Gas Channels Workshop

1. Gas Channels Workshop September 7, 2012 Cleveland, Ohio Mathematical Modeling of Gas Movements in an Oocyte Rossana Occhipinti, Ph.D. Department of Physiology & Biophysics Case Western Reserve University School of Medicine 10900 Euclid Avenue Cleveland, OH 44106-4906

2. [CO2]S 7.7 pHS 7.5 2 min pH 7.3 7.0 Xenopus oocyte:pH Changes Caused by CO2 Influx pHi H+ H2O CO2 CO2 HCO3– CO2 H2O [HCO3–] H+ 1.5% CO2 / 10 mM HCO3– HCO3– HCO3– pHS Bulk Extracellular Fluid (BECF) pHi (data kindly provided by Dr. Musa-Aziz)

3. An appropriate mathematical model should include… 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

4. d CO2 Free Diffusion H2O H2O CO2 CO2 CO2 H2O H2O H2O H2O H2O H2O H2CO3 H2CO3 H2CO3 H2CO3 HCO3- + Intracellular Fluid (ICF) HCO3- + HCO3- + HCO3- + Bulk Extracellular Fluid (BECF) + H+ + + + H+ H+ H+ Extracellular Unconvected Fluid (EUF) TheMathematical Model Somersalo, Occhipinti, Boron, Calvetti, J Theor Biol, 2012

5. 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

6. d CO2 Free Diffusion H2O H2O CO2 CO2 CO2 H2O H2O H2O H2O H2O H2O H2CO3 H2CO3 H2CO3 H2CO3 HCO3- + Intracellular Fluid (ICF) HCO3- + HCO3- + HCO3- + Bulk Extracellular Fluid (BECF) + H+ + + + H+ H+ H+ Extracellular Unconvected Fluid (EUF)

7. R∞ BECF EUF R Oocyte Assuming spherical symmetry, we write a reaction-diffusion equation for each species j, with r distance from the center of the oocyte Reaction term (law of mass action) Diffusion term (Fick’s second law)

8. R∞ R R Methodof Lines t r R∞ = rN R rj r1 r2 r3 r0= 0 Intracellular fluid (ICF) Extracellular Unconvected Fluid (EUF) Center of Cell Somersalo, Occhipinti, Boron, Calvetti, J Theor Biol, 2012

9. Numerical Experiments Assumptions • 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 mobilenon-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

10. (B) (C) (A) (D) (E) (F) Results Extracellular concentration-time profiles for solutes

11. (B) (C) (A) (D) (E) (F) Intracellular concentration-time profiles for solutes

12. Effects of Decreasing CO2 Membrane Permeability -3 x10 (A) 8 (C) 7.508 6 7.20 7.506 max ) 4 7.15 pHS -3 7.504 x 10 D pHS ( pHi 2 7.10 3 7.502 0 7.05 2 7.500 PM,CO PM,CO (cm/sec) (cm/sec) 0 200 400 600 800 1000 1200 dpHi/dt -( ) 2 2 7.00 max 1 Time (sec) 0 200 400 600 800 1000 1200 Time (sec) (B) (D) -2 0 2 -4 10 10 10 10 0 -4 -2 0 2 10 10 10 10

13. 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

14. Effects of Changing the Width of the EUF The EUF is a generalization of the concept of unstirred layer (UL) R∞ BECF 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) EUF d R 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 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

15. (A) (C) 7.515 m d = 150 m 7.20 m d = 100 m m d = 50 m 7.15 7.510 m d = 25 m max ) m pHS pHi d = 10 m pHS 7.10 m d = 5 m D ( 7.505 m d = 1 m 7.05 7.500 7.00 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Time (sec) Time (sec) dpHi/dt -( ) max (B) (D) -3 x 10 0.015 8 7 0.010 6 5 0.005 4 3 0 0 50 100 150 0 50 100 150 m m d ( m) d ( m)

16. pHS Implications There is competition between diffusion and reaction in replenishing the lost CO2 near the outer surface of the oocyte diffusion CO2 H2O – H+ HCO3 • We quantify this competition by introducing the diffusion reaction ratio (DRR) pH electrode DRR rises as the width d of the EUF decreases

17. 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

18. No Vit Membrane =1/4 =0.25 g g 1/ 1/ = 1/16 = 0.06 g g 1/ 1/ g 1/ = 0.03 g g 1/ 1/ = 0.12 = 1/8 g g 1/ 1/ = 1/2 = 0.50 7.56 No VitMemb 0.06 max 7.54 ) 0.04 pHS pHS D ( 7.52 0.02 g 1/ = 1/32 7.50 0 0 0.5 1 0 200 400 600 800 g 1/ Time (sec) 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

19. pHS 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 pHS spike Because the pHS electrode creates a special environment with restricted diffusion,our implementation of the vitelline membrane somehow mimics this environment diffusion CO2 CO2 pHS electrode H2O HCO3- H+ diffusion CO2

20. Conclusions The model can reproduce the pH transients observed experimentally • The simulations predict that: • The background permeability of the oocyte membrane must be very low • 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

21. 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 pHSelectrode

22. Acknowledgments Principal Investigator Walter F. Boron, M.D., Ph.D. Collaborators ErkkiSomersalo, Ph. D.(CWRU) Daniela Calvetti, Ph. D. (CWRU) Raif Musa-Aziz, Ph.D. (University ofSao Paulo)