[Epimetheus]

I've been looking at the Michaleis-Menten constant, Km, being zero from a mathematical perspective and it doesn't make any sense - i.e it breaks the Michaleis-Menten equation (the velocity of a reaction becomes undefined).

But does having a Km=0 make any physical sense? Is it possible  - does the model need amending to take into account zero values?

Thanks for any insight.

[Babcock_Hall]

Consider a very simple system involving E, ES, and three rate constants.  The definition of K_m in terms of rate constants is that it is equal to (k_-1 + k₂)/k₁.

[Epimetheus]

Indeed, so K_m=0 would imply K_-1=-K_cat.

However, deriving the MM equation at various stages requires us dividing by K_m, and so K_m must strictly not be zero for the derivation to make sense..

Also if we let K_m=S (as we would to find that K_m is half V_max) then the velocity is also undefined.

Hence, i cannot make sense of it mathematically. But does K_m=0 make sense physically?

[Babcock_Hall]

Indeed, so K_m=0 would imply K_-1=-K_cat.

This does not sound correct.  Both k_-1 and k₂would have to be zero, in order for K_m to be zero.

[Epimetheus]

How so?

We have:

K_m = (K_-1 + K₂)/K₁ =0

Hence we only need the numerator to sum to zero for this to be true, i.e: K_-1 + K₂ = 0.

Which implies K_-1=-K₂.

K_-1 = -K₂ = 0 is just one possibility.

[Babcock_Hall]

Rate constants cannot be negative numbers.

[Epimetheus]

Ah yes. Excellent.

Then K_m cannot be zero as there wouldn't be a MM reaction taking place at all.