Analysis of nonlinear dynamics on arbitrary geometries with the Virtual Cell

Multi-condensate state as a functional strategy to optimize the cell signaling output

The existence of multiple biomolecular condensates inside living cells is a peculiar phenomenon not compatible with the predictions of equilibrium statistical mechanics. In this work, we address the problem of multiple condensates state (MCS) from a functional perspective. We combine Langevin dynamics, reaction-diffusion simulation, and dynamical systems theory to demonstrate that MCS can indeed be a function optimization strategy. Using Arp2/3 mediated actin nucleation pathway as an example, we show that actin polymerization is maximum at an optimal number of condensates. For a fixed amount of Arp2/3, MCS produces a greater response compared to its single condensate counterpart. Our analysis reveals the functional significance of the condensate size distribution which can be mapped to the recent experimental findings. Given the spatial heterogeneity within condensates and non-linear nature of intracellular networks, we envision MCS to be a generic functional solution, so that structures of network motifs may have evolved to accommodate such configurations.

Multi-condensate state as a functional strategy to optimize the cell signaling output

The existence of multiple biomolecular condensates inside living cells is a peculiar phenomenon not compatible with the predictions of equilibrium statistical mechanics. In this work, we address the problem of multiple condensates state (MCS) from a functional perspective. We combined Langevin dynamics, reaction-diffusion simulation, and dynamical systems theory to demonstrate that MCS can indeed be a function optimization strategy. Using Arp2/3 mediated actin nucleation pathway as an example, we show that actin polymerization is maximum at an optimal number of condensates. For a fixed amount of Arp2/3, MCS produces a greater response compared to its single condensate counterpart. Our analysis reveals the functional significance of the condensate size distribution which can be mapped to the recent experimental findings. Given the spatial heterogeneity within condensates and non-linear nature of intracellular networks, we envision MCS to be a generic functional solution, so that structures of network motifs may have evolved to accommodate such configurations.

Heineken, Tsuchiya and Aris on the mathematical status of the pseudo-steady state hypothesis: A classic from volume 1 of Mathematical Biosciences

Volume 1, Issue 1 of Mathematical Biosciences was the venue for a now-classic paper on the application of singular perturbation theory in enzyme kinetics, "On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics" by F. G. Heineken, H. M. Tsuchiya and R. Aris. More than 50 years have passed, and yet this paper continues to be studied and mined for insights. This perspective discusses both the strengths and weaknesses of the work presented in this paper. For many, the justification of the pseudo-steady-state approximation using singular perturbation theory is the main achievement of this paper. However, there is so much more material here, which laid the foundation for a great deal of research in mathematical biochemistry in the intervening decades. The parameterization of the equations, construction of the first-order uniform singular-perturbation solution, and an attempt to apply similar principles to the pseudo-equilibrium approximation are discussed in particular detail.

Compartmental and Spatial Rule-Based Modeling with Virtual Cell

In rule-based modeling, molecular interactions are systematically specified in the form of reaction rules that serve as generators of reactions. This provides a way to account for all the potential molecular complexes and interactions among multivalent or multistate molecules. Recently, we introduced rule-based modeling into the Virtual Cell (VCell) modeling framework, permitting graphical specification of rules and merger of networks generated automatically (using the BioNetGen modeling engine) with hand-specified reaction networks. VCell provides a number of ordinary differential equation and stochastic numerical solvers for single-compartment simulations of the kinetic systems derived from these networks, and agent-based network-free simulation of the rules. In this work, compartmental and spatial modeling of rule-based models has been implemented within VCell. To enable rule-based deterministic and stochastic spatial simulations and network-free agent-based compartmental simulations, the BioNetGen and NFSim engines were each modified to support compartments. In the new rule-based formalism, every reactant and product pattern and every reaction rule are assigned locations. We also introduce the rule-based concept of molecular anchors. This assures that any species that has a molecule anchored to a predefined compartment will remain in this compartment. Importantly, in addition to formulation of compartmental models, this now permits VCell users to seamlessly connect reaction networks derived from rules to explicit geometries to automatically generate a system of reaction-diffusion equations. These may then be simulated using either the VCell partial differential equations deterministic solvers or the Smoldyn stochastic simulator.

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.

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.

Modeling and Simulation of Intracellular Drug Transport and Disposition Pathways with Virtual Cell

The development of computational approaches for modeling the spatiotemporal dynamics of intracellular, small molecule drug concentrations has become an increasingly important area of pharmaceutical research. For systems pharmacology, the system dynamics of subcellular transport can be coupled to downstream pharmacological effects on biochemical pathways that impact cell structure and function. Here, we demonstrate how a widely used systems biology modeling package - Virtual Cell - can also be used to model the intracellular, passive transport pathways of small druglike molecules. Using differential equations to represent passive drug transport across cellular membranes, spatiotemporal changes in the intracellular distribution and concentrations of exogenous chemical agents in specific subcellular organelles were simulated for weakly acidic, neutral, and basic molecules, as a function of the molecules' lipophilicity and ionization potentials. In addition, we simulated the transport properties of small molecule chemical agents in the presence of a homogenous extracellular concentration or a transcellular concentration gradient. We also simulated the effects of cell type-dependent variations in the intracellular microenvironments on the distribution and accumulation of small molecule chemical agents in different organelles over time, under influx and efflux conditions. Lastly, we simulated the transcellular transport of small molecule chemical agents, in the presence of different apical and basolateral microenvironments. By incorporating existing models of drug permeation and subcellular distribution, our results indicate that Virtual Cell can provide a user-friendly, open, online computational modeling platform for systems pharmacology and biopharmaceutics research, making mathematical models and simulation results accessible to a broad community of users, without requiring advanced computer programming knowledge.

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.

Modeling capping protein FRAP and CALI experiments reveals in vivo regulation of actin dynamics

To gain insights on cellular mechanisms regulating actin polymerization, we used the Virtual Cell to model fluorescence recovery after photobleaching (FRAP) and chromophore-assisted laser inactivation (CALI) experiments on EGFP-capping protein (EGFP-CP). Modeling the FRAP kinetics demonstrated that the in vivo rate for the dissociation of CP from actin filaments is much faster (approximately 0.1 s(-1)) than that measured in vitro (0.01-0.0004 s(-1)). The CALI simulation revealed that in order to induce sustainable changes in cell morphology after CP inactivation, the cells should exhibit anticapping ability. We included the VASP protein as the anticapping agent in the modeling scheme. The model predicts that VASP affinity for barbed ends has a cooperative dependence on the concentration of VASP-barbed end complexes. This dependence produces a positive feedback that stabilizes the complexes and allows sustained growth at clustered filament tips. We analyzed the range of laser intensities that are sufficient to induce changes in cell morphology. This analysis demonstrates that FRAP experiments with EGFP-CP can be performed safely without changes in cell morphology, because, the intensity of the photobleaching beam is not high enough to produce the critical concentration of free barbed ends that will induce filament growth before diffusional replacement of EGFP-CP occurs.

A Cut Cell Method for Simulating Spatial Models of Biochemical Reaction Networks in Arbitrary Geometries

Cells use signaling networks consisting of multiple interacting proteins to respond to changes in their environment. In many situations, such as chemotaxis, spatial and temporal information must be transmitted through the network. Recent computational studies have emphasized the importance of cellular geometry in signal transduction, but have been limited in their ability to accurately represent complex cell morphologies. We present a finite volume method that addresses this problem. Our method uses Cartesian cut cells and is second order in space and time. We use our method to simulate several models of signaling systems in realistic cell morphologies obtained from live cell images and examine the effects of geometry on signal transduction.

SIMULATING BIOCHEMICAL SIGNALING NETWORKS IN COMPLEX MOVING GEOMETRIES

Signaling networks regulate cellular responses to environmental stimuli through cascades of protein interactions. External signals can trigger cells to polarize and move in a specific direction. During migration, spatially localized activity of proteins is maintained. To investigate the effects of morphological changes on intracellular signaling, we developed a numerical scheme consisting of a cut cell finite volume spatial discretization coupled with level set methods to simulate the resulting advection-reaction-diffusion system. We then apply the method to several biochemical reaction networks in changing geometries. We found that a Turing instability can develop exclusively by cell deformations that maintain constant area. For a Turing system with a geometry-dependent single or double peak solution, simulations in a dynamically changing geometry suggest that a single peak solution is the only stable one, independent of the oscillation frequency. The method is also applied to a model of a signaling network in a migrating fibroblast.

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.

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

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

Diffusion in cytoplasm: effects of excluded volume due to internal membranes and cytoskeletal structures

The intricate geometry of cytoskeletal networks and internal membranes causes the space available for diffusion in cytoplasm to be convoluted, thereby affecting macromolecule diffusivity. We present a first systematic computational study of this effect by approximating intracellular structures as mixtures of random overlapping obstacles of various shapes. Effective diffusion coefficients are computed using a fast homogenization technique. It is found that a simple two-parameter power law provides a remarkably accurate description of effective diffusion over the entire range of volume fractions and for any given composition of structures. This universality allows for fast computation of diffusion coefficients, once the obstacle shapes and volume fractions are specified. We demonstrate that the excluded volume effect alone can account for a four-to-sixfold reduction in diffusive transport in cells, relative to diffusion in vitro. The study lays the foundation for an accurate coarse-grain formulation that would account for cytoplasm heterogeneity on a micron scale and binding of tracers to intracellular structures.

An open model of actin dendritic nucleation

The availability of quantitative experimental data on the kinetics of actin assembly has enabled the construction of many mathematical models focused on explaining specific behaviors of this complex system. However these ad hoc models are generally not reusable or accessible by the large community of actin biologists. In this work, we present a comprehensive model that integrates and unifies much of the in vitro data on the components of the dendritic nucleation mechanism for actin dynamics. More than 300 simulations have been run based on compartmental and three-dimensional spatial versions of this model. Several key findings are highlighted, including an explanation for the sharp boundary between actin assembly and disassembly in the lamellipodia of migrating cells. Because this model, with the simulation results, is "open source", in the sense that it is publicly available and editable through the Virtual Cell database (http://vcell.org), it can be accessed, analyzed, modified, and extended.

Virtual Cell modelling and simulation software environment

The Virtual Cell (VCell; http://vcell.org/) is a problem solving environment, built on a central database, for analysis, modelling and simulation of cell biological processes. VCell integrates a growing range of molecular mechanisms, including reaction kinetics, diffusion, flow, membrane transport, lateral membrane diffusion and electrophysiology, and can associate these with geometries derived from experimental microscope images. It has been developed and deployed as a web-based, distributed, client-server system, with more than a thousand world-wide users. VCell provides a separation of layers (core technologies and abstractions) representing biological models, physical mechanisms, geometry, mathematical models and numerical methods. This separation clarifies the impact of modelling decisions, assumptions and approximations. The result is a physically consistent, mathematically rigorous, spatial modelling and simulation framework. Users create biological models and VCell will automatically (i) generate the appropriate mathematical encoding for running a simulation and (ii) generate and compile the appropriate computer code. Both deterministic and stochastic algorithms are supported for describing and running non-spatial simulations; a full partial differential equation solver using the finite volume numerical algorithm is available for reaction-diffusion-advection simulations in complex cell geometries including 3D geometries derived from microscope images. Using the VCell database, models and model components can be reused and updated, as well as privately shared among collaborating groups, or published. Exchange of models with other tools is possible via import/export of SBML, CellML and MatLab formats. Furthermore, curation of models is facilitated by external database binding mechanisms for unique identification of components and by standardised annotations compliant with the MIRIAM standard. VCell is now open source, with its native model encoding language (VCML) being a public specification, which stands as the basis for a new generation of more customised, experiment-centric modelling tools using a new plug-in based platform.

Modeling molecular regulatory networks with JigCell and PET

We demonstrate how to model macromolecular regulatory networks with JigCell and the Parameter Estimation Toolkit (PET). These software tools are designed specifically to support the process typically used by systems biologists to model complex regulatory circuits. A detailed example illustrates how a model of the cell cycle in frog eggs is created and then refined through comparison of simulation output with experimental data. We show how parameter estimation tools automatically generate rate constants that fit a model to experimental data.

Depolymerization-driven flow in nematode spermatozoa relates crawling speed to size and shape

Cell crawling is an inherently physical process that includes protrusion of the leading edge, adhesion to the substrate, and advance of the trailing cell body. Research into advance of the cell body has focused on actomyosin contraction, with cytoskeletal disassembly regarded as incidental, rather than causative; however, extracts from nematode spermatozoa, which use Major Sperm Protein rather than actin, provide at least one example where cytoskeletal disassembly apparently generates force in the absence of molecular motors. To test whether depolymerization can explain force production during nematode sperm crawling, we constructed a mathematical model that simultaneously describes the dynamics of both the cytoskeleton and the cytosol. We also performed corresponding experiments using motile Caenorhabditis elegans spermatozoa. Our experiments reveal that crawling speed is an increasing function of both cell size and anterior-posterior elongation. The quantitative, depolymerization-driven model robustly predicts that cell speed should increase with cell size and yields a cytoskeletal disassembly rate that is consistent with previous measurements. Notably, the model requires anisotropic elasticity, with the cell being stiffer along the direction of motion, to accurately reproduce the dependence of speed on elongation. Our simulations also predict that speed should increase with cytoskeletal anisotropy and disassembly rate.

Synthetic Turing protocells: vesicle self-reproduction through symmetry-breaking instabilities

The reproduction of a living cell requires a repeatable set of chemical events to be properly coordinated. Such events define a replication cycle, coupling the growth and shape change of the cell membrane with internal metabolic reactions. Although the logic of such process is determined by potentially simple physico-chemical laws, modelling of a full, self-maintained cell cycle is not trivial. Here we present a novel approach to the problem that makes use of so-called symmetry breaking instabilities as the engine of cell growth and division. It is shown that the process occurs as a consequence of the breaking of spatial symmetry and provides a reliable mechanism of vesicle growth and reproduction. Our model opens the possibility of a synthetic protocell lacking information but displaying self-reproduction under a very simple set of chemical reactions.

Diffusion on a Curved Surface Coupled to Diffusion in the Volume: Application to Cell Biology

An algorithm is presented for solving a diffusion equation on a curved surface coupled to diffusion in the volume, a problem often arising in cell biology. It applies to pixilated surfaces obtained from experimental images and performs at low computational cost. In the method, the Laplace-Beltrami operator is approximated locally by the Laplacian on the tangential plane and then a finite volume discretization scheme based on a Voronoi decomposition is applied. Convergence studies show that mass conservation built in the discretization scheme and cancellation of sampling error ensure convergence of the solution in space with an order between 1 and 2. The method is applied to a cell-biological problem where a signaling molecule, G-protein Rac, cycles between the cytoplasm and cell membrane thus coupling its diffusion in the membrane to that in the cell interior. Simulations on realistic cell geometry are performed to validate, and determine the accuracy of, a recently proposed simplified quantitative analysis of fluorescence loss in photobleaching. The method is implemented within the Virtual Cell computational framework freely accessible at www.vcell.org.

Evolving a lingua franca and associated software infrastructure for computational systems biology: the Systems Biology Markup Language (SBML) project

Biologists are increasingly recognising that computational modelling is crucial for making sense of the vast quantities of complex experimental data that are now being collected. The systems biology field needs agreed-upon information standards if models are to be shared, evaluated and developed cooperatively. Over the last four years, our team has been developing the Systems Biology Markup Language (SBML) in collaboration with an international community of modellers and software developers. SBML has become a de facto standard format for representing formal, quantitative and qualitative models at the level of biochemical reactions and regulatory networks. In this article, we summarise the current and upcoming versions of SBML and our efforts at developing software infrastructure for supporting and broadening its use. We also provide a brief overview of the many SBML-compatible software tools available today.

Protocell self-reproduction in a spatially extended metabolism-vesicle system

Cellular life requires the presence of a set of biochemical mechanisms in order to maintain a predictable process of growth and division. Several attempts have been made towards the building of minimal protocells from a top-down approach, i.e. by using available biomolecules. This type of synthetic approach has so far been only partially successful, and appropriate models of the synthetic protocell cycle might be needed to guide future experiments. In this paper, we present a simple biochemically and physically feasible model of cell replication involving a discrete semi-permeable vesicle with an internal minimal metabolism involving two reactive centers. It is shown that such a system can effectively undergo a whole cell replication cycle. The model can be used as a basic framework to model whole protocell dynamics including more complex sets of reactions. The possible implementation of our design in future synthetic protocells is outlined.

Systems analysis of PKA-mediated phosphorylation gradients in live cardiac myocytes

Compartmentation and dynamics of cAMP and PKA signaling are important determinants of specificity among cAMP's myriad cellular roles. Both cardiac inotropy and the progression of heart disease are affected by spatiotemporal variations in cAMP/PKA signaling, yet the dynamic patterns of PKA-mediated phosphorylation that influence differential responses to agonists have not been characterized. We performed live-cell imaging and systems modeling of PKA-mediated phosphorylation in neonatal cardiac myocytes in response to G-protein coupled receptor stimuli and UV photolysis of "caged" cAMP. cAMP accumulation was rate-limiting in PKA-mediated phosphorylation downstream of the beta-adrenergic receptor. Prostaglandin E1 stimulated higher PKA activity in the cytosol than at the sarcolemma, whereas isoproterenol triggered faster sarcolemmal responses than cytosolic, likely due to restricted cAMP diffusion from submembrane compartments. Localized UV photolysis of caged cAMP triggered gradients of PKA-mediated phosphorylation, enhanced by phosphodiesterase activity and PKA-mediated buffering of cAMP. These findings indicate that combining live-cell FRET imaging and mechanistic computational models can provide quantitative understanding of spatiotemporal signaling.

A framework for whole-cell mathematical modeling

The default framework for modeling biochemical processes is that of a constant-volume reactor operating under steady-state conditions. This is satisfactory for many applications, but not for modeling growth and division of cells. In this study, a whole-cell modeling framework is developed that assumes expanding volumes and a cell-division cycle. A spherical newborn cell is designed to grow in volume during the growth phase of the cycle. After 80% of the cycle period, the cell begins to divide by constricting about its equator, ultimately affording two spherical cells with total volume equal to twice that of the original. The cell is partitioned into two regions or volumes, namely the cytoplasm (Vcyt) and membrane (Vmem), with molecular components present in each. Both volumes change during the cell cycle; Vcyt changes in response to osmotic pressure changes as nutrients enter the cell from the environment, while Vmem changes in response to this osmotic pressure effect such that membrane thickness remains invariant. The two volumes change at different rates; in most cases, this imposes periodic or oscillatory behavior on all components within the cell. Since the framework itself rather than a particular set of reactions and components is responsible for this behavior, it should be possible to model various biochemical processes within it, affording stable periodic solutions without requiring that the biochemical process itself generates oscillations as an inherent feature. Given that these processes naturally occur in growing and dividing cells, it is reasonable to conclude that the dynamics of component concentrations will be more realistic than when modeled within constant-volume and/or steady-state frameworks. This approach is illustrated using a symbolic whole cell model.

Simulating cellular dynamics through a coupled transcription, translation, metabolic model

In order to predict cell behavior in response to changes in its surroundings or to modifications of its genetic code, the dynamics of a cell are modeled using equations of metabolism, transport, transcription and translation implemented in the Karyote software. Our methodology accounts for the organelles of eukaryotes and the specialized zones in prokaryotes by dividing the volume of the cell into discrete compartments. Each compartment exchanges mass with others either through membrane transport or with a time delay effect associated with molecular migration. Metabolic and macromolecular reactions take place in user-specified compartments. Coupling among processes are accounted for and multiple scale techniques allow for the computation of processes that occur on a wide range of time scales. Our model is implemented to simulate the evolution of concentrations for a user-specifiable set of molecules and reactions that participate in cellular activity. The underlying equations integrate metabolic, transcription and translation reaction networks and provide a framework for simulating whole cells given a user-specified set of reactions. A rate equation formulation is used to simulate transcription from an input DNA sequence while the resulting mRNA is used via ribosome-mediated polymerization kinetics to accomplish translation. Feedback associated with the creation of species necessary for metabolism by the mRNA and protein synthesis modifies the rates of production of factors (e.g. nucleotides and amino acids) that affect the dynamics of transcription and translation. The concentrations of predicted proteins are compared with time series or steady state experiments. The expression and sequence of the predicted proteins are compared with experimental data via the construction of synthetic tryptic digests and associated mass spectra. We present the mathematical model showing the coupling of transcription, translation and metabolism in Karyote and illustrate some of its unique characteristics.

Toward Automated Cell Model Development through Information Theory†

The objective of this paper is to present a methodology for developing and calibrating models of complex reaction/transport systems. In particular, the complex network of biochemical reaction/transport processes and their spatial organization make the development of a predictive model of a living cell a grand challenge for the 21st century. However, advances in reaction/transport modeling and the exponentially growing databases of genomic, proteomic, metabolic, and bioelectric data make cell modeling feasible, if these two elements can be automatically integrated in an unbiased fashion. In this paper, we present a procedure to integrate data with a new cell model, Karyote, that accounts for many of the physical processes needed to attain the goal of predictive modeling. Our integration methodology is based on the use of information theory. The model is integrated with a variety of types and qualities of experimental data using an objective error assessment approach. Data that can be used in this approach include NMR, spectroscopy, microscopy, and electric potentiometry. The approach is demonstrated on the well-studied Trypanosoma brucei system. A major obstacle for the development of a predictive cell model is that the complexity of these systems makes it unlikely that any model presently available will soon be complete in terms of the set of processes accounted for. Thus, one is faced with the challenge of calibrating and running an incomplete model. We present a probability functional method that allows the integration of experimental data and soft information such as choice of error measure, a priori information, and physically motivated regularization to address the incompleteness challenge.

Kinetic analysis of receptor-activated phosphoinositide turnover

We studied the bradykinin-induced changes in phosphoinositide composition of N1E-115 neuroblastoma cells using a combination of biochemistry, microscope imaging, and mathematical modeling. Phosphatidylinositol-4,5-bisphosphate (PIP2) decreased over the first 30 s, and then recovered over the following 2-3 min. However, the rate and amount of inositol-1,4,5-trisphosphate (InsP3) production were much greater than the rate or amount of PIP2 decline. A mathematical model of phosphoinositide turnover based on this data predicted that PIP2 synthesis is also stimulated by bradykinin, causing an early transient increase in its concentration. This was subsequently confirmed experimentally. Then, we used single-cell microscopy to further examine phosphoinositide turnover by following the translocation of the pleckstrin homology domain of PLCdelta1 fused to green fluorescent protein (PH-GFP). The observed time course could be simulated by incorporating binding of PIP2 and InsP3 to PH-GFP into the model that had been used to analyze the biochemistry. Furthermore, this analysis could help to resolve a controversy over whether the translocation of PH-GFP from membrane to cytosol is due to a decrease in PIP2 on the membrane or an increase in InsP3 in cytosol; by computationally clamping the concentrations of each of these compounds, the model shows how both contribute to the dynamics of probe translocation.

Computational cell biology: spatiotemporal simulation of cellular events

The field of computational cell biology has emerged within the past 5 years because of the need to apply disciplined computational approaches to build and test complex hypotheses on the interacting structural, physical, and chemical features that underlie intracellular processes. To meet this need, newly developed software tools allow cell biologists and biophysicists to build models and generate simulations from them. The construction of general-purpose computational approaches is especially challenging if the spatial complexity of cellular systems is to be explicitly treated. This review surveys some of the existing efforts in this field with special emphasis on a system being developed in the authors' laboratory, Virtual Cell. The theories behind both stochastic and deterministic simulations are discussed. Examples of respective applications to cell biological problems in RNA trafficking and neuronal calcium dynamics are provided to illustrate these ideas.

The Virtual Cell: a software environment for computational cell biology

The newly emerging field of computational cell biology requires software tools that address the needs of a broad community of scientists. Cell biological processes are controlled by an interacting set of biochemical and electrophysiological events that are distributed within complex cellular structures. Computational modeling is familiar to researchers in fields such as molecular structure, neurobiology and metabolic pathway engineering, and is rapidly emerging in the area of gene expression. Although some of these established modeling approaches can be adapted to address problems of interest to cell biologists, relatively few software development efforts have been directed at the field as a whole. The Virtual Cell is a computational environment designed for cell biologists as well as for mathematical biologists and bioengineers. It serves to aid the construction of cell biological models and the generation of simulations from them. The system enables the formulation of both compartmental and spatial models, the latter with either idealized or experimentally derived geometries of one, two or three dimensions.