[balthazargabka]

Hi, I'm a mathematician and I want to describe the dynamical changes in HCO₃^-and H⁺ due to transmembrane ionic fluxes and Carbonic Anhydrase activity.
The reaction is

H₂O + CO₂⇌H₂CO₃→HCO₃^- +  H⁺

I've found some papers, including one titled *Catalytic Properties of Murine Carbonic Anhydrase VII*, but I don't know how to use the rates they describe to get sets of coupled ordinary differential equations.

The paper states in it's into.
> Unlike the other isozymes of this class of CA, for
> which kcat/Km is described by the single ionization of zinc-bound water, CA VII exhibits a pH profile for
> kcat/Km for CO2 hydration described by two ionizations at pKa 6.2 and 7.5, with a maximum approaching
> 8  107 M-1 s-1. The pH dependence of kcat/Km for the hydrolysis of 4-nitrophenyl acetate could also
> be described by these two ionizations, yielding a maximum of 71 M-1 s-1 at pH >9. Using a novel
> method that compares rates of 18O exchange and dehydration of HCO3
> -, we assigned values for the apparent
> pKa at 6.2 to the zinc-bound water and the pKa of 7.5 to His 64. The magnitude of kcat, its pH profile,18O-exchange data for both wild-type and a H64A mutant, and inhibition by CuSO4 and acrolein suggest
> that the histidine at position 64 is functioning as a proton-transfer group and is responsible for one of the
> observed ionizations.

[Babcock_Hall]

I am not sure I can help much, but I will do what I can.  This enzyme produces bicarbonate when the red blood cell (RBC) is in respiring tissue and to produce carbon dioxide when the RBC is in the lungs.  The chloride/bicarbonate exchanger on the surface of the RBC is also an integral part of the process of the transport of carbon dioxide.  I am guessing that the terminology such as k_cat/K_m is unfamiliar to you.  If so, perhaps we could discuss that.  Or perhaps you were wondering about the effect of pH on the rate.

[balthazargabka1]

Hi Babcock_Hall,
I'm writing from a different account because I lost the log-in data.
I conceptually understand what the enzyme does but I don't understand the terminology and therefore can simulate the process. I'm working on this for a summer school so I don't have the time to properly read the chemistry literature.
I would appreciate if you could help me understand the terminology and also explain if the process is self limiting, by which I mean: given a certain pH and CO2 and HCO3 concentration, the activity of Carbonic Anhydrase should come to a steady state. Such a reaction wouldn't be described by Michaelis-Menten dynamics, and I'm not sure how what the underlying differential relations are.

[Babcock_Hall]

I am still not entirely following you, but let me make some general comments.  One, when Michaelis and Menten first derived their equation, they assumed a pseudo-equlibrium condition.  But when Briggs and Haldane reexamined the question, they found that a steady-state assumption also leads to the same saturation kinetics and the same equation relating rate to concentration of substrate.  There have been many papers on this.  Here is one that uses some differential equations within a tutorial to help students visualize the pre-steady state, from the Journal of Chemical Education:  http://eric.ed.gov/?id=EJ820736  Under normal conditions an enzyme rapidly (less than one second) reaches steady state.

However, what you may be interested in is a case where the reaction is reversible and the process therefore comes to thermodynamic equilibrium, which chemists differentiate from the steady-state.  I would assume that carbonic anhydrase is working under something close to equilibrium conditions, something that is not true for every enzyme.  I am not sure that my assumption is correct, but there might be some literature available on this subject.  I think that you might need a form of the Michaelis-Menten equation that explicitly includes the concentration of product, and allows for the reverse reaction.  The enzyme kinetics textbook by Athel Cornish-Bowden may have this equation.

[balthazargabka1]

My problem is that I have 2 processes in a neuron that affect bicarbonate concentrations, and pH. I assume the CO₂ concentration is constant, because of the time scales I want to consider. The activity of the anhydrase, should then balance the efflux of bicarbonate and the pH buffering. I think that in order to capture those dynamics the reaction has to be reversible.
1) I don't understand if the Carbonic Anhydrase is reversible.
2) If it is, how do incorporate the HCO₃ and pH dependence. Intuitively the reaction should tend to an equlibrium HCO₃ concentration of 13mMolar.

[Babcock_Hall]

I think that the reversibility boils down to the particulars of bicarbonate and carbon dioxide concentrations at the neural cell.  I don't know the answer myself.  What I would suggest is using a reliable search engine such as PubMed.  It might be that there exists a decent review article that would provide some information, for example:  Carbonic anhydrase: chemistry, physiology, and inhibition, T. H. Maren
Physiological ReviewsPublished 1 October 1967, Vol. 47no. 595-781.  This one can be obtained for free:
Physiol Rev. 2003 Oct;83(4):1183-221.  Regulation and modulation of pH in the brain.  Chesler M.  I found 44 review articles using the search terms neuron AND carbonic anhydrase.  For example:  Subcell Biochem. 2014;75:271-90. doi: 10.1007/978-94-007-7359-2_14.
Carbonic anhydrases and brain pH in the control of neuronal excitability.  Ruusuvuori E1, Kaila K.