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Gas Channels WorkshopSeptember 7, 2012
Cleveland, Ohio
Mathematical Modeling of Gas Movements in an Oocyte
Department of Physiology & BiophysicsCase Western Reserve University School of Medicine
10900 Euclid AvenueCleveland, OH 44106-4906
Rossana Occhipinti, Ph.D.
CO2CO2
HCO3–
H+
H2O
HCO3–
CO2
H2OHCO3
–
H+
pHS
[CO2]S
Bulk Extracellular Fluid (BECF) 2 min
pH7.5
7.7
7.3
7.0
1.5% CO2 / 10 mM HCO3
–
pHS
pHi
pHi
(data kindly provided by Dr. Musa-Aziz)
[HCO3–]
Xenopus oocyte:pH Changes Caused by CO2 Influx
• A spherical cell• Transport of CO2 across the plasma membrane
• Reactions of a multitude of extra- and intracellular buffers• Diffusion of solutes through the extra- and intracellular
spaces• Temporal and spatial variations of solute concentrations• Carbonic anhydrase (CA) activity at specific loci
An appropriate mathematical model should include…
Intracellular Fluid(ICF)
HCO3-
+
A
HA
H+
+
CO2
H2OH2O
1k1k
2k 2k
2k 2k
H2CO3
HCO3-
+
A
HA
+
CO2
H2OH2O
1k1k
2k 2k
2k 2k
H2CO3
H+
HCO3-
+
A
HA
H+
+
CO2
H2OH2O
1k1k
2k 2k
2k 2k
H2CO3
HCO3-
+
A
HA
+
CO2
H2OH2O
1k1k
2k 2k
2k 2k
H2CO3
H+
Extracellular Unconvected Fluid (EUF)
Free Diffusion
Bu
lk E
xtra
cellu
lar
Flu
id (
BE
CF
)
d
The Mathematical Model
Somersalo, Occhipinti, Boron, Calvetti, J Theor Biol, 2012
The Key Components of the Model
Bulk extracellular fluid (BECF)Infinite reservoir where convection could occur but not reaction or diffusion
Extracellular unconvected fluid (EUF) Thin layer adjacent to the surface of the oocyte where no convection occurs, but reactions and diffusion do occur
Plasma membrane Infinitely thin and permeable only to CO2
In both EUF and intracellular fluid (ICF) Slow equilibration of the CO2 hydration/dehydration reactions
Competing equilibria among the CO2/HCO3– and a multitude of
non-CO2/HCO3– buffers
Intracellular Fluid(ICF)
HCO3-
+
A
HA
H+
+
CO2
H2OH2O
1k1k
2k 2k
2k 2k
H2CO3
HCO3-
+
A
HA
+
CO2
H2OH2O
1k1k
2k 2k
2k 2k
H2CO3
H+
HCO3-
+
A
HA
H+
+
CO2
H2OH2O
1k1k
2k 2k
2k 2k
H2CO3
HCO3-
+
A
HA
+
CO2
H2OH2O
1k1k
2k 2k
2k 2k
H2CO3
H+
Extracellular Unconvected Fluid (EUF)
Free Diffusion
Bu
lk E
xtra
cellu
lar
Flu
id (
BE
CF
)
d
Assuming spherical symmetry, we write a reaction-diffusion equation for each species j,
with r distance from the center of the oocyte
Diffusion term(Fick’s second law)
Reaction term(law of mass action)
2
2
1
,1
1( , ) (( , ) , ),j jj
L
jL
C r t D r S rrt
tr r
tC r 0 ,r R R
R
R∞
Oocyte
EUFBECF
R
R∞
R
r0= 0 Rr
t
r1 r2 rj R∞ = rN
( , )jc t r
Method of Lines
r3
Intracellular fluid (ICF) ExtracellularUnconvectedFluid (EUF)Center of
Cell
Somersalo, Occhipinti, Boron, Calvetti, J Theor Biol, 2012
Numerical Experiments
• The BECF, EUF, ICF and plasma membrane have same properties as water • The EUF has thickness d = 100 µm • Small CA-like activity uniformly distributed inside the oocyte and on the surface
of the plasma membrane • The BECF and EUF - contain 1.5% CO2/9.9 mM HCO3
– / pH 7.50
- have a single mobile non-CO2/HCO3– buffer with pK = 7.5 (e.g., HEPES) and
[TA] = 5mM • The ICF
- has initial pHi = 7.20
- [CO2] = [H2CO3] = [HCO3– ] = 0 mM
- has a single mobile non-CO2/HCO3 – buffer with pK = 7.10 and [TA] ≈ 27.31mM
Assumptions
ResultsExtracellular concentration-time profiles for solutes
(A) (B) (C)
0 200 400 600 800 100012001.27
1.28
1.29
1.3
1.31x 10
-3H
2CO
3
Time (sec)
Con
cent
ratio
n (m
M)
0 200 400 600 800 10001200
9.87
9.88
9.89
9.9
9.91
HCO3-
Time (sec)
Con
cent
ratio
n (m
M)
(F)(D) (E)
0 200 400 600 800 1000 12007.5
7.502
7.504
7.506
7.508
7.51pH
Time (sec)0 200 400 600 800 10001200
2.475
2.48
2.485
2.49
2.495
2.5
2.505
HA1
Time (sec)
Con
cent
ratio
n (m
M)
0 200 400 600 800 100012002.5
2.505
2.51
2.515
2.52
2.525
A-1
Time (sec)
Con
cent
ratio
n (m
M)
0 200 400 600 800 1000 12000
0.1
0.2
0.3
0.4
0.5
CO2
Time (sec)
Con
cent
ratio
n (m
M)
r = 651 mr = 670 mr = 690 mr = 710 mr = 730 mr = 750 m
2 2 2 3 3
1 1
CO H O H CO HCO H
HA A H
ƒ ƒ
ƒ
(F)(D) (E)
0 200 400 600 800 1000 12007
7.05
7.1
7.15
7.2
pH
Time (sec)0 200 400 600 800 10001200
12
13
14
15
HA1
Time (sec)
Con
cent
ratio
n (m
M)
0 200 400 600 800 1000120012
13
14
15
A-1
Time (sec)C
once
ntra
tion
(mM
)
(A) (B) (C)
0 200 400 600 800 1000 12000
0.5
1
1.5x 10
-3H
2CO
3
Time (sec)
Con
cent
ratio
n (m
M)
0 200 400 600 800 1000 12000
1
2
3
HCO3-
Time (sec)
Con
cent
ratio
n (m
M)
0 200 400 600 800 1000 12000
0.1
0.2
0.3
0.4
0.5
CO2
Time (sec)
Con
cent
ratio
n (m
M)
r 8 mr 160 mr 320 mr 480 mr 640 mr = 650 m
Intracellular concentration-time profiles for solutes 2 2 2 3 3
1 1
CO H O H CO HCO H
HA A H
ƒ ƒ
ƒ
0 200 400 600 800 1000 12007.500
7.502
7.504
7.506
7.508
Time (sec)
pHS
2M,CO 34.2 cm/secP =
2
1M,CO /10P
2
2M,CO /10P
2
3M,CO /10P
2
4M,CO /10P
2
4M,CO / 2.5 10P ×
2
4M,CO / 5.0 10P ×
2
4M,CO / 7.5 10P ×
2
5M,CO /10P
(A)
0 200 400 600 800 1000 12007.00
7.05
7.10
7.15
7.20
Time (sec)
pHi
(C)
10-4
10-2
100
1020
2
4
6
8
(DpHS)max
PM,CO 2 (cm/sec)
x 10-3 (B) (D)
0
x 10-3
10-4
10-2
100
102
1
2
3
-(dpHi/dt )max
PM,CO 2 (cm/sec)
Effects of Decreasing CO2 Membrane Permeability
Implications
The background permeability of the membrane (i.e., in the absence of gas channels) must be very low
Given a sufficiently small PM,CO2, gas channels could contribute to CO2 permeability even in the presence of a large d (in our numerical experiments d = 100µm)
With additional refinements to the model, we ought to be able to estimate absolute permeabilities
ULs are thin, diffuse layers of fluid, always present near the surface of solid bodies immersed in a fluid, where molecules move predominantly via diffusion (Dainty and House, J Physiol, 1966; Korjamo et al, J Pharm Sci, 2009)
The EUF is a generalization of the concept of unstirred layer (UL)
R
R∞EUF
BECF
d
Oocyte
For a particular solute, the width of the UL ( ) is defined as
where D is the diffusion constant and P is the empirically measured permeability
D
P
Effects of Changing the Width of the EUF
The width of the UL:1. A steady-state concept2. Solute-dependent3. Ignores the effects of chemical reactions
It is because our system is dynamic, involves multiples solutes, and solutes can react in the “UL”, that we decided to define the EUF
(A)
0 200 400 600 800 1000 1200Time (sec)
7.500
7.505
7.510
7.515
pHS
d = 150 mmd = 100 mm
d = 50 mm
d = 25 mm
d = 10 mm
d = 5 mmd = 1 mm
0 50 100 150d (mm)
0
0.005
0.010
0.015
(DpHS)max
0 200 400 600 800 1000 1200Time (sec)
7.00
7.05
7.10
7.15
7.20
pHi
0 50 100 1503
4
5
6
7
8 x 10-3
d (mm)
-(dpHi/dt)max
(B) (D)
(C)
pHS
H+
CO2
H2O
–HCO3
diffusion
pH electrode
Implications
There is competition between diffusion and reaction in replenishing the lost CO2 near the outer surface of the oocyte
DRR rises as the width d of the EUF decreases
We quantify this competition by introducing the diffusion reaction ratio (DRR)
2
2
rate of CO replenisced by diffusionDRR=
rate of CO produced by reaction
The Vitelline Membrane: pHS Spike
Additional diffusion barrier to the movement of solutes
Implemented by reducing the mobility D of each solute near the outer surface of the oocyte by the same factor γ, i.e., D* = D/γ
As we increase γ, the maximal height of the pHS spike, (ΔpHS)max, increases
Implementation of the vitelline membrane reduces the contribution of diffusion and enhances the contribution of reaction at the surface
1/g = 0.031/g = 0.06
1/g = 0.121/g = 0.25
1/g = 0.50
No Vit Membrane
0 200 400 600
7.50
7.52
800
7.54
7.56
Time (sec)0 0.5 1
0
0.02
0.04
0.06
1/g
1/g = 1/321/g = 1/16
1/g = 1/81/g = 1/4
1/g = 1/2
No Vit Memb
pHS(DpHS
)max
ImplicationsImplementation of the vitelline membrane – which reduces the contribution of diffusion and enhances the contribution of the reaction – can explain the height of the pHS spike
Because the pHS electrode creates a special environment with restricted diffusion, our implementation of the vitelline membrane somehow mimics this environment
diffusion
H+
CO2H2O
HCO3-
CO2
CO2
pHSdiffusion
pHS electrode
Conclusions
The model can reproduce the pH transients observed experimentally
The simulations predict that:
1. The background permeability of the oocyte membrane must be very low
2. Given a sufficiently small PM,CO2, gas channels could contribute to CO2 permeability even with a large EUF
The model provides new insights into the competition between diffusion and reaction processes near the outer surface of the plasma membrane
Future Directions
Apply the model to investigate the movements of ammonia and ammonium across the plasma membrane
Model the pHS electrode’s touching on the oocyte surface to explore the special environment underneath the pHS electrode
Acknowledgments
Principal InvestigatorWalter F. Boron, M.D., Ph.D.
CollaboratorsErkki Somersalo, Ph. D. (CWRU)Daniela Calvetti, Ph. D. (CWRU)Raif Musa-Aziz, Ph.D. (University of Sao Paulo)