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Details
Environment
The system is superimposed on a 3-dimensional cubic lattice which is used for two purposes: Defining the local environments and storing the local concentrations of each small molecule species. Each cube in the lattice will be configured to represent one environment type: cytosol or cell membrane for example. Each environment in turn determines the physical propertiessuch as diffusion coefficients of the various substances through that location, and effects how it will be displayed.
Complex concentration gradients in Cell++ are modeled through the calculation of each small molecule species' concentrations in every lattice unit throughout the simulation.
In this example, a membrane divides an intracellular compartment from the rest of the cytosol. The local environment affects (A) the speed of diffusion, (B) the behaviour of interacting components and (C) creates barriers.
Large Molecules
Movement of discrete components, such as enzymes and signaling molecules is modeled off lattice, by a random walk approximation of Brownian motion. At the beginning of each simulation time-step, a displacement vector with a random direction is chosen for each particle. The magnitude of this displacement vector is determined based on the molecule's relative mobility.
Particles encounter barriers in the form of prohibited environment types andare reflected off them in such a way as to preserve their energy. The particle is moved to the location where its displacement vector intersects with the inaccessible environment and the distance it has travelled is recorded. A new vector is chosen with a magnitude equal to the difference between the original vector's length and the actual distance travelled. In this way, the total path length for that particle is kept the same as was originally chosen for the first displacement vector.
Small Molecules
Small molecule flux is determined by both the distance to neighbouring lattice sites and the type of local environment. Concentration changes each iteration are determined by the diffusion coefficient of each chemical in each environment. Impermeable boundaries are modelled as lattice units with a diffusion coefficient of zero to prohibit metabolites from diffusing into them.
Chemical Reactions
Large particle enzymes can catalyze the metabolites to chemically react. Kinetic parameters associated with each enzyme species and the concentrations of the metabolites are used to determine the rate of such reactions and generate appropriate differential equations.
Differential Equations
Solutions to the differential equations are approximated in Cell++ with the Euler Forward Method. This technique is relatively simple to implement and has the mathematical benefits of being 2nd order accurate in time, 2nd order accurate in space and numerically stable over the values used in Cell++. It is an explicit method, meaning that the derivatives obtained at the beginning of the time step are used throughout that interval.
Enzyme-catalyzed metabolite chemical reactions are computed using this technique. The metabolite concentrations and enzyme copy numbers are considered at the beginning of the reaction time step and are used to calculate how quickly metabolites will be chemically converted. This rate of change is multiplied by the length of the time step to determine the concentration changes for each iteration.
Small molecule diffusion is also implemented this way. The concentration gradients for each pair of neighbouring lattice cells are determined at the start of the diffusion time step. Along with the diffusion coefficients, those gradients determine the flow rates. Multiplying by the timescale determines how much of each metabolite moves between each lattice unit.
