'pH' in a small volume?

tivol at tethys.ph.albany.edu tivol at tethys.ph.albany.edu
Fri Sep 23 10:40:18 EST 1994


Dear Peter,
	The time average for pH is, indeed, different for different parts of a
small volume in the sense that the pH within an active site of a protein is
different from that in the solvent.  However, the pH within the solvent will
be the same for any part of the solvent--at least, my intuitive view of what
goes on in free water (with solutes, possibly) will lead to a uniform time
average of conditions.  All bets are off for bound water, etc., but the time
average still makes sense.
	The parameter most directly connected to the rate and/or direction of a
chemical process is the chemical potential (or equivalently, the free en-
ergy).  These parameters can be evaluated by "taking a snapshot" of the vol-
ume at many times and counting the numbers of product and reactant molecules
seen.  The ratios give a measure of the rates, equilibrium constants, etc.
	Mapping these values near proteins and membranes is certainly quite
valuable.  After all, these maps are the keys to understanding the catalytic
mechanisms.  E.g., for an isomerase which translocates a proton, the proton 
must leave a group on a reactant, reside in an intermediate, and become bound
to the appropriate location on a product.  In the case of delta-5-3-ketoster-
oid isomerase, the proton leaves C4 (as I recall) and binds to a glutamic
acid residue.  For this to occur rapidly, the pH of the Glu must be unusually
high.  The Glu then is thought to rotate so that the proton is translocated
to a position near C5, and then the proton binds to C5.  Under these condi-
tions, the pH of the Glu either changes, or the pH of C5 is much higher than
C4.  In either case, the free energy favors the proton on C5 where two double
bonds become conjugated.
	As far as proton transport across a membrane is concerned, the time av-
erage must be over a period large enough so that any pH changes don't see it.
If the small volume is at steady state, it makes no sense to define pH as a
varying quantity.  In steady state, there will be fluctuations as protons are
transported and other metabolic events occur, but the consequence of a proton
transported inward is to drive reactions in a direction which results in a
proton being bound or exiting the volume, thus restoring the initial condi-
tions--this is essentially the definition of steady state.  The time averages
for steady state, therefore, must be over large enough times so that these
fluctuations are averaged out.  If the system is not at or near steady state,
the thermodynamic parameters have no definition.
	In fact, the definitions of pH, temperature, etc. are strictly for equ-
ilibrium.  They can be extended to steady state and adiabatic transitions, 
but no further.  The way to tell whether a transition is adiabatic is to see
whether it procedes through a succession of quasi-equilibrium states.  Thus
a process is adiabatic if it stays near equilibrium where the thermodynamic
quantities are defined and vice versa.
	It is, of course, true that active transport is stochiometric; however,
the rate at which any given transport protein molecule cycles depends on when
a proton collides with the outside of the pore, when a molecule of ATP col-
lides with the binding site, the fraction of these collisions which lead to
binding, etc.  So, if there are fewer protons outside or more protons inside,
the rate will slow (if enough protons are inside, they will synthesize ATP 
and be transported outside).
	Once again, all these events are fluctuations, and for the concept of
pH to be valid, they must be averaged out.
	To answer your last point, the relative rates of transport vs catalysis
depend both on the specific enzymes involved and on the conditions within the
volume under consideration.  Some enzymes are diffusion limited, whereas
others have their turnover rates limited be some step in the catalytic pro-
cess.  For complex coupled systems, both the enzymes and the conditions can
be modified by components outside the small volume, so generalizations are
difficult.  The most useful concept is that of equilibrium/steady-state/
adiabatic-transition with fluctuations.
					Yours,
					Bill Tivol



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