# Structural Biochemistry/Enzyme/Kcat/Km

k$_{cat}$,k$_{dand}$K$_M$ == k$_{cat}$,k$_d$ and K$_M$ are terms helpful in the description of an enzyme that follows the Michaelis-Menten kinetics.
K$_d$ is dissociation constant. which describe how affinite two reactants are in a reaction. The following reaction is an example to show dissociation constant:
Where A and B are the two reactant, AB is the formed complex, k$_{-1}$ is the reverse constant rate, and k$_1$ is the forward constant rate. The dissociation constant is defined as: k$_d$=k$_{-1}$/k$_1$. The smaller the dissociation constant is, the better two reactants can combine. Since the affinity of enzyme with substrate determines how favorable the reaction can form enzyme-substrate complex, k$_d$ is often studied in Michaelis-Menten equation.
Deriving from Michaelis-Menten equation: k$_M$=(k$_{-1}$+k$_{cat}$)/k$_1$ Since K$_M$, which is also referred as Michaelis constant, is an important constant to study the ability of catalysis reaction of enzyme with specific substrate. k$_M$ can be separated into two parts: a.k$_d$ The first step of catalysis kinetic is the binding between substrate and enzyme, which is also the rate determine step in the reaction. the better enzyme bind to substrate, the smaller k$_d$is, thus the smaller k$_M$ is. b.k$_{cat}$ The second step of catalysis kinetic is the forming of product. The larger k$_{cat}$ is, the more favorable the reaction towards product, and the larger k$_M$ is. There seems to be a contradiction between k$_d$ and k$_{cat}$ in the Michelis constant equation: the better enzyme to the specific substrate, the smaller k$_d$ is, and the larger k$_{cat}$ is. However, what determine the performance of catalysis reaction is dissociation constant k$_d$, because the first step of the reaction--binding is the rate determine step, forming enzyme-substrate complex is the essential step to form product, thus k$_d$ is the major factor to determine k$_M$ Together they show an enzymes preference for different substrates. k$_{cat}$/K$_M$ results in the rate constant that measures catalytic efficiency. This measure of efficiency is helpful in determining whether the rate is limited by the creation of product or the amount of substrate in the environment.
In situations where k$_{-1}$ (the rate at which substrate unbinds from the enzyme) is much greater than k$_2$ (the rate at which substrate converts to product), if the rate of efficiency is:
* HIGH, k$_{cat}$ is much larger than K$_M$, and the enzyme complex converts a greater proportion of the substrate it binds into product. This increased conversion can be seen in one of two ways -- either substrate binds more firmly to the enzyme, a consequence of relatively low K$_M$, or a greater proportion of the substrate that is bound is converted before it dissociates, due to a large turnover rate k$_{cat}$.
* LOW, k$_{cat}$ is much smaller than K$_M$, and the complex converts a lesser proportion of the substrate it binds into product.
k$_{cat}$/K$_M$ measures the catalytic efficiency, though, only when the substrate concentration is much lower than the K$_M$. Looking at the enzyme/substrate catalytic reaction equation,

#### E+S↔ES->E+P

with the rate going towards ES being k$_1$, the rate going back towards E+S being k$_{-1}$, and the rate going towards product formation (E+P) being k$_2$ or k$_{cat}$, it is evident from

#### k$_{cat}$/K$_M$=[k$_{cat}$/(k$_{-1}$ + k$_{cat}$)]k$_1$

that even if k$_{cat}$ is much greater than k$_{-1}$ (much product is forming) and there is great efficiency, the equation will still be limited by k$_1$, which is the rate of ES formation. This tells us that k$_{cat}$/K$_M$ has a limit placed on efficiency in that it cannot be faster than the diffusion controlled encounter of an enzyme and its substrate (k1). Therefore, enzymes that have high k$_{cat}$/K$_M$ ratios have essentially attained kinetic perfection because they have come very close to reaching complete efficiency only being limited by the rate at which they encounter the substrate in solution.
In cases near the limit, there may be attractive electrostatic forces on the enzyme that entice the substrate to the active site, known as Circe effects. Diffusion in solution can be partly overcome by confining substrates and products in the limited volume of a multienzyme complex. Some series of enzymes are associated into organized assemblies so that the product of one enzyme is rapidly found by the next enzyme.

## References

Berg, Jeremy M. 2007. Biochemistry. Sixth Ed. New York: W.H. Freeman.