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Virtual Cell How to model reaction diffusion systems When a systems model is not enough • 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 Live cell intracellular calcium using fluorescent Colocalization of intracellular ankyrin-B and β2biosensors 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 Reaction Diffusion Models [C] source sink .( DC [C]) J i J j Rm t i j m J isource f i (C,t, x, y, z) J sink f j (C,t, x, y, z) j [m] Rm km ([B] [m])[C] km [m] t m B C where: [C] intracellular species concentration DC diffusion constant for C Jisource – source i for C Jjsink –sink j for C km+, km- - kinetic rates Finite Difference Method for 1-D Heat Equation u 2u D 2 0 t x Applying the finite difference approximations to the 1-D heat equation: uik 1 uik uik1 2uik uik1 2 4 D O t ,x t x 2 t k- increment in time k 1 k ui ui D 2 uik1 2uik uik1 i- increment in space x This explicit (backward Euler ) scheme is stable if: t 1 D 2 2 x Forward Euler Scheme ¶u ¶2u +D =0 2 ¶t ¶x Instead of: Dt k +1 k k k k ui = ui + D u 2 u + u i i -1 Dx 2 i +1 Accuracy and stability can be ( ) improved with an implicit scheme: Dt k +1 k k +1 k +1 k +1 é ù ui - ui = D u 2 u + u i i -1 û 2 ë i +1 Dx * 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 Simple Shapes 2-D Image of a cell from a Microscope 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 Re-Segment Import-Vcell Smoothing Advantages of Vcell Can define patches of AC and PDE on a membrane Simplistic Geometry (still several hours to solve with complicated reations) Create a diffusional map in segments 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 • Each membrane can have defined voltages Creating diffusional and capacitances membranes • 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 Creating Patch Clamping a cellbound either membrane voltage clamp or current clamp the cell reactions 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 [Ca2+]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: http://vcell.org/education/frap_tutorial/FRAPmovie/FRAP_0505_h264.mp4 Simple FRAP: http://vcell.org/webstart/Rel/user_docs/Tutorial01_SimpleFRAP.pdf FRAP with Binding: http://vcell.org/webstart/Rel/user_docs/Tutorial02_FRAPBinding.pdf 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