Proceedings of the 2006 Winter Simulation Conference L. F. Perrone, F. P. Wieland, J. Liu, B. G. Lawson, D. M. Nicol, and R. M. Fujimoto, eds.
THINK SIMULATION – THINK EXPERIMENT: THE VIRTUAL CELL PARADIGM
Ion I. Moraru James C. Schaff Leslie M. Loew
Dept. of Cell Biology University of Connecticut Health Center Farmington, CT 06030-3505, U.S.A.
ABSTRACT gene knockouts, drug targeting). Perhaps more useful, how- ever, is when the model is unable to predict the observed bi- The Virtual Cell modeling and simulation framework is the ology, implying that the elements of the model are either in- product of interdisciplinary research in biology that applies correct or incomplete. Analysis of such faulty models can the diverse strengths and experiences of individuals from directly motivate the discovery, via new experiments, of engineering, the physical sciences, the biological sciences, previously unknown critical biochemical or structural fea- and mathematics. A key feature is the separation of layers tures required for the cellular process under investigation. (core technologies and abstractions) representing biologi- However, the difficulties associated with the formula- cal models, physical mechanisms, geometry, mathematical tion of mathematical models and the generation of simula- models and numerical methods. This reduces software tions from them has impeded the adoption of this disci- complexity, allowing independent development and verifi- plined and quantitative approach to research in traditional cation, but most importantly it clarifies the impact of mod- cell biology, as well as slowed down the process of realiz- eling decisions, assumptions, and approximations. The re- ing the ultimate potential of the new, large scale, “omic” sult is a physically consistent, mathematically rigorous, datasets, keeping the rapidly growing area of systems biol- spatial modeling and simulation framework for cell biol- ogy at a level of mostly qualitative, top-down approaches ogy. The Virtual Cell has a rich, interactive user interface (Moraru and Loew 2005). Because biologists rarely have which connects to remote services providing scalable ac- sufficient training in the mathematics and physics required cess to a modeling database and a large dedicated cluster to build quantitative models, modeling has been largely the for shared computation and storage. In addition to new purview of theoreticians who have the appropriate training modeling capabilities, future developments will emphasize but little experience in the laboratory. With the notable ex- data and tool interoperability, extensibility, and experimen- ception of areas with rich traditions of biophysical research tally oriented model analysis tools. (such as cardiac and neuronal electrophysiology), this dis- connection to the laboratory has limited the impact of 1 INTRODUCTION mathematical modeling in cell biology and, in some quar- ters, has even given modeling a poor reputation. The accelerating progress in cataloguing the critical molecu- lar and structural elements responsible for cell function has 2 ABSTRACTING BIOLOGY FROM led to the hope that cell biological processes can be analyzed SIMULATION and understood in terms of the interactions of these compo- nents. Besides the obvious need for the acquisition and or- The Virtual Cell project (
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chenko et al. 2003). The current layered, modular design of the Virtual Cell was motivated by several factors: manag- ing complexity, facilitating model reuse, providing bio- logically and mathematically oriented tools, and building a distributed architecture. One of the innovative features in the design of the Vir- tual Cell is the introduction of three distinct layers of ab- stractions in the modeling process, which are following a paradigm similar to hypothesis-driven experimental stud- ies. Figure 1 schematizes the way in which the modeling process is structured within the Virtual Cell. This hierar- chical layered structure starts with the system Physiology – essentially the mechanistic hypothesis to be investigated. It includes the description of the molecular species and hy- pothesized interactions localized to cellular structures. Cellular structures need only be specified as the topologi- cal arrangement of membranes and membrane bounded Figure 1: Decomposing the Modeling Process in the Vir- compartments. Interactions can be biochemical reactions tual Cell BioModel Workspace defined either within volumetric compartments of the cell or in membranes; molecular fluxes across membranes; and Finally, as shown in Figure 1, several simulations can electrical currents. Additionally, the Physiology captures be spawned off of a given Application. That is, the mathe- the biochemical and biophysical properties of these inter- matics can be solved with multiple choices of numerical actions as rate laws for reactions and fluxes rate (that can solver, time step, mesh size for spatial simulations, time be determined as an explicit function of the local environ- duration and overrides of the default initial conditions or ment, e.g., concentrations, surface densities, membrane po- parameter values to permit parameter scans. A local sensi- tential, and of one or more kinetic parameters, e.g., kon and tivity analysis service is also available at the simulation koff). Mass action and Michaelis-Menten rate laws are level to aid in parameter estimation and to determine which available automatically, and arbitrary user-defined general features of the model are most critical in determining its kinetic expressions can also be used. Membrane transport overall behavior. A parameter estimation tool can fit non- kinetics can be specified with expressions for molecular spatial models to time series experimental data. A particu- flux or, for ions, the electric current. The transport kinetics lar simulation specification is, in turn, sufficient to describe can be described in terms of standard electrophysiological the software requirements for numerically calculating solu- formulas (e.g., Goldman-Hodgkin-Katz permeability or tions and the Virtual Cell thus automatically generates the Nernst conductance) or as user defined molecular flux or appropriate C++ code to produce simulation results. current. Such a decomposition of modeling concepts makes The Physiology can then have several Applications – possible the effective reuse of the model in multiple con- essentially the “virtual experiment” environment, the set of texts. To allow a single quantitative mechanistic hypothe- conditions under which one or more simulations will be sis to be validated under multiple experimental conditions performed. A particular Application would specify a spe- at different scales and using different physical approxima- cific geometry (e.g., size and shape of a cell), the boundary tions, the abstract physiological modeling concept is util- conditions, default initial concentrations and parameter ized. For example, a receptor binding reaction localized to values, and whether any of the reactions are sufficiently a membrane can have many different mathematical forms fast to permit a pseudo-steady state approximation. Also at while the quantitative mechanistic description is un- the Application level, individual reactions can be disabled changed. Its form depends on whether the membrane is as an aid in determining the proper initial conditions for a mapped to a spatially resolved surface, or mapped volu- prestimulus stable state. An Application of a Physiology is metrically as a distributed surface for an unresolved organ- sufficient to completely describe the governing mathemat- elle, or as non-spatial representation in a compartmental ics of the model and the mathematical representation of the model. The mathematical form also depends on additional system is automatically generated at this point by the soft- modeling assumptions, such as whether the reaction is ware (stored in a proprietary, but quite intuitive, mathe- “knocked out” or considered to be “fast” (pseudo-steady matical description language, VCMDL, that can be viewed state approximation). by the user). The Virtual Cell is designed to maintain a We will now use a study of calcium dynamics in a separation between this mathematical description, gener- neuronal cell (Fink et al. 1999, Fink et al. 2000) as a con- ated either via a BioModel or a MathModel, and the details crete illustration of this paradigm. That study started from of how the simulations are implemented. the experimental observation that the neuromodulator bra-
1714 Moraru, Schaff, and Loew dykinin applied to the cells produced a calcium wave that the neurite and too low in the soma compared to the ex- starts in the neurite and spreads to the soma and growth perimental calcium dynamics. This prompted us to investi- cones (the calcium wave was recorded by digital micro- gate the InsP3 receptor distribution by immunofluorescence scope imaging of a fluorescent calcium indicator). The hy- revealing that the density in the soma was twice as high as pothesis was that interaction of bradykinin with its receptor the neurite; the simulation resulting from this new elabora- on the plasma membrane activated production of inositol- tion of the model provided an excellent match to the ex- 1,4,5-trisphosphate (InsP3) that diffused to its receptor on perimental calcium wave. Furthermore, a model (or hy- the endoplasmic reticulum (ER) leading to calcium release. pothesis) should lead to predictions that can be further A model was constructed in the Virtual Cell software, tested with new experiments. For example, our model pre- where the Physiology component contained the biochemi- dicted that if bradykinin were applied to different regions cal interactions decribing the kinetics of InsP3 production, of the cell, rather than globally, application to the proximal the transport of calcium through the InsP3 receptor calcium neurite would be sufficient to produce a calcium wave; but channel, the SERCA pump activity, and calcium buffering activation of bradykinin receptors in only the growth cone (all based on direct experimental studies, new or from ex- or soma was predicted to produce an aborted calcium isting literature). At the Application level, the specifics of wave. These predictions were subsequently borne out by the calcium wave experiments were then implemented: the experiments in which bradykinin was regionally applied actual geometry to be used for simulations was created via pressure application through a micropipet (Fink et al. from real microscope images of the cultured cells, details 1999). of the relevant receptor distributions were added (obtained from quantitative immunofluorescence experiments), and 3 REACTION/DIFFUSION, FLOW AND initial conditions were specified (some directly from ex- ELECTRODIFFUSION IN ARBITRARY perimental measurements, other, unknown, from separate GEOMETRIES compartmental simulations that were run until a steady- state equilibrium was reached), as well as parameters rele- The design approach to the Virtual Cell client interface de- vant to the experimental stimulus. Several Applications scribed in the previous section also inspired and facilitated were spawned off to test individual experimental condi- the creation of an overall modular software structure, that tions, and several simulations in each Application were de- allows independent development and validation of individ- fined and run to test for varying time courses and parame- ual services, which are loosely coupled in an understand- ter values. The mathematical equations generated by every able overall architecture. For example, the solvers only particular combination of hypothesis, experimental condi- know mathematics, not physics or biology. This allows a tions, and parameter values, were then solved to produce a progressive introduction of new capabilities, starting with simulation of the spatiotemporal pattern of calcium that development and validation of numerical tools that are de- could be directly compared to the experiment. The power signed to handle certain new classes of mathematical prob- of this approach is evident once one realizes that all the lems (e.g., partial and ordinary differential equations of different Applications and Simulations were reusing the certain form as described below). When the supported same mechanistic hypothesis of the intracellular interaction class of mathematics is expanded, this new capability is in- network controlling calcium dynamics described in the stantly available to a mathematically oriented user. The Physiology. On the other hand, when some of the elements richer mathematical form then provides new flexibility for of the mechanistic hypothesis were being questioned, and mapping physiological mechanisms and geometry to a corresponding changes were being made to the Physiology, concrete mathematical form required for simulation. they were seamlessly propagated to all Applications and At its most fundamental level, a cell biological process their Simulations, which could simply be rerun with a push can be described as the consequence of a complex series of of a button under the new hypothesis, all the while preserv- chemical transformations. To understand the process, the ing the details of the experimental conditions to be simu- relevant molecules have to be identified and their time- lated. varying concentrations and spatial distributions have to be This example is also useful to illustrate the conceptual determined. A model, at this molecular level, chooses all aspects of applying the scientific method in a quantitative the presumed chemical species, assigns them initial con- cell biology modeling study. Building a biological model centrations and spatial distributions and connects them in the Virtual Cell is an iterative process that should start with appropriate kinetic expressions. A simulation that with a simple model that is incrementally validated against predicts the spatiotemporal behavior of this system has to simulations and experimental observations. In our calcium solve a class of problems known as reaction/diffusion dynamics study, for example, an initial assumption was equations. The mathematical problem is summarized by that the InsP3 receptor was distributed uniformly through- the equations: out the cell. The simulations produced under this assump- = − ∇ − μ ∇Φ − μ = Di F tion gave calcium amplitudes that were much too high in Fi Di Ci zi i Ci CiVi i RT
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d[]i fluctuations can become important if the number of mole- R = = k [][]k j − k− []i i dt 1 1 cules involved in a process is relatively small. For fully stochastic problems in which the number of particles in a ∂ reaction/diffusion system is too small to solve with nu- Ci = − + divFi Ri merical solutions of PDEs, diffusion can be described as ∂t Brownian random walks of individual particles and chemi- cal kinetics is simulated as stochastic reaction events. We The first line is a modified form of the familiar Nernst- also need to consider hybrid systems of stochastic differen- Planck equation that describes the flux, Fi , of a molecule i, tial equations where one can combine the numerical tech- ∇ driven by its concentration gradient, Ci, and, if it has an niques commonly applied to regular differential equations ∇Φ and Monte Carlo methods employing random number gen- ionic charge zi , the electric field in the system . The diffusion coefficient, D , and the mobility μ , are the pro- erators. We have recently prototyped an efficient algorithm i i in which the probabilities of each reaction are calculated portionality constants for these driving forces. The Virtual from rate constants and numbers of substrate molecules Cell has recently added the capability of accounting for (Gillespie 1977, 2001). A stochastic method is used to de- flow, as indicated by the term CV , where V is the velocity i i i termine which reaction will occur based on their relative of molecular species i. The second line portrays a typical probabilities. The time step is then adjusted to match the reaction that produces molecule i (while consuming j and particular reaction that occurs. After the reaction is com- k). The mass action ordinary differential equation for the plete the numbers of substrate molecules are readjusted rate of change of i, Ri, depends on the concentrations of prior to the next cycle. We have also prototyped a module the reactants and products. In general, Ri can depend on to simulate Brownian dynamics of particles in arbitrary ge- the concentrations of any of the molecules in the system ometries where the particles can react with continuously and may have a more complex form than the mass action distributed species. Although these approaches have been expression shown here. The third line combines the fluxes implemented in our C++ library and have been applied to and reactions into a system of partial differential equations problems on the dynamics of RNA granule trafficking that must be integrated to simulate the behavior of the mo- (Carson et al. 2001), generalized stochastic modeling ca- lecular species. pabilities are not yet fully developed. A particular chal- The fact that the Virtual Cell is designed to handle any lenge is the efficient computation of models that consider reaction system in any geometry, precludes the formulation rare events involving few molecules in the presence of of a general analytical solution for the problem. There are very frequent events among a large population of mole- two generic approaches to numerical solutions – stochastic cules. and continuous. The continuous approach provides a de- terministic description in terms of average species concen- 4 VIRTUAL CELL SOFTWARE COMPONENTS tration. This approach is effective and accurate so long as the number of molecules in a system is large, such that The Virtual Cell is a continuously evolving, large-scale thermal stochastic fluctuations around average values can software project where complexity has been managed by be ignored. We have found that the finite volume method decomposing the software into a collection of loosely cou- (Patankar 1980) for discretization of a system of PDEs is pled services interacting with conceptually simple data ab- especially well suited for our problem domain – that is re- stractions, which is critical for the robustness and flexibil- action/diffusion equations in arbitrary geometries (Choi et ity of the distributed architecture of the platform (remote al. 1999, Schaff et al. 1997, Schaff et al. 2001). Of course, clients/database server/compute nodes/simulation data the software can also solve non-spatial problems corre- storage, etc.) illustrated in Figure 2. sponding to systems of ODEs describing reactions within The server-side components were originally deployed well stirred compartments and fluxes across the mem- as RMI servers (Java Remote Message Interface) with the branes that separate the compartments. The software pro- appropriate RMI protocol wrappers. New protocol wrap- vides a choice of several solvers for such compartmental pers were developed to adapt these services to an architec- problems including a stiff solver. For both spatial and ture based on a message-oriented middleware solution compartmental problems, we have implemented an auto- (components became asynchronous, stateless, and transac- mated pseudo-steady approximation that can be invoked by tional). The Database Service (which internally consists of the user when a subsystem of reactions equilibrates rapidly multiple database layers) provides object-relational map- on the timescale of the overall process of interest (Slep- ping, object caching, transactions, connection manage- chenko et al. 2000). ment, and object access control. The component interface The current version of Virtual Cell handles only con- consists of simple object persistence operations such as tinuous solutions, and alternate stochastic solutions are cur- load, save, update, delete, annotate, and query. The Simu- rently in the development and testing phase. Stochastic lation Data Service provides simulation results retrieval,
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caching, data reduction, exporting and access control. The 5 EXCHANGES WITH EXTERNAL RESOURCES Simulation Dispatch Service provides batch scheduling, enforces quotas, monitors running simulations with check- The Virtual Cell provides mechanisms to link to external pointing and restarting and load balancing. The Simulation resources to facilitate the sharing of models across various Worker Service normally runs on the compute nodes of our computational cell biology and systems biology tools. This cluster and is responsible for code generation, compilation, is achieved by supporting the 2 prevalent XML languages running the simulations, reporting progress, and storing that cover these fields: SBML (
JMS data to users. To define a molecular species unambigu- Message ously, we have provided the ability to bind to a dictionary Broker Simulation Dispatcher of species derived from KEGG (their Compound database for small molecules, metabolites, lipids, etc. and their En- Database Service SimData Service Simulation Services SimulationSimulation Services Services zyme database for enzymes) and SwissProt (for proteins). This also allows the user to search his/her own models File PDE Solver ODE Solver Oracle server (‘My Models’) or other public VC models (‘All Models’) Library Library for molecules that have already been bound. Virtual Cell Figure 2: Existing Virtual Cell Software Components also imports data from external databases to automate the process of defining and binding new species and generat- All of these services are capable of being deployed ing fragments of reaction schemes. During this import (without the protocol wrappers) in a standalone application process, in addition to inserting this enzymatic reaction, all context. A Math Generation service was originally de- the newly created species are automatically bound to the ployed as a server-side service but is now deployed locally respective external database definitions. on the client. This service generates a declarative Mathe- matical Description from an Application. This transforma- 6 CONCLUSION tion is invoked before saving or running a simulation, and serves to decouple the biological representation from the We have described above the design principles and overall mathematical representation. There were also client-side structure of the Virtual Cell, a unique software platform for components developed to simplify the user interface design modeling and simulation in the field of cell biology, which by decoupling the user interface components (editors, etc.) is based on a paradigm similar to the classical scientific from remote services and any data management issues. method applied in experimental, “wet lab” studies. This approach, combined with features specifically targeting experimentalists (intuitive biological user interface, auto- matic math generation), with the fact that it is freely acces- sible via the web, does not require specialized local soft-
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ware or hardware resources, and allows collaborative Journal of Physiology – Endocrinology and Metabo- work, has led to an increasing popularity among cell biolo- lism 285: E138-154. gists. There are currently hundreds of active users and nu- Gillespie, D. T. 1977. Exact stochastic simulation of cou- merous papers have been recently published describing pled chemical reactions. Journal of Physical Chemis- combined experimental and theoretical studies using the try 81:2340-2361. Virtual Cell, addressing a quite wide variety of cell bio- Gillespie, D. T. 2001. Approximate accelerated stochastic logical processes, such as intracellular calcium dynamics simulation of chemically reacting systems. Journal of (Fink et al. 1999, 2000, Xu et al. 2003, Fridlyand et al. Chemical Physics 115:1716 -1733. 2003, Hernjak et al. 2005), nuclear transport (Smith et al. Schneider, I. C., and J. M. Haugh. 2005. Quantitative elu- 2002, Kalab et al. 2006), membrane electrophysiology cidation of a distinct spatial gradient-sensing mecha- (Suh et al. 2004, Horowitz et al. 2005), cell cycle regula- nism in fibroblasts. Journal of Cell Biology 171:883- tion (Slepchenko and Terasaki 2003), gradient sensing 892. mechanisms (Ma et al. 2004, Schneider and Haugh 2005), Hernjak, N., B. M. Slepchenko, K. Fernald, C. C. Fink, D. etc. Fortin, I. I. Moraru, J. Watras, and L. M. Loew. 2005. Last but not least, since being made publicly available Modeling and analysis of calcium signaling events six years ago, the Virtual Cell framework has been con- leading to long-term depression in cerebellar Purkinje tinuously enhanced, and many new capabilities are cur- cells. Biophysical Journal 89:3790-3806. rently under development, such as new solvers (stochastic Horowitz, L. F., W. Hirdes, B. C. Suh, D. W. Hilgemann, and hybrid), new model analysis features (parameter scan- K. Mackie, and B. Hille. 2005. Phospholipase C in liv- ning and optimization), new Application tools for explicit ing cells: activation, inhibition, Ca2+ requirement, and representation of experimental data, protocols and virtual regulation of M current. Journal of General Physiol- instruments, new abstractions for encapsulation of model ogy 126:243-262. components and hierarchical assembly of large models, Kalab, P., A. Pralle, E. Y. Isacoff, R. Heald, and K. Weis.. and more. An up-to-date list of features as well as links to 2006. Analysis of a RanGTP-regulated gradient in mi- public models and published papers is available at the Vir- totic somatic cells. Nature 440: 697-701. tual Cell homepage (
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signaling by Gq. Journal of General Physiology 123:663-683. Xu, C., J. Watras, and L. M. Loew. 2003. Kinetic analysis of receptor-activated phosphoinositide turnover. Jour- nal of Cell Biology 161:779-791.
AUTHOR BIOGRAPHIES
ION I. MORARU is Assistant Professor of Cell Biology at the University of Connecticut Health Center. He is head- ing the Computational Facility of the Richard D. Berlin Center for Cell Analysis and Modeling. His e-mail address is
JAMES C. SCHAFF is Assistant Professor of Cell Biol- ogy at the University of Connecticut Health Center. He is the lead software developer for the Virtual Cell project. His e-mail address is
LESLIE M. LOEW is Professor of Cell Biology and Computer Science at the University of Connecticut Health Center. He is the Director of the Richard D. Berlin Center for Cell Analysis and Modeling, an NIH National Tech- nology Center in Networks and Pathways. His e-mail ad- dress is
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