When using Michaelis-Menten kinetics to model enzyme kinetics of a simple reaction such as E + S <==k[sub]f[/sub]/k[sub]r[/sub]==> ES --k[sub]cat[/sub]–> E + P, researchers will often measure reaction rate at different substrate concentrations and extract V[sub]max[/sub] and K[sub]M[/sub], which fit in the M-M equation: v = V[sub]max[/sub][S] / (K[sub]M[/sub] + [S])
Enzyme activity often changes in different environments. E.g. different pH, different solvent (e.g. aqueous/organic mix), small molecules that bind the enzyme but don’t necessarily compete with the substrate, enzyme immobilization on a solid support or in a gel, etc.
For each different environment, the researcher can determine k[sub]cat[/sub] and K[sub]M[/sub]. Thus, it’s easy to see how the different environment affects k[sub]cat[/sub]. However, it’s difficult to see how the different environment affects the substrate binding equilibrium, k[sub]f[/sub]/k[sub]r[/sub]. Unless we assume k[sub]cat[/sub] is very small with respect to k[sub]r[/sub], in which case the ratio is just K[sub]M[/sub][sup]-1[/sup]. But I don’t know if that’s always a safe assumption.
My question:
Is there a simple and accurate way to determine the substrate binding equilibrium constant, either by varying substrate concentration or by some other means?
Yup! Just use a Lineweaver-Burkeplot, which rearranges the M-M equation to give 1/V[sub]0[/sub] as a function of 1/[S]. That gives a linear equation such that the Y-intercept is 1/V[sub]max[/sub] and the X-intercept is 1/K[sub]M[/sub]. Just plug in your measurements of V[sub]0[/sub] at a range of substrate concentrations [S].
That’s how it was done before computers could fit equations faster than we can pick up a pencil and graph paper. Now you can just fit the M-M equation to your V[sub]0[/sub] measurements. Still, the Lineweaver-Burke plot makes a handy teaching and visualization tool, particularly for determining (say) mechanisms of enzyme inhibition.
ETA: The more general problem of measuring K[sub]m[/sub] without handy measurements of an enzyme catalyzed reaction is a lot trickier. There’s no way to just measure simple protein-protein binding that I trust completely…
I’ll try to rephrase. I understand how you get the numbers from the L-B plot (although as you mention, it’s better to fit the curve.) But K[sub]M[/sub] includes the rate constants that govern the initial substrate/enzyme equilibrium and k[sub]cat[/sub]. So unless k[sub]cat[/sub] is very small compared to the rate constants that govern the initial equilibrium, the classic experiment doesn’t tell us much about that initial equilibrium.
Or do bio folks always assume k[sub]cat[/sub] is (comparatively) small?
Ah, sorry, misunderstood your question. Yes, M-M kinetics assumes that K[sub]cat[/sub] is negligible compared to binding and dissociation rates (the “rapid equilibrium” assumption). That’s reasonable enough, often enough, even though biochemists don’t “always” assume it is true. I’m aware that there are ways to model cases with relatively high K[sub]cat[/sub]. However, beyond invoking the “Briggs-Haldane” after glancing through a biochem text, you’ve reached the limit of my knowledge…