Lecture 4: Diffusion and the Fokker-Planck equation

Name: Lecture 4: Diffusion and the Fokker-Planck equation
Uploaded: 2017-10-13T12:43:08+00:00
Duration: PTM28S42
Channel: Grace Holland
Description: Lecture 4: Diffusion and the 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

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:

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

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 equation initial condition

diffusion equation initial condition solution:

Drift current and Fokker-Planck equation
Drift (convective) current:

Drift (convective) current: Combining drift and diffusion: Fokker-Planck equation:

Drift (convective) current: Combining drift and diffusion: Fokker-Planck equation: Slightly more generally, D can depend on x:

Drift (convective) current: Combining drift and diffusion: Fokker-Planck equation: Slightly more generally, D can depend on x: =>

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

Solution (with no boundaries):

Solution (with no boundaries): Stationary case:

Solution (with no boundaries): Stationary case: Gas of Brownian particles in gravitational field: u0 = μF = -μmg

Solution (with no boundaries): Stationary case: Gas of Brownian particles in gravitational field: u0 = μF = -μmg μ =mobility

Solution (with no boundaries): Stationary case: Gas of Brownian particles in gravitational field: u0 = μF = -μmg μ =mobility Boundary conditions (bottom of container, stationarity):

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

But from equilibrium stat mech we know So D = μT

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

Pumps maintain different inside and outside concentrations of ions Voltage diff (“membrane potential”) between inside and outside of cell

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

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?

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 outside inside

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 outside inside V(x) Vm

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 outside inside V(x) Vm

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 outside inside V(x) Vm

Reversal potential If there is no current, equilibrium
=> ρin/ρout=exp(-βV)

=> ρin/ρout=exp(-βV) This defines the reversal potential at which J = 0.

=> 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 outside inside 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 outside inside

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 outside inside

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 outside inside

Steady-state FP equation

Use Einstein relation:

Use Einstein relation: Solution:

Use Einstein relation: Solution: We are given ρ(0) and ρ(d). Use this to solve for J:

GHK current, another way
Start from

Start from Note

Start from Note Integrate from 0 to d:

Start from Note Integrate from 0 to d: (as before)

Start from Note Integrate from 0 to d: (as before) Note: J = 0 at Vm= Vr

GHK current is nonlinear
(using z, Vr for Ca++) J V

Kramers escape Rate of escape from a potential well due to thermal fluctuations P2(x) P1(x) V1(x) V2(x)

Kramers escape (2) V(x) a b c

Kramers escape (2) V(x) J  a b c

Kramers escape (2) Basic assumption: (V(b) – V(a))/T >> 1 V(x)
J  a b c Basic assumption: (V(b) – V(a))/T >> 1

Fokker-Planck equation
Conservation (continuity):

Conservation (continuity): Use Einstein relation:

Conservation (continuity): Use Einstein relation: Current:

Conservation (continuity): Use Einstein relation: Current: If equilibrium, J = 0,

Conservation (continuity): Use Einstein relation: Current: If equilibrium, J = 0, Here: almost equilibrium, so use this P(x)

Calculating the current
(J is constant)

(J is constant) integrate:

(J is constant) (P(c) very small) integrate:

(J is constant) (P(c) very small) integrate: If p is probability to be in the well, J = pr, where r = escape rate

calculating escape rate
In integral integrand is peaked near x = b

calculating escape rate
In integral integrand is peaked near x = b ________

More about drift current
Notice: If u(x) is not constant, the probability cloud can shrink or spread even if there is no diffusion

Notice: If u(x) is not constant, the probability cloud can shrink or spread even if there is no diffusion (like density of cars on a road where the speed limit varies)

Notice: If u(x) is not constant, the probability cloud can shrink or spread even if there is no diffusion (like density of cars on a road where the speed limit varies) Demo: initial P: Gaussian centered at x = 2 u(x) = x

Derivation from master equation

(1st argument of r: starting point; 2nd argument: step size)

(1st argument of r: starting point; 2nd argument: step size) Small steps assumption: r(x;s) falls rapidly to zero with increasing |s| on the scale on which it varies with x or the scale on which P varies with x.

(1st argument of r: starting point; 2nd argument: step size) Small steps assumption: r(x;s) falls rapidly to zero with increasing |s| on the scale on which it varies with x or the scale on which P varies with x. x s

Derivation from master equation (2)
expand:

expand: Kramers-Moyal expansion

expand: Kramers-Moyal expansion Fokker-Planck eqn if drop terms of order >2

expand: Kramers-Moyal expansion Fokker-Planck eqn if drop terms of order >2 rn(x)Δt = nth moment of distribution of step size in time Δt

Lecture 4: Diffusion and the Fokker-Planck equation

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Lecture 4: Diffusion and the Fokker-Planck equation

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