Purpose

The modeling framework is a major component of the Virtual Cell distributed software system. This goal of the modeling framework is to provide the biological abstractions necessary to represent models of cellular physiology. This framework, in turn, uses the services of the Mathematics Framework for the simulation of it's models.

System Level Interface

Modeling Language

A declarative modeling language has been developed to concisely describe a class of physiological models that has been encountered in the Virtual Cell project. This language defines molecular species, cellular structures, biochemical reactions, and cellular geometry.

Testability

A substantial advantage of separating the math framework from the modeling framework is the improved ability to verify the correctness of the two frameworks from high level system interfaces. The Modeling Framework can be tested by specifying simple physiological models, requesting a specific simulation, and observing the resulting mathematical description generated.

Framework Design

The current implementation of the cell model description involves the manipulation of abstract modeling objects that reside in the Modeling Framework as Java objects. These modeling objects can be edited, viewed, stored in a remote database, and analyzed using the WWW-based user interface (see User Interface section). These objects are categorized as Models, Geometry, and Application objects. This adopts the naming convention used in the current Modeling Framework software.

Physiological Models

A Model object represents the physiological model of the cell system under study. Each Model is defined as a collection of Species (e.g. calcium, ATP), Reactions (e.g. enzyme kinetics, receptor binding, membrane fluxes), and Structures (e.g. ER, cytosol).

Structures

The Structure objects define mutually exclusive compartments within the cells, and the membranes that isolate them. Using this definition, the extracellular region is separated from the cytosol (and hence the ER, nucleus, etc.) by the plasma membrane. These structures contain Species and a collection of Reactions that describe the biochemical behavior of that compartment. Average geometric information, such as surface to volume ratios for the appropriate Feature complete the necessary information for a single point compartmental simulation.

Species

The Species are objects that identify molecular species, and are classified as being either Membrane Species or Volume Species. The Membrane Species are described by a surface density for each membrane structure. The Volume Species are described by a concentration and a diffusion constant.

Reactions

The Reactions are objects that represent complete descriptions of reaction kinetics. Reactions are collections of related Reaction Steps (e.g. membrane receptor binding, calcium buffering), and Membrane Fluxes (e.g. flux through an ion channel). The fluxes and reaction rates are represented by arbitrary algebraic expressions. These expression objects are capable of basic algebraic simplification, partial differentiation, and binding to the appropriate Parameters and Species, and numeric evaluation.

Cellular Geometry

The Geometry objects represent the cellular geometry (based on segmented images) and can be mapped directly to the corresponding cellular features. The geometry can currently be specified as 2-D or 3-D segmented images with the appropriate scaling information to properly define a simulation domain.

Application

The Application objects represent the context of a particular simulation as a specific mapping of the Model objects to the Geometry objects. This mapping specifies the Species and Reactions present in each Feature within the corresponding region in the Geometry. With the addition of initial conditions and boundary conditions for each Species, a particular simulation is completely specified. This context specifies the generation of the ordinary and partial differential equations of the system. The computational mesh (orthogonal grid) is sampled from the Geometry. These equations are represented symbolically within the Modeling Framework using expression objects and are translated into a corresponding mathematical description. The resulting mathematical description can be sent to the Mathematical Framework for simulation and analysis.

Structure Mapping

The design of the biological to mathematical mapping allows separate use of biology and math interfaces. Simulations may be based upon single point approximation or spatially resolved models. A compartmental model is assumed for a single point approximation. All cellular structures defining the geometry are mapped to a single compartment. Ordinary differential equations representing the reactions kinetics are generated and passed to an interpreted ODE solver within the client applet. Partial differential equations, that correspond to diffusive species, and ODE's for non-diffusive species are generated for the solution of a complete spatial simulation. Each mutually exclusive volumetric region in the geometry is mapped to a single compartment. A compartment that is not spatially resolved in the geometry may be considered continuously distributed within the geometric region of its parent compartment. The simulation is executed on a remote server and the results are displayed in an applet.

Model Analysis

It is important to determine the sensitivity of model behavior to the choice of which physiological mechanisms are incorporated and their parameter values. It is informative to determine the relative change in model behavior due to a relative change in parameter value. For compartmental models, the software computes the sensitivity of any species concentration to any parameter as a function of time evaluated at the nominal solution.