Engineering physics/ matlab

Electrical current through a pore in a membrane

Artificial stimulation of action potential:

Ion flux due to electro-chemical gradient:

[Na+] = 145 mM

[Ca2+] = 1.2 mM

[Cl-] = 116 mM

[K+] = 4.5 mM

[Na+] = 15 mM

[Ca2+] = 0.0001 mM

[Cl-] = 20 mM

[K+] = 120 mM

outsideinside

𝐽𝐽𝑗𝑗

𝐽𝐽𝑗𝑗 = −𝐷𝐷 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

+ 𝑧𝑧𝑧𝑧𝑑𝑑 𝑅𝑅𝑅𝑅

𝑑𝑑Ψ 𝑑𝑑𝑑𝑑

Nernst-Planck equation

Assumptions:

• Pore is large enough for diffusion coefficient to be same as in bulk

• Ψ changes linearly across the membrane

Ion pore

Ψ𝑖𝑖 Ψ𝑜𝑜

𝑑𝑑𝑖𝑖 𝑑𝑑𝑜𝑜

𝑑𝑑Ψ 𝑑𝑑𝑑𝑑

= ∆Ψ 𝛿𝛿

𝛿𝛿

Ionic flux for a single ion: outsideinside

𝐽𝐽𝑗𝑗

𝐽𝐽𝑗𝑗 = −𝐷𝐷 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

+ 𝑧𝑧𝑧𝑧𝑑𝑑 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿

Ion pore

Ψ𝑖𝑖 Ψ𝑜𝑜

𝑑𝑑𝑖𝑖 𝑑𝑑𝑜𝑜

−𝐽𝐽𝑗𝑗 𝐷𝐷

= 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

+ 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿

𝑑𝑑

𝑄𝑄 = 𝑑𝑑𝑑𝑑(𝑑𝑑) 𝑑𝑑𝑑𝑑

+ 𝑃𝑃 𝑑𝑑 𝑑𝑑(𝑑𝑑)

𝑑𝑑(𝑑𝑑) 𝑒𝑒𝑑𝑑𝑒𝑒 �𝑃𝑃 𝑑𝑑 𝑑𝑑𝑑𝑑 = �𝑄𝑄 𝑒𝑒𝑑𝑑𝑒𝑒 �𝑃𝑃 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

(Integral tables)

(rearrange)

𝛿𝛿

𝑑𝑑(𝑑𝑑) 𝑒𝑒𝑑𝑑𝑒𝑒 � 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑𝑑𝑑 = �

−𝐽𝐽𝑗𝑗 𝐷𝐷

𝑒𝑒𝑑𝑑𝑒𝑒 � 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

Ionic flux for a single ion:

𝑒𝑒𝑑𝑑𝑒𝑒 � 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑𝑑𝑑 = 𝑒𝑒𝑑𝑑𝑒𝑒

𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑

𝑑𝑑(𝑑𝑑) 𝑒𝑒𝑑𝑑𝑒𝑒 � 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑𝑑𝑑 = �

−𝐽𝐽𝑗𝑗 𝐷𝐷

𝑒𝑒𝑑𝑑𝑒𝑒 � 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

𝑑𝑑 𝑑𝑑 𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑 =

−𝐽𝐽𝑗𝑗 𝐷𝐷

�𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑 𝑑𝑑𝑑𝑑 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

�𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑 𝑑𝑑𝑑𝑑 =

𝑅𝑅𝑅𝑅𝛿𝛿 𝑧𝑧𝑧𝑧∆Ψ

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑

𝑑𝑑 𝑑𝑑 𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑 =

−𝐽𝐽𝑗𝑗 𝐷𝐷

𝑅𝑅𝑅𝑅𝛿𝛿 𝑧𝑧𝑧𝑧∆Ψ

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

Ionic flux for a single ion:

𝑑𝑑 𝑑𝑑 𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑 =

−𝐽𝐽𝑗𝑗 𝐷𝐷

𝑅𝑅𝑅𝑅𝛿𝛿 𝑧𝑧𝑧𝑧∆Ψ

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

𝑑𝑑 𝑑𝑑 𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑 =

−𝐽𝐽𝑗𝑗 𝐷𝐷

𝑅𝑅𝑅𝑅𝛿𝛿 𝑧𝑧𝑧𝑧∆Ψ

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

0

𝛿𝛿

�𝑑𝑑 𝑑𝑑 𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑

0

𝛿𝛿

= � −𝐽𝐽𝑗𝑗 𝐷𝐷

𝑅𝑅𝑅𝑅𝛿𝛿 𝑧𝑧𝑧𝑧∆Ψ

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧 𝑅𝑅𝑅𝑅

∆Ψ 𝛿𝛿 𝑑𝑑

0

𝛿𝛿

𝑑𝑑 𝛿𝛿 𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

− 𝑑𝑑 0 = −𝐽𝐽𝑗𝑗 𝐷𝐷

𝑅𝑅𝑅𝑅𝛿𝛿 𝑧𝑧𝑧𝑧∆Ψ

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

− 1

𝐽𝐽𝑗𝑗 = − 𝑧𝑧𝑧𝑧𝐷𝐷∆Ψ 𝑅𝑅𝑅𝑅𝛿𝛿

𝑑𝑑 𝛿𝛿 𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ𝑅𝑅𝑅𝑅 − 𝑑𝑑 0

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ𝑅𝑅𝑅𝑅 − 1

Ionic flux for a single ion: outsideinside

𝐽𝐽𝑗𝑗

Ion pore

Ψ𝑖𝑖 Ψ𝑜𝑜

𝑑𝑑𝑖𝑖 𝑑𝑑𝑜𝑜

𝑑𝑑(𝛿𝛿) 𝑑𝑑(0)

𝐽𝐽𝑗𝑗 = − 𝑧𝑧𝑧𝑧𝐷𝐷∆Ψ 𝑅𝑅𝑅𝑅𝛿𝛿

𝑑𝑑 𝛿𝛿 𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ𝑅𝑅𝑅𝑅 − 𝑑𝑑 0

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ𝑅𝑅𝑅𝑅 − 1

𝑑𝑑 𝛿𝛿 = 𝛽𝛽𝑑𝑑𝑖𝑖

𝑑𝑑 0 = 𝛽𝛽𝑑𝑑0 𝛽𝛽 – partition coefficient

𝐽𝐽𝑗𝑗 = − 𝐷𝐷𝛽𝛽 𝛿𝛿

𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

𝑑𝑑𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅 − 𝑑𝑑0

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ𝑅𝑅𝑅𝑅 − 1

𝑒𝑒𝑗𝑗 – permeability constant for ion 𝑗𝑗

𝐼𝐼𝑗𝑗 = 𝑧𝑧𝑧𝑧𝐽𝐽𝑗𝑗 = −𝑒𝑒𝑗𝑗 𝑧𝑧𝑧𝑧 2

𝑅𝑅𝑅𝑅 ∆Ψ

𝑑𝑑𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅 − 𝑑𝑑0

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ𝑅𝑅𝑅𝑅 − 1

Ionic current

Ionic flux

Membrane at equilibrium:

𝐼𝐼𝑁𝑁𝑁𝑁 = −𝑒𝑒𝑁𝑁𝑁𝑁 𝑧𝑧𝑧𝑧 2

𝑅𝑅𝑅𝑅 ∆Ψ

𝑁𝑁𝑁𝑁 𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅 − 𝑁𝑁𝑁𝑁 0

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ𝑅𝑅𝑅𝑅 − 1 𝐼𝐼𝐾𝐾 = −𝑒𝑒𝐾𝐾

𝑧𝑧𝑧𝑧 2

𝑅𝑅𝑅𝑅 ∆Ψ

𝐾𝐾 𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅 − 𝐾𝐾 0

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ𝑅𝑅𝑅𝑅 − 1

[Na+] = 145 mM

[Ca2+] = 1.2 mM

[Cl-] = 116 mM

[K+] = 4.5 mM

[Na+] = 15 mM

[Ca2+] = 0.0001 mM

[Cl-] = 20 mM

[K+] = 120 mM

K+

Na+

𝐼𝐼𝐾𝐾 + 𝐼𝐼𝑁𝑁𝑁𝑁 = 0

Equilibrium, if sum of all ionic currents is zero:

𝑒𝑒𝐾𝐾 𝐾𝐾 𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

− 𝐾𝐾 0 + 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

− 𝑁𝑁𝑁𝑁 0 = 0

Membrane equilibrium potential:

[Na+] = 145 mM

[Ca2+] = 1.2 mM

[Cl-] = 116 mM

[K+] = 4.5 mM

[Na+] = 15 mM

[Ca2+] = 0.0001 mM

[Cl-] = 20 mM

[K+] = 120 mM

K+

Na+

𝐼𝐼𝐾𝐾 + 𝐼𝐼𝑁𝑁𝑁𝑁 = 0

Equilibrium, if sum of all ionic currents is zero:

𝑒𝑒𝐾𝐾 𝐾𝐾 𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

− 𝐾𝐾 0 + 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

− 𝑁𝑁𝑁𝑁 0 = 0

𝑒𝑒𝐾𝐾 𝐾𝐾 𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

− 𝑒𝑒𝐾𝐾 𝐾𝐾 0 + 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 𝑖𝑖𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

− 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 0 = 0

𝑒𝑒𝑑𝑑𝑒𝑒 𝑧𝑧𝑧𝑧∆Ψ 𝑅𝑅𝑅𝑅

= 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 0 + 𝑒𝑒𝐾𝐾 𝐾𝐾 0 𝑒𝑒𝐾𝐾 𝐾𝐾 𝑖𝑖 + 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 𝑖𝑖

∆Ψ = 𝑅𝑅𝑅𝑅 𝑧𝑧𝑧𝑧

𝑙𝑙𝑐𝑐 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 0 + 𝑒𝑒𝐾𝐾 𝐾𝐾 0 𝑒𝑒𝐾𝐾 𝐾𝐾 𝑖𝑖 + 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 𝑖𝑖

Membrane equilibrium potential:

Goldman-Hodgkin-Katz model:

Ψ𝑖𝑖𝑖𝑖 − Ψ𝑜𝑜𝑜𝑜𝑜𝑜 = ∆Ψ = − 𝑅𝑅𝑅𝑅 𝑧𝑧𝑧𝑧

𝑙𝑙𝑐𝑐 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁+ 𝑖𝑖𝑖𝑖 + 𝑒𝑒𝐾𝐾 𝐾𝐾+ 𝑖𝑖𝑖𝑖 𝑒𝑒𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁+ 𝑜𝑜𝑜𝑜𝑜𝑜 + 𝑒𝑒𝐾𝐾 𝐾𝐾+ 𝑜𝑜𝑜𝑜𝑜𝑜

𝑒𝑒𝑗𝑗 = 𝐷𝐷𝑗𝑗 𝛿𝛿

𝐷𝐷𝑗𝑗 – diffusion coefficient for ion 𝑗𝑗, 𝛿𝛿 – membrane thickness

• Cl- is in electrochemical equilibrium and its flux is omitted

• Ca2+ flux is relatively small and is commonly ignored for simplicity

Homework assignment:

• Exam #3 has been posted – due the day before Thanksgiving!

PLEASE MAKE SURE YOUR CODE RUNS BEFORE

SUBMITTING!!!