of the development. In our study, we have chosen to evaluate three stochastic simulation tools: Dizzy, Systems Biology Toolbox, and Copasi, using our own MATLAB implementation as a benchmark. The Gillespie stochastic simulation algorithm is used in all tests. With all the tools, we are able to simulate stochastically the behavior of the selected test case and to produce similar results as our own MATLAB implementation. However, it is not possible to use time-dependent inputs in stochastic simulations in Systems Biology Toolbox and Copasi. The present study is one of the first evaluations of stochastic simulation tools for realistic signal transduction pathways.

Shaping dots and lines: adding modularity into protein interaction networks using structural information

Determining protein interaction networks and generating models to simulate network changes in time and space are crucial for understanding a biological system and for predicting the effect of mutants found in diseases. In this review we discuss the great potential of using structural information together with computational tools towards reaching this goal: the prediction of new protein interactions, the estimation of affinities and kinetic rate constants between protein complexes, and finally the determination of which interactions are compatible with each other and which interactions are exclusive. The latter one will be important to reorganize large scale networks into functional modular networks.

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

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

Modelling reaction kinetics inside cells

In the past decade, advances in molecular biology such as the development of non-invasive single molecule imaging techniques have given us a window into the intricate biochemical activities that occur inside cells. In this chapter we review four distinct theoretical and simulation frameworks: (i) non-spatial and deterministic, (ii) spatial and deterministic, (iii) non-spatial and stochastic and (iv) spatial and stochastic. Each framework can be suited to modelling and interpreting intracellular reaction kinetics. By estimating the fundamental length scales, one can roughly determine which models are best suited for the particular reaction pathway under study. We discuss differences in prediction between the four modelling methodologies. In particular we show that taking into account noise and space does not simply add quantitative predictive accuracy but may also lead to qualitatively different physiological predictions, unaccounted for by classical deterministic models.

Structures in systems biology

Oil and water do not normally mix, and apparently structural biology and systems biology look like two different universes. It can be argued that structural biology could play a very important role in systems biology. Although at the final stage of understanding a signal transduction pathway, a cell, an organ or a living system, structures could be obviated, we need them to be able to reach that stage. Structures of macromolecules, especially molecular machines, could provide quantitative parameters, help to elucidate functional networks or enable rational designed perturbation experiments for reverse engineering. The role of structural biology in systems biology should be to provide enough understanding so that macromolecules can be translated into dots or even into equations devoid of atoms.

The hard cell: From proteomics to a whole cell model

Proteomics has provided a wealth of data related to the nature of the proteome, including subcellular location, copy number, interaction partners and protein complexes. This raises the question of whether it is feasible to combine these data, together with other data related to overall cellular structure, to construct a static picture of the cell. In this minireview, we discuss these data, and the issues of turning them into whole cell models.

Multinomial tau-leaping method for stochastic kinetic simulations

We introduce the multinomial tau-leaping (MtauL) method for general reaction networks with multichannel reactant dependencies. The MtauL method is an extension of the binomial tau-leaping method where efficiency is improved in several ways. First, tau-leaping steps are determined simply and efficiently using a priori information and Poisson distribution-based estimates of expectation values for reaction numbers over a tentative tau-leaping step. Second, networks are partitioned into closed groups of reactions and corresponding reactants in which no group reactant set is found in any other group. Third, product formation is factored into upper-bound estimation of the number of times a particular reaction occurs. Together, these features allow larger time steps where the numbers of reactions occurring simultaneously in a multichannel manner are estimated accurately using a multinomial distribution. Furthermore, we develop a simple procedure that places a specific upper bound on the total reaction number to ensure non-negativity of species populations over a single multiple-reaction step. Using two disparate test case problems involving cellular processes--epidermal growth factor receptor signaling and a lactose operon model--we show that the tau-leaping based methods such as the MtauL algorithm can significantly reduce the number of simulation steps thus increasing the numerical efficiency over the exact stochastic simulation algorithm by orders of magnitude.

Oscillations in the immune system

Oscillations are surprisingly common in the immune system, both in its healthy state and in disease. The most famous example is that of periodic fevers caused by the malaria parasite. A number of hereditary disorders, which also cause periodic fevers, have also been known for a long time. Various reports of oscillations in cytokine concentrations following antigen challenge have been published over at least the past three decades. Oscillations can also occur at the intracellular level. Calcium oscillations following T-cell activation are familiar to all immunologists, and metabolic and reactive oxygen species oscillations in neutrophils have been well documented. More recently, oscillations in nuclear factor kappaB activity following stimulation by tumor necrosis factor alpha have received considerable publicity. However, despite all of these examples, oscillations in the immune system still tend to be considered mainly as pathological aberrations, and their causes and significance remained largely unknown. This is partly because of a lack of awareness within the immunological community of the appropriate theoretical frameworks for describing and analyzing such behavior. We provide an introduction to these frameworks and give a survey of the currently known oscillations that occur within the immune system.

Computational systems biology in drug discovery and development: methods and applications

Computational systems biology is an emerging field in biological simulation that attempts to model or simulate intra- and intercellular events using data gathered from genomic, proteomic or metabolomic experiments. The need to model complex temporal and spatiotemporal processes at many different scales has led to the emergence of numerous techniques, including systems of differential equations, Petri nets, cellular automata simulators, agent-based models and pi calculus. This review provides a brief summary and an assessment of most of these approaches. It also provides examples of how these methods are being used to facilitate drug discovery and development.

Diffusion of transcription factors can drastically enhance the noise in gene expression

We study by Green's Function Reaction Dynamics the effect of the diffusive motion of repressor molecules on the noise in mRNA and protein levels for a gene that is under the control of a repressor. We find that spatial fluctuations due to diffusion can drastically enhance the noise in gene expression. After dissociation from the operator, a repressor can rapidly rebind to the DNA. Our results show that the rebinding trajectories are so short that, on this timescale, the RNA polymerase (RNAP) cannot effectively compete with the repressor for binding to the promoter. As a result, a dissociated repressor molecule will on average rebind many times, before it eventually diffuses away. These rebindings thus lower the effective dissociation rate, and this increases the noise in gene expression. Another consequence of the timescale separation between repressor rebinding and RNAP association is that the effect of spatial fluctuations can be described by a well-stirred, zero-dimensional, model by renormalizing the reaction rates for repressor-DNA (un) binding. Our results thus support the use of well-stirred, zero-dimensional models for describing noise in gene expression. We also show that for a fixed repressor strength, the noise due to diffusion can be minimized by increasing the number of repressors or by decreasing the rate of the open complex formation. Lastly, our results emphasize that power spectra are a highly useful tool for studying the propagation of noise through the different stages of gene expression.

Probability-based model of protein-protein interactions on biological timescales

BACKGROUND

Simulation methods can assist in describing and understanding complex networks of interacting proteins, providing fresh insights into the function and regulation of biological systems. Recent studies have investigated such processes by explicitly modelling the diffusion and interactions of individual molecules. In these approaches, two entities are considered to have interacted if they come within a set cutoff distance of each other.

RESULTS

In this study, a new model of bimolecular interactions is presented that uses a simple, probability-based description of the reaction process. This description is well-suited to simulations on timescales relevant to biological systems (from seconds to hours), and provides an alternative to the previous description given by Smoluchowski. In the present approach (TFB) the diffusion process is explicitly taken into account in generating the probability that two freely diffusing chemical entities will interact within a given time interval. It is compared to the Smoluchowski method, as modified by Andrews and Bray (AB).

CONCLUSION

When implemented, the AB & TFB methods give equivalent results in a variety of situations relevant to biology. Overall, the Smoluchowski method as modified by Andrews and Bray emerges as the most simple, robust and efficient method for simulating biological diffusion-reaction processes currently available.

From in vivo to in silico biology and back

The massive acquisition of data in molecular and cellular biology has led to the renaissance of an old topic: simulations of biological systems. Simulations, increasingly paired with experiments, are being successfully and routinely used by computational biologists to understand and predict the quantitative behaviour of complex systems, and to drive new experiments. Nevertheless, many experimentalists still consider simulations an esoteric discipline only for initiates. Suspicion towards simulations should dissipate as the limitations and advantages of their application are better appreciated, opening the door to their permanent adoption in everyday research.

Cell++—simulating biochemical pathways

MOTIVATION

With the generation of a wealth of information, detailing cellular components, their functions and interactions, there is a growing need for the development of new computational tools capable of interpreting these data within spatial and dynamic contexts. Here, we introduce Cell++, a novel stochastic simulation environment with the capacity to study a wide variety of biochemical processes within a spatial context.

RESULTS

Focusing on three case studies, we highlight the potential impact of spatial organization in the evolution and engineering of signaling and metabolic pathways. In addition to altering signaling and metabolic efficiency, simulations also demonstrated features consistent with the phenomenon of metabolic channeling.

AVAILABILITY

Cell++ is licensed under the GNU general public license (GPL) and has been successfully implemented under Linux and IRIX operating systems. Source code together with a simple tutorial is available at http://www.compsysbio.org/CellSim/.

Accommodating space, time and randomness in network simulation

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

Synthetic biology--putting engineering into biology

Synthetic biology is interpreted as the engineering-driven building of increasingly complex biological entities for novel applications. Encouraged by progress in the design of artificial gene networks, de novo DNA synthesis and protein engineering, we review the case for this emerging discipline. Key aspects of an engineering approach are purpose-orientation, deep insight into the underlying scientific principles, a hierarchy of abstraction including suitable interfaces between and within the levels of the hierarchy, standardization and the separation of design and fabrication. Synthetic biology investigates possibilities to implement these requirements into the process of engineering biological systems. This is illustrated on the DNA level by the implementation of engineering-inspired artificial operations such as toggle switching, oscillating or production of spatial patterns. On the protein level, the functionally self-contained domain structure of a number of proteins suggests possibilities for essentially Lego-like recombination which can be exploited for reprogramming DNA binding domain specificities or signaling pathways. Alternatively, computational design emerges to rationally reprogram enzyme function. Finally, the increasing facility of de novo DNA synthesis-synthetic biology's system fabrication process-supplies the possibility to implement novel designs for ever more complex systems. Some of these elements have merged to realize the first tangible synthetic biology applications in the area of manufacturing of pharmaceutical compounds.

Noise in transcription negative feedback loops: simulation and experimental analysis

Negative feedback loops have been invoked as a way to control and decrease transcriptional noise. Here, we have built three circuits to test the effect of negative feedback loops on transcriptional noise of an autoregulated gene encoding a transcription factor (TF) and a downstream gene (DG), regulated by this TF. Experimental analysis shows that self-repression decreases noise compared to expression from a non-regulated promoter. Interestingly enough, we find that noise minimization by negative feedback loop is optimal within a range of repression strength. Repression values outside this range result in noise increase producing a U-shaped behaviour. This behaviour is the result of external noise probably arising from plasmid fluctuations as shown by simulation of the network. Regarding the target gene of a self-repressed TF (sTF), we find a strong decrease of noise when repression by the sTF is strong and a higher degree of noise anti-correlation between sTF and its target. Simulations of the circuits indicate that the main source of noise in these circuits could come from plasmid variation and therefore that negative feedback loops play an important role in suppressing both external and internal noise. An important observation is that DG expression without negative feedback exhibits bimodality at intermediate TF repression values. This bimodal behaviour seems to be the result of external noise as it can only be found in those simulations that include plasmid variation.

Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli

MOTIVATION

Many biochemical networks involve reactions localized on the cell membrane. This can give rise to spatial gradients of the concentration of cytosolic species. Moreover, the number of membrane molecules can be small and stochastic effects can become relevant. Pathways usually consist of a complex interaction network and are characterized by a large set of parameters. The inclusion of spatial and stochastic effects is a major challenge in developing quantitative and dynamic models of pathways.

RESULTS

We have developed a particle-based spatial stochastic method (GMP) to simulate biochemical networks in space, including fluctuations from the diffusion of particles and reactions. Gradients emerging from membrane reactions can be resolved. As case studies for the GMP method we used a simple gene expression system and the phosphoenolpyruvate:glucose phosphotransferase system pathway.

AVAILABILITY

The source code for the GMP method is available at http://www.science.uva.nl/research/scs/CellMath/GMP.

Noise-induced Min phenotypes in E. coli

The spatiotemporal oscillations of the Escherichia coli proteins MinD and MinE direct cell division to the region between the chromosomes. Several quantitative models of the Min system have been suggested before, but no one of them accounts for the behavior of all documented mutant phenotypes. We analyzed the stochastic reaction-diffusion kinetics of the Min proteins for several E. coli mutants and compared the results to the corresponding deterministic mean-field description. We found that wild-type (wt) and filamentous (ftsZ⁻) cells are well characterized by the mean-field model, but that a stochastic model is necessary to account for several of the characteristics of the spherical (rodA⁻) and phospathedylethanolamide-deficient (PE⁻) phenotypes. For spherical cells, the mean-field model is bistable, and the system can get trapped in a non-oscillatory state. However, when the intrinsic noise is considered, only the experimentally observed oscillatory behavior remains. The stochastic model also reproduces the change in oscillation directions observed in the spherical phenotype and the occasional gliding of the MinD region along the inner membrane. For the PE⁻ mutant, the stochastic model explains the appearance of randomly localized and dense MinD clusters as a nucleation phenomenon, in which the stochastic kinetics at low copy number causes local discharges of the high MinD^ATP to MinD^ADP potential. We find that a simple five-reaction model of the Min system can explain all documented Min phenotypes, if stochastic kinetics and three-dimensional diffusion are accounted for. Our results emphasize that local copy number fluctuation may result in phenotypic differences although the total number of molecules of the relevant species is high.

Many molecules inside a living cell do not have time to diffuse through the whole cell in-between reactions. Furthermore, the chemical reactions are random and discrete events. In this study, the authors study an example in which these aspects of intracellular chemistry need to be considered when we try to understand how a biological system works.

The authors have investigated the spatial oscillation patterns that are displayed by the Min system of Escherichia coli. In wild-type E. coli, the Min proteins oscillate back and forth between the cell poles to help the bacterium find its middle before cell division. The authors used computer simulations to explain why the oscillation patterns change the way they do in different mutants of E. coli. They find that two of the mutant phenotypes can only be explained if one considers the randomness and discreteness of chemical reactions in addition to the spatial characteristics of the process. Particularly interesting is the phospathedylethanolamide-deficient phenotype, in which large dense clusters of MinD protein appear for some time at random locations on the membrane. The authors believe that this phenotype is due to a nucleation phenomenon, in which the stochastic kinetics at low copy number is amplified to macroscopic proportions.

Multiscale Hy3S: hybrid stochastic simulation for supercomputers

Background

Stochastic simulation has become a useful tool to both study natural biological systems and design new synthetic ones. By capturing the intrinsic molecular fluctuations of "small" systems, these simulations produce a more accurate picture of single cell dynamics, including interesting phenomena missed by deterministic methods, such as noise-induced oscillations and transitions between stable states. However, the computational cost of the original stochastic simulation algorithm can be high, motivating the use of hybrid stochastic methods. Hybrid stochastic methods partition the system into multiple subsets and describe each subset as a different representation, such as a jump Markov, Poisson, continuous Markov, or deterministic process. By applying valid approximations and self-consistently merging disparate descriptions, a method can be considerably faster, while retaining accuracy. In this paper, we describe Hy3S, a collection of multiscale simulation programs.

Results

Building on our previous work on developing novel hybrid stochastic algorithms, we have created the Hy3S software package to enable scientists and engineers to both study and design extremely large well-mixed biological systems with many thousands of reactions and chemical species. We have added adaptive stochastic numerical integrators to permit the robust simulation of dynamically stiff biological systems. In addition, Hy3S has many useful features, including embarrassingly parallelized simulations with MPI; special discrete events, such as transcriptional and translation elongation and cell division; mid-simulation perturbations in both the number of molecules of species and reaction kinetic parameters; combinatorial variation of both initial conditions and kinetic parameters to enable sensitivity analysis; use of NetCDF optimized binary format to quickly read and write large datasets; and a simple graphical user interface, written in Matlab, to help users create biological systems and analyze data. We demonstrate the accuracy and efficiency of Hy3S with examples, including a large-scale system benchmark and a complex bistable biochemical network with positive feedback. The software itself is open-sourced under the GPL license and is modular, allowing users to modify it for their own purposes.

Conclusion

Hy3S is a powerful suite of simulation programs for simulating the stochastic dynamics of networks of biochemical reactions. Its first public version enables computational biologists to more efficiently investigate the dynamics of realistic biological systems.

Current challenges in quantitative modeling of epidermal growth factor signaling

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

Stochastic reaction-diffusion simulation with MesoRD

MesoRD is a tool for stochastic simulation of chemical reactions and diffusion. In particular, it is an implementation of the next subvolume method, which is an exact method to simulate the Markov process corresponding to the reaction-diffusion master equation.

AVAILABILITY

MesoRD is free software, written in C++ and licensed under the GNU general public license (GPL). MesoRD runs on Linux, Mac OS X, NetBSD, Solaris and Windows XP. It can be downloaded from http://mesord.sourceforge.net.

CONTACT

johan.elf@icm.uu.se; johan.hattne@embl-hamburg.de

SUPPLEMENTARY INFORMATION

'MesoRD User's Guide' and other documents are available at http://mesord.sourceforge.net.

Space in systems biology of signaling pathways--towards intracellular molecular crowding in silico

How cells utilize intracellular spatial features to optimize their signaling characteristics is still not clearly understood. The physical distance between the cell-surface receptor and the gene expression machinery, fast reactions, and slow protein diffusion coefficients are some of the properties that contribute to their intricacy. This article reviews computational frameworks that can help biologists to elucidate the implications of space in signaling pathways. We argue that intracellular macromolecular crowding is an important modeling issue, and describe how recent simulation methods can reproduce this phenomenon in either implicit, semi-explicit or fully explicit representation.

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.