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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. Lecture 4: Diffusion and the Fokker-Planck equation. Outline: intuitive treatment

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Lecture 4: Diffusion and the Fokker-Planck equation

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  1. Lecture 4: Diffusion and the Fokker-Planck equation • Outline: • intuitive treatment • Diffusion as flow down a concentration gradient • Drift current and Fokker-Planck equation

  2. 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:

  3. 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

  4. 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

  5. Diffusion Fick’s law:

  6. Diffusion Fick’s law: cf Ohm’s law

  7. Diffusion Fick’s law: cf Ohm’s law conservation:

  8. Diffusion Fick’s law: cf Ohm’s law conservation: =>

  9. Diffusion Fick’s law: cf Ohm’s law conservation: => diffusion equation

  10. Diffusion Fick’s law: cf Ohm’s law conservation: => diffusion equation initial condition

  11. Diffusion Fick’s law: cf Ohm’s law conservation: => diffusion equation initial condition solution:

  12. Diffusion Fick’s law: cf Ohm’s law conservation: => diffusion equation initial condition solution: http://www.nbi.dk/~hertz/noisecourse/gaussspread.m

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

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

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

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

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

  18. 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

  19. Examples: constant drift velocity http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m

  20. Examples: constant drift velocity http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m Solution (with no boundaries):

  21. Examples: constant drift velocity http://www.nbi.dk/~hertz/noisecourse/gaussspreadmove.m Solution (with no boundaries): Stationary case:

  22. 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

  23. 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

  24. 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):

  25. 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

  26. Einstein relation FP equation:

  27. Einstein relation FP equation: Solution:

  28. Einstein relation FP equation: Solution: But from equilibrium stat mech we know

  29. Einstein relation FP equation: Solution: But from equilibrium stat mech we know So D = μT

  30. Einstein relation FP equation: Solution: But from equilibrium stat mech we know So D = μT Einstein relation

  31. Constant current: Goldman-Hodgkin-Katz model of an (open) ion channel Pumps maintain different inside and outside concentrations of ions

  32. 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

  33. 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

  34. 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?

  35. 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

  36. 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

  37. 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

  38. 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

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

  40. Reversal potential If there is no current, equilibrium => ρin/ρout=exp(-βV) This defines the reversal potential at which J = 0.

  41. 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

  42. 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

  43. 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

  44. 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

  45. 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

  46. 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

  47. Steady-state FP equation

  48. Steady-state FP equation

  49. Steady-state FP equation Use Einstein relation:

  50. Steady-state FP equation Use Einstein relation:

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