Engineering physics/ matlab

Ions, cell membranes, signal propogation

Movement accross membranes:

• Biological membranes serve as (selectively permeable) barriers between cytoplasm and outside world

• Major ways of getting across the lipid bilayer: – Simple diffusion (small, hydrophobic, lipophilic molecules) – Protein-mediated transport (hydrophilic, polar, charged molecules) – Endocytosis/exocytosis (proteins, viruses, microparticles, iron)

Simple diffusion across membranes:

• Hydrophobic substances (gases, steroid hormones) diffuse across membranes easily

• Polar molecules are repelled by the hydrophobic interior of the bilayer and do not diffuse across easily (unless very small and uncharged)

• Lipid bilayers are virtually impermeable to ions (ions are surrounded by a cage of water)

*pKa is the pH at which 50% of a given substance is protonated

Passive protein-mediated transport:

• Passive transport of molecules down the concentration gradient

• Channel proteins create a pore through the lipid bilayer • Carrier proteins bind to molecules and physically transport

them across the lipid bilayer

Active protein-mediated transport:

• Active transport allows to moving molecules against concentration gradients • Coupled transporters use energy stored in gradient of one ion to transport the

other • ABC (ATP-binding cassette) transporters mostly work on getting foreign

substances (drugs and other toxins) out of cells

Selectively permeable membranes at rest:

[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

Lipid bilayer

(Extracellular space)

3×Na+

2×K+

Na+

K+

resting channel

Na/K pump

Cl-

(fewer for Na+)

(many more for K+) Cl-

(K+) coupled transporter

Resting 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

ΔΨ

• Lipid bilayer is a decent insulator (resistivity 103 - 109 Ω·cm) • Unequal distribution of ions between inside and outside of

the cell creates membrane (resting, unstimulated) potential

ΔΨ erythrocyte

= −10 mV

ΔΨ T−lymphocyte

= −50 mV

ΔΨ muscle cell

= −80 mV

ΔΨ neuron

= −70 mV

~ wet wood

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 𝐽𝐽𝑐𝑐

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𝐽𝐽𝑒𝑒

𝐶𝐶 – concentration 𝑧𝑧 – valence 𝑓𝑓𝑠𝑠 – frictional resistance (𝑓𝑓𝑠𝑠 = 6𝜋𝜋𝜋𝜋𝜋𝜋 for spherical molecules) 𝑁𝑁𝐴𝐴= 6.0221 × 1023 [mol-1] 𝐹𝐹 = 96485 [C mol-1] 𝑅𝑅 = 8.314 [J mol−1 K−1]

𝐽𝐽 = 𝐽𝐽𝑐𝑐 + 𝐽𝐽𝑒𝑒 = −𝐷𝐷 𝑑𝑑𝐶𝐶 𝑑𝑑𝑑𝑑

− 𝑧𝑧𝐹𝐹𝐶𝐶 𝑓𝑓𝑠𝑠𝑁𝑁𝐴𝐴

𝑑𝑑Ψ 𝑑𝑑𝑑𝑑 𝐷𝐷 =

𝑅𝑅𝑅𝑅 𝑓𝑓𝑠𝑠𝑁𝑁𝐴𝐴

𝐽𝐽 = −𝐷𝐷 𝑑𝑑𝐶𝐶 𝑑𝑑𝑑𝑑

− 𝐷𝐷 𝑧𝑧𝐹𝐹𝐶𝐶 𝑅𝑅𝑅𝑅

𝑑𝑑Ψ 𝑑𝑑𝑑𝑑

= −𝐷𝐷 𝑑𝑑𝐶𝐶 𝑑𝑑𝑑𝑑

+ 𝑧𝑧𝐹𝐹𝐶𝐶 𝑅𝑅𝑅𝑅

𝑑𝑑Ψ 𝑑𝑑𝑑𝑑

concentration gradient gradient of electric field

Equilibrium potential for one ion:

[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 𝐽𝐽𝑐𝑐

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𝐽𝐽𝑒𝑒

𝐽𝐽 = −𝐷𝐷 𝑑𝑑𝐶𝐶 𝑑𝑑𝑑𝑑

+ 𝑧𝑧𝐹𝐹𝐶𝐶 𝑅𝑅𝑅𝑅

𝑑𝑑Ψ 𝑑𝑑𝑑𝑑

= 0

−𝐷𝐷𝐶𝐶 𝑑𝑑𝑑𝑑

𝑑𝑑𝐶𝐶 𝐶𝐶

+ 𝑧𝑧𝐹𝐹 𝑅𝑅𝑅𝑅

𝑑𝑑Ψ = 0

� Ψ𝑜𝑜𝑜𝑜𝑜𝑜

Ψ𝑖𝑖𝑖𝑖

𝑑𝑑Ψ = − 𝑅𝑅𝑅𝑅 𝑧𝑧𝐹𝐹

� 𝐶𝐶𝑜𝑜𝑜𝑜𝑜𝑜

𝐶𝐶𝑖𝑖𝑖𝑖 1 𝐶𝐶 𝑑𝑑𝐶𝐶

Ψ𝑖𝑖𝑖𝑖 − Ψ𝑜𝑜𝑜𝑜𝑜𝑜 = − 𝑅𝑅𝑅𝑅 𝑧𝑧𝐹𝐹

𝑙𝑙𝑙𝑙 𝐶𝐶𝑖𝑖𝑖𝑖 𝐶𝐶𝑜𝑜𝑜𝑜𝑜𝑜

Ψ𝑖𝑖𝑖𝑖 Ψ𝑜𝑜𝑜𝑜𝑜𝑜

𝐶𝐶𝑖𝑖𝑖𝑖 𝐶𝐶𝑜𝑜𝑜𝑜𝑜𝑜

Nernst equation:

Ψ𝑖𝑖𝑖𝑖 − Ψ𝑜𝑜𝑜𝑜𝑜𝑜 = − 𝑅𝑅𝑅𝑅 𝑧𝑧𝐹𝐹

𝑙𝑙𝑙𝑙 𝐶𝐶𝑖𝑖𝑖𝑖 𝐶𝐶𝑜𝑜𝑜𝑜𝑜𝑜

Action potential:

• Action potential is caused by localized depolarization of cell membrane

• Propagation of action potentials along nerve cells conveys information in the nervous system

[e.g. from -70mV to +40mV]

Membrane depolirization:

• (K+ flux is balanced by Na+ flux => constant membrane potential) • Membrane depolarizes pass threshold • Voltage-gated Na+ channels open (rapidly) • Na+ influx increases, causing further depolarization • (Slow) opening of K+ channels repolarizes membrane

Ψ𝑖𝑖𝑖𝑖 − Ψ𝑜𝑜𝑜𝑜𝑜𝑜 = − 𝑅𝑅𝑅𝑅 𝑧𝑧𝐹𝐹

𝑙𝑙𝑙𝑙 𝐶𝐶𝑖𝑖𝑖𝑖 𝐶𝐶𝑜𝑜𝑜𝑜𝑜𝑜

∆Ψ𝑁𝑁𝑁𝑁= +60 𝑚𝑚𝑚𝑚

∆Ψ𝐾𝐾= −90 𝑚𝑚𝑚𝑚[resting phase]

Propagation of action potentials:

Propagation in uninsulated cable:

Cable Equation:

Propagation of action potentials:

Propagation in insulated cable:

faster propagation!

Homework assignment:

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