Virtual Cell How to model reaction diffusion systems

When a systems model is not enough Live cell intracellular calcium using fluorescent biosensors Systems models ignore convection and diffusion as well as non-uniform concentration in a cell Systems models also ignore stochastic events in a reaction pathway Colocalization of intracellular ankyrin-B and β 2 - spectrin in neonatal cardiomyocytes

Different Chemical Kinetic Theories Fokker-Planck equation (Kolmogorov forward) – Describes the time evolution of the probability density function of the velocity of a particle, and can be generalized to other observables as well Kolmogorov backward equation – PDE’s that arise from the assumption of continuous-time and continuous-state Markov processes

where: [C] intracellular species concentration D C diffusion constant for C J i source – source i for C J j sink –sink j for C k m +, k m - - kinetic rates Reaction Diffusion Models

Finite Difference Method for 1-D Heat Equation k- increment in time i- increment in space

Forward Euler Scheme * Virtual Cell also can convert determinist models to stochastic models using the Gibson approximation

Defining the Biomodel Each reaction is defined as a function of its flux

Defining a Geometry: 2-D 2-D Image of a cell from a Microscope 2-D Simple Shapes Defining several different domains

Defining a Geometry Define a geometry using mathematical equations Import a geometry using microscopy images (2-D) or using a z-stack for 3-D images

Importing a geometry into Vcell Segmentation Smoothing Re-SegmentImport-Vcell

Advantages of Vcell Can define patches of AC and PDE on a membrane Create a diffusional map in segments Simplistic Geometry (still several hours to solve with complicated reations) Can accept Immuno-gold data in TEM’s

Defining Compartments Defining the geometry of the compartments with differing sizes Creating diffusional membranes Creating membrane bound reactions

Electrical Mapping Defining the geometry of the compartments with differing sizes Creating diffusional membranes Creating membrane bound reactions Each membrane can have defined voltages and capacitances A membrane voltage can be defined to trigger an event (such as calcium release) The current or voltage of the membrane can be clamped to better mimic experimental conditions Most experiments when Patch Clamping a cell either voltage clamp or current clamp the cell

Defining the Spatial and Temporal Mesh Mesh units in the x, y, and z directions as well as in time An error tolerance can also be defined which defines the maximum allowed step change

Viewing the Results Can track concentration changes in space or time Can graph reaction rates as well

Virtual Cell Experiment and simulation of calcium dynamics following Bradykinin ( BK) stimulation of a neuroblastoma cell. A 250 nM solution of BK was applied at time 0, and the [Ca 2+ ] cyt is monitored with fura-2 to produce the experimental record (left) obtained at 15 frames/sec. Representative frames are shown, and the change in calcium in the neurite (green box) and soma (yellow box) are plotted in the inset. The Virtual Cell simulation shown in the next column provides a good match to the experiment. The third and fourth columns display the simulation results for [InsP3] cyt and Po, the open probability of the InsP3-sensitive calcium channel in the ER membrane (Slepchenko BM et al. Annu Rev Biophys Biomol Struct 2002;31:423-41)

FRAP Fluorescence Recovery After Photobleaching FRAP Video: Simple FRAP: FRAP with Binding:

HW Take a screen shot of your results for the simple frap experiment showing diffusion into the blocked region Take a screen show of your reaction diagram for the FRAP with binding and of your final results again showing diffusion into the blocked region