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University of Utah Mathematical Biology the Imagine Possibilities Introduction to Physiology II: Control of Cell Volume and Membrane Potential J. P. Keener Mathematics Department University of Utah Math Physiology – p.1/24

Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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Page 1: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

Introduction to PhysiologyII: Control of Cell Volume and Membrane

Potential

J. P. Keener

Mathematics Department

University of Utah

Math Physiology – p.1/24

Page 2: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

Basic Problem

• The cell is full of stuff: Proteins,ions, fats, etc.

• The cell membrane issemipermeable, and thesesubstances create osmoticpressures, sucking water intothe cell.

• The cell membrane is like soapfilm, has no structural strengthto resist bursting.

Math Physiology – p.2/24

Page 3: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Basic Solution

• Carefully regulate the intracellular ionic concentrations sothat there are no net osmotic pressures.

• As a result, the major ions (Na+, K+, Cl− and Ca++) havedifferent intracellular and extracellular concentrations.

• Consequently, there is an electrical potential differenceacross the cell membrane, the membrane potential.

Math Physiology – p.3/24

Page 4: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

Membrane Transporters

+

+ ++KNa

+

ATP

Na

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

O CO2 2

GlucoseH O2

+CaK

• transmembrane diffusion - carbon dioxide, oxygen

• transporters - glucose, sodium-calcium exchanger

• pores - water

• ion-selective, gated channels - sodium, potassium, calcium

• ATPase exchangers - sodium-potassium ATPase, SERCA

Math Physiology – p.4/24

Page 5: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

Membrane Transporters

+

++Na Na+

ADP

K

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

O CO2 2

GlucoseH O2

+CaK

+

• transmembrane diffusion - carbon dioxide, oxygen

• transporters - glucose, sodium-calcium exchanger

• pores - water

• ion-selective, gated channels - sodium, potassium, calcium

• ATPase exchangers - sodium-potassium ATPase, SERCA

Math Physiology – p.4/24

Page 6: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

Membrane Transporters

+

+ ++KNa

+

ATP

Na

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

O CO2 2

GlucoseH O2

+CaK

• transmembrane diffusion - carbon dioxide, oxygen

• transporters - glucose, sodium-calcium exchanger

• pores - water

• ion-selective, gated channels - sodium, potassium, calcium

• ATPase exchangers - sodium-potassium ATPase, SERCA

Math Physiology – p.4/24

Page 7: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

Membrane Transporters

+

++Na Na+

ADP

K

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

O CO2 2

GlucoseH O2

+CaK

+

• transmembrane diffusion - carbon dioxide, oxygen

• transporters - glucose, sodium-calcium exchanger

• pores - water

• ion-selective, gated channels - sodium, potassium, calcium

• ATPase exchangers - sodium-potassium ATPase, SERCA

Math Physiology – p.4/24

Page 8: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

Membrane Transporters

+

+ ++KNa

+

ATP

Na

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

O CO2 2

GlucoseH O2

+CaK

• transmembrane diffusion - carbon dioxide, oxygen

• transporters - glucose, sodium-calcium exchanger

• pores - water

• ion-selective, gated channels - sodium, potassium, calcium

• ATPase exchangers - sodium-potassium ATPase, SERCA

Math Physiology – p.4/24

Page 9: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

How Things Move

Most molecules move by a random walk:

0 10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

time

posi

tion

0 10 20 30 40 50 60 70 80 90 100−30

−20

−10

0

10

20

30

time

posi

tion

Fick’s law: When there are a large number of these molecules,their motion can be described by

Math Physiology – p.5/24

Page 10: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

How Things Move

Most molecules move by a random walk:

0 10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

time

posi

tion

0 10 20 30 40 50 60 70 80 90 100−30

−20

−10

0

10

20

30

time

posi

tion

Fick’s law: When there are a large number of these molecules,their motion can be described by

J = − D∂C

∂x

Math Physiology – p.5/24

Page 11: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

How Things Move

Most molecules move by a random walk:

0 10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

time

posi

tion

0 10 20 30 40 50 60 70 80 90 100−30

−20

−10

0

10

20

30

time

posi

tion

Fick’s law: When there are a large number of these molecules,their motion can be described by

J = − D∂C

∂x

molecular flux,Math Physiology – p.5/24

Page 12: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

How Things Move

Most molecules move by a random walk:

0 10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

time

posi

tion

0 10 20 30 40 50 60 70 80 90 100−30

−20

−10

0

10

20

30

time

posi

tion

Fick’s law: When there are a large number of these molecules,their motion can be described by

J = − D∂C

∂x

molecular flux, diffusion coefficient,Math Physiology – p.5/24

Page 13: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

How Things Move

Most molecules move by a random walk:

0 10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

time

posi

tion

0 10 20 30 40 50 60 70 80 90 100−30

−20

−10

0

10

20

30

time

posi

tion

Fick’s law: When there are a large number of these molecules,their motion can be described by

J = − D∂C

∂x

molecular flux, diffusion coefficient, concentration gradient.Math Physiology – p.5/24

Page 14: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Conservation Law

Conservation:∂C

∂t+

∂J

∂x= 0

leading to the Diffusion Equation

∂C

∂t=

∂x(D

∂C

∂x).

Math Physiology – p.6/24

Page 15: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Basic Consequences - I

Diffusion in a tube fed by a reservoir

C(x, t) = f(x2

Dt)

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

C

Math Physiology – p.7/24

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theImagine Possibilities

Basic Consequences - II

Diffusion time:t = x2

Dfor hydrogen (D = 10−5cm2/s).

x t Example

10 nm 100 ns cell membrane

1 µm 1 ms mitochondrion

10 µm 100 ms mammalian cell

100 µm 10 s diameter of muscle fiber

250 µm 60 s radius of squid giant axon

1 mm 16.7 min half-thickness of frog sartorius muscle

2 mm 1.1h half-thickness of lens in the eye

5 mm 6.9 h radius of mature ovarian follicle

2 cm 2.6 d thickness of ventricular myocardium

1 m 31.7 yrs length of sciatic nerve

Math Physiology – p.8/24

Page 17: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Basic Consequences - Ohm’s Law

Diffusion across a membrane

J = AD

L(C1 − C2)

��

����

����

����

���

V

C2

1 V2L

C1

J

Flux changes as things like C1, C2 and L change.

Math Physiology – p.9/24

Page 18: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Quorum Sensing

Quorum sensing: The ability of bacteria to respond to theirpopulation size. Question: How do bacteria conduct a census?

EA

Solution: Autoinducer (HSL)- a freely diffusing chemical withauto-catalytic (positive feedback) production.

3−oxo−C12−HSL

LasR

rsaL

LasI

RsaL

LasR

C4−HSL

LasR

Vfr

rhlR

RhlR

RhlR

RhlI

rhlI

lasR

lasI

Math Physiology – p.10/24

Page 19: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

University of UtahMathematical Biology

theImagine Possibilities

Quorum Sensing

Quorum sensing: The ability of bacteria to respond to theirpopulation size. Question: How do bacteria conduct a census?

E

A

E

A

Solution: Autoinducer (HSL)- a freely diffusing chemical withauto-catalytic (positive feedback) production.

3−oxo−C12−HSL

LasR

rsaL

LasI

RsaL

LasR

C4−HSL

LasR

Vfr

rhlR

RhlR

RhlR

RhlI

rhlI

lasR

lasI

Math Physiology – p.10/24

Page 20: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Quorum Sensing-II

E

A

E

A

dA

dt= F (A) + δ(E −A)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)

Math Physiology – p.11/24

Page 21: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Quorum Sensing-II

E

A

E

A

dA

dt= F (A) + δ(E −A)

rate of change of A,

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)

Math Physiology – p.11/24

Page 22: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Quorum Sensing-II

E

A

E

A

dA

dt= F (A) + δ(E −A)

rate of change of A, production rate,

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)

Math Physiology – p.11/24

Page 23: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Quorum Sensing-II

E

A

E

A

dA

dt= F (A) + δ(E −A)

rate of change of A, production rate, diffusive loss.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)

Math Physiology – p.11/24

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theImagine Possibilities

Quorum Sensing-III

Extracellular Autoinducer E:

E

A

dE

dt= − kEE + δ(A− E)

Math Physiology – p.12/24

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theImagine Possibilities

Quorum Sensing-III

Extracellular Autoinducer E:

E

A

dE

dt= − kEE + δ(A− E)

rate of change of E,

Math Physiology – p.12/24

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theImagine Possibilities

Quorum Sensing-III

Extracellular Autoinducer E:

E

A

dE

dt= − kEE + δ(A− E)

rate of change of E, degradation rate,

Math Physiology – p.12/24

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theImagine Possibilities

Quorum Sensing-III

Extracellular Autoinducer E:

E

A

dE

dt= − kEE + δ(A− E)

rate of change of E, degradation rate, diffusive source,

Math Physiology – p.12/24

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theImagine Possibilities

Quorum Sensing-III

Extracellular Autoinducer E:

E

A

(1− ρ) (dE

dt+ KEE) = ρ δ(A− E)

rate of change of E, degradation rate, diffusive source,density dependence.

Math Physiology – p.12/24

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theImagine Possibilities

Quorum Sensing-IV

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

4

5

Internal Autoinducer

Ext

erna

l aut

oind

ucer

A nullclineE nullcline

dA

dt= F (A) + δ(E −A), E = A−

1

δF (A)

dE

dt= −kEE +

ρ

1− ρδ(A−E) A = (

1− ρ

ρδkE + 1)E

Math Physiology – p.13/24

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theImagine Possibilities

Carrier Mediated Diffusion

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

Ce

−→

← Ci

Se + Ce

← Pe

← Pi

← Si + Ci

For this system,

J = Jmax

Se − Si

(Se + Ke)(Si + Ki).

Math Physiology – p.14/24

Page 31: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Carrier Mediated Diffusion

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

Ce

−→

← Ci

Se + Ce

← Pe

← Pi

← Si + Ci

For this system,

J = Jmax

Se − Si

(Se + Ke)(Si + Ki).

Math Physiology – p.14/24

Page 32: Introduction to Physiology II: Control of Cell Volume …keener/lectures/Cambridge_workshop/...University of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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theImagine Possibilities

Carrier Mediated Diffusion

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

Ce

−→

← Ci

Se + Ce

← Pe

← Pi

← Si + Ci

For this system,

J = Jmax

Se − Si

(Se + Ke)(Si + Ki).

Math Physiology – p.14/24

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Carrier Mediated Diffusion

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

Ce

−→

← Ci

Se + Ce

← Pe

← Pi

← Si + Ci

For this system,

J = Jmax

Se − Si

(Se + Ke)(Si + Ki).

Math Physiology – p.14/24

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theImagine Possibilities

Carrier Mediated Diffusion

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

Ce

−→

← Ci

Se + Ce

← Pe

← Pi

← Si + Ci

For this system,

J = Jmax

Se − Si

(Se + Ke)(Si + Ki).

Math Physiology – p.14/24

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theImagine Possibilities

Carrier Mediated Diffusion

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

Ce

−→

← Ci

Se + Ce

← Pe

← Pi

← Si + Ci

For this system,

J = Jmax

Se − Si

(Se + Ke)(Si + Ki).

Math Physiology – p.14/24

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University of UtahMathematical Biology

theImagine Possibilities

Carrier Mediated Diffusion

� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �

Ce

−→

← Ci

Se + Ce

← Pe

← Pi

← Si + Ci

For this system,

J = Jmax

Se − Si

(Se + Ke)(Si + Ki).

Math Physiology – p.14/24

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Ion Movement

Ions move according to the Nernst-Planck equation

J = −D(∇C +Fz

RT∇φ)

Consequently, at equilibrium

VN = Vi − Ve =RT

zFln

(

[C]e[C]i

)

i

Ve Vi

extracellular intracellular

[C] [C]e

This is called the Nernst Potential or Reversal Potential.

Math Physiology – p.15/24

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Ion Current Models

The two most popular ion current models

Iion = g(V − VN ) Linear Model

Iion = PF 2

RTV

(

[C]i − [C]e exp(−zV F

RT)

1− exp(−zV F

RT

)

, GHK Model

Both of these have the same reversal potential, as they must.

Math Physiology – p.16/24

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Sodium-Potassium ATPase

Math Physiology – p.17/24

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Osmotic Pressure and Flux

rQ = P1 − P2 − π1 + π2

πi = kTCi

osmolite

P P1 2

C C1 2

Math Physiology – p.18/24

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Volume Control; Pump-Leak Model

Na+ is pumped out, K+ is pumped in, Cl− moves passively, negativelycharged macromolecules are trapped in the cell.

Math Physiology – p.19/24

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theImagine Possibilities Charge Balance and Osmotic

Balance

• Inside and outside are both electrically neutral,macromolecules have negative charge zx.

qw(Ni+Ki−Ci)+zxqX = qw(Ne+Ke−Ce) = 0, (charge balance)

• Total amount of osmolyte is the same on each side.

Ni + Ki + Ci +X

w= Ne + Ke + Ce (osmotic balance)

Math Physiology – p.20/24

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The Solution

The resulting system of algebraic equations is readily solved5

4

3

2

1

0

Ce

ll vo

lum

e

14121086420

Pump rate

-100

-50

0

50

100

Po

ten

tia

l (m

V)

14121086420

Pump rate

VNa

VK

V

• If the pump stops, the cell bursts, as expected.

• The minimal volume gives approximately correct membranepotential (although there are MANY deficiencies with thismodel.) Math Physiology – p.21/24

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theImagine Possibilities Interesting Problems (suitable for

projects)

• How do organism (e.g., T. Californicus living in tidal basins)adjust to dramatic environmental changes?

• How do plants in arid, salty regions, prevent dehydration?(They make proline)

• How do fish (e.g., salmon) adjust to both freshwater and saltwater?

• What happens to a cell and its environment when there isischemia (loss of ATP)?

• How do cell in high salt environments (epithelial cell inkidney) maintain constant volume?

Math Physiology – p.22/24