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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.1/23

Introduction to Physiology II: Control of Cell Volume …keener/lectures/MBI/mbi_2_homeo.pdfUniversity of Utah Mathematical Biology Imagine the Possibilities Introduction to Physiology

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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.1/23

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.2/23

University of UtahMathematical Biology

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.3/23

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.4/23

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.4/23

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.4/23

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.4/23

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.4/23

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.5/23

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.5/23

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.5/23

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.5/23

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.5/23

University of UtahMathematical Biology

theImagine Possibilities

Conservation Law

Conservation:∂C

∂t+

∂J

∂x= 0

leading to the Diffusion Equation

∂C

∂t=

∂x(D

∂C

∂x).

Math Physiology – p.6/23

University of UtahMathematical Biology

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.7/23

University of UtahMathematical Biology

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.8/23

University of UtahMathematical Biology

theImagine Possibilities

Basic Consequences - Ohm’s Law

Diffusion across a membrane

J = ADL

(C1 − C2)

��

����

����

����

���

V

C2

1 V2L

C1

J

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

Math Physiology – p.9/23

University of UtahMathematical Biology

theImagine Possibilities

Carrier Mediated Diffusion

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

Ce−→

← Ci

Se + Ce→

← Pe→

← Pi→

← Si + Ci

For this system,

J = JmaxSe − Si

(Se + Ke)(Si + Ki).

.... a saturating Fick’s law

Math Physiology – p.10/23

University of UtahMathematical Biology

theImagine Possibilities

Carrier Mediated Diffusion

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

Ce−→

← Ci

Se + Ce→

← Pe→

← Pi→

← Si + Ci

For this system,

J = JmaxSe − Si

(Se + Ke)(Si + Ki).

.... a saturating Fick’s law

Math Physiology – p.10/23

University of UtahMathematical Biology

theImagine Possibilities

Carrier Mediated Diffusion

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

Ce−→

← Ci

Se + Ce→

← Pe→

← Pi→

← Si + Ci

For this system,

J = JmaxSe − Si

(Se + Ke)(Si + Ki).

.... a saturating Fick’s law

Math Physiology – p.10/23

University of UtahMathematical Biology

theImagine Possibilities

Carrier Mediated Diffusion

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

Ce−→

← Ci

Se + Ce→

← Pe

← Pi→

← Si + Ci

For this system,

J = JmaxSe − Si

(Se + Ke)(Si + Ki).

.... a saturating Fick’s law

Math Physiology – p.10/23

University of UtahMathematical Biology

theImagine Possibilities

Carrier Mediated Diffusion

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

Ce−→

← Ci

Se + Ce→

← Pe→

← Pi

← Si + Ci

For this system,

J = JmaxSe − Si

(Se + Ke)(Si + Ki).

.... a saturating Fick’s law

Math Physiology – p.10/23

University of UtahMathematical Biology

theImagine Possibilities

Carrier Mediated Diffusion

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

Ce−→

← Ci

Se + Ce→

← Pe→

← Pi→

← Si + Ci

For this system,

J = JmaxSe − Si

(Se + Ke)(Si + Ki).

.... a saturating Fick’s law

Math Physiology – p.10/23

University of UtahMathematical Biology

theImagine Possibilities

Carrier Mediated Diffusion

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

Ce−→

← Ci

Se + Ce→

← Pe→

← Pi→

← Si + Ci

For this system,

J = JmaxSe − Si

(Se + Ke)(Si + Ki).

.... a saturating Fick’s lawMath Physiology – p.10/23

University of UtahMathematical Biology

theImagine Possibilities

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.11/23

University of UtahMathematical Biology

theImagine Possibilities

Ion Current Models

There are many different possible Models of Iionic.

• Barrier models, binding models, saturating models, PNPequations, etc.

• Constant field assumption:

Iion = PF 2

RTV

(

[C]i − [C]e exp(−zV FRT

)

1− exp(−zV FRT

)

, GHK Model

• Long Channel limit (used by HH)

Iion = g(V − VN ) Linear Model

All of these have the same reversal potential, as they must.Math Physiology – p.12/23

University of UtahMathematical Biology

theImagine Possibilities

Electrodiffusion Models

x=0 x=L

[S1+] = [S2

-] = ci[S1

+] = [S2-] = ce

S1

S2

Inside Outside

φ (0) = V φ (L) = 0

cell membrane

d2φ

dx2= −λ2(c1 − c2), Poisson Equation

J1 = −D1(dc1

dx+

F

RTc1

dx), Nernst Planck Equations

J2 = −D2(dc2

dx−

F

RTc2

dx),

Math Physiology – p.13/23

University of UtahMathematical Biology

theImagine Possibilities

Short Channel Limit

If the channel is short, then L ≈ 0⇒ λ ≈ 0. Then d2φdx2 = 0 implies

the field is constant:

dx= v ⇒

dc1

dx− vc1 = −J1

⇒ J1 = vci − cee

−v

1− e−v

⇒ Iion = PF 2

RTV

(

[C]i − [C]e exp(−zV FRT

)

1− exp(−zV FRT

)

This is the Goldman-Hodgkin-Katz equation.

Math Physiology – p.14/23

University of UtahMathematical Biology

theImagine Possibilities

Long Channel Limit

If the channel is long, then 1

L≈ 0⇒ 1

λ≈ 0. Then c1 ≈ c2

throughout the channel:

c1 = c2 ⇒ 2dc1

dx= −J1 − J2

⇒ c1 = c2 + (ce − ci)x

⇒ φ = −v

v1

ln

(

ci

ce+ (1−

ci

ce)x

)

v1 = Nernst potential

⇒ J1 =ce − ci

v1

(v − v1)

This is the linear I-V curve used by Hodgkin and Huxley.

Math Physiology – p.15/23

University of UtahMathematical Biology

theImagine Possibilities

Sodium-Potassium ATPase

Math Physiology – p.16/23

University of UtahMathematical Biology

theImagine Possibilities

Osmotic Pressure and Flux

rQ = P1 − P2 − π1 + π2

πi = kTCi

osmolite

P P1 2

C C1 2

Math Physiology – p.17/23

University of UtahMathematical Biology

theImagine Possibilities

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.18/23

University of UtahMathematical Biology

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.19/23

University of UtahMathematical Biology

theImagine Possibilities

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.20/23

University of UtahMathematical Biology

theImagine Possibilities

Volume Control and Ion Transport

• How can epithelial cellstransport ions and waterwhile maintaining constantcell volume under widelyvarying conditions?

• Spatial separation of leaksand pumps?

• Other intricate controlmechanisms are needed.

• Lots of interesting problems(A. Weinstein, BMB 54, 537,1992.)

3 Na+

2 K+

Mucosal

side

Serosal

side

Na+

Cl-

Cl-

Math Physiology – p.21/23

University of UtahMathematical Biology

theImagine Possibilities

Inner Meduullary Collecting Duct

• Real cells are far morecomplicated

• Notice the large Na+ fluxfrom the lumen.

• cf. A. Weinstein, Am. J.Physiol. 274, F841-F855,1998.

Math Physiology – p.22/23

University of UtahMathematical Biology

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.23/23