modeling neurobiological systems, a mathematical approach weizmann institute 2004, d. holcman
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Modeling Neurobiological systems, a mathematical
approach
Weizmann Institute 2004, D. Holcman
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Examples
• Where are the mathematical problems?
• Synaptic plasticity: Receptors movements
• Sensor cells: Photo-transduction
• Dynamics of transient process
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Synaptic plasticity:
Receptor trafficking
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Synapse
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Receptor trafficking
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Mathematical Modeling
How long it takes to escape from micro-domains
How to compute a coarse-grained diffusion constant?
Answers:
Formulate a stochastic equation and solve the associated Partial Differential equations
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Exit from a small opening
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Photo-transduction
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diffusion in a single cone
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Geometry of the cone outer-segment
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Response curves of photon detection
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Dark noise in the outer-segment of photo receptor cells
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Two dimensional random walk of a Rhodopsin molecules
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Mathematical modeling
• How to model amplification:1-Photon change at the cellular level.2-Single photon response-curve
• Amplification, how to model 1-chemical reactions, diffusion
2-Noise 3- explain cone rods difference.
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Mathematical tools
• What is a chemical reaction at a molecular level. Computation of chemical constant: forward a backward binding rate
• Reaction-Diffusion equations
• Analyze the role of the cell-geometry
Noise analysis: solve PDE and stochastic PDE
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Dynamics in microstructures:
dendritic spines
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Dendritic spines
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Calcium dynamics in a spine
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Model transient dynamics
• Model effect of few ions:
1-Chemical reactions
2-effect of the geometry
3-find coarse-grained approach
• Produce a simulation, based at a molecular level
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Simulation of Ca dynamics in a dendritic spine
D.Holcman et.al, Biophysical J. 2004
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Conclusion
Purpose of the class Describe microbiological systems and predict the function.
Organization of the class
• Stochastic, Brownian motion• Stochastic equations, Ito calculus.• PDE( elliptic and parabolic, linear and nonlinear) • Asymptotic analysis examples: compute Chemical reaction constants• Neurobiological examples