# Notation in MM kinetics.

1. Jan 27, 2012

### nobahar

1. The problem statement, all variables and given/known data

For the reaction between enzyme and substrate:
E + S <-> ES -> E + P
E = enzyme, S = substrate, ES = enzyme-substrate complex, and P = product.
Where E + S -> ES rate constant = $$k_{1}$$
For ES -> E + S = $$k_{-1}$$
For ES -> E + P = $$k_{2}$$

2. Relevant equations

My textbook states that:

The rate of formation of ES:
$$\frac{d[ES]}{dt} = k_{1}[E]$$

The rate of breakdown of ES:
$$-\frac{d[ES]}{dt} = k_{-1}[ES] + k_{2}[ES]$$

For Michaelis-Menten kinetics, there is the steady state assumption that the concentration of ES is constant, so that:
$$\frac{d[ES]}{dt} = -\frac{d[ES]}{dt}$$

3. The attempt at a solution
I don't understand how these represent formation and breakdown of [ES], respectively:
$$\frac{d[ES]}{dt}$$
$$-\frac{d[ES]}{dt}$$

In my opinion, $$\frac{d[ES]}{dt}$$ means the rate of change of the concentration of the enzyme-substrate complex with time. That's it. Not the formation, but actually any factor that contributes to the change in ES concentration should be considered in this equation. I don't understand the negative derivative; to me, it says simply the negative of the change in ES concentration. It doesn't even seem to be possible to argue that the textbook is saying that the rate of breakdown is equal in magnitude but opposite in sign to the rate of formation, because, although this is true under steady state conditions, the notation ITSELF doesn't make sense, AND the notation is supposed to extend beyond these conditions (hence why it is used to say that WHEN d[ES]/dt = -d[ES]/dt, then steady state is achieved; which, as I say, doesn't even seem to make sense).

In my opinion, it should be:
$$\frac{d[ES]}{dt} = k_{1}[E] - k_{-1}[ES] - k_{2}[ES]$$
I took k-1 and k2 to be positive values, since they could be interpreted as referring to the 'formation of E + S from ES and E + P from ES, respectively; they don't 'know' they are being used in reference to ES activity, which is the reverse of their direction. If that makes sense.

There may be something I am missing, since all the textbooks seem to say it.
Any help appreciated.

Last edited: Jan 27, 2012