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Lecture 4: Diffusion and the Fokker-Planck equation

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## Lecture 4: Diffusion and the Fokker-Planck equation

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**Lecture 4: Diffusion and the Fokker-Planck equation**• Outline: • intuitive treatment • Diffusion as flow down a concentration gradient • Drift current and Fokker-Planck equation**Lecture 4: Diffusion and the Fokker-Planck equation**• Outline: • intuitive treatment • Diffusion as flow down a concentration gradient • Drift current and Fokker-Planck equation • examples: • No current: equilibrium, Einstein relation • Constant current, out of equilibrium:**Lecture 4: Diffusion and the Fokker-Planck equation**• Outline: • intuitive treatment • Diffusion as flow down a concentration gradient • Drift current and Fokker-Planck equation • examples: • No current: equilibrium, Einstein relation • Constant current, out of equilibrium: • Goldman-Hodgkin-Katz equation • Kramers escape over an energy barrier**Lecture 4: Diffusion and the Fokker-Planck equation**• Outline: • intuitive treatment • Diffusion as flow down a concentration gradient • Drift current and Fokker-Planck equation • examples: • No current: equilibrium, Einstein relation • Constant current, out of equilibrium: • Goldman-Hodgkin-Katz equation • Kramers escape over an energy barrier • derivation from master equation**Diffusion**Fick’s law:**Diffusion**Fick’s law: cf Ohm’s law**Diffusion**Fick’s law: cf Ohm’s law conservation:**Diffusion**Fick’s law: cf Ohm’s law conservation: =>**Diffusion**Fick’s law: cf Ohm’s law conservation: => diffusion equation**Diffusion**Fick’s law: cf Ohm’s law conservation: => diffusion equation initial condition**Diffusion**Fick’s law: cf Ohm’s law conservation: => diffusion equation initial condition solution:**Diffusion**Fick’s law: cf Ohm’s law conservation: => diffusion equation initial condition solution: http://www.nbi.dk/~hertz/noisecourse/gaussspread.m**Drift current and Fokker-Planck equation**Drift (convective) current:**Drift current and Fokker-Planck equation**Drift (convective) current: Combining drift and diffusion: Fokker-Planck equation:**Drift current and Fokker-Planck equation**Drift (convective) current: Combining drift and diffusion: Fokker-Planck equation:**Drift current and Fokker-Planck equation**Drift (convective) current: Combining drift and diffusion: Fokker-Planck equation: Slightly more generally, D can depend on x:**Drift current and Fokker-Planck equation**Drift (convective) current: Combining drift and diffusion: Fokker-Planck equation: Slightly more generally, D can depend on x: =>**Drift current and Fokker-Planck equation**Drift (convective) current: Combining drift and diffusion: Fokker-Planck equation: Slightly more generally, D can depend on x: => First term alone describes probability cloud moving with velocity u(x) Second term alone describes diffusively spreading probability cloud**Examples: constant drift velocity**http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m**Examples: constant drift velocity**http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m Solution (with no boundaries):**Examples: constant drift velocity**http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m Solution (with no boundaries): Stationary case:**Examples: constant drift velocity**http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m Solution (with no boundaries): Stationary case: Gas of Brownian particles in gravitational field: u0 = μF = -μmg**Examples: constant drift velocity**http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m Solution (with no boundaries): Stationary case: Gas of Brownian particles in gravitational field: u0 = μF = -μmg μ =mobility**Examples: constant drift velocity**http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m Solution (with no boundaries): Stationary case: Gas of Brownian particles in gravitational field: u0 = μF = -μmg μ =mobility Boundary conditions (bottom of container, stationarity):**Examples: constant drift velocity**http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m Solution (with no boundaries): Stationary case: Gas of Brownian particles in gravitational field: u0 = μF = -μmg μ =mobility Boundary conditions (bottom of container, stationarity): drift and diffusion currents cancel**Einstein relation**FP equation:**Einstein relation**FP equation: Solution:**Einstein relation**FP equation: Solution: But from equilibrium stat mech we know**Einstein relation**FP equation: Solution: But from equilibrium stat mech we know So D = μT**Einstein relation**FP equation: Solution: But from equilibrium stat mech we know So D = μT Einstein relation**Constant current: Goldman-Hodgkin-Katz model of an (open)**ion channel Pumps maintain different inside and outside concentrations of ions**Constant current: Goldman-Hodgkin-Katz model of an (open)**ion channel • Pumps maintain different inside and outside concentrations of ions • Voltage diff (“membrane potential”) between inside and outside of cell**Constant current: Goldman-Hodgkin-Katz model of an (open)**ion channel • Pumps maintain different inside and outside concentrations of ions • Voltage diff (“membrane potential”) between inside and outside of cell • Can vary membrane potential experimentally by adding external field**Constant current: Goldman-Hodgkin-Katz model of an (open)**ion channel • Pumps maintain different inside and outside concentrations of ions • Voltage diff (“membrane potential”) between inside and outside of cell • Can vary membrane potential experimentally by adding external field • Question: At a given Vm, what current flows through the channel?**Constant current: Goldman-Hodgkin-Katz model of an (open)**ion channel • Pumps maintain different inside and outside concentrations of ions • Voltage diff (“membrane potential”) between inside and outside of cell • Can vary membrane potential experimentally by adding external field • Question: At a given Vm, what current flows through the channel? x=0 x=d x inside outside**Constant current: Goldman-Hodgkin-Katz model of an (open)**ion channel • Pumps maintain different inside and outside concentrations of ions • Voltage diff (“membrane potential”) between inside and outside of cell • Can vary membrane potential experimentally by adding external field • Question: At a given Vm, what current flows through the channel? x=0 x=d Vout= 0 x inside outside V(x) Vm**Constant current: Goldman-Hodgkin-Katz model of an (open)**ion channel • Pumps maintain different inside and outside concentrations of ions • Voltage diff (“membrane potential”) between inside and outside of cell • Can vary membrane potential experimentally by adding external field • Question: At a given Vm, what current flows through the channel? x=0 x=d ρout ρin Vout= 0 x inside outside V(x) Vm**Constant current: Goldman-Hodgkin-Katz model of an (open)**ion channel • Pumps maintain different inside and outside concentrations of ions • Voltage diff (“membrane potential”) between inside and outside of cell • Can vary membrane potential experimentally by adding external field • Question: At a given Vm, what current flows through the channel? x=0 x=d ρout ? ρin Vout= 0 x inside outside V(x) Vm**Reversal potential**If there is no current, equilibrium => ρin/ρout=exp(-βV)**Reversal potential**If there is no current, equilibrium => ρin/ρout=exp(-βV) This defines the reversal potential at which J = 0.**Reversal potential**If there is no current, equilibrium => ρin/ρout=exp(-βV) This defines the reversal potential at which J = 0. For Ca++, ρout>> ρin => Vr >> 0**GHK model (2)**Vm< 0: both diffusive current and drift current flow in x=0 x=d ρout ? ρin Vout= 0 x inside outside V(x) Vm**GHK model (2)**Vm< 0: both diffusive current and drift current flow in Vm= 0: diffusive current flows in, no drift current x=0 x=d ρout ? ρin Vout= 0 V(x) x inside outside**GHK model (2)**Vm< 0: both diffusive current and drift current flow in Vm= 0: diffusive current flows in, no drift current Vm> 0: diffusive current flows in, drift current flows out x=0 x=d ρout ? Vm V(x) ρin Vout= 0 x inside outside**GHK model (2)**Vm< 0: both diffusive current and drift current flow in Vm= 0: diffusive current flows in, no drift current Vm> 0: diffusive current flows in, drift current flows out At Vm= Vr they cancel x=0 x=d ρout ? Vm V(x) ρin Vout= 0 x inside outside**GHK model (2)**Vm< 0: both diffusive current and drift current flow in Vm= 0: diffusive current flows in, no drift current Vm> 0: diffusive current flows in, drift current flows out At Vm= Vr they cancel x=0 x=d ρout ? Vm V(x) ρin Vout= 0 x inside outside**Steady-state FP equation**Use Einstein relation:**Steady-state FP equation**Use Einstein relation: