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

Bul

k Ex

trac

ellu

lar F

luid

(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

Bul

k Ex

trac

ellu

lar F

luid

(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)

22

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 10-3 H2CO

3

Time (sec)

Con

cent

ratio

n (m

M)

0 200 400 600 800 10001200

9.87

9.88

9.89

9.9

9.91HCO

3-

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

2.505HA

1

Time (sec)

Con

cent

ratio

n (m

M)

0 200 400 600 800 100012002.5

2.505

2.51

2.515

2.52

2.525A-

1

Time (sec)

Con

cent

ratio

n (m

M)

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5CO

2

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 ƒƒ

(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

1.5 x 10-3 H2CO

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

0.5CO

2

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 ƒƒ

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

DP

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 md = 100 md = 50 md = 25 md = 10 md = 5 md = 1 m

0 50 100 150d (m)

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 (m)

-(dpHi/dt)max

(B) (D)

(C)

pHS

H+

CO2H2O

–HCO3

diffusion

pH electrode

ImplicationsThere 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 = 0.061/g = 0.121/g = 0.251/g = 0.50No 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 = 1/161/g = 1/81/g = 1/41/g = 1/2No 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

ConclusionsThe 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

AcknowledgmentsPrincipal 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)