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 outside inside

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 outside inside 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 outside inside 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 outside inside 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 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

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

Steady-state FP equation

Steady-state FP equation Use Einstein relation:

Steady-state FP equation Use Einstein relation:

Steady-state FP equation Use Einstein relation: Solution:

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

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

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

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

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

GHK current, another way Start from

GHK current, another way Start from

GHK current, another way Start from

GHK current, another way Start from Note

GHK current, another way Start from Note Integrate from 0 to d:

GHK current, another way Start from Note Integrate from 0 to d:

GHK current, another way Start from Note Integrate from 0 to d:

GHK current, another way Start from Note Integrate from 0 to d:

GHK current, another way Start from Note Integrate from 0 to d: (as before)

GHK current, another way 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

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

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

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

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

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

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) www.nbi.dk/hertz/noisecourse/demos/Pseq.mat www.nbi.dk/hertz/noisecourse/demos/runseq.m

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):

Fokker-Planck equation Conservation (continuity):

Fokker-Planck equation Conservation (continuity): Use Einstein relation:

Fokker-Planck equation Conservation (continuity): Use Einstein relation: Current:

Fokker-Planck equation Conservation (continuity): Use Einstein relation: Current: If equilibrium, J = 0,

Fokker-Planck equation Conservation (continuity): Use Einstein relation: Current: If equilibrium, J = 0,

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

Calculating the current (J is constant)

Calculating the current (J is constant) integrate:

Calculating the current (J is constant) (P(c) very small) integrate:

Calculating the current (J is constant) (P(c) very small) integrate:

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

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

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

Calculating the current (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

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

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

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

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

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

More about drift current 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)

More about drift current 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) = .00015x http://www.nbi.dk/~hertz/noisecourse/driftmovie.m

Derivation from master equation

Derivation from master equation (1st argument of r: starting point; 2nd argument: step size)

Derivation from master equation (1st argument of r: starting point; 2nd argument: step size)

Derivation from master equation (1st argument of r: starting point; 2nd argument: step size)

Derivation from master equation (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.

Derivation from master equation (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:

Derivation from master equation (2) expand:

Derivation from master equation (2) expand:

Derivation from master equation (2) expand:

Derivation from master equation (2) expand: Kramers-Moyal expansion

Derivation from master equation (2) expand: Kramers-Moyal expansion Fokker-Planck eqn if drop terms of order >2

Derivation from master equation (2) expand: Kramers-Moyal expansion Fokker-Planck eqn if drop terms of order >2

Derivation from master equation (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

Derivation from master equation (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