Team:Imperial College London/Drylab/Enzyme/Analysis/Detailed

The behaviour of the system (the single-substrate mechanism) can be described with four ordinary differential equations (ODEs). Each component is described by a simple rate term for each of the four species involved in the mechanism. The rate term is a direct consequence of the application of the law of mass action.

We adopt the following notations for the kinetic rate constants involved in the single-substrate mechanism

The Enzyme Mechanism

In substrate binding, 1 molecule of Enzyme (E) binds with 1 molecule of Substrate (S) in a reversible reaction to form the Enzyme-Substrate complex (ES). From an experimental point of view, this step is blind: we are unable to measure this step. In catalysis, the Enzyme-Substrate complex (ES) produces the unchanged enzyme (E) and the product (P) in an irreversible reaction. This is the step that can be measured experimentally. When [S] becomes high enough, the second step of the enzyme kinetics becomes the rate determining step, controlled by k3. (link to simulation results)

Equation 1: Rate of change of Enzyme

Equation 2: Rate of change of substrate

Without MM assumption

Equation 3: Rate of change of enzyme-substrate complex

Equation 4: Rate of change of product

With Michaelis-Menten Assumptions

Consider the enzyme mechanism again:

From the model, we want to derive an equation that describes the rate of enzyme activity (amount of product formed per time interval) as a function of substrate concentration.

Consider equation 4: the rate of product formation. This equation is essentially useless as we cannot measure [ES] experimentally, therefore, to correctly determine d[P]/dt, we need to solve for ES in terms of the other quantities. Therefore, we need to make some approximations as mentioned previously. (hyperlink to assumptions part) At steady state, equation 3 is reduced to: When this result is used together with equation 4, a new differential equation for rate of change of product is obtained:

Equation 4-MM: New rate of change of product

Michaelis-Menten assumption:

• The Michaelis-Menten (MM) assumption is a simplifying assumption that is commonly made with enzymatic reactions. It says that since substrate binding is very fast compared to catalysis, the enzyme complex ES is always at quasi-steady state, ie: , during the time of the experiment.

• It is straightforward to prove that combining the MM assumption with the conservation equation [E₀]=[E]+[ES] yields a new form for the fourth equation , where K_M is the Michaelis-Menten constant.

Equation (4-MM) seems more complex but is in practice more useful exploitable than Equation (4). Indeed, [S] can often be measured during an experiment, whereas [ES] cannot. Furthermore when the initial concentration of substrate [S₀] is large compared to K_M, then [S] remains large compared to K_M over a certain range of time (that includes the start of the assay) and (4-MM) simplifies to over this range of time. Simply put, for these times P is produced at a constant rate at the start of the assay. Because of its close association to one of the possible behaviours of the system, the K_M is one of the important pieces of information that can be given about an enzymatic reaction.

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References