# General Enzymatic Kinetics

Enzymes are catalysts, most are proteins, that bind temporarily to one or more of the reactants of the reaction they catalyze. In doing so, they lower the amount of activation energy needed and thus speed up the reaction.

### Introduction

Enzymes are Agents of Metabolic Function. A given Enzyme is quite selective, both in the substances with which it interacts and in the reaction it catalyzes. The substances upon which an enzyme acts are called Substrates. The selective qualities of an enzyme are recognized as its specificity.The specific site on the enzyme where substrates bind and where catalysis occurs is called the Active site

• Catalysts lower the Free energy of Activation for a reaction given by the Arrhenius Equation: $$k = Ae^{-E_a/RT}$$

### Can the rate of an Enzyme-catalyzed Reaction be defined in a mathematical way?

Enzymes Kinetics exhibit the following rates of chemical reactions:

We seek to determine the maximum reaction velocity that an enzyme can attain and its binding affinity for substrates and inhibitors. When working with enzyme catalyzed reactions, generally the enzyme is less than the total substrate, so that it is a valid assumption that the concentration of free substrate concentration is equal to the TOTAL AMOUNT OF SUBSTRATE

$[E] << [S]_{total}$

$[S]_{free} = [S]_{total}$

We will use this assumption throughout the discussion of enzyme kinetics.

• Most frequently one takes the phrase "enzyme kinetics" refers to the analysis of the "steady state" kinetics of enzymatic reactions. The term "steady state" has a specific meaning. The steady state is the phase of a reaction in which reactive intermediates are both formed and decomposed at the same rate so that their concentrations are essentially constant.

### Analogies are always helpful for an intuitive understanding

A good analogy is that an animal population can be in a Steady-state Approximation

• This occurs when the rate of birth and death are equal.

The "reactive intermediates" (the live animals in the population) are then in a 'steady state', with the most "stable" states being those of pre-birth and death. Consider the following chemical reaction:

$A + B \leftrightharpoons AB^* \leftrightharpoons C + D$

• If the $$AB^*$$ complex is a high energy species relative to the $$A + B$$ and $$C + D$$ ground states then, as a good approximation, as soon as it is formed it will either go to $$A + B$$ or $$C + D$$. This means that it's concentration will be very low and nearly constant over a very large part of the reaction.
• The idea of steady state kinetics applied to enzyme catalyzed reactions is that the enzyme is generally present only in very low concentrations relative to the total substrate concentration, and that the reactive intermediate corresponding to $$AB^*$$ above is the $$E^*S$$ complex.

### Basic Equations of Enzyme Kinetics

$$E + S \leftrightharpoons E^*S \rightleftharpoons E + P$$

The typical mechanism for enzyme kinetics:

Enzyme + Substrate goes to the Enzyme Substrate Complex, and further yields Enzyme and Substrate!

• If $$[S]_{total} >> [E]_{total}$$, as is usually the case, then $$[S]$$ in the $$E^*S$$ complex is a very small portion or fraction of $$[S]_{total}$$.

This is a necessary condition for the steady state assumption to be valid: the reactive intermediate {{ math.formula("ES\) must be at a very low and constant concentration compared to the reactants {{ math.formula("S\).

• We generally miss the pre-steady state part of enzymatic reactions when we mix enzyme catalyzed reactions by hand in a cuvette and stick it in a spectrophotometer to measure it. By the time we start taking data, the reaction is already in the steady state because rate constants for enzyme catalyzed reactions are generally large.
• In 1913, Michaelis and Menten published the idea that enzymes and substrates formed reasonably stable complexes with their substrates that then subsequently undergo a reaction.

#### The BIG, BIG EQUATION FOR ENZYME KINETICS

The Michealis-Menten Equation

$E + S \leftrightharpoons E^*S \rightleftharpoons E + P$

The mechanism involves constants: KM (Michealis Constant) and kcat ( turnover number)

$v_o = \left(\dfrac{dP}{dt} \right )_o \dfrac{V_{max}[S]}{Km}{K_m + [S]}$

### References

1. Atkins, Peter and de Paula, Julio, 2006, Physical Chemistry for the Life Sciences, Oxford University Press.
2. Segel, Irwin H. , 1976, Biochemical Calculations: How to Solve Mathematical Problems in General Biochemistry, 2nd Edition
3. Segel, Irwin H., Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems (Wiley Classics Library)

• Jai Pal