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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.1/24
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
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.3/24
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
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
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
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
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
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
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
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
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
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
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.6/24
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.7/24
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.8/24
University of UtahMathematical Biology
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
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?
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
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
University of UtahMathematical Biology
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
University of UtahMathematical Biology
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
University of UtahMathematical Biology
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
University of UtahMathematical Biology
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
University of UtahMathematical Biology
theImagine Possibilities
Quorum Sensing-III
Extracellular Autoinducer E:
E
A
dE
dt= − kEE + δ(A− E)
Math Physiology – p.12/24
University of UtahMathematical Biology
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
University of UtahMathematical Biology
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
University of UtahMathematical Biology
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
University of UtahMathematical Biology
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
University of UtahMathematical Biology
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
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
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
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
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
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
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
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
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.15/24
University of UtahMathematical Biology
theImagine Possibilities
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
University of UtahMathematical Biology
theImagine Possibilities
Sodium-Potassium ATPase
Math Physiology – p.17/24
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.18/24
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.19/24
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.20/24
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.21/24
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.22/24