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

Dissecting the general mechanisms of protein cage self-assembly by coarse-grained simulations

The development of artificial protein cages has recently gained massive attention due to their promising application prospect as novel delivery vehicles for therapeutics. These nanoparticles are formed through a process called self-assembly, in which individual subunits spontaneously arrange into highly ordered patterns via non-covalent but specific interactions. Therefore, the first step toward the design of novel engineered protein cages is to understand the general mechanisms of their self-assembling dynamics. Here we have developed a new computational method to tackle this problem. Our method is based on a coarse-grained model and a diffusion-reaction simulation algorithm. Using a tetrahedral cage as test model, we showed that self-assembly of protein cage requires of a seeding process in which specific configurations of kinetic intermediate states are identified. We further found that there is a critical concentration to trigger self-assembly of protein cages. This critical concentration allows that cages can only be successfully assembled under a persistently high concentration. Additionally, phase diagram of self-assembly has been constructed by systematically testing the model across a wide range of binding parameters. Finally, our simulations demonstrated the importance of protein's structural flexibility in regulating the dynamics of cage assembly. In summary, this study throws lights on the general principles underlying self-assembly of large cage-like protein complexes and thus provides insights to design new nanomaterials.

Close agreement between deterministic versus stochastic modeling of first-passage time to vesicle fusion

Ca2+-dependent cell processes, such as neurotransmitter or endocrine vesicle fusion, are inherently stochastic due to large fluctuations in Ca2+ channel gating, Ca2+ diffusion, and Ca2+ binding to buffers and target sensors. However, previous studies revealed closer-than-expected agreement between deterministic and stochastic simulations of Ca2+ diffusion, buffering, and sensing if Ca2+ channel gating is not Ca2+ dependent. To understand this result more fully, we present a comparative study complementing previous work, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Specifically, we compare deterministic (mean-field/mass-action) and stochastic simulations of vesicle exocytosis latency, quantified by the probability density of the first-passage time (FPT) to the Ca2+-bound state of a vesicle fusion sensor, following a brief Ca2+ current pulse. We show that under physiological constraints, the discrepancy between FPT densities obtained using the two approaches remains small even if as few as ∼50 Ca2+ ions enter per single channel-vesicle release unit. Using a reduced two-compartment model for ease of analysis, we illustrate how this close agreement arises from the smallness of correlations between fluctuations of the reactant molecule numbers, despite the large magnitude of fluctuation amplitudes. This holds if all relevant reactions are heteroreaction between molecules of different species, as is the case for bimolecular Ca2+ binding to buffers and downstream sensor targets. In this case, diffusion and buffering effectively decorrelate the state of the Ca2+ sensor from local Ca2+ fluctuations. Thus, fluctuations in the Ca2+ sensor's state underlying the FPT distribution are only weakly affected by the fluctuations in the local Ca2+ concentration around its average, deterministically computable value.

Hybrid modelling of biological systems: current progress and future prospects

Integrated modelling of biological systems is becoming a necessity for constructing models containing the major biochemical processes of such systems in order to obtain a holistic understanding of their dynamics and to elucidate emergent behaviours. Hybrid modelling methods are crucial to achieve integrated modelling of biological systems. This paper reviews currently popular hybrid modelling methods, developed for systems biology, mainly revealing why they are proposed, how they are formed from single modelling formalisms and how to simulate them. By doing this, we identify future research requirements regarding hybrid approaches for further promoting integrated modelling of biological systems.

Understanding the functional role of membrane confinements in TNF-mediated signaling by multiscale simulations

The interaction between TNFα and TNFR1 is essential in maintaining tissue development and immune responses. While TNFR1 is a cell surface receptor, TNFα exists in both soluble and membrane-bound forms. Interestingly, it was found that the activation of TNFR1-mediated signaling pathways is preferentially through the soluble form of TNFα, which can also induce the clustering of TNFR1 on plasma membrane of living cells. We developed a multiscale simulation framework to compare receptor clustering induced by soluble and membrane-bound ligands. Comparing with the freely diffusive soluble ligands, we hypothesize that the conformational dynamics of membrane-bound ligands are restricted, which affects the clustering of ligand-receptor complexes at cell-cell interfaces. Our simulation revealed that only small clusters can form if TNFα is bound on cell surface. In contrast, the clustering triggered by soluble TNFα is more dynamic, and the size of clusters is statistically larger. We therefore demonstrated the impact of membrane-bound ligand on dynamics of receptor clustering. Moreover, considering that larger TNFα-TNFR1 clusters is more likely to provide spatial platform for downstream signaling pathway, our studies offer new mechanistic insights about why the activation of TNFR1-mediated signaling pathways is not preferred by membrane-bound form of TNFα.

Mechanistic dissection of spatial organization in NF-κB signaling pathways by hybrid simulations

The nuclear factor kappa-light-chain-enhancer of activated B cells (NF-κB) is one of the most important transcription factors involved in the regulation of inflammatory signaling pathways. Inappropriate activation of these pathways has been linked to autoimmunity and cancers. Emerging experimental evidences have been showing the existence of elaborate spatial organizations for various molecular components in the pathways. One example is the scaffold protein tumor necrosis factor receptor associated factor (TRAF). While most TRAF proteins form trimeric quaternary structure through their coiled-coil regions, the N-terminal region of some members in the family can further be dimerized. This dimerization of TRAF trimers can drive them into higher-order clusters as a response to receptor stimulation, which functions as a spatial platform to mediate the downstream poly-ubiquitination. However, the molecular mechanism underlying the TRAF protein clustering and its functional impacts are not well-understood. In this article, we developed a hybrid simulation method to tackle this problem. The assembly of TRAF-based signaling platform at the membrane-proximal region is modeled with spatial resolution, while the dynamics of downstream signaling network, including the negative feedbacks through various signaling inhibitors, is simulated as stochastic chemical reactions. These two algorithms are further synchronized under a multiscale simulation framework. Using this computational model, we illustrated that the formation of TRAF signaling platform can trigger an oscillatory NF-κB response. We further demonstrated that the temporal patterns of downstream signal oscillations are closely regulated by the spatial factors of TRAF clustering, such as the geometry and energy of dimerization between TRAF trimers. In general, our study sheds light on the basic mechanism of NF-κB signaling pathway and highlights the functional importance of spatial regulation within the pathway. The simulation framework also showcases its potential of application to other signaling pathways in cells.

A multiscale study on the mechanisms of spatial organization in ligand-receptor interactions on cell surfaces

The binding of cell surface receptors with extracellular ligands triggers distinctive signaling pathways, leading into the corresponding phenotypic variation of cells. It has been found that in many systems, these ligand-receptor complexes can further oligomerize into higher-order structures. This ligand-induced oligomerization of receptors on cell surfaces plays an important role in regulating the functions of cell signaling. The underlying mechanism, however, is not well understood. One typical example is proteins that belong to the tumor necrosis factor receptor (TNFR) superfamily. Using a generic multiscale simulation platform that spans from atomic to subcellular levels, we compared the detailed physical process of ligand-receptor oligomerization for two specific members in the TNFR superfamily: the complex formed between ligand TNFα and receptor TNFR1 versus the complex formed between ligand TNFβ and receptor TNFR2. Interestingly, although these two systems share high similarity on the tertiary and quaternary structural levels, our results indicate that their oligomers are formed with very different dynamic properties and spatial patterns. We demonstrated that the changes of receptor’s conformational fluctuations due to the membrane confinements are closely related to such difference. Consistent to previous experiments, our simulations also showed that TNFR can preassemble into dimers prior to ligand binding, while the introduction of TNF ligands induced higher-order oligomerization due to a multivalent effect. This study, therefore, provides the molecular basis to TNFR oligomerization and reveals new insights to TNFR-mediated signal transduction. Moreover, our multiscale simulation framework serves as a prototype that paves the way to study higher-order assembly of cell surface receptors in many other bio-systems.

A computational study of co-inhibitory immune complex assembly at the interface between T cells and antigen presenting cells

The activation and differentiation of T-cells are mainly directly by their co-regulatory receptors. T lymphocyte-associated protein-4 (CTLA-4) and programed cell death-1 (PD-1) are two of the most important co-regulatory receptors. Binding of PD-1 and CTLA-4 with their corresponding ligands programed cell death-ligand 1 (PD-L1) and B7 on the antigen presenting cells (APC) activates two central co-inhibitory signaling pathways to suppress T cell functions. Interestingly, recent experiments have identified a new cis-interaction between PD-L1 and B7, suggesting that a crosstalk exists between two co-inhibitory receptors and the two pairs of ligand-receptor complexes can undergo dynamic oligomerization. Inspired by these experimental evidences, we developed a coarse-grained model to characterize the assembling of an immune complex consisting of CLTA-4, B7, PD-L1 and PD-1. These four proteins and their interactions form a small network motif. The temporal dynamics and spatial pattern formation of this network was simulated by a diffusion-reaction algorithm. Our simulation method incorporates the membrane confinement of cell surface proteins and geometric arrangement of different binding interfaces between these proteins. A wide range of binding constants was tested for the interactions involved in the network. Interestingly, we show that the CTLA-4/B7 ligand-receptor complexes can first form linear oligomers, while these oligomers further align together into two-dimensional clusters. Similar phenomenon has also been observed in other systems of cell surface proteins. Our test results further indicate that both co-inhibitory signaling pathways activated by B7 and PD-L1 can be down-regulated by the new cis-interaction between these two ligands, consistent with previous experimental evidences. Finally, the simulations also suggest that the dynamic and the spatial properties of the immune complex assembly are highly determined by the energetics of molecular interactions in the network. Our study, therefore, brings new insights to the co-regulatory mechanisms of T cell activation.

The activation of a T cell can be regulated by the receptors on its surface, such as CTLA-4 and PD-1. People used to think that these two receptors inhibit T cell activation through distinct pathways. However, recent experiments discovered that the ligands of these two receptors, B7 and PD-L1, can interact with each other on the same surface of antigen presenting cells. Here we utilized computational simulations to investigate functional roles of this newly discovered interaction in T cell coregulation. The specific environment of interface between T cell and antigen presenting cell has been taken into account of our model. Ligand and receptors randomly diffuse within this interface area. They further involve in different types of interactions, with each other from the same side or the opposite side of cell surface. Using this method, we found ligands and receptors can not only form complexes, but also aggregate into large-scale clusters. We also demonstrated that the engagement between B7 and PD-L1 can reduce the interactions with their corresponding receptors. This study, therefore, offers new insights to our understanding of signal regulation in T cells.

A Multiscale and Comparative Model for Receptor Binding of 2019 Novel Coronavirus and the Implication of its Life Cycle in Host Cells

The respiratory syndrome caused by a new type of coronavirus has been emerging from China and caused more than one million death globally since December 2019. This new virus, called severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) uses the same receptor called Angiotensin-converting enzyme 2 (ACE2) to attack humans as the coronavirus that caused the severe acute respiratory syndrome (SARS) seventeen years ago. Both viruses recognize ACE2 through the spike proteins (S-protein) on their surfaces. It was found that the S-protein from the SARS coronavirus (SARS-CoV) bind stronger to ACE2 than SARS-CoV-2. However, function of a bio-system is often under kinetic, rather than thermodynamic, control. To address this issue, we constructed a structural model for complex formed between ACE2 and the S-protein from SARS-CoV-2, so that the rate of their association can be estimated and compared with the binding of S-protein from SARS-CoV by a multiscale simulation method. Our simulation results suggest that the association of new virus to the receptor is slower than SARS, which is consistent with the experimental data obtained very recently. We further integrated this difference of association rate between virus and receptor into a mathematical model which describes the life cycle of virus in host cells and its interplay with the innate immune system. Interestingly, we found that the slower association between virus and receptor can result in longer incubation period, while still maintaining a relatively higher level of viral concentration in human body. Our computational study therefore provides, from the molecular level, one possible explanation that this new pandemic by far spread much faster than SARS.

Computer-Aided Whole-Cell Design: Taking a Holistic Approach by Integrating Synthetic With Systems Biology

Computer-aided design (CAD) for synthetic biology promises to accelerate the rational and robust engineering of biological systems. It requires both detailed and quantitative mathematical and experimental models of the processes to (re)design biology, and software and tools for genetic engineering and DNA assembly. Ultimately, the increased precision in the design phase will have a dramatic impact on the production of designer cells and organisms with bespoke functions and increased modularity. CAD strategies require quantitative models of cells that can capture multiscale processes and link genotypes to phenotypes. Here, we present a perspective on how whole-cell, multiscale models could transform design-build-test-learn cycles in synthetic biology. We show how these models could significantly aid in the design and learn phases while reducing experimental testing by presenting case studies spanning from genome minimization to cell-free systems. We also discuss several challenges for the realization of our vision. The possibility to describe and build whole-cells in silico offers an opportunity to develop increasingly automatized, precise and accessible CAD tools and strategies.

Gibbs free energy change of a discrete chemical reaction event

In modeling the interior of cells by simulating a reaction-diffusion master equation over a grid of compartments, one employs the assumption that the copy numbers of various chemical species are small, discrete quantities. We show that, in this case, textbook expressions for the change in Gibbs free energy accompanying a chemical reaction or diffusion between adjacent compartments are inaccurate. We derive exact expressions for these free energy changes for the case of discrete copy numbers and show how these expressions reduce to traditional expressions under a series of successive approximations leveraging the relative sizes of the stoichiometric coefficients and the copy numbers of the solutes and solvent. Numerical results are presented to corroborate the claim that if the copy numbers are treated as discrete quantities, then only these more accurate expressions lead to correct behavior. Thus, the newly derived expressions are critical for correctly computing entropy production in mesoscopic simulations based on the reaction-diffusion master equation formalism.

A computational model for understanding the oligomerization mechanisms of TNF receptor superfamily

By recognizing members in the tumor necrosis factor (TNF) receptor superfamily, TNF ligand proteins function as extracellular cytokines to activate various signaling pathways involved in inflammation, proliferation, and apoptosis. Most ligands in TNF superfamily are trimeric and can simultaneously bind to three receptors on cell surfaces. It has been experimentally observed that the formation of these molecular complexes further triggers the oligomerization of TNF receptors, which in turn regulate the intracellular signaling processes by providing transient compartmentalization in the membrane proximal regions of cytoplasm. In order to decode the molecular mechanisms of oligomerization in TNF receptor superfamily, we developed a new computational method that can physically simulate the spatial-temporal process of binding between TNF ligands and their receptors. The simulations show that the TNF receptors can be organized into hexagonal oligomers. The formation of this spatial pattern is highly dependent not only on the molecular properties such as the affinities of trans and cis binding, but also on the cellular factors such as the concentration of TNF ligands in the extracellular area or the density of TNF receptors on cell surfaces. Moreover, our model suggests that if TNF receptors are pre-organized into dimers before ligand binding, these lateral interactions between receptor monomers can play a positive role in stabilizing the ligand-receptor interactions, as well as in regulating the kinetics of receptor oligomerization. Altogether, this method throws lights on the mechanisms of TNF ligand-receptor interactions in cellular environments.

Multiscale simulation unravel the kinetic mechanisms of inflammasome assembly

In the innate immune system, the host defense from the invasion of external pathogens triggers the inflammatory responses. Proteins involved in the inflammatory pathways were often found to aggregate into supramolecular oligomers, called 'inflammasome', mostly through the homotypic interaction between their domains that belong to the death domain superfamily. Although much has been known about the formation of these helical molecular machineries, the detailed correlation between the dynamics of their assembly and the structure of each domain is still not well understood. Using the filament formed by the PYD domains of adaptor molecule ASC as a test system, we constructed a new multiscale simulation framework to study the kinetics of inflammasome assembly. We found that the filament assembly is a multi-step, but highly cooperative process. Moreover, there are three types of binding interfaces between domain subunits in the ASC^PYD filament. The multiscale simulation results suggest that dynamics of domain assembly are rooted in the primary protein sequence which defines the energetics of molecular recognition through three binding interfaces. Interface I plays a more regulatory role than the other two in mediating both the kinetics and the thermodynamics of assembly. Finally, the efficiency of our computational framework allows us to design mutants on a systematic scale and predict their impacts on filament assembly. In summary, this is, to the best of our knowledge, the first simulation method to model the spatial-temporal process of inflammasome assembly. Our work is a useful addition to a suite of existing experimental techniques to study the functions of inflammasome in innate immune system.

Multiscale Stochastic Reaction-Diffusion Algorithms Combining Markov Chain Models with Stochastic Partial Differential Equations

Two multiscale algorithms for stochastic simulations of reaction-diffusion processes are analysed. They are applicable to systems which include regions with significantly different concentrations of molecules. In both methods, a domain of interest is divided into two subsets where continuous-time Markov chain models and stochastic partial differential equations (SPDEs) are used, respectively. In the first algorithm, Markov chain (compartment-based) models are coupled with reaction-diffusion SPDEs by considering a pseudo-compartment (also called an overlap or handshaking region) in the SPDE part of the computational domain right next to the interface. In the second algorithm, no overlap region is used. Further extensions of both schemes are presented, including the case of an adaptively chosen boundary between different modelling approaches.

A Multiscale Computational Model for Simulating the Kinetics of Protein Complex Assembly

Proteins fulfill versatile biological functions by interacting with each other and forming high-order complexes. Although the order in which protein subunits assemble is important for the biological function of their final complex, this kinetic information has received comparatively little attention in recent years. Here we describe a multiscale framework that can be used to simulate the kinetics of protein complex assembly. There are two levels of models in the framework. The structural details of a protein complex are reflected by the residue-based model, while a lower-resolution model uses a rigid-body (RB) representation to simulate the process of complex assembly. These two levels of models are integrated together, so that we are able to provide the kinetic information about complex assembly with both structural details and computational efficiency.

Simulating biological processes: stochastic physics from whole cells to colonies

The last few decades have revealed the living cell to be a crowded spatially heterogeneous space teeming with biomolecules whose concentrations and activities are governed by intrinsically random forces. It is from this randomness, however, that a vast array of precisely timed and intricately coordinated biological functions emerge that give rise to the complex forms and behaviors we see in the biosphere around us. This seemingly paradoxical nature of life has drawn the interest of an increasing number of physicists, and recent years have seen stochastic modeling grow into a major subdiscipline within biological physics. Here we review some of the major advances that have shaped our understanding of stochasticity in biology. We begin with some historical context, outlining a string of important experimental results that motivated the development of stochastic modeling. We then embark upon a fairly rigorous treatment of the simulation methods that are currently available for the treatment of stochastic biological models, with an eye toward comparing and contrasting their realms of applicability, and the care that must be taken when parameterizing them. Following that, we describe how stochasticity impacts several key biological functions, including transcription, translation, ribosome biogenesis, chromosome replication, and metabolism, before considering how the functions may be coupled into a comprehensive model of a 'minimal cell'. Finally, we close with our expectation for the future of the field, focusing on how mesoscopic stochastic methods may be augmented with atomic-scale molecular modeling approaches in order to understand life across a range of length and time scales.

Computational Systems Biology in Cancer: Modeling Methods and Applications

In recent years it has become clear that carcinogenesis is a complex process, both at the molecular and cellular levels. Understanding the origins, growth and spread of cancer, therefore requires an integrated or system-wide approach. Computational systems biology is an emerging sub-discipline in systems biology that utilizes the wealth of data from genomic, proteomic and metabolomic studies to build computer simulations of intra and intercellular processes. Several useful descriptive and predictive models of the origin, growth and spread of cancers have been developed in an effort to better understand the disease and potential therapeutic approaches. In this review we describe and assess the practical and theoretical underpinnings of commonly-used modeling approaches, including ordinary and partial differential equations, petri nets, cellular automata, agent based models and hybrid systems. A number of computer-based formalisms have been implemented to improve the accessibility of the various approaches to researchers whose primary interest lies outside of model development. We discuss several of these and describe how they have led to novel insights into tumor genesis, growth, apoptosis, vascularization and therapy.

A Computational Model for Kinetic Studies of Cadherin Binding and Clustering

Cadherin is a cell-surface transmembrane receptor that mediates calcium-dependent cell-cell adhesion and is a major component of adhesive junctions. The formation of intercellular adhesive junctions is initiated by trans binding between cadherins on adjacent cells, which is followed by the clustering of cadherins via the formation of cis interactions between cadherins on the same cell membranes. Moreover, classical cadherins have multiple glycosylation sites along their extracellular regions. It was found that aberrant glycosylation affects the adhesive function of cadherins and correlates with metastatic phenotypes of several cancers. However, a mechanistic understanding of cadherin clustering during cell adhesion and the role of glycosylation in this process is still lacking. Here, we designed a kinetic model that includes multistep reaction pathways for cadherin clustering. We further applied a diffusion-reaction algorithm to numerically simulate the clustering process using a recently developed coarse-grained model. Using experimentally measured rates of trans binding between soluble E-cadherin extracellular domains, we conducted simulations of cadherin-mediated cell-cell binding kinetics, and the results are quantitatively comparable to experimental data from micropipette experiments. In addition, we show that incorporating cadherin clustering via cis interactions further increases intercellular binding. Interestingly, a two-phase kinetic profile was derived under the assumption that glycosylation regulates the kinetic rates of cis interactions. This two-phase profile is qualitatively consistent with experimental results from micropipette measurements. Therefore, our computational studies provide new, to our knowledge, insights into the molecular mechanism of cadherin-based cell adhesion.

Understanding the Functional Roles of Multiple Extracellular Domains in Cell Adhesion Molecules with a Coarse-Grained Model

Intercellular contacts in multicellular organisms are maintained by membrane receptors called cell adhesion molecules (CAMs), which are expressed on cell surfaces. One interesting feature of CAMs is that almost all of their extracellular regions contain repeating copies of structural domains. It is not clear why so many extracellular domains need to be evolved through natural selection. We tackled this problem by computational modeling. A generic model of CAMs was constructed based on the domain organization of neuronal CAM, which is engaged in maintaining neuron-neuron adhesion in central nervous system. By placing these models on a cell-cell interface, we developed a Monte-Carlo simulation algorithm that incorporates both molecular factors including conformational changes of CAMs and cellular factor including fluctuations of plasma membranes to approach the physical process of CAM-mediated adhesion. We found that the presence of multiple domains at the extracellular region of a CAM plays a positive role in regulating its trans-interaction with other CAMs from the opposite side of cell surfaces. The trans-interaction can further be facilitated by the intramolecular contacts between different extracellular domains of a CAM. Finally, if more than one CAM is introduced on each side of cell surfaces, the lateral binding (cis-interactions) between these CAMs will positively correlate with their trans-interactions only within a small energetic range, suggesting that cell adhesion is an elaborately designed process in which both trans- and cis-interactions are fine-tuned collectively by natural selection. In short, this study deepens our general understanding of the molecular mechanisms of cell adhesion.

Master equations and the theory of stochastic path integrals

This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.

Elucidating the Functional Roles of Spatial Organization in Cross-Membrane Signal Transduction by a Hybrid Simulation Method

The ligand-binding of membrane receptors on cell surfaces initiates the dynamic process of cross-membrane signal transduction. It is an indispensable part of the signaling network for cells to communicate with external environments. Recent experiments revealed that molecular components in signal transduction are not randomly mixed, but spatially organized into distinctive patterns. These patterns, such as receptor clustering and ligand oligomerization, lead to very different gene expression profiles. However, little is understood about the molecular mechanisms and functional impacts of this spatial-temporal regulation in cross-membrane signal transduction. In order to tackle this problem, we developed a hybrid computational method that decomposes a model of signaling network into two simulation modules. The physical process of binding between receptors and ligands on cell surfaces are simulated by a diffusion-reaction algorithm, while the downstream biochemical reactions are modeled by stochastic simulation of Gillespie algorithm. These two processes are coupled together by a synchronization framework. Using this method, we tested the dynamics of a simple signaling network in which the ligand binding of cell surface receptors triggers the phosphorylation of protein kinases, and in turn regulates the expression of target genes. We found that spatial aggregation of membrane receptors at cellular interfaces is able to either amplify or inhibit downstream signaling outputs, depending on the details of clustering mechanism. Moreover, by providing higher binding avidity, the co-localization of ligands into multi-valence complex modulates signaling in very different ways that are closely related to the binding affinity between ligand and receptor. We also found that the temporal oscillation of the signaling pathway that is derived from genetic feedback loops can be modified by the spatial clustering of membrane receptors. In summary, our method demonstrates the functional importance of spatial organization in cross-membrane signal transduction. The method can be applied to any specific signaling pathway in cells.

Elucidating the general principles of cell adhesion with a coarse-grained simulation model

Coarse-grained simulation of interplay between cell adhesion and cell signaling.

Decoy receptors block TRAIL sensitivity at a supracellular level: the role of stromal cells in controlling tumour TRAIL sensitivity

Tumour necrosis factor-related apoptosis-inducing ligand (TRAIL) is a death ligand cytokine known for its cytotoxic activity against malignantly transformed cells. TRAIL induces cell death through binding to death receptors DR4 and DR5. The inhibitory decoy receptors (DcR1 and DcR2) co-expressed with death receptor 4 (DR4)/DR5 on the same cell can block the transmission of the apoptotic signal. Here, we show that DcRs also regulate TRAIL sensitivity at a supracellular level and thus represent a mechanism by which the microenvironment can diminish tumour TRAIL sensitivity. Mathematical modelling and layered or spheroid stroma-extracellular matrix-tumour cultures were used to model the tumour microenvironment. By engineering TRAIL to escape binding by DcRs, we found that DcRs do not only act in a cell-autonomous or cis-regulatory manner, but also exert trans-cellular regulation originating from stromal cells and affect tumour cells, highlighting the potent inhibitory effect of DcRs in the tumour tissue and the necessity of selective targeting of the two death-inducing TRAIL receptors to maximise efficacy.

The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion

Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, owing to recent advances in computational power, the simulation and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction-diffusion systems in which regions of low copy numbers exist. The specific stochastic models with which we shall be concerned in this manuscript are referred to as 'compartment-based' or 'on-lattice'. These models are characterized by a discretization of the computational domain into a grid/lattice of 'compartments'. Within each compartment, particles are assumed to be well mixed and are permitted to react with other particles within their compartment or to transfer between neighbouring compartments. Stochastic models provide accuracy, but at the cost of significant computational resources. For models that have regions of both low and high concentrations, it is often desirable, for reasons of efficiency, to employ coupled multi-scale modelling paradigms. In this work, we develop two hybrid algorithms in which a PDE in one region of the domain is coupled to a compartment-based model in the other. Rather than attempting to balance average fluxes, our algorithms answer a more fundamental question: 'how are individual particles transported between the vastly different model descriptions?' First, we present an algorithm derived by carefully redefining the continuous PDE concentration as a probability distribution. While this first algorithm shows very strong convergence to analytical solutions of test problems, it can be cumbersome to simulate. Our second algorithm is a simplified and more efficient implementation of the first, it is derived in the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by comparing the averaged simulations with analytical solutions of PDEs for mean concentrations.

Methodologies for the modeling and simulation of biochemical networks, illustrated for signal transduction pathways: a primer

Biochemical networks depict the chemical interactions that take place among elements of living cells. They aim to elucidate how cellular behavior and functional properties of the cell emerge from the relationships between its components, i.e. molecules. Biochemical networks are largely characterized by dynamic behavior, and exhibit high degrees of complexity. Hence, the interest in such networks is growing and they have been the target of several recent modeling efforts. Signal transduction pathways (STPs) constitute a class of biochemical networks that receive, process, and respond to stimuli from the environment, as well as stimuli that are internal to the organism. An STP consists of a chain of intracellular signaling processes that ultimately result in generating different cellular responses. This primer presents the methodologies used for the modeling and simulation of biochemical networks, illustrated for STPs. These methodologies range from qualitative to quantitative, and include structural as well as dynamic analysis techniques. We describe the different methodologies, outline their underlying assumptions, and provide an assessment of their advantages and disadvantages. Moreover, publicly and/or commercially available implementations of these methodologies are listed as appropriate. In particular, this primer aims to provide a clear introduction and comprehensive coverage of biochemical modeling and simulation methodologies for the non-expert, with specific focus on relevant literature of STPs.

A coarse-grained model for the simulations of biomolecular interactions in cellular environments

The interactions of bio-molecules constitute the key steps of cellular functions. However, in vivo binding properties differ significantly from their in vitro measurements due to the heterogeneity of cellular environments. Here we introduce a coarse-grained model based on rigid-body representation to study how factors such as cellular crowding and membrane confinement affect molecular binding. The macroscopic parameters such as the equilibrium constant and the kinetic rate constant are calibrated by adjusting the microscopic coefficients used in the numerical simulations. By changing these model parameters that are experimentally approachable, we are able to study the kinetic and thermodynamic properties of molecular binding, as well as the effects caused by specific cellular environments. We investigate the volumetric effects of crowded intracellular space on bio-molecular diffusion and diffusion-limited reactions. Furthermore, the binding constants of membrane proteins are currently difficult to measure. We provide quantitative estimations about how the binding of membrane proteins deviates from soluble proteins under different degrees of membrane confinements. The simulation results provide biological insights to the functions of membrane receptors on cell surfaces. Overall, our studies establish a connection between the details of molecular interactions and the heterogeneity of cellular environments.

Modelling and simulating reaction-diffusion systems using coloured Petri nets

Reaction-diffusion systems often play an important role in systems biology when developmental processes are involved. Traditional methods of modelling and simulating such systems require substantial prior knowledge of mathematics and/or simulation algorithms. Such skills may impose a challenge for biologists, when they are not equally well-trained in mathematics and computer science. Coloured Petri nets as a high-level and graphical language offer an attractive alternative, which is easily approachable. In this paper, we investigate a coloured Petri net framework integrating deterministic, stochastic and hybrid modelling formalisms and corresponding simulation algorithms for the modelling and simulation of reaction-diffusion processes that may be closely coupled with signalling pathways, metabolic reactions and/or gene expression. Such systems often manifest multiscaleness in time, space and/or concentration. We introduce our approach by means of some basic diffusion scenarios, and test it against an established case study, the Brusselator model.

Stochastic analysis of reaction-diffusion processes

Reaction and diffusion processes are used to model chemical and biological processes over a wide range of spatial and temporal scales. Several routes to the diffusion process at various levels of description in time and space are discussed and the master equation for spatially discretized systems involving reaction and diffusion is developed. We discuss an estimator for the appropriate compartment size for simulating reaction-diffusion systems and introduce a measure of fluctuations in a discretized system. We then describe a new computational algorithm for implementing a modified Gillespie method for compartmental systems in which reactions are aggregated into equivalence classes and computational cells are searched via an optimized tree structure. Finally, we discuss several examples that illustrate the issues that have to be addressed in general systems.

ReaDDy--a software for particle-based reaction-diffusion dynamics in crowded cellular environments

We introduce the software package ReaDDy for simulation of detailed spatiotemporal mechanisms of dynamical processes in the cell, based on reaction-diffusion dynamics with particle resolution. In contrast to other particle-based reaction kinetics programs, ReaDDy supports particle interaction potentials. This permits effects such as space exclusion, molecular crowding and aggregation to be modeled. The biomolecules simulated can be represented as a sphere, or as a more complex geometry such as a domain structure or polymer chain. ReaDDy bridges the gap between small-scale but highly detailed molecular dynamics or Brownian dynamics simulations and large-scale but little-detailed reaction kinetics simulations. ReaDDy has a modular design that enables the exchange of the computing core by efficient platform-specific implementations or dynamical models that are different from Brownian dynamics.

A simulator for spatially extended kappa models

Spatial Kappa is a simulator of models written in a variant of the rule-based stochastic modelling language Kappa, with spatial extensions.

Improved spatial direct method with gradient-based diffusion to retain full diffusive fluctuations

The spatial direct method with gradient-based diffusion is an accelerated stochastic reaction-diffusion simulation algorithm that treats diffusive transfers between neighboring subvolumes based on concentration gradients. This recent method achieved a marked improvement in simulation speed and reduction in the number of time-steps required to complete a simulation run, compared with the exact algorithm, by sampling only the net diffusion events, instead of sampling all diffusion events. Although the spatial direct method with gradient-based diffusion gives accurate means of simulation ensembles, its gradient-based diffusion strategy results in reduced fluctuations in populations of diffusive species. In this paper, we present a new improved algorithm that is able to anticipate all possible microscopic fluctuations due to diffusive transfers in the system and incorporate this information to retain the same degree of fluctuations in populations of diffusing species as the exact algorithm. The new algorithm also provides a capability to set the desired level of fluctuation per diffusing species, which facilitates adjusting the balance between the degree of exactness in simulation results and the simulation speed. We present numerical results that illustrate the recovery of fluctuations together with the accuracy and efficiency of the new algorithm.

URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries

BACKGROUND

Experiments in silico using stochastic reaction-diffusion models have emerged as an important tool in molecular systems biology. Designing computational software for such applications poses several challenges. Firstly, realistic lattice-based modeling for biological applications requires a consistent way of handling complex geometries, including curved inner- and outer boundaries. Secondly, spatiotemporal stochastic simulations are computationally expensive due to the fast time scales of individual reaction- and diffusion events when compared to the biological phenomena of actual interest. We therefore argue that simulation software needs to be both computationally efficient, employing sophisticated algorithms, yet in the same time flexible in order to meet present and future needs of increasingly complex biological modeling.

RESULTS

We have developed URDME, a flexible software framework for general stochastic reaction-transport modeling and simulation. URDME uses Unstructured triangular and tetrahedral meshes to resolve general geometries, and relies on the Reaction-Diffusion Master Equation formalism to model the processes under study. An interface to a mature geometry and mesh handling external software (Comsol Multiphysics) provides for a stable and interactive environment for model construction. The core simulation routines are logically separated from the model building interface and written in a low-level language for computational efficiency. The connection to the geometry handling software is realized via a Matlab interface which facilitates script computing, data management, and post-processing. For practitioners, the software therefore behaves much as an interactive Matlab toolbox. At the same time, it is possible to modify and extend URDME with newly developed simulation routines. Since the overall design effectively hides the complexity of managing the geometry and meshes, this means that newly developed methods may be tested in a realistic setting already at an early stage of development.

CONCLUSIONS

In this paper we demonstrate, in a series of examples with high relevance to the molecular systems biology community, that the proposed software framework is a useful tool for both practitioners and developers of spatial stochastic simulation algorithms. Through the combined efforts of algorithm development and improved modeling accuracy, increasingly complex biological models become feasible to study through computational methods. URDME is freely available at http://www.urdme.org.

Spatial simulations in systems biology: from molecules to cells

Cells are highly organized objects containing millions of molecules. Each biomolecule has a specific shape in order to interact with others in the complex machinery. Spatial dynamics emerge in this system on length and time scales which can not yet be modeled with full atomic detail. This review gives an overview of methods which can be used to simulate the complete cell at least with molecular detail, especially Brownian dynamics simulations. Such simulations require correct implementation of the diffusion-controlled reaction scheme occurring on this level. Implementations and applications of spatial simulations are presented, and finally it is discussed how the atomic level can be included for instance in multi-scale simulation methods.

Intrinsic noise analyzer: a software package for the exploration of stochastic biochemical kinetics using the system size expansion

The accepted stochastic descriptions of biochemical dynamics under well-mixed conditions are given by the Chemical Master Equation and the Stochastic Simulation Algorithm, which are equivalent. The latter is a Monte-Carlo method, which, despite enjoying broad availability in a large number of existing software packages, is computationally expensive due to the huge amounts of ensemble averaging required for obtaining accurate statistical information. The former is a set of coupled differential-difference equations for the probability of the system being in any one of the possible mesoscopic states; these equations are typically computationally intractable because of the inherently large state space. Here we introduce the software package intrinsic Noise Analyzer (iNA), which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen's system size expansion of the Chemical Master Equation. iNA is platform independent and supports the popular SBML format natively. The present implementation is the first to adopt a complementary approach that combines state-of-the-art analysis tools using the computer algebra system Ginac with traditional methods of stochastic simulation. iNA integrates two approximation methods based on the system size expansion, the Linear Noise Approximation and effective mesoscopic rate equations, which to-date have not been available to non-expert users, into an easy-to-use graphical user interface. In particular, the present methods allow for quick approximate analysis of time-dependent mean concentrations, variances, covariances and correlations coefficients, which typically outperforms stochastic simulations. These analytical tools are complemented by automated multi-core stochastic simulations with direct statistical evaluation and visualization. We showcase iNA's performance by using it to explore the stochastic properties of cooperative and non-cooperative enzyme kinetics and a gene network associated with circadian rhythms. The software iNA is freely available as executable binaries for Linux, MacOSX and Microsoft Windows, as well as the full source code under an open source license.

Reaction-diffusion master equation in the microscopic limit

Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice reaction-diffusion master equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. Here we give a general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model and lets us quantify this limit in two and three dimensions. In this light we review and discuss recent work in which the RDME has been modified in different ways in order to better agree with the microscale model for very small voxel sizes.

Kinetics in signal transduction pathways involving promiscuous oligomerizing receptors can be determined by receptor specificity: apoptosis induction by TRAIL

Here we show by computer modeling that kinetics and outcome of signal transduction in case of hetero-oligomerizing receptors of a promiscuous ligand largely depend on the relative amounts of its receptors. Promiscuous ligands can trigger the formation of nonproductive receptor complexes, which slows down the formation of active receptor complexes and thus can block signal transduction. Our model predicts that increasing the receptor specificity of the ligand without changing its binding parameters should result in faster receptor activation and enhanced signaling. We experimentally validated this hypothesis using the cytokine tumor necrosis factor-related apoptosis-inducing ligand (TRAIL) and its four membrane-bound receptors as an example. Bypassing ligand-induced receptor hetero-oligomerization by receptor-selective TRAIL variants enhanced the kinetics of receptor activation and augmented apoptosis. Our results suggest that control of signaling pathways by promiscuous ligands could result in apparent slow biological kinetics and blocking signal transmission. By modulating the relative amount of the different receptors for the ligand, signaling processes like apoptosis can be accelerated or decelerated and even inhibited. It also implies that more effective treatments using protein therapeutics could be achieved simply by altering specificity.

Stochastic modeling of cellular networks

Noise and stochasticity are fundamental to biology because they derive from the nature of biochemical reactions. Thermal motions of molecules translate into randomness in the sequence and timing of reactions, which leads to cell-cell variability ("noise") in mRNA and protein levels even in clonal populations of genetically identical cells. This is a quantitative phenotype that has important functional repercussions, including persistence in bacterial subpopulations challenged with antibiotics, and variability in the response of cancer cells to drugs. In this chapter, we present the modeling of such stochastic cellular behaviors using the formalism of jump Markov processes, whose probability distributions evolve according to the chemical master equation (CME). We also discuss the techniques used to solve the CME. These include kinetic Monte Carlo simulations techniques such as the stochastic simulation algorithm (SSA) and method closure techniques such as the linear noise approximation (LNA).

The two-regime method for optimizing stochastic reaction-diffusion simulations

Spatial organization and noise play an important role in molecular systems biology. In recent years, a number of software packages have been developed for stochastic spatio-temporal simulation, ranging from detailed molecular-based approaches to less detailed compartment-based simulations. Compartment-based approaches yield quick and accurate mesoscopic results, but lack the level of detail that is characteristic of the computationally intensive molecular-based models. Often microscopic detail is only required in a small region (e.g. close to the cell membrane). Currently, the best way to achieve microscopic detail is to use a resource-intensive simulation over the whole domain. We develop the two-regime method (TRM) in which a molecular-based algorithm is used where desired and a compartment-based approach is used elsewhere. We present easy-to-implement coupling conditions which ensure that the TRM results have the same accuracy as a detailed molecular-based model in the whole simulation domain. Therefore, the TRM combines strengths of previously developed stochastic reaction-diffusion software to efficiently explore the behaviour of biological models. Illustrative examples and the mathematical justification of the TRM are also presented.

Quantification of mRNA and protein and integration with protein turnover in a bacterium

Determination of the average cellular copy number of 400 proteins under different growth conditions and integration with protein turnover and absolute mRNA levels reveals the dynamics of protein expression in the genome-reduced bacterium Mycoplasma pneumoniae.

Our study provides a fine-grained, quantitative picture to unprecedented detail in an established model organism for systems-wide studies.

Our integrative approach reveals a novel, dynamic view on the processes, interactions and regulations underlying the central dogma pathway and the composition of protein complexes.

Simulations using our quantitative data on mRNA, protein and turnover show how an organism copes with stochastic noise in gene expression in vivo.

Our data serve as an important resource for colleagues both within our field of research and in related disciplines.

A hallmark of Systems Biology is the integration of diverse, large quantitative data sets with the aim to gain novel insights into how biological processes work. We measured individual mRNA and protein abundances as well as protein turnover in the bacterium Mycoplasma pneumoniae. This human pathogen is an ideal model organism for organism-wide studies. It can be readily cultured under laboratory conditions and it has a very small genome with only 690 protein-coding genes. This comparably low complexity allows for the exhaustive analysis of major cellular biomolecules avoiding constrains introduced by limitations of available analysis techniques.

Using a recently developed mass spectrometry-based approach, we determined the average cellular copy number for over 400 individual proteins under different growth and stress conditions. The 20 most abundant proteins, including Elongation factor Tu, cellular chaperones, and proteins involved in metabolizing glucose, the major energy source of M. pneumoniae account for nearly 44% of the total cellular protein mass. We observed abundance changes of many expected and several unexpected proteins in response to cellular stress, such as heat shock, DNA damage and osmotic stress, as well as along batch culture growth over 4 days.

Integration of the protein abundance data with quantitative mRNA measurements revealed a modest correlation between these two classes of biomolecules. However, for several classical stress-induced proteins, we observed a correlated induction of mRNA and protein in response to heat shock. A focused analysis of mRNA–protein abundance dynamics during batch culture growth suggested that the regulation of gene expression is largely decoupled from protein dynamics in M. pneumoniae, indicating extensive post-transcriptional and post-translational regulation influencing the cellular mRNA–protein ratios.

To investigate the factors influencing the cellular protein abundance, we measured individual protein turnover rates by mass spectrometry using a label-chase approach involving stable isotope-labelled amino acids. The average half-life of a protein in M. pneumoniae is 23 h. Based on the measured quantitative mRNA data, the protein abundances and their half-lives, we established an ordinary differential equations model for the estimation of individual in vivo protein degradation and translation efficiency rates. We found out that translation efficiency rather than protein turnover is the dominating factor influencing protein abundance. Using our abundance and turnover data, we additionally performed stochastic simulations of gene expression. We observed that long protein half-life and low translational efficiency buffers gene expression noise propagating from low cellular mRNA levels in vivo.

We compared the abundance ratios of proteins associating into complexes in vivo with their expected functional stoichiometries. We observed that for stable protein complexes, such as the GroEL/ES chaperonin or DNA gyrase, our measured abundance ratios reflected the expected subunit stoichiometries. More dynamic protein complexes, such as the DnaK/J/GrpE chaperone system or RNA polymerase, showed several unusual subunit ratios, pointing towards transient interaction of sub-stoichiometric subunits for function. A detailed, quantitative analysis of the ribosome, the largest cellular protein complex, revealed large abundance differences of the 51 subunits. This observation indicates a multi-functionality for several, abundant ribosomal proteins.

Finally, a comparison of the determined average cellular protein abundances with a different pathogenic bacterium, Leptospira interrogans, revealed that cellular protein abundances closely reflect their respective lifestyles.

Our study represents an organism-wide, quantitative analysis of cellular protein abundances. Integrating our proteomics data with determined mRNA levels and protein turnover rates reveals insights into the dynamic interplay and regulation of mRNA and proteins, the central biomolecules of a cell.

Biological function and cellular responses to environmental perturbations are regulated by a complex interplay of DNA, RNA, proteins and metabolites inside cells. To understand these central processes in living systems at the molecular level, we integrated experimentally determined abundance data for mRNA, proteins, as well as individual protein half-lives from the genome-reduced bacterium Mycoplasma pneumoniae. We provide a fine-grained, quantitative analysis of basic intracellular processes under various external conditions. Proteome composition changes in response to cellular perturbations reveal specific stress response strategies. The regulation of gene expression is largely decoupled from protein dynamics and translation efficiency has a higher regulatory impact on protein abundance than protein turnover. Stochastic simulations using in vivo data show how low translation efficiency and long protein half-lives effectively reduce biological noise in gene expression. Protein abundances are regulated in functional units, such as complexes or pathways, and reflect cellular lifestyles. Our study provides a detailed integrative analysis of average cellular protein abundances and the dynamic interplay of mRNA and proteins, the central biomolecules of a cell.

Systems biology in immunology: a computational modeling perspective

Systems biology is an emerging discipline that combines high-content, multiplexed measurements with informatic and computational modeling methods to better understand biological function at various scales. Here we present a detailed review of the methods used to create computational models and to conduct simulations of immune function. We provide descriptions of the key data-gathering techniques employed to generate the quantitative and qualitative data required for such modeling and simulation and summarize the progress to date in applying these tools and techniques to questions of immunological interest, including infectious disease. We include comments on what insights modeling can provide that complement information obtained from the more familiar experimental discovery methods used by most investigators and the reasons why quantitative methods are needed to eventually produce a better understanding of immune system operation in health and disease.

STSE: Spatio-Temporal Simulation Environment Dedicated to Biology

Background

Recently, the availability of high-resolution microscopy together with the advancements in the development of biomarkers as reporters of biomolecular interactions increased the importance of imaging methods in molecular cell biology. These techniques enable the investigation of cellular characteristics like volume, size and geometry as well as volume and geometry of intracellular compartments, and the amount of existing proteins in a spatially resolved manner. Such detailed investigations opened up many new areas of research in the study of spatial, complex and dynamic cellular systems. One of the crucial challenges for the study of such systems is the design of a well stuctured and optimized workflow to provide a systematic and efficient hypothesis verification. Computer Science can efficiently address this task by providing software that facilitates handling, analysis, and evaluation of biological data to the benefit of experimenters and modelers.

Results

The Spatio-Temporal Simulation Environment (STSE) is a set of open-source tools provided to conduct spatio-temporal simulations in discrete structures based on microscopy images. The framework contains modules to digitize, represent, analyze, and mathematically model spatial distributions of biochemical species. Graphical user interface (GUI) tools provided with the software enable meshing of the simulation space based on the Voronoi concept. In addition, it supports to automatically acquire spatial information to the mesh from the images based on pixel luminosity (e.g. corresponding to molecular levels from microscopy images). STSE is freely available either as a stand-alone version or included in the linux live distribution Systems Biology Operational Software (SB.OS) and can be downloaded from http://www.stse-software.org/. The Python source code as well as a comprehensive user manual and video tutorials are also offered to the research community. We discuss main concepts of the STSE design and workflow. We demonstrate it's usefulness using the example of a signaling cascade leading to formation of a morphological gradient of Fus3 within the cytoplasm of the mating yeast cell Saccharomyces cerevisiae.

Conclusions

STSE is an efficient and powerful novel platform, designed for computational handling and evaluation of microscopic images. It allows for an uninterrupted workflow including digitization, representation, analysis, and mathematical modeling. By providing the means to relate the simulation to the image data it allows for systematic, image driven model validation or rejection. STSE can be scripted and extended using the Python language. STSE should be considered rather as an API together with workflow guidelines and a collection of GUI tools than a stand alone application. The priority of the project is to provide an easy and intuitive way of extending and customizing software using the Python language.

An accelerated algorithm for discrete stochastic simulation of reaction-diffusion systems using gradient-based diffusion and tau-leaping

Stochastic simulation of reaction-diffusion systems enables the investigation of stochastic events arising from the small numbers and heterogeneous distribution of molecular species in biological cells. Stochastic variations in intracellular microdomains and in diffusional gradients play a significant part in the spatiotemporal activity and behavior of cells. Although an exact stochastic simulation that simulates every individual reaction and diffusion event gives a most accurate trajectory of the system's state over time, it can be too slow for many practical applications. We present an accelerated algorithm for discrete stochastic simulation of reaction-diffusion systems designed to improve the speed of simulation by reducing the number of time-steps required to complete a simulation run. This method is unique in that it employs two strategies that have not been incorporated in existing spatial stochastic simulation algorithms. First, diffusive transfers between neighboring subvolumes are based on concentration gradients. This treatment necessitates sampling of only the net or observed diffusion events from higher to lower concentration gradients rather than sampling all diffusion events regardless of local concentration gradients. Second, we extend the non-negative Poisson tau-leaping method that was originally developed for speeding up nonspatial or homogeneous stochastic simulation algorithms. This method calculates each leap time in a unified step for both reaction and diffusion processes while satisfying the leap condition that the propensities do not change appreciably during the leap and ensuring that leaping does not cause molecular populations to become negative. Numerical results are presented that illustrate the improvement in simulation speed achieved by incorporating these two new strategies.

Exploring the spatial and temporal organization of a cell's proteome

To increase our current understanding of cellular processes, such as cell signaling and division, knowledge is needed about the spatial and temporal organization of the proteome at different organizational levels. These levels cover a wide range of length and time scales: from the atomic structures of macromolecules for inferring their molecular function, to the quantitative description of their abundance, and spatial distribution in the cell. Emerging new experimental technologies are greatly increasing the availability of such spatial information on the molecular organization in living cells. This review addresses three fields that have significantly contributed to our understanding of the proteome's spatial and temporal organization: first, methods for the structure determination of individual macromolecular assemblies, specifically the fitting of atomic structures into density maps generated from electron microscopy techniques; second, research that visualizes the spatial distributions of these complexes within the cellular context using cryo electron tomography techniques combined with computational image processing; and third, methods for the spatial modeling of the dynamic organization of the proteome, specifically those methods for simulating reaction and diffusion of proteins and complexes in crowded intracellular fluids. The long-term goal is to integrate the varied data about a proteome's organization into a spatially explicit, predictive model of cellular processes.

Monte Carlo simulations of single- and multistep enzyme-catalyzed reaction sequences: effects of diffusion, cell size, enzyme fluctuations, colocalization, and segregation

Following the discovery of slow fluctuations in the catalytic activity of an enzyme in single-molecule experiments, it has been shown that the classical Michaelis-Menten (MM) equation relating the average enzymatic velocity and the substrate concentration may hold even for slowly fluctuating enzymes. In many cases, the average velocity is that given by the MM equation with time-averaged values of the fluctuating rate constants and the effect of enzyme fluctuations is simply averaged out. The situation is quite different for a sequence of reactions. For colocalization of a pair of enzymes in a sequence to be effective in promoting reaction, the second must be active when the first is active or soon after. If the enzymes are slowly varying and only rarely active, the product of the first reaction may diffuse away before the second enzyme is active, and colocalization may have little value. Even for single-step reactions the interplay of reaction and diffusion with enzyme fluctuations leads to added complexities, but for multistep reactions the interplay of reaction and diffusion, cell size, compartmentalization, enzyme fluctuations, colocalization, and segregation is far more complex than for single-step reactions. In this paper, we report the use of stochastic simulations at the level of whole cells to explore, understand, and predict the behavior of single- and multistep enzyme-catalyzed reaction systems exhibiting some of these complexities. Results for single-step reactions confirm several earlier observations by others. The MM relationship, with altered constants, is found to hold for single-step reactions slowed by diffusion. For single-step reactions, the distribution of enzymes in a regular grid is slightly more effective than a random distribution. Fluctuations of enzyme activity, with average activity fixed, have no observed effects for simple single-step reactions slowed by diffusion. Two-step sequential reactions are seen to be slowed by segregation of the enzymes for each step, and results of the calculations suggest limits for cell size. Colocalization of enzymes for a two-step sequence is seen to promote reaction, and rates fall rapidly with increasing distance between enzymes. Low frequency fluctuations of the activities of colocalized enzymes, with average activities fixed, can greatly reduce reaction rates for sequential reactions.

Stochastic reaction-diffusion kinetics in the microscopic limit

Quantitative analysis of biochemical networks often requires consideration of both spatial and stochastic aspects of chemical processes. Despite significant progress in the field, it is still computationally prohibitive to simulate systems involving many reactants or complex geometries using a microscopic framework that includes the finest length and time scales of diffusion-limited molecular interactions. For this reason, spatially or temporally discretized simulations schemes are commonly used when modeling intracellular reaction networks. The challenge in defining such coarse-grained models is to calculate the correct probabilities of reaction given the microscopic parameters and the uncertainty in the molecular positions introduced by the spatial or temporal discretization. In this paper we have solved this problem for the spatially discretized Reaction-Diffusion Master Equation; this enables a seamless and physically consistent transition from the microscopic to the macroscopic frameworks of reaction-diffusion kinetics. We exemplify the use of the methods by showing that a phosphorylation-dephosphorylation motif, commonly observed in eukaryotic signaling pathways, is predicted to display fluctuations that depend on the geometry of the system.

Approximating intrinsic noise in continuous multispecies models

In small-scale chemical reaction networks, the local density of molecules is changed by discrete jumps owing to reactive collisions, and through transport. A systematic perturbation scheme is developed to analytically characterize these non-equilibrium intrinsic fluctuations in a multispecies spatially varying system. The method is illustrated on a variety of model systems. In all cases, the continuous approximation method is corroborated with extensive stochastic simulation. As an example of our technique applied to a spatially varying steady state, we demonstrate that a model for embryonic patterning mediated by regulatory mRNA is surprisingly robust to intrinsic fluctuations.

Programming cells: towards an automated 'Genetic Compiler'

One of the visions of synthetic biology is to be able to program cells using a language that is similar to that used to program computers or robotics. For large genetic programs, keeping track of the DNA on the level of nucleotides becomes tedious and error prone, requiring a new generation of computer-aided design (CAD) software. To push the size of projects, it is important to abstract the designer from the process of part selection and optimization. The vision is to specify genetic programs in a higher-level language, which a genetic compiler could automatically convert into a DNA sequence. Steps towards this goal include: defining the semantics of the higher-level language, algorithms to select and assemble parts, and biophysical methods to link DNA sequence to function. These will be coupled to graphic design interfaces and simulation packages to aid in the prediction of program dynamics, optimize genes, and scan projects for errors.