[Sosi]

Suppose I have a protein with 2 active sites (thus, represented as PiPi) that reacts with a substrate X according to the following reactions.

(1) PiPi + X  PXPi
(2) PXPi + X  PXPX

(3) PXPX  PXPi
(4) PXPi  PiPi

Where X is the substrate, Pi is an unbound active site, and PX is a bound active site.

Reactions (1) and (2) occur with a rate constant ka, the rate constant for reaction of X with one of the active sites. Reactions (3) and (4) with a rate constant kb, the rate constant for "uncoupling" of one of the active sites. I want to write down the differential equations for all species and thus have the following questions:

A. The probability that X reacts with either active site in PiPi (reaction 1) is equal. This thus allows us to assume that the rate of consumption of X according to (1) is twofold higher than the rate of consumption of PiPi. Is this correct?

B. This would imply that the reaction rate of consumption of X according to (1) v1X = 2*ka*PiPi*X because X may react with either reactive sites (though it reacts with only one of the reactive sites at a time)? Conversely, in (2) X is consumed with reaction rate v2X = ka*PXPi*X because X only reacts with one active site in this reaction;

C. The reaction rate of consumption of PiPi according to (1) is v1P = ka*PiPi*X and according to (2) the consumption of PXPi is v2P = ka*PXPi*X?

D. Because in (3) either active site may "uncouple", the rate of consumption of PXPX is v3P = 2*kb*PXPX?

E. Conversely, the rate of consumption of PXPi according to (4) is v4P = kb*PXPi because X only uncouples from one active site.

I appreciate all the help in understanding these questions. Thank you so much

(also, I hope this isn't in the wrong section)

[Corribus]

The section is fine but your nomenclature and symbol choices are really hard to follow.  If the rates are the same regardless of whether the enzyme has bound an X or not, then it just comes down to a matter of probability.  The similar k values will simplify your overall rate expressions to some degree, but you still have to be careful to keep (1) and (2) functionally separate.

[Sosi]

@Corribus - Indeed, I need to keep (1) and (2) separate. Could you please comment on the veracity of the rates that I expressed?

Also, I will try to clear up the formatting in my post.

[Corribus]

Basically you're asking what the effect is with an enzyme that has two reactive sites - does that double the effective rate?

It's an interesting question and honestly I'm not sure what the answer is.  However, here is my thought: it comes down to what the rate constant is specified for.

On the one hand, the rate constant is usually specified for a specific overall reaction and is not conditional for the reaction happening at a specific site or in a specific way.  What I mean by that is that the probability of reactants colliding in a manner condusive to a reactive event occuring is already incorporated into the value of the rate constant (Arrhenius expression, if you want to model it that way).  Practically what this means is that the possibility of the reaction happening at site 1 or site 2 on the enzyme would already be incorproated into the rate constant expression, and so incorporating a factor of two in your rate expression would be redundant and erroneous.  The is probably one reason why enzymes which bind more than one substrate typically have different rate constants for the first and second binding events (not the only reason, though - binding of a substrate can induce structural changes that making binding of additional substrates more favorable).

That said, if the rate constant is specified for a single, independent binding site rather than the enzyme as a whole, then it would make sense to incorporate a factor of two to reflect the fact that the concentration of binding sites is effectively double in (1) than (2).  However, in this case it would be better to restate your reactions to reflect this - that is, ignore the fact that it is an enzyme with multiple binding sites and just speak of reactions between binding sites.  In this case, though, having two separate reactions would be unnecessary because we'd be treating the binding sites as wholly independent of each other.  This probably isn't a very accurate picture, but again it's something that would probably be implicitly incorporated into the rate constant.

[Sosi]

(...) what this means is that the possibility of the reaction happening at site 1 or site 2 on the enzyme would already be incorproated into the rate constant expression, and so incorporating a factor of two in your rate expression would be redundant and erroneous.  The is probably one reason why enzymes which bind more than one substrate typically have different rate constants for the first and second binding events (not the only reason, though - binding of a substrate can induce structural changes that making binding of additional substrates more favorable).(...)

Indeed, I also expect that substrate binding to an enzyme should occur with a different rate constant in many cases. That would be a cooperative process, either positive or negative cooperativity. Yet, in my specific case, I will consider the substrate binds any of the sites with a similar rate constant.

 

That said, if the rate constant is specified for a single, independent binding site rather than the enzyme as a whole, then it would make sense to incorporate a factor of two to reflect the fact that the concentration of binding sites is effectively double in (1) than (2).  However, in this case it would be better to restate your reactions to reflect this - that is, ignore the fact that it is an enzyme with multiple binding sites and just speak of reactions between binding sites.  In this case, though, having two separate reactions would be unnecessary because we'd be treating the binding sites as wholly independent of each other.  This probably isn't a very accurate picture, but again it's something that would probably be implicitly incorporated into the rate constant.

Indeed, in a previous analysis that's what I did: I considered that all the reactions occur based on concentration of active site, not of enzyme. Yet, now I'm interested in considering the two active sites because, although the reactions involving them occur with similar rate constants (this is an assumption I'm taking), I want to distinguish between PXPX and PXPi forms during my modeling of the system.


I have discussed this issue with a few colleagues, and was able to come to a different conclusion that I will refer after giving some prior input.

Consider that you want to study only reaction (1) above, but instead consider that you want to distinguish between the two active sites where X may bind. Then, consider the reactions:

[tex](5) AiBi + X  \rightarrow  AXBi[/tex]
[tex](6) AiBi + X \rightarrow AiBX[/tex]

Where Ai and Bi represent two distinct unbound active sites; X represents the substrate; and AX and BX represent bound active sites. Reactions (5) and (6) occur with same rate constant k. If I then want to write the differential equations of AiBi according to equations (5) and (6) I would get
[tex]\frac{dAiBi}{dt}=-k*AiBi*X-k*AiBi*X=-2*k*AiBi*X[/tex]

The same for X:
[tex]\frac{dX}{dt}=-k*AiBi*X-k*AiBi*X=-2*k*AiBi*X[/tex]

Finally, the equations of AXBi and AiBX:
[tex]\frac{dAXBi}{dt}=k*AiBi*X=k*AiBi*X[/tex]
[tex]\frac{dAiBX}{dt}=k*AiBi*X=k*AiBi*X[/tex]

Because (to me) it is indifferent whether X binds A or B, I can simplify the system above and consider that having AXBi = AiBX. I will thus compute
[tex]\frac{dAiBX}{dt}+\frac{dAXBi}{dt}=k*AiBi*X+k*AiBi*X=2*k*AiBi*X[/tex]

[Corribus]

I can't find any fault in the reasoning, assuming the k values are indeed the same for the two reactions.  Of course, most likely the k value here will be about half of what the rate constant would be if the reactions were not considered distinct, so it's sort of a trivial difference to my mind.  Either the value of two is put in front of the equation, or it's incorporated into the rate constant itself - the effect in both cases would be the same.