## Sunday, December 18, 2011

### It's Christmas!

December 18, 2011 10:37 am

It's Christmas, term is winding down at the University, the students are going home and that means spare time to do something other than official work. Here is a small Windows App I wrote that simulates a simple enzyme mechanism. The point of the simulation is to illustrate when the quasi-steady-state assumption is reasonable or not. In the simulation, you can change the different kinetic binding and unbinding constants but also by way of sliders change the initial total enzyme and substrate concentrations. The point to observe is that when the total substrate is low compared to the total enzyme, it is possible that the steady-state assumption is no longer reasonable and the derivation of the classic Briggs-Haldane equation (Often incorrectly referred to as the Michaelis-Menten equation) is no longer valid - it also depends on the Km of the enzyme (See Quasi-Steady-State Index). The first plot below shows the case when the steady-state assumption does not readily hold. Over a given timescale the enzyme-substrate complex reaches a peak then quickly declines (Steady-state index = 1.43, anything << 1.0 means that we can reasonably assume the quasi-steady state, see Murray, 2002).

The second plot shows the case where the total enzyme is now much smaller than the total substrate and this time the concentration of enzyme-substrate complex (shown in blue) remains relatively steady. The time axis in both plots is the same but this time the steady-state index = 0.07 which is much less than 1.0.

The Windows application that made these plots can be downloaded here. This is a single exe, no need to install anything, just run the executable and play with the sliders and settings. I've zipped the exe to save space, so unzip the file to get at the executable.

[latexpage]
Quasi-Steady-State Index, if $\epsilon \ll 1$ then the steady state assumption is reasonable:

$$\epsilon = \frac{E_o}{S_o + K_m}$$

where $E_o$ is the total amount of enzyme and $S_o$ the total amount of substrate.

Murray, J.D. (2002). Mathematical Biology: I. An Introduction (3 ed.). Springer.ISBN 978-0387952239., equation 6.18.