Gas Channels Workshop September 7, 2012 - PowerPoint Presentation

Slide1

Gas Channels Workshop

September 7, 2012Cleveland, Ohio

Mathematical Modeling of Gas Movements in an Oocyte

Department of Physiology & BiophysicsCase Western Reserve University School of Medicine10900 Euclid AvenueCleveland, OH 44106-4906

Rossana Occhipinti, Ph.D.

CO

2

CO

2

HCO

3

–

H

+

H

2

O

HCO

3

–

CO

2

H

2

O

HCO

3

–

H

+

pH

S

[CO

2

]

S

Bulk Extracellular Fluid (BECF)

2 min

pH

7.5

7.7

7.3

7.0

1.5% CO

2

/ 10 mM HCO

3

–

pH

S

pH

i

pH

i

(data kindly provided by Dr. Musa-Aziz)



[

HCO

3

–

]

Xenopus

oocyte:

pH Changes Caused by CO

2

Influx

A spherical cellTransport of CO2 across the plasma membrane

Reactions of a multitude of extra- and intracellular buffersDiffusion of solutes through the extra- and intracellular spacesTemporal and spatial variations of solute concentrationsCarbonic anhydrase (CA) activity at specific loci

An appropriate mathematical model should include…

Intracellular

Fluid

(ICF)

HCO

3

-

+

H

+

+

CO

2

H

2

O

H

2

O

H

2

CO

3

HCO

3

-

+

+

CO

2

H

2

O

H

2

O

H

2

CO

3

H

+

HCO

3

-

+

H

+

+

CO

2

H

2

O

H

2

O

H

2

CO

3

HCO

3

-

+

+

CO

2

H

2

O

H

2

O

H

2

CO

3

H

+

Extracellular Unconvected Fluid

(EUF)

Free Diffusion

Bulk Extracellular Fluid

(BECF)

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 diffusionExtracellular unconvected fluid (EUF) Thin layer adjacent to the surface of the oocyte where no convection occurs, but reactions and diffusion do occurPlasma membrane

Infinitely thin and permeable only to CO2In both EUF and intracellular fluid (ICF) Slow equilibration of the CO2 hydration/dehydration reactionsCompeting equilibria among the CO

2/HCO3– and a multitude of non-CO2/HCO3

–

buffers

Intracellular

Fluid

(ICF)

HCO

3

-

+

H

+

+

CO

2

H

2

O

H

2

O

H

2

CO

3

HCO

3

-

+

+

CO

2

H

2

O

H

2

O

H

2

CO

3

H

+

HCO

3

-

+

H

+

+

CO

2

H

2

O

H

2

O

H

2

CO

3

HCO

3

-

+

+

CO

2

H

2

O

H

2

O

H

2

CO

3

H

+

Extracellular Unconvected Fluid

(EUF)

Free Diffusion

Bulk Extracellular Fluid

(BECF)

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)

R

R

∞

Oocyte

EUF

BECF

R

R

∞

R

r

0

=

0

R

r

t

r

1

r

2

r

j

R

∞

=

r

N

Method

of Lines

r

3

Intracellular fluid (ICF)

Extracellular

Unconvected

Fluid (EUF)

Center of Cell

Somersalo, Occhipinti, Boron, Calvetti,

J Theor Biol

, 2012

Numerical ExperimentsThe 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

/

HCO

3

–

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

Results

Extracellular concentration-time profiles for solutes

(A)

(B)

(C)

(F)

(D)

(E)

(F)

(D)

(E)

(A)

(B)

(C)

Intracellular concentration-time profiles for solutes

0

200

400

600

800

1000

1200

7.500

7.502

7.504

7.506

7.508

Time (sec)

pH

S

(A)

0

200

400

600

800

1000

1200

7.00

7.05

7.10

7.15

7.20

Time (sec)

pH

i

(C)

10

-4

10

-2

10

0

10

2

0

2

4

6

8

(

D

pH

S

)

max

P

M,CO

2

(cm/sec)

x

10

-

3

(B)

(D)

0

x 10

-3

10

-4

10

-2

10

0

10

2

1

2

3

-(

dpH

i

/dt

)

max

P

M,CO

2

(cm/sec)

Effects of Decreasing CO

2

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

Effects of Changing the Width

of

the EUF

The width of the

UL:

A steady-state concept

Solute-dependent

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

1200

Time (sec)

7.500

7.505

7.510

7.515

pH

S

d = 150

m

m

d = 100

m

m

d = 50

m

m

d = 25

m

m

d = 10

m

m

d = 5

m

m

d = 1

m

m

0

50

100

150

d (

m

m)

0

0.005

0.010

0.015

(

D

pH

S

)

max

0

200

400

600

800

1000

1200

Time (sec)

7.00

7.05

7.10

7.15

7.20

pH

i

0

50

100

150

3

4

5

6

7

8

x 10

-

3

d (

m

m)

-(

dpH

i

/dt

)

max

(B)

(D)

(C)

pH

S

H

+

CO

2

H

2

O

–

HCO

3

diffusion

pH electrode

Implications

There is competition between diffusion and reaction in replenishing the lost CO

2

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

)

The Vitelline Membrane: pHS Spike

Additional diffusion

barrier to the movement of solutesImplemented 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, increasesImplementation of the vitelline membrane reduces the contribution of diffusion and enhances the contribution of reaction at the surface

1/

g

= 0.03

1/

g

= 0.06

1/

g

= 0.12

1/

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

=

1/32

1/

g

=

1/16

1/

g

=

1/8

1/

g

=

1/4

1/

g

=

1/2

No

Vit

Memb

pH

S

(

D

pH

S

)

max

Implications

Implementation of the

vitelline

membrane – which reduces the contribution of diffusion and enhances the contribution of the reaction – can explain the height of the pH

S

spike

Because the

pH

S

electrode creates a special environment with restricted diffusion,

our implementation of the

vitelline

membrane somehow mimics this environment

diffusion

H

+

CO

2

H

2

O

HCO

3

-

CO

2

CO

2

pH

S

diffusion

pH

S

electrode

Conclusions

The model can reproduce the pH transients observed experimentally

The simulations predict that:

The background permeability of the oocyte membrane must be very lowGiven a sufficiently small

P

M,CO2

, gas channels could contribute to CO

2

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 Investigator

Walter 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)