Computational cell biology: spatiotemporal simulation of cellular events

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.

Understanding the impacts of cellular environments on ligand binding of membrane receptors by computational simulations

Binding of cell surface receptors with their extracellular ligands initiates various intracellular signaling pathways. However, our understanding of the cellular functions of these receptors is very limited due to the fact that in vivo binding between ligands and receptors has only been successfully measured in a very small number of cases. In living cells, receptors are anchored on surfaces of the plasma membrane, which undergoes thermal undulations. Moreover, it has been observed in various systems that receptors can be organized into oligomers prior to ligand binding. It is not well understood how these cellular factors play roles in regulating the dynamics of ligand-receptor interactions. Here, we tackled these problems by using a coarse-grained kinetic Monte Carlo simulation method. Using this method, we demonstrated that the membrane undulations cause a negative effect on ligand-receptor interactions. We further found that the preassembly of membrane receptors on the cell surface can not only accelerate the kinetics of ligand binding but also reduce the noises during the process. In general, our study highlights the importance of membrane environments in regulating the function of membrane receptors in cells. The simulation method can be potentially applied to specific receptor systems involved in cell signaling.

Understand the Functions of Scaffold Proteins in Cell Signaling by a Mesoscopic Simulation Method

Scaffold proteins are central players in regulating the spatial-temporal organization of many important signaling pathways in cells. They offer physical platforms to downstream signaling proteins so that their transient interactions in a crowded and heterogeneous environment of cytosol can be greatly facilitated. However, most scaffold proteins tend to simultaneously bind more than one signaling molecule, which leads to the spatial assembly of multimeric protein complexes. The kinetics of these protein oligomerizations are difficult to quantify by traditional experimental approaches. To understand the functions of scaffold proteins in cell signaling, we developed a, to our knowledge, new hybrid simulation algorithm in which both spatial organization and binding kinetics of proteins were implemented. We applied this new technique to a simple network system that contains three molecules. One molecule in the network is a scaffold protein, whereas the other two are its binding targets in the downstream signaling pathway. Each of the three molecules in the system contains two binding motifs that can interact with each other and are connected by a flexible linker. By applying the new simulation method to the model, we show that the scaffold proteins will promote not only thermodynamics but also kinetics of cell signaling given the premise that the interaction between the two signaling molecules is transient. Moreover, by changing the flexibility of the linker between two binding motifs, our results suggest that the conformational fluctuations in a scaffold protein play a positive role in recruiting downstream signaling molecules. In summary, this study showcases the capability of computational simulation in understanding the general principles of scaffold protein functions.

Validation of Modeling and Simulation Methods in Computational Biology

In recent years, a highly sophisticated array of modeling and simulation tools in all areas of biological and biomedical research has been developed. These tools have the potential to provide new insights into biological mechanisms integrating subcellular, cellular, tissue, organ, and potentially whole organism levels. Current research is focused on how to use these methods for translational medical research, such as for disease diagnosis and understanding, as well as drug discovery. In addition, these approaches enhance the ability to use human-derived data and to contribute to the refinement of high-cost experimental-based research. Additionally, the conflicting conceptual frameworks and conceptions of modeling and simulation methods from the broad public of users could have a significant impact on the successful implementation of aforementioned applications. This in turn could result in successful collaborations across academic, clinical, and industrial sectors. To that end, this study provides an overview of the frameworks and disciplines used for validation of computational methodologies in biomedical sciences.

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.

"Essentially, all models are wrong, but some are useful"-a cross-disciplinary agenda for building useful models in cell biology and biophysics

Intuition alone often fails to decipher the mechanisms underlying the experimental data in Cell Biology and Biophysics, and mathematical modeling has become a critical tool in these fields. However, mathematical modeling is not as widespread as it could be, because experimentalists and modelers often have difficulties communicating with each other, and are not always on the same page about what a model can or should achieve. Here, we present a framework to develop models that increase the understanding of the mechanisms underlying one's favorite biological system. Development of the most insightful models starts with identifying a good biological question in light of what is known and unknown in the field, and determining the proper level of details that are sufficient to address this question. The model should aim not only to explain already available data, but also to make predictions that can be experimentally tested. We hope that both experimentalists and modelers who are driven by mechanistic questions will find these guidelines useful to develop models with maximum impact in their field.

Parameter uncertainty quantification using surrogate models applied to a spatial model of yeast mating polarization

A common challenge in systems biology is quantifying the effects of unknown parameters and estimating parameter values from data. For many systems, this task is computationally intractable due to expensive model evaluations and large numbers of parameters. In this work, we investigate a new method for performing sensitivity analysis and parameter estimation of complex biological models using techniques from uncertainty quantification. The primary advance is a significant improvement in computational efficiency from the replacement of model simulation by evaluation of a polynomial surrogate model. We demonstrate the method on two models of mating in budding yeast: a smaller ODE model of the heterotrimeric G-protein cycle, and a larger spatial model of pheromone-induced cell polarization. A small number of model simulations are used to fit the polynomial surrogates, which are then used to calculate global parameter sensitivities. The surrogate models also allow rapid Bayesian inference of the parameters via Markov chain Monte Carlo (MCMC) by eliminating model simulations at each step. Application to the ODE model shows results consistent with published single-point estimates for the model and data, with the added benefit of calculating the correlations between pairs of parameters. On the larger PDE model, the surrogate models allowed convergence for the distribution of 15 parameters, which otherwise would have been computationally prohibitive using simulations at each MCMC step. We inferred parameter distributions that in certain cases peaked at values different from published values, and showed that a wide range of parameters would permit polarization in the model. Strikingly our results suggested different diffusion constants for active versus inactive Cdc42 to achieve good polarization, which is consistent with experimental observations in another yeast species S. pombe.

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.

Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology

Hybrid deterministic-stochastic methods provide an efficient alternative to a fully stochastic treatment of models which include components with disparate levels of stochasticity. However, general-purpose hybrid solvers for spatially resolved simulations of reaction-diffusion systems are not widely available. Here we describe fundamentals of a general-purpose spatial hybrid method. The method generates realizations of a spatially inhomogeneous hybrid system by appropriately integrating capabilities of a deterministic partial differential equation solver with a popular particle-based stochastic simulator, Smoldyn. Rigorous validation of the algorithm is detailed, using a simple model of calcium 'sparks' as a testbed. The solver is then applied to a deterministic-stochastic model of spontaneous emergence of cell polarity. The approach is general enough to be implemented within biologist-friendly software frameworks such as Virtual Cell.

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.

Simulated single molecule microscopy with SMeagol

Summary: SMeagol is a software tool to simulate highly realistic microscopy data based on spatial systems biology models, in order to facilitate development, validation and optimization of advanced analysis methods for live cell single molecule microscopy data.

Availability and implementation: SMeagol runs on Matlab R2014 and later, and uses compiled binaries in C for reaction–diffusion simulations. Documentation, source code and binaries for Mac OS, Windows and Ubuntu Linux can be downloaded from http://smeagol.sourceforge.net.

Contact:johan.elf@icm.uu.se

Supplementary information: Supplementary data are available at Bioinformatics online.

Unbiased Rare Event Sampling in Spatial Stochastic Systems Biology Models Using a Weighted Ensemble of Trajectories

The long-term goal of connecting scales in biological simulation can be facilitated by scale-agnostic methods. We demonstrate that the weighted ensemble (WE) strategy, initially developed for molecular simulations, applies effectively to spatially resolved cell-scale simulations. The WE approach runs an ensemble of parallel trajectories with assigned weights and uses a statistical resampling strategy of replicating and pruning trajectories to focus computational effort on difficult-to-sample regions. The method can also generate unbiased estimates of non-equilibrium and equilibrium observables, sometimes with significantly less aggregate computing time than would be possible using standard parallelization. Here, we use WE to orchestrate particle-based kinetic Monte Carlo simulations, which include spatial geometry (e.g., of organelles, plasma membrane) and biochemical interactions among mobile molecular species. We study a series of models exhibiting spatial, temporal and biochemical complexity and show that although WE has important limitations, it can achieve performance significantly exceeding standard parallel simulation—by orders of magnitude for some observables.

Stochastic simulations (simulations where randomness plays a role) of even simple biological systems are often so computationally intensive that it is impossible, in practice, to simulate them exhaustively and gather good statistics about the likelihood of different outcomes. The difficulty is compounded for the observation of rare events in these simulations; unfortunately, rare events, such as state transitions and barrier crossings, are often those of particular interest. Using the weighted ensemble (WE) method, we are able to enhance the characterization of rare events in cell biology simulations, but in such a way that the statistics for these events remain unbiased. The histogram of outcomes that WE produces has the same shape as a naive one, but the resolution of events in the tails of the histogram is greatly improved. This improved resolution in rare event statistics can be used to infer unbiased estimates of long timescale dynamics from short simulations, and we show that using a weighted ensemble can result in a reduction in total simulation time needed to sample certain events of interest in spatial, stochastic models of biological systems.

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

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

Regulation of Spatiotemporal Patterns by Biological Variability: General Principles and Applications to Dictyostelium discoideum

Spatiotemporal patterns often emerge from local interactions in a self-organizing fashion. In biology, the resulting patterns are also subject to the influence of the systematic differences between the system’s constituents (biological variability). This regulation of spatiotemporal patterns by biological variability is the topic of our review. We discuss several examples of correlations between cell properties and the self-organized spatiotemporal patterns, together with their relevance for biology. Our guiding, illustrative example will be spiral waves of cAMP in a colony of Dictyostelium discoideum cells. Analogous processes take place in diverse situations (such as cardiac tissue, where spiral waves occur in potentially fatal ventricular fibrillation) so a deeper understanding of this additional layer of self-organized pattern formation would be beneficial to a wide range of applications. One of the most striking differences between pattern-forming systems in physics or chemistry and those in biology is the potential importance of variability. In the former, system components are essentially identical with random fluctuations determining the details of the self-organization process and the resulting patterns. In biology, due to variability, the properties of potentially very few cells can have a driving influence on the resulting asymptotic collective state of the colony. Variability is one means of implementing a few-element control on the collective mode. Regulatory architectures, parameters of signaling cascades, and properties of structure formation processes can be "reverse-engineered" from observed spatiotemporal patterns, as different types of regulation and forms of interactions between the constituents can lead to markedly different correlations. The power of this biology-inspired view of pattern formation lies in building a bridge between two scales: the patterns as a collective state of a very large number of cells on the one hand, and the internal parameters of the single cells on the other.

Pattern formation is abundant in nature—from the rich ornaments of sea shells and the diversity of animal coat patterns to the myriad of fractal structures in biology and pattern-forming colonies of bacteria. Particularly fascinating are patterns changing with time, spatiotemporal patterns, like propagating waves and aggregation streams. Bacteria form large branched and nested aggregation-like patterns to immobilize themselves against water flow. The individual amoeba in Dictyostelium discoideum colonies initiates a transition to a collective multicellular state via a quorum-sensing form of communication—a cAMP signal propagating through the community in the form of spiral waves—and the subsequent chemotactic response of the cells leads to branch-like aggregation streams. The theoretical principle underlying most of these spatial and spatiotemporal patterns is self-organization, in which local interactions lead to patterns as large-scale collective”modes” of the system. Over more than half a century, these patterns have been classified and analyzed according to a”physics paradigm,” investigating such questions as how parameters regulate the transitions among patterns, which (types of) interactions lead to such large-scale patterns, and whether there are "critical parameter values" marking the sharp, spontaneous onset of patterns. A fundamental discovery has been that simple local interaction rules can lead to complex large-scale patterns. The specific pattern "layouts" (i.e., their spatial arrangement and their geometric constraints) have received less attention. However, there is a major difference between patterns in physics and chemistry on the one hand and patterns in biology on the other: in biology, patterns often have an important functional role for the biological system and can be considered to be under evolutionary selection. From this perspective, we can expect that individual biological elements exert some control on the emerging patterns. Here we explore spiral wave patterns as a prominent example to illustrate the regulation of spatiotemporal patterns by biological variability. We propose a new approach to studying spatiotemporal data in biology: analyzing the correlation between the spatial distribution of the constituents’ properties and the features of the spatiotemporal pattern. This general concept is illustrated by simulated patterns and experimental data of a model organism of biological pattern formation, the slime mold Dictyostelium discoideum. We introduce patterns starting from Turing (stripe and spot) patterns, together with target waves and spiral waves. The biological relevance of these patterns is illustrated by snapshots from real and theoretical biological systems. The principles of spiral wave formation are first explored in a stylized cellular automaton model and then reproduced in a model of Dictyostelium signaling. The shaping of spatiotemporal patterns by biological variability (i.e., by a spatial distribution of cell-to-cell differences) is demonstrated, focusing on two Dictyostelium models. Building up on this foundation, we then discuss in more detail how the nonlinearities in biological models translate the distribution of cell properties into pattern events, leaving characteristic geometric signatures.

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.

Targeting the biophysical properties of the myeloma initiating cell niches: a pharmaceutical synergism analysis using multi-scale agent-based modeling

Multiple myeloma, the second most common hematological cancer, is currently incurable due to refractory disease relapse and development of multiple drug resistance. We and others recently established the biophysical model that myeloma initiating (stem) cells (MICs) trigger the stiffening of their niches via SDF-1/CXCR4 paracrine; The stiffened niches then promote the colonogenesis of MICs and protect them from drug treatment. In this work we examined in silico the pharmaceutical potential of targeting MIC niche stiffness to facilitate cytotoxic chemotherapies. We first established a multi-scale agent-based model using the Markov Chain Monte Carlo approach to recapitulate the niche stiffness centric, pro-oncogenetic positive feedback loop between MICs and myeloma-associated bone marrow stromal cells (MBMSCs), and investigated the effects of such intercellular chemo-physical communications on myeloma development. Then we used AMD3100 (to interrupt the interactions between MICs and their stroma) and Bortezomib (a recently developed novel therapeutic agent) as representative drugs to examine if the biophysical properties of myeloma niches are drugable. Results showed that our model recaptured the key experimental observation that the MBMSCs were more sensitive to SDF-1 secreted by MICs, and provided stiffer niches for these initiating cells and promoted their proliferation and drug resistance. Drug synergism analysis suggested that AMD3100 treatment undermined the capability of MICs to modulate the bone marrow microenvironment, and thus re-sensitized myeloma to Bortezomib treatments. This work is also the first attempt to virtually visualize in 3D the dynamics of the bone marrow stiffness during myeloma development. In summary, we established a multi-scale model to facilitate the translation of the niche-stiffness centric myeloma model as well as experimental observations to possible clinical applications. We concluded that targeting the biophysical properties of stem cell niches is of high clinical potential since it may re-sensitize tumor initiating cells to chemotherapies and reduce risks of cancer relapse.

Modeling and simulation of biological systems from image data

This essay provides an introduction to the terminology, concepts, methods, and challenges of image-based modeling in biology. Image-based modeling and simulation aims at using systematic, quantitative image data to build predictive models of biological systems that can be simulated with a computer. This allows one to disentangle molecular mechanisms from effects of shape and geometry. Questions like "what is the functional role of shape" or "how are biological shapes generated and regulated" can be addressed in the framework of image-based systems biology. The combination of image quantification, model building, and computer simulation is illustrated here using the example of diffusion in the endoplasmic reticulum.

Multilevel complexity of calcium signaling: Modeling angiogenesis

Intracellular calcium signaling is a universal, evolutionary conserved and versatile regulator of cell biochemistry. The complexity of calcium signaling and related cell machinery can be investigated by the use of experimental strategies, as well as by computational approaches. Vascular endothelium is a fascinating model to study the specific properties and roles of calcium signals at multiple biological levels. During the past 20 years, live cell imaging, patch clamp and other techniques have allowed us to detect and interfere with calcium signaling in endothelial cells (ECs), providing a huge amount of information on the regulation of vascularization (angiogenesis) in normal and tumoral tissues. These data range from the spatiotemporal dynamics of calcium within different cell microcompartments to those in entire multicellular and organized EC networks. Beside experimental strategies, in silico endothelial models, specifically designed for simulating calcium signaling, are contributing to our knowledge of vascular physiology and pathology. They help to investigate and predict the quantitative features of proangiogenic events moving through subcellular, cellular and supracellular levels. This review focuses on some recent developments of computational approaches for proangiogenic endothelial calcium signaling. In particular, we discuss the creation of hybrid simulation environments, which combine and integrate discrete Cellular Potts Models. They are able to capture the phenomenological mechanisms of cell morphological reorganization, migration, and intercellular adhesion, with single-cell spatiotemporal models, based on reaction-diffusion equations that describe the agonist-induced intracellular calcium events.

Rise and demise of bioinformatics? Promise and progress

The field of bioinformatics and computational biology has gone through a number of transformations during the past 15 years, establishing itself as a key component of new biology. This spectacular growth has been challenged by a number of disruptive changes in science and technology. Despite the apparent fatigue of the linguistic use of the term itself, bioinformatics has grown perhaps to a point beyond recognition. We explore both historical aspects and future trends and argue that as the field expands, key questions remain unanswered and acquire new meaning while at the same time the range of applications is widening to cover an ever increasing number of biological disciplines. These trends appear to be pointing to a redefinition of certain objectives, milestones, and possibly the field itself.

A framework for modeling the relationship between cellular steady-state and stimulus-responsiveness

In cell signaling systems, the abundances of signaling molecules are generally thought to determine the response to stimulation. However, the kinetics of molecular processes, for example receptor trafficking and protein turnover, may also play an important role. Few studies have systematically examined this relationship between the resting state and stimulus-responsiveness. Fewer still have investigated the relative contribution of steady-state concentrations and reaction kinetics. Here we describe a mathematical framework for modeling the resting state of signaling systems. Among other things, this framework allows steady-state concentration measurements to be used in parameterizing kinetic models, and enables comprehensive characterization of the relationship between the resting state and the cellular response to stimulation.

Spatial modeling of cell signaling networks

The shape of a cell, the sizes of subcellular compartments, and the spatial distribution of molecules within the cytoplasm can all control how molecules interact to produce a cellular behavior. This chapter describes how these spatial features can be included in mechanistic mathematical models of cell signaling. The Virtual Cell computational modeling and simulation software is used to illustrate the considerations required to build a spatial model. An explanation of how to appropriately choose between physical formulations that implicitly or explicitly account for cell geometry and between deterministic versus stochastic formulations for molecular dynamics is provided, along with a discussion of their respective strengths and weaknesses. As a first step toward constructing a spatial model, the geometry needs to be specified and associated with the molecules, reactions, and membrane flux processes of the network. Initial conditions, diffusion coefficients, velocities, and boundary conditions complete the specifications required to define the mathematics of the model. The numerical methods used to solve reaction-diffusion problems both deterministically and stochastically are then described and some guidance is provided in how to set up and run simulations. A study of cAMP signaling in neurons ends the chapter, providing an example of the insights that can be gained in interpreting experimental results through the application of spatial modeling.

Virtual Cell: computational tools for modeling in cell biology

The Virtual Cell (VCell) is a general computational framework for modeling physicochemical and electrophysiological processes in living cells. Developed by the National Resource for Cell Analysis and Modeling at the University of Connecticut Health Center, it provides automated tools for simulating a wide range of cellular phenomena in space and time, both deterministically and stochastically. These computational tools allow one to couple electrophysiology and reaction kinetics with transport mechanisms, such as diffusion and directed transport, and map them onto spatial domains of various shapes, including irregular three-dimensional geometries derived from experimental images. In this article, we review new robust computational tools recently deployed in VCell for treating spatially resolved models.

Assessment of cellular mechanisms contributing to cAMP compartmentalization in pulmonary microvascular endothelial cells

Cyclic AMP signals encode information required to differentially regulate a wide variety of cellular responses; yet it is not well understood how information is encrypted within these signals. An emerging concept is that compartmentalization underlies specificity within the cAMP signaling pathway. This concept is based on a series of observations indicating that cAMP levels are distinct in different regions of the cell. One such observation is that cAMP production at the plasma membrane increases pulmonary microvascular endothelial barrier integrity, whereas cAMP production in the cytosol disrupts barrier integrity. To better understand how cAMP signals might be compartmentalized, we have developed mathematical models in which cellular geometry as well as total adenylyl cyclase and phosphodiesterase activities were constrained to approximate values measured in pulmonary microvascular endothelial cells. These simulations suggest that the subcellular localizations of adenylyl cyclase and phosphodiesterase activities are by themselves insufficient to generate physiologically relevant cAMP gradients. Thus, the assembly of adenylyl cyclase, phosphodiesterase, and protein kinase A onto protein scaffolds is by itself unlikely to ensure signal specificity. Rather, our simulations suggest that reductions in the effective cAMP diffusion coefficient may facilitate the formation of substantial cAMP gradients. We conclude that reductions in the effective rate of cAMP diffusion due to buffers, structural impediments, and local changes in viscosity greatly facilitate the ability of signaling complexes to impart specificity within the cAMP signaling pathway.

Developing models in virtual cell

This Teaching Resource provides lecture notes, slides, and a student assignment for a two-part lecture on mathematical modeling using the Virtual Cell environment. The lectures discuss the steps involved in developing and running simulations using Virtual Cell, with particular focus on spatial partial differential equation models. We discuss how to construct both ordinary differential equation models, in which the cytoplasm is considered a well-mixed cellular compartment, and partial differential equation models, which calculate how chemical species change as a function of both time and location. The Virtual Cell environment is especially well suited for models that explore spatial specificity of cellular reactions. Partial differential equation models in Virtual Cell can give rise to simulations using predefined cellular geometries, which enable direct comparison with imaging data. These models address questions regarding the regulatory capability arising from spatial organization of the cell. Examples are provided of studies that have successfully exploited the Virtual Cell software to address the spatial contribution to signaling.

Models at the single cell level

Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell-cell communication are among the many phenomena that require fine-grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model.

Quantitative understanding of cell signaling: the importance of membrane organization

Systems biology modeling of signal transduction pathways traditionally employs ordinary differential equations, deterministic models based on the assumptions of spatial homogeneity. However, this can be a poor approximation for certain aspects of signal transduction, especially its initial steps: the cell membrane exhibits significant spatial organization, with diffusion rates approximately two orders of magnitude slower than those in the cytosol. Thus, to unravel the complexities of signaling pathways, quantitative models must consider spatial organization as an important feature of cell signaling. Furthermore, spatial separation limits the number of molecules that can physically interact, requiring stochastic simulation methods that account for individual molecules. Herein, we discuss the need for mathematical models and experiments that appreciate the importance of spatial organization in the membrane.

ENVIRONMENT: a computational platform to stochastically simulate reacting and self-reproducing lipid compartments

'ENVIRONMENT' is a computational platform that has been developed in the last few years with the aim to simulate stochastically the dynamics and stability of chemically reacting protocellular systems. Here we present and describe some of its main features, showing how the stochastic kinetics approach can be applied to study the time evolution of reaction networks in heterogeneous conditions, particularly when supramolecular lipid structures (micelles, vesicles, etc) coexist with aqueous domains. These conditions are of special relevance to understand the origins of cellular, self-reproducing compartments, in the context of prebiotic chemistry and evolution. We contrast our simulation results with real lab experiments, with the aim to bring together theoretical and experimental research on protocell and minimal artificial cell systems.

A mitotic kinesin-6, Pav-KLP, mediates interdependent cortical reorganization and spindle dynamics in Drosophila embryos

We investigated the role of Pav-KLP, a kinesin-6, in the coordination of spindle and cortical dynamics during mitosis in Drosophila embryos. In vitro, Pav-KLP behaves as a dimer. In vivo, it localizes to mitotic spindles and furrows. Inhibition of Pav-KLP causes defects in both spindle dynamics and furrow ingression, as well as causing changes in the distribution of actin and vesicles. Thus, Pav-KLP stabilizes the spindle by crosslinking interpolar microtubule bundles and contributes to actin furrow formation possibly by transporting membrane vesicles, actin and/or actin regulatory molecules along astral microtubules. Modeling suggests that furrow ingression during cellularization depends on: (1) a Pav-KLP-dependent force driving an initial slow stage of ingression; and (2) the subsequent Pav-KLP-driven transport of actin- and membrane-containing vesicles to the furrow during a fast stage of ingression. We hypothesize that Pav-KLP is a multifunctional mitotic motor that contributes both to bundling of interpolar microtubules, thus stabilizing the spindle, and to a biphasic mechanism of furrow ingression by pulling down the furrow and transporting vesicles that deliver new material to the descending furrow.

Use of virtual cell in studies of cellular dynamics

The Virtual Cell (VCell) is a unique computational environment for modeling and simulation of cell biology. It has been specifically designed to be a tool for a wide range of scientists, from experimental cell biologists to theoretical biophysicists. The models created with VCell can range from the simple, to evaluate hypotheses or to interpret experimental data, to complex multilayered models used to probe the predicted behavior of spatially resolved, highly nonlinear systems. In this chapter, we discuss modeling capabilities of VCell and demonstrate representative examples of the models published by the VCell users.

Persistent cAMP-signals triggered by internalized G-protein-coupled receptors

Real-time monitoring of G-protein-coupled receptor (GPCR) signaling in native cells suggests that the receptor for thyroid stimulating hormone remains active after internalization, challenging the current model for GPCR signaling.

G-protein–coupled receptors (GPCRs) are generally thought to signal to second messengers like cyclic AMP (cAMP) from the cell surface and to become internalized upon repeated or prolonged stimulation. Once internalized, they are supposed to stop signaling to second messengers but may trigger nonclassical signals such as mitogen-activated protein kinase (MAPK) activation. Here, we show that a GPCR continues to stimulate cAMP production in a sustained manner after internalization. We generated transgenic mice with ubiquitous expression of a fluorescent sensor for cAMP and studied cAMP responses to thyroid-stimulating hormone (TSH) in native, 3-D thyroid follicles isolated from these mice. TSH stimulation caused internalization of the TSH receptors into a pre-Golgi compartment in close association with G-protein α_s-subunits and adenylyl cyclase III. Receptors internalized together with TSH and produced downstream cellular responses that were distinct from those triggered by cell surface receptors. These data suggest that classical paradigms of GPCR signaling may need revision, as they indicate that cAMP signaling by GPCRs may occur both at the cell surface and from intracellular sites, but with different consequences for the cell.

Cells respond to many environmental cues through the activity of cell surface receptor proteins, which sense these cues and convey that information to signaling molecules inside the cell. G-protein–coupled receptors (GPCRs) form the largest eukaryotic family of plasma membrane receptors. They convert the information provided by extracellular stimuli into intracellular second messengers, like cyclic AMP (cAMP). After prolonged stimulation, they are internalized inside cells, an event that to date has been thought to terminate the production of second messengers. Though many of the key steps of GPCR signaling are known in detail, precisely how signaling and termination actually occur in time and space (i.e., in subcellular compartments or microdomains) is still largely unexplored. To observe GPCR signaling in living cells, we generated mice expressing a fluorescent sensor that allows monitoring the intracellular levels of cAMP with a microscope. We utilized this system to study, directly in native thyroid follicles, the signal sent by the receptor for thyroid-stimulating hormone (TSH). Our findings indicate that TSH receptors are internalized rapidly after activation but continue to stimulate cAMP production inside cells and that this sustained, cAMP production is apparently required for localized activation of downstream components. These data challenge the current model of the GPCR-cAMP pathway by suggesting the existence of previously unrecognized intracellular site(s) for cAMP generation and of differential signaling outcomes as a result of intracellular GPCR signaling. Such intracellular site(s) may provide specialized signaling platforms, thus contributing to the spatiotemporal regulation of cAMP production and to signaling specificity within the GPCR family.

Programming with models: modularity and abstraction provide powerful capabilities for systems biology

Mathematical models are increasingly used to understand how phenotypes emerge from systems of molecular interactions. However, their current construction as monolithic sets of equations presents a fundamental barrier to progress. Overcoming this requires modularity, enabling sub-systems to be specified independently and combined incrementally, and abstraction, enabling generic properties of biological processes to be specified independently of specific instances. These, in turn, require models to be represented as programs rather than as datatypes. Programmable modularity and abstraction enables libraries of modules to be created, which can be instantiated and reused repeatedly in different contexts with different components. We have developed a computational infrastructure that accomplishes this. We show here why such capabilities are needed, what is required to implement them and what can be accomplished with them that could not be done previously.

Illoura: a software tool for analysis, visualization and semantic querying of cellular and other spatial biological data

New high-resolution approaches for mapping ultrastructure of cells in 3D are leading to unprecedented quantities of spatial data. Here we present Illoura, a software tool for the integrated management, analysis and visualization of these data within a semantic context, and illustrate its capability by analysis of spatial relationships in mammalian beta cells.

AVAILABILITY

http://www.visiblecell.com/illoura.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

The secret marriage between calcium and tumor angiogenesis

Endothelial cell biochemistry and responsiveness to a wide variety of external stimula is regulated by intracellular calcium concentration. During the last twenty years, electrophysiology and functional imaging based on the use of fluorescent probes provided several informations about the dynamics and role of calcium at the single cell level: highly diverse extracellular agonists, such as proangiogenic growth factors and vasoactive compounds, trigger increases in intracellular calcium and specific informations are transduced for proliferation, differentiation, death, movement in physiological and pathological conditions. Obviously, the investigation at multicellular and tissutal levels is much more complex. In this review we discuss the potential specific roles of calcium signaling in tumor angiogenesis progression trying to address two key questions: (i) how can this ion play specific roles in the angiogenesis regulation; and (ii) could it be used as a target to interfere with or prevent tumor vascularization?

Towards a quantitative representation of the cell signaling mechanisms of hallucinogens: measurement and mathematical modeling of 5-HT1A and 5-HT2A receptor-mediated ERK1/2 activation

Through a multidisciplinary approach involving experimental and computational studies, we address quantitative aspects of signaling mechanisms triggered in the cell by the receptor targets of hallucinogenic drugs, the serotonin 5-HT2A receptors. To reveal the properties of the signaling pathways, and the way in which responses elicited through these receptors alone and in combination with other serotonin receptors' subtypes (the 5-HT1AR), we developed a detailed mathematical model of receptor-mediated ERK1/2 activation in cells expressing the 5-HT1A and 5-HT2A subtypes individually, and together. In parallel, we measured experimentally the activation of ERK1/2 by the action of selective agonists on these receptors expressed in HEK293 cells. We show here that the 5-HT1AR agonist Xaliproden HCl elicited transient activation of ERK1/2 by phosphorylation, whereas 5-HT2AR activation by TCB-2 led to higher, and more sustained responses. The 5-HT2AR response dominated the MAPK signaling pathway when co-expressed with 5-HT1AR, and diminution of the response by the 5-HT2AR antagonist Ketanserin could not be rescued by the 5-HT1AR agonist. Computational simulations produced qualitative results in good agreement with these experimental data, and parameter optimization made this agreement quantitative. In silico simulation experiments suggest that the deletion of the positive regulators PKC in the 5-HT2AR pathway, or PLA2 in the combined 5-HT1A/2AR model greatly decreased the basal level of active ERK1/2. Deletion of negative regulators of MKP and PP2A in 5-HT1AR and 5-HT2AR models was found to have even stronger effects. Under various parameter sets, simulation results implied that the extent of constitutive activity in a particular tissue and the specific drug efficacy properties may determine the distinct dynamics of the 5-HT receptor-mediated ERK1/2 activation pathways. Thus, the mathematical models are useful exploratory tools in the ongoing efforts to establish a mechanistic understanding and an experimentally testable representation of hallucinogen-specific signaling in the cellular machinery, and can be refined with quantitative, function-related information.

Cell shape and negative links in regulatory motifs together control spatial information flow in signaling networks

The role of cell size and shape in controlling local intracellular signaling reactions, and how this spatial information originates and is propagated, is not well understood. We have used partial differential equations to model the flow of spatial information from the beta-adrenergic receptor to MAPK1,2 through the cAMP/PKA/B-Raf/MAPK1,2 network in neurons using real geometries. The numerical simulations indicated that cell shape controls the dynamics of local biochemical activity of signal-modulated negative regulators, such as phosphodiesterases and protein phosphatases within regulatory loops to determine the size of microdomains of activated signaling components. The model prediction that negative regulators control the flow of spatial information to downstream components was verified experimentally in rat hippocampal slices. These results suggest a mechanism by which cellular geometry, the presence of regulatory loops with negative regulators, and key reaction rates all together control spatial information transfer and microdomain characteristics within cells.

Closed-loop control of cellular functions using combinatory drugs guided by a stochastic search algorithm

A mixture of drugs is often more effective than using a single effector. However, it is extremely challenging to identify potent drug combinations by trial and error because of the large number of possible combinations and the inherent complexity of the underlying biological network. With a closed-loop optimization modality, we experimentally demonstrate effective searching for potent drug combinations for controlling cellular functions through a large parametric space. Only tens of iterations out of one hundred thousand possible trials were needed to determine a potent combination of drugs for inhibiting vesicular stomatitis virus infection of NIH 3T3 fibroblasts. In addition, the drug combination reduced the required dosage by approximately 10-fold compared with individual drugs. In another example, a potent mixture was identified in thirty iterations out of a possible million combinations of six cytokines that regulate the activity of nuclear factor kappa B in 293T cells. The closed-loop optimization approach possesses the potential of being an effective approach for manipulating a wide class of biological systems.

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.

Stochastic binding of Ca2+ ions in the dyadic cleft; continuous versus random walk description of diffusion

Ca(2+) signaling in the dyadic cleft in ventricular myocytes is fundamentally discrete and stochastic. We study the stochastic binding of single Ca(2+) ions to receptors in the cleft using two different models of diffusion: a stochastic and discrete Random Walk (RW) model, and a deterministic continuous model. We investigate whether the latter model, together with a stochastic receptor model, can reproduce binding events registered in fully stochastic RW simulations. By evaluating the continuous model goodness-of-fit for a large range of parameters, we present evidence that it can. Further, we show that the large fluctuations in binding rate observed at the level of single time-steps are integrated and smoothed at the larger timescale of binding events, which explains the continuous model goodness-of-fit. With these results we demonstrate that the stochasticity and discreteness of the Ca(2+) signaling in the dyadic cleft, determined by single binding events, can be described using a deterministic model of Ca(2+) diffusion together with a stochastic model of the binding events, for a specific range of physiological relevant parameters. Time-consuming RW simulations can thus be avoided. We also present a new analytical model of bimolecular binding probabilities, which we use in the RW simulations and the statistical analysis.

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.

Toward realistic modeling of dynamic processes in cell signaling: quantification of macromolecular crowding effects

One of the major factors distinguishing molecular processes in vivo from biochemical experiments in vitro is the effect of the environment produced by macromolecular crowding in the cell. To achieve a realistic modeling of processes in the living cell based on biochemical data, it becomes necessary, therefore, to consider such effects. We describe a protocol based on Brownian dynamics simulation to characterize and quantify the effect of various forms of crowding on diffusion and bimolecular association in a simple model of interacting hard spheres. We show that by combining the elastic collision method for hard spheres and the mean field approach for hydrodynamic interaction (HI), our simulations capture the correct dynamics of a monodisperse system. The contributions from excluded volume effect and HI to the crowding effect are thus quantified. The dependence of the results on size distribution of each component in the system is illustrated, and the approach is applied as well to the crowding effect on electrostatic-driven association in both neutral and charged environments; values for effective diffusion constants and association rates are obtained for the specific conditions. The results from our simulation approach can be used to improve the modeling of cell signaling processes without additional computational burdens.

Synthetic protocell biology: from reproduction to computation

Cells are the building blocks of biological complexity. They are complex systems sustained by the coordinated cooperative dynamics of several biochemical networks. Their replication, adaptation and computational features emerge as a consequence of appropriate molecular feedbacks that somehow define what life is. As the last decades have brought the transition from the description-driven biology to the synthesis-driven biology, one great challenge shared by both the fields of bioengineering and the origin of life is to find the appropriate conditions under which living cellular structures can effectively emerge and persist. Here, we review current knowledge (both theoretical and experimental) on possible scenarios of artificial cell design and their future challenges.

Developing Itô stochastic differential equation models for neuronal signal transduction pathways

Mathematical modeling and simulation of dynamic biochemical systems are receiving considerable attention due to the increasing availability of experimental knowledge of complex intracellular functions. In addition to deterministic approaches, several stochastic approaches have been developed for simulating the time-series behavior of biochemical systems. The problem with stochastic approaches, however, is the larger computational time compared to deterministic approaches. It is therefore necessary to study alternative ways to incorporate stochasticity and to seek approaches that reduce the computational time needed for simulations, yet preserve the characteristic behavior of the system in question. In this work, we develop a computational framework based on the Itô stochastic differential equations for neuronal signal transduction networks. There are several different ways to incorporate stochasticity into deterministic differential equation models and to obtain Itô stochastic differential equations. Two of the developed models are found most suitable for stochastic modeling of neuronal signal transduction. The best models give stable responses which means that the variances of the responses with time are not increasing and negative concentrations are avoided. We also make a comparative analysis of different kinds of stochastic approaches, that is the Itô stochastic differential equations, the chemical Langevin equation, and the Gillespie stochastic simulation algorithm. Different kinds of stochastic approaches can be used to produce similar responses for the neuronal protein kinase C signal transduction pathway. The fine details of the responses vary slightly, depending on the approach and the parameter values. However, when simulating great numbers of chemical species, the Gillespie algorithm is computationally several orders of magnitude slower than the Itô stochastic differential equations and the chemical Langevin equation. Furthermore, the chemical Langevin equation produces negative concentrations. The Itô stochastic differential equations developed in this work are shown to overcome the problem of obtaining negative values.