570 Review TRENDS in Cell Biology Vol.13 No.11 November 2003

Quantitative cell biology with the Virtual Cellq

Boris M. Slepchenko1, James C. Schaff1, Ian Macara2 and Leslie M. Loew1

1Center For Biomedical Imaging Technology and National Resource for Cell Analysis and Modeling, Department of Physiology, University of Connecticut Health Center, Farmington, CT 06030, USA 2Center for Cell Signaling and Department of Pharmacology, University of Virginia, Charlottesville, VA 22908, USA

Cell biological processes are controlled by an interact- techniques such as patch-clamp single-channel recording. ing set of biochemical and electrophysiological events Once such data are acquired, a crucial and obvious that are distributed within complex cellular structures. challenge is to determine how these, often disparate and Computational models, comprising quantitative data complex, details can explain the cellular process under on the interacting molecular participants in these events, investigation. The ideal way to meet this challenge is to provide a means for applying the scientific method to integrate and organize the data into a predictive model. these complex systems. The Virtual Cell is a compu- Indeed, the value of computational approaches in cell tational environment designed for cell biologists, to biology has been increasingly recognized as demonstrated facilitate the construction of models and the generation by several recent reviews [4–6] and a book [37]. However, of predictive simulations from them. This review sum- for cell biologists, formulating a mathematical model marizes how a Virtual Cell model is assembled and based on quantitative data and solving its equations to describes the physical principles underlying the calcu- generate simulations can present significant technical lations that are performed. Applications to problems in challenges. Overcoming this barrier, in order to complete nucleocytoplasmic transport and intracellular calcium the cycle of hypothesis–prediction–experiment, is required dynamics will illustrate the power of this paradigm for to understand the biophysical mechanisms underlying the elucidating cell biology. function of a cellular subsystem. To meet these needs for quantitative cell biology, we have developed a software- Fundamental cell biological processes, such as growth, modeling environment called the Virtual Cell. [7–9]. signaling, differentiation and death, are extraordinarily The Virtual Cell provides a formal framework for complex, depending upon an enormous number of different modeling biochemical, electrophysiological, and transport molecules and molecular interactions. High-throughput phenomena while considering the subcellular localization methods for analysis of DNA and RNA (genomics) and of the molecules that take part in them. This localization protein (proteomics) are now providing an overwhelming can take the form of a three-dimensional (3D) arbitrarily amount of qualitative data, cataloguing the molecules shaped cell, where the molecular species might be hetero- involved in these complex pathways. The field of ‘systems geneously distributed. The geometry of the cell, including biology’ [1–3] uses computational analyses to sift through the locations and shapes of subcellular organelles, can be the large volume of data, to identify unique sets of mol- imported directly from microscope images. Such a model ecules involved in particular cellular responses, and to explicitly considers the diffusion of the molecules within develop statistical relationships between molecules that the geometry, as well as their reactions. Simulations from suggest causal interactions. Such largely qualitative such a ‘spatial’ model require the solution of partial dif- data and analyses can offer significant insights into the ferential equations. Alternatively, the localization can be operation of large, complex biochemical systems, such as more crudely represented by considering the average those involved in metabolic and signaling networks. concentrations of molecules within well-mixed compart- To develop a full understanding of the mechanisms ments that correspond to the cellular and subcellular underlying a cell biological event, however, the quantitat- structures. This corresponds to the assumption that dif- ive physical–chemical details must be elucidated. As a fusion is fast compared with the timescales of all the first prerequisite, this requires measurements of the biochemical transformations, so that any diffusible mol- concentrations, locations, biochemical reaction rates and ecule will be uniformly distributed as soon as it is produced membrane transport kinetics for the involved molecules. and no matter how convoluted the geometry of its com- Such data can come from traditional biochemical methods, partment; such a nonspatial model will result in ordinary but are also available through direct in vivo measurements differential equations that can be numerically solved much on live cells, using modern microscope imaging with more efficiently. Thus, these two classes of models result in fluorescent probes and indicators and electrophysiological very different mathematical forms, although the original quantitative mechanistic hypothesis could be the same. q Supplementary data associated with this article can be found at doi: 10.1016/j.tcb. 2003.09.002 Simulation results on the dynamic distributions of the Corresponding author: Leslie M. Loew ([email protected]). molecules controlling a cellular function are displayed http://ticb.trends.com 0962-8924/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.tcb.2003.09.002 Review TRENDS in Cell Biology Vol.13 No.11 November 2003 571

format also cannot capture the quantitative relationships. Box 1. Access to the Virtual Cell and its database Model features are, therefore, most effectively expressed The Virtual Cell software is maintained at a central server within the in terms of diagrams linked to mathematical expressions. National Resource for Cell Analysis and Modeling (NRCAM) at the Figure 1a shows diagrams taken directly from the Virtual University of Connecticut Health Center, and is available to users Cell user interface for the compartmental distribution of worldwide through JAVA technology on a web browser. This web- the molecules, transport processes in the ER membrane, based client-server architecture has several advantages: it obviates and the assignment of compartments to the spatial geo- the task of software distribution and installation on heterogeneous hardware platforms – the Virtual Cell runs equally well on Windows, metry of a neuroblastoma cell. Furthermore, generating Linux and Macintosh machines; enhancements and bug fixes are quantitative predictions from a model requires solving available to the entire user community as soon as they are deployed; mathematical equations (i.e. simulations). Computational the computationally intensive calculations, which might be unac- tools such as the Virtual Cell facilitate both the process of ceptably slow on the computer of a user, are optimized for a model construction and the generation of simulations. dedicated cluster of fast computers at NRCAM; and, most impor- tantly, a central database of user models, maintained at NRCAM, facilitates the online sharing of model components, collaboration, Building a model in the Virtual Cell and interactive publication of models. Building a biological model in the Virtual Cell is an iterative process that should start with a simple model that is incrementally validated against simulations and experimental observations. In our calcium dynamics illu- within the software environment and can also be exported stration, for example, an initial assumption was that the as arrays of numerical values, sets of images or digital Ins(1,4,5)P3 receptor was distributed uniformly through- movies. Box 1 provides details of the software and how to out the cell. The simulations produced under this assump- access it through the Internet. tion gave calcium amplitudes that were much too high in the neurite and too low in the soma compared with Quantitative cell biology the experimental calcium dynamics. This prompted us The paradigm that we term quantitative cell biology is to investigate the Ins(1,4,5)P3 receptor distribution by described in Figure 1, where a study of calcium dynamics immunofluorescence, which revealed that the calcium in a neuronal cell [10,11] is used as a concrete illustration. density in the soma was twice as high as that in the In this study, the neuromodulator bradykinin applied to neurite; the simulation resulting from this new elabora- the cells produced a calcium wave that starts in the neurite tion of the model provided an excellent match to the and spreads to the soma and growth cones. The calcium experimental calcium wave. Furthermore, a model (or wave was monitored using digital microscope imaging of a hypothesis) should lead to predictions that can be further fluorescent calcium indicator. The hypothesis was that tested with new experiments. For example, our model interaction of bradykinin with its receptor on the plasma predicted that if bradykinin were applied to different membrane induced the production of inositol (1,4,5)- regions of the cell, rather than globally, application to the trisphosphate (Ins(1,4,5)P ) that diffused to its receptor proximal neurite would be sufficient to produce a calcium 3 wave; but activation of bradykinin receptors in only the on the endoplasmic reticulum (ER), leading to calcium growth cone or soma would produce an aborted calcium release. The model contained details of the relevant recep- wave. These predictions were subsequently borne out by tor distributions (through immunofluorescence analysis) experiments in which bradykinin was regionally applied within the cell geometry, the kinetics of Ins(1,4,5)P pro- 3 through pressure application, using a micropipette [10]. duction (through biochemical analysis of Ins(1,4,5)P in 3 A Virtual Cell biological model is composed of three cell populations and photorelease of caged Ins(1,4,5)P in 3 parts in an expanding tree structure: a single ‘physio- individual cells); the transport of calcium through the logical model’ that captures the mechanistic hypothesis; Ins(1,4,5)P3 receptor calcium channel and the SERCA one or more ‘applications’, where experimental conditions, pump (from literature studies of single-channel kinetics geometry and modeling approximations are introduced and radioligand flux); and calcium buffering by both to pose a concrete mathematical problem; and one or endogenous proteins and the fluorescent indicator (from more ‘simulations’, where each represents a numerical confocal measurements of indicator concentrations). The solution to the mathematical problem posed by an mathematical equations generated by this combination of application (Figure 2). molecular distributions and reaction and membrane A physiological model consists of molecular species and transport kinetics was then solved to produce a simulation mechanisms (reactions and fluxes) localized to cellular of the spatiotemporal pattern of calcium that could be structures. Cellular structures are created by specifying directly compared with the experiment. the topological arrangement of membranes and mem- The procedure described in Figure 1 is just a restate- brane-bound compartments. Biochemical reactions are ment of the scientific method, where ‘model’ is simply defined within volumetric compartments of the cell, as well defined as a complex quantitative ‘hypothesis’. Practically, as in membranes; molecular fluxes and electrical currents however, a model is typically composed of several hypo- are defined across membranes. The reaction and flux theses, where even the choice of which of the hypothetical rates are determined as an explicit function of the local components to include can in itself be a hypothesis. environment (e.g. concentrations, surface densities and Because of the complexity of models, it is difficult to membrane potential) and one or more kinetic parameters formulate them as a string of verbal hypotheses; a text (e.g. kf and kr, the forward and reverse mass action rate http://ticb.trends.com 572 Review TRENDS in Cell Biology Vol.13 No.11 November 2003

Quantitative cell biology

Experiment Hypothesis (Model)

Predictions

¥ What are the initial concentrations, diffusion coefficients and locations of all the implicated molecules? Concentrations of all molecular species as a function of ¥ What are the rate laws and rate constants time and space for all the biochemical transformations?

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Figure 1. Quantitative cell biology is dependent on computational modeling. Experiments provide image, biochemical and electrophysiological data on the initial concentrations, kinetic rates, and transport properties of the molecules and cellular structures, which are presumed to be the key components of a cellular event. If the system is perturbed by addition of a ligand, electrical stimulus or other experimental intervention, the model should be capable of predicting changes in the spatiotemporal distribution of all the molecules. This cycle is illustrated by reference to a study of calcium waves evoked by activation of the bradykinin receptor in the plasma membrane of a neuronal cell. [10,11] Inputs to the model in the Hypothesis panel include the identity of the molecules hypothetically involved in calcium release, and the cellular compartment in which they are located (upper left); the reactions and transport processes between theses molecules (illustrated with a scheme for transport and reactions within the endoplasmic reticulum (ER) membrane); the assignment of compartments and molecular locations to a specific cell geometry (illustrated with the geometry of a neuroblastoma cell in the center); and specification of initial concentrations, boundary conditions and diffusion coefficients. This information is sufficient to generate simulations of the calcium dynamics (illustrated in the Predictions panel) that can be directly compared with the experimentally observed calcium dynamics (illustrated in the Experiment panel). If the prediction and the experiment do not agree, the assumptions underlying model inputs will need to be reexamined. The model can also be used to perform ‘virtual’ experiments in advance of a real experiment and can simulate the behavior of molecules that are not easily visualized experimentally.

constants). Mass action and Michaelis–Menten rate laws steady-state (basal) behavior of the model. A steady state is are available automatically, but user-defined general achieved when concentrations and membrane potentials kinetic expressions are also possible. Membrane transport do not change over time. If the relative surface areas and kinetics can be specified with expressions for molecular volumes of the cellular structures are specified, based on flux or, for ions, the electric current. The transport kinetics the true 3D cellular geometry, the steady state compart- can be described in terms of standard electrophysiological mental solution will generally also apply to the full spatial formulas (e.g. Goldman–Hodgkin–Katz permeability or problem. It is worth noting that most physiological models Nernst conductance) or as user-defined molecular flux are not ‘closed’ systems, and that they will exchange mass or current. and energy with the environment. For example, in a After the physiological model has been defined, a non- typical physiological state a model of glucose metabolism spatial application should be constructed to investigate the would continuously consume glucose, oxygen and ADP and http://ticb.trends.com Review TRENDS in Cell Biology Vol.13 No.11 November 2003 573

Physiology

Simulation Applications

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Figure 2. A Virtual-Cell biological model consists of a single mechanistic physiological model that encodes the general structure and function of the cellular system, and multiple applications that allow specific geometries, initial concentrations, protocols and modeling assumptions required to apply the model to various experimental contexts. Each application, in turn, can spawn multiple simulations with different numerical methods and varying parameters.

continuously create ATP and carbonic acid. If this steady idealized shapes (e.g. spheres, hemispheres and cylin- state consumption and production were not considered, ders), or on a segmented experimental image that has been the model would consume all metabolites and not achieve a classified according to the cell morphology (e.g. an image physiological steady state. Therefore, the model bound- or image stack with one color for cytosol, another for aries are set with either a fixed concentration (equivalent extracellular, and yet another for the nucleus as in to an infinitely large bathing medium) or a fixed rate of flux Figure 2). Inhomogeneous initial conditions might be across the boundary, which, when set to zero, corresponds specified as an explicit function of spatial coordinates. to a closed barrier. Boundary conditions are then specified to determine how Next, the response of the system to external stimuli the diffusive transport will behave at the edge of the might be considered. Typical perturbations include: hor- simulation domain. This realistic spatial representation mone injected into the extracellular media, voltage- or allows for the direct comparison of simulation results current-clamp protocols, and photorelease of a caged with data obtained from tools such as fluorescent compound within the cytosol. It is essential, however, microscopy. This technology allows modeling to investi- that the steady-state behavior be characterized first, so gate the impact of cell morphology and localization on that the response to the stimuli can be distinguished cellular physiology. from a spontaneous change caused by nonequilibrium An application is sufficient to fully specify the math- initial conditions. ematics to be passed to a simulation. As fully described in After characterizing the steady state and transient the next section, several simulations, using different behaviors of the nonspatial system, one or more spatial numerical solver algorithms and algorithmic parameters applications can be created to investigate the impact of cell (e.g. time steps and spatial mesh resolutions), can be geometry. Spatial simulations are necessary to capture the generated from a single application. Model parameter behavior of the model when the distributions of molecular variations and sensitivity analysis can also be performed species are spatially inhomogeneous and, on the timescale at the level of the simulations. Results are displayed in a of the biochemical transformations, diffusion is slow. convenient window that allows for visualization of tem- A cellular geometry must be defined based on either poral and spatial variations of any concentration, flux, http://ticb.trends.com 574 Review TRENDS in Cell Biology Vol.13 No.11 November 2003 current or membrane potential, and for the export of the mesh size and, generally, as they decrease, the solution spreadsheet data, images or movies. of the algebraic system should converge to that of the original problem. The Virtual Cell provides tools for both spatial and Under the hood: the mathematics and physics for the nonspatial modeling. Solving a problem both in time and Virtual Cell space (known as spatial modeling) is a computationally Once the biological model is formulated, it has to be intensive process, requiring solution of partial differen- translated into a mathematical model. This is done tial equations. Additional complexity is associated with automatically in the Virtual Cell based on the fact that defining and meshing geometry [9]. However, when dif- mathematically, the range of the mechanisms accommo- fusion is fast compared with reactions, the approximation dated by the Virtual Cell – diffusion, reactions, molecular of a ‘well-mixed’ compartment is physically justified and and ionic transport across membranes – can be generally the problem reduces to finding the average concentration described by means of reaction–diffusion equations (Box 2, within each compartment for each species as a function see supplementary data online). The most recent exten- of time (nonspatial, or compartmental, modeling). Non- sion of the software allows a user to study the effects of spatial models generate ordinary differential equations, membrane potential on spatiotemporal changes of mol- which can be numerically solved much more efficiently. ecular species and vice versa, even for processes as The Virtual Cell also offers a variety of robust numerical fast as action potentials in cells where the space-clamp algorithms for solving equations (solvers) in the nonspatial conditions apply (for electrical modeling of neurons with case. Some of them assure precision within the prescribed complex geometries, there are other excellent software tolerance without asking a user to specify an integration packages [12,13]). time-step. As mentioned above, even if the ultimate goal is The spatiotemporal changes to the concentrations of spatial modeling, the non-spatial approximation is a good molecular species are governed by a mass conservation starting point for determining the initial steady state and law: the rate of concentration change of a molecular sensitivities of the model to parameter changes. species inside a volume element is caused by all the If the model contains drastically different time scales – reactions that affect this species and diffusion fluxes some reactions are expected to be much faster than other coming in and out of the element. The dynamics of the reactions or diffusion – integration with very short time- membrane potential are governed by conservation of steps will be required even if only the long timescale electrical charge: the total cross-membrane electrical behavior of the system is of interest. To alleviate this, one current is the sum of the capacitative current and the can use a procedure in the Virtual Cell that automatically ‘direct’ current associated with cross-membrane fluxes implements the pseudo-steady approximation with respect of ions and charged molecules. The mass and charge to any reactions labeled as ‘fast’ [14]. This algorithm will conservation laws, which form the physical basis in the update all the variables in two substeps: first, those asso- Virtual Cell, are mathematically expressed by a system of ciated with the slow processes using an appropriately long differential equations (Box 2, equations (2) and (3), see time-step, and then solving the minimal system of non- supplementary data online). Equation (2) also requires linear algebraic equations for instantaneous equilibration additional conditions to be specified at the membranes of fast reactions. This can improve the computational and at the boundaries of the computational domain times required for a simulation by many orders of mag- (boundary conditions). nitude and make it practical to include explicitly the effects Any realistic biological model usually results in a of rapid buffers and fluorescent indicators on the behavior system of nonlinear differential equations with multiple of molecules to which they bind. variables. Because no general algorithms exist that would lead to an explicit analytical solution for such a system, particularly if it were mapped on irregular geometry taken A case study for quantitative cell biology with the Virtual from an experimental image, the Virtual Cell solves the Cell: nucleocytoplasmic transport mathematical model numerically. This means that the The Virtual Cell has been successfully applied to several solution of the Virtual Cell to the mathematical model is research projects in our laboratory and, since its appear- approximate and obtained only for a finite set of time and ance online ,three years ago, in several other labora- spatial points. The results are then presented in the form tories. Several of these have involved the modeling of of tables of numbers that appear as graphs or images. calcium dynamics [10,11,15–18], where the availability of Thus, numerical modeling requires sampling of the simu- fluorescent indicators for calcium enables direct compari- lated time interval and the spatial computational domain. son of model predictions with experimental studies of The differential equations are then approximated by finite spatiotemporal patterns. Analysis of photorelease experi- differences with respect to selected grid and time points; ments, the translocation of fluorescently labeled proteins essentially, the original equations are replaced with an and markers, and restricted diffusion within highly con- algebraic system of equations from which the values of fined volumes, such as the crista of mitochondria [19–24], variables at those points can be found. Although the are other areas of application of the Virtual Cell software. algebraic system is usually solved with high precision, it is Nucleocytoplasmic transport, and in particular the important to realize that the solution is an approximation transport of the small GTPase Ran, serves as a particu- to that of the original system of differential equations. The larly good example of the application of the Virtual Cell to magnitude of the error depends on the time step and quantitative cell biology [25]. Nuclear transport is in many http://ticb.trends.com Review TRENDS in Cell Biology Vol.13 No.11 November 2003 575 ways an ideal system to which computational cell biology independent reaction had to be added for the tagged methods can be applied: it involves only two cellular Ran protein. An example of a 3D spatial simulation is compartments; the components are soluble, mixing within shown in Figure 3, in which three time-points are shown each compartment is rapid compared with transport following a simulated injection of fluorescent Ran into the between them, and communication between the two cytoplasm of a cell. compartments occurs largely by facilitated diffusion The Virtual Cell model was able to simulate qualita- through pores in the nuclear envelope. Moreover, detailed tively and quantitatively Ran transport over a range of in vitro kinetics have been studied for many of the conditions. It presented the first estimates for the steady- components that mediate transport. Lastly, because the state flux of Ran across the nuclear envelope, which in turn nuclear and cytoplasmic compartments are large, it is provided a lower estimate for the total nucleocytoplasmic relatively easy to obtain quantitative data on transport flux (,20 million macromolecules per cell per minute). rates, so predictions from computational models can be It also predicted that a very high (,500-fold) gradient of rigorously tested. Ran–GTP exists between the nucleus and cytoplasm – a Ran (in its GDP-bound state) is carried into the prediction that has since been verified experimentally nucleus in a complex with the nuclear transport factor 2 using a fluorescence resonance energy transfer (FRET) (NTF2). Within the nucleus, a Ran guanine nucleotide biosensor [26]. Lastly, the model also predicted that exchange factor (RanGEF) catalyzes GTP/GDP exchange, the concentration of uncomplexed Ran–GTP within the producing Ran–GTP that is released from NTF2. NTF2 nucleus is very low (,10 nM) – lower than the measured then cycles back through the nuclear pores to the cyto- equilibrium binding-constants for Ran–GTP binding to plasm. Ran–GTP binds to members of a family of proteins certain karyopherins [27]. This concentration would be called karyopherins. These proteins facilitate the import too low to trigger the release of bound cargo from these or export of specific classes of cargo (either proteins or karyopherins, and it indicates that cofactors, in addition to RNAs) that need to be moved from one compartment to the Ran–GTP might be required to facilitate cargo release. other. Importins can only bind their cargo in the absence of Several important questions, about nuclear transport Ran–GTP (i.e. in the cytoplasm). Once the importin–cargo and Ran that are amenable to computational approaches, complex arrives in the nucleus, Ran–GTP displaces the remain unanswered. Open issues include the role of adap- cargo to produce an importin–Ran–GTP complex that tor proteins for binding cargo to karyopherins, the details cycles back out to the cytoplasm. There a GTPase- of Ran transport through nuclear pores, and the function activating protein (RanGAP) hydrolyzes the GTP to pro- of Ran in the organization of chromosomes during mitosis. duce Ran–GDP, which is released from importin to com- Gratifyingly, several groups are now using computational plete the transport cycle. We included each of these steps approaches to address these questions [28–30]. in the Virtual Cell model for Ran transport, using kinetic parameters that had been obtained by several other groups for individual steps in the cycle. Because the rela- Concluding remarks tive abundance of the various karyopherins in the cell is Although the Virtual Cell has very broad applicability to not known, a ‘generic’ karyopherin, and an estimated cell biological problems, there are areas that it does not yet concentration of 15 mM in the cell were used. Additionally, tackle. For example, if the number of molecules involved a simplified version of the kinetics of the RanGEF was in a cellular event is small (i.e. ,100), a probabilistic used. This basic model produced the expected gradient treatment involving stochastic reactions and Brownian of Ran across the nuclear envelope. To test the model motion might be required to properly simulate the biology; experimentally, however, we needed to simulate the micro- the Virtual Cell is currently only able to treat deterministic injection of fluorescent Ran into the cytoplasm and the processes. Also, large biochemical networks with partially subsequent movement of this protein into the nucleus. known reaction rates or initial concentrations would be This requirement added a second layer of complexity to cumbersome to model using the Virtual Cell. There are the model, because for each reaction involving Ran an other software systems that have complementary strengths

Figure 3. the Virtual Cell simulation of Ran transport. Left to right shows three frames from a simulation of Ran, microinjected into the cytoplasm, over a period of ,5 min. Concentrations of the injected Ran are represented in pseudocolors from red (high concentration) to blue (low concentration). http://ticb.trends.com 576 Review TRENDS in Cell Biology Vol.13 No.11 November 2003 in areas such as parameter optimization for metabolic and 11 Fink, C.C. et al. (2000) An image-based model of calcium waves in signaling pathways [31], graphic representation of com- differentiated neuroblastoma cells. Biophys. J. 79, 163–183 plex networks, [32] stochastic spatial simulations in small 12 Bower, J.M. and Beeman, D. (1998) The Book of GENESIS: Exploring Realistic Neural Models with the General Neural Simulation System, volumes, [33] or whole-cell metabolic modeling. [34] Indeed, Springer Verlag software for computational cell biology is rapidly expand- 13 Hines, M.L. and Carnevale, N.T. 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