ã 1999 GraphPad Software Inc.
Introduction to enzyme kinetics
From curvefit.com. Copyright 1999 by GraphPad Software, Inc. All Rights Reserved.
Living systems depend on chemical reactions which, on their own, would occur at extremely slow rates. Enzymes are catalysts which reduce the needed activation energy so these reactions proceed at rates that are useful to the cell.
Product accumulation is often linear with time
In most cases, an enzyme converts one chemical (the substrate), into another (the product). A graph of product concentration vs. time follows three phases as shown in the following graph.
At very early time points, the rate of product accumulation increases over time. Special techniques are needed to study the early kinetics of enzyme action, since this transient phase usually lasts less than a second (the figure greatly exaggerates the first phase).
For an extended period of time, the product concentration increases linearly with time.
At later times, the substrate is depleted, so the curve starts to level off. Eventually the concentration of product reaches a plateau and doesn't change with time.
It is difficult to fit a curve to a graph of product as a function of time, even if you use a simplified model that ignores the transient phase and assumes that the reaction is irreversible. The model simply cannot be solved to an equation that expresses product concentration as a function of time. To fit these kind of data (called an enzyme progress curve) you need to use a program that can fit data to a model defined by differential equations or by an implicit equation. Prism cannot do this. For more details, see RG Duggleby, "Analysis of Enzyme Reaction Progress Curves by Nonlinear Regression", Methods in Enzymology, 249: page 60-, 1995.
Rather than fit the enzyme progress curve, most analyses of enzyme kinetics fit the initial velocity of the enzyme reaction as a function of substrate concentration. The velocity of the enzyme reaction is the slope of the linear phase, expressed as amount of product formed per time. If the initial transient phase is very short, you can simply measure product formed at a single time, and define the velocity to be the concentration divided by the time interval.
This chapter considers data collected only in the second phase. The terminology describing these phases can be confusing. The second phase is often called the "initial rate", ignoring the short transient phase that precedes it. It is also called "steady state", because the concentration of enzyme-substrate complex doesn't change. However, the concentration of product accumulates, so the system is not truly at steady state until, much later, the concentration of product truly doesn't change over time.
Enzyme velocity as a function of substrate concentration
If you measure enzyme velocity at many different concentrations of substrate, the graph generally looks like this:
Enzyme velocity as a function of substrate concentration often follows the Michaelis-Menten equation:
Vmax is the limiting velocity as substrate concentrations get very large. Vmax (and V) are expressed in units of product formed per time. If you know the molar concentration of enzyme, you can divide the observed velocity by the concentration of enzyme sites in the assay, and express Vmax as units of moles of product formed per second per mole of enzyme sites. This is the turnover number, the number of molecules of substrate converted to product by one enzyme site per second. In defining enzyme concentration, distinguish the concentration of enzyme molecules and concentration of enzyme sites (if the enzyme is a dimer with two active sites, the molar concentration of sites is twice the molar concentration of enzyme).
KM is expressed in units of concentration, usually in Molar units. KM is the concentration of substrate that leads to half-maximal velocity. To prove this, set [S] equal to KM in the equation above. Cancel terms and you'll see that V=Vmax/2.
The meaning of KM
To understand the meaning of Km, you need to have a model of enzyme action. The simplest model is the classic model of Michaelis and Menten, which has proven useful with many kinds of enzymes.
The substrate (S) binds reversibly to the enzyme (E) in the first reaction. In most cases, you can't measure this step. What you measure is production of product (P), created by the second reaction.
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.
The rate of product formation equals the rate at which ES turns into E+P, which equals k2 times [ES]. This equation isn't helpful, because we don't know ES. We need to solve for ES in terms of the other quantities. This calculation can be greatly simplified by making two reasonable assumptions. First, we assume that the concentration of ES is steady during the time intervals used for enzyme kinetic work. That means that the rate of ES formation, equals the rate of ES dissociation (either back to E+S or forward to E+P). Second, we assume that the reverse reaction (formation of ES from E+P) is negligible, because we are working at early time points where the concentration of product is very low.
We also know that the total concentration of enzyme, Etotal, equals ES plus E. So the equation can be rewritten.
Solving for ES:
The velocity of the enzyme reaction therefore is:
Finally, define Vmax (the velocity at maximal concentrations of substrate) as k2 times Etotal, and KM, the Michaelis-Menten constant, as (k2+k-1)/k1. Substituting:
Note that Km is not a binding constant that measures the strength of binding between the enzyme and substrate. Its value includes the affinity of substrate for enzyme, but also the rate at which the substrate bound to the enzyme is converted to product. Only if k2 is much smaller than k-1 will KM equal a binding affinity.
The Michaelis-Menten model is too simple for many purposes. The Briggs-Haldane model has proven more useful:
Under the Briggs-Haldane model, the graph of enzyme velocity vs. substrate looks the same as under the Michaelis-Menten model, but KM is defined as a combination of all five of the rate constants in the model.
Assumptions of enzyme kinetic analyses
Standard analyses of enzyme kinetics (the only kind discussed here) assume: