Michaelis-Menten Kinetics (Tifft)

Michaelis-Menten Kinetics (Overview)

Michaelis-Menten kinetics is a model used to describe enzyme kinetics.

This greaph shows the amount of product formed over time in an enzyme catalyzed reaction for four different substrate concentrations.

Michaelis-Menten model is based on measurement of product formation at very early timepoints following the mixing of substrate and enzyme.

The initial reaction velocity or V0 is the slope of the line starting at time zero. Note that the initial velocity is measured before significant product accumulates, before there is any products being converted into substrate and before the reaction reaches equilibrium.

Initial velocities are measured at multiple starting concentrations of substrate. The values of initial velocity from multiple substrate concentrations can be plotted on a graph to show the relationship between initial velocity and substrate concentration.

Generic Reaction and Rate Constants

The Michaelis-Menten model relies on several important assumptions about initial velocity that simplify the situation enough to create an equation to model enzyme behavior.

Let's consider a generic reaction between substrate S and product P that is catalyzed by enzyme E. E and S bind to form ES.

K₁ is the rate of binding between E and S and K_-1 is the rate of unbinding of E and S. ES can also be converted to E + P.

K₂ is the rate constant for product formation.

The product can bind to enzyme and be convert to substrate, resulting in the ES complex. The rate constant K_-2 represent the binding and conversion of P to S.

Assumption 1: Substrate Concentration Does Not Change

The first assumption of Michaelis-Menten kinetics is that the amount of substrate does not change during the time that the initial velocity is measured. Although the substrate is depleted during product formation, the change in substrate concentration will be negligible at early timepoints if there is enough excess substrate present.

Assumption 2: ES Complex Reaches Steady State

The second assumption of Michaelis-Menten kinetics is that the amount of ES does not change during the time that the initial velocity is measured.

If E and S are mixed together, there will be a time delay before the population of E and S reach a steady state with the complex ES.

Michaelis-Menten assumes that a steady state is reached instantaneously and therefore the amount of ES is stable.

Assumption 3: No Conversion of Product to Substrate

The third assumption of Michaelis-Menten kinetics is that there is no conversion of product to substrate occurring when initial velocity is measured.

If E and S are mixed together, there will be a time delay before sufficient product builds up to have the reverse reaction begin to occur. Therefore, the rate constant K_-2 does not need to be considered for the Michaelis-Menten model.

Initial Velocity, Kcat, and Vmax

In the Michaelis-Menten the initial velocity of product formation or V₀ depends on the concentration of ES and the rate constant k2.

K2 is the rate constant for the conversion of ES to E + P. k2 is essentially the rate of catalysis and therefore also called k_cat. The initial velocity of product formation is equal to [ES[*k2.

maximum initial velocity V_max

The maximum initial velocity for a given amount of enzyme occurs at substrate concentrations when all enzymes are bound by substrate.

Adding more substrate cannot increase the rate of product formation if there are no enzymes without substrate bound. The maximum initial velocity of product formation or V_max is equal to the total concentration of enzyme times the catalytic rate constant k2.

[ES] depends on

binding affinity and [S]

The concentration of ES depends on the binding affinity between E and S and the concentration of S. Although the ES complex can be converted to both E+S and E+P we can think of the binding between E and S as very similar to protein binding to a ligand, because k2 for the catalysis step is typically much slower than k-1 for the unbinding of substrate.

Km

The binding affinity of ES is described by the Michaelis constant, which is abbreviated Km.

The Km for an enzyme and substrate is very similar to the Kd for a protein and ligand.

The Kd is the ligand concentration when binding is 50% saturated and the concentration of unbound P is equal to bound PL.

The Km is the concentration of substrate when the initial velocity is 50% of the maximum velocity or Vmax and the concentration of enzyme equals the concentration of the enzyme substrate complex.

The equation we use for Kd is Kd divided by the concentration of L equals the concentration of P (unbound protein) divided by the concentration of PL (or bound protein).

SImilarly the Km divided by substrate concentration is equal to concnetration of enzyme not bound to substrate divided by concentration of enzyme substrate complex

Equations for Initial Velocity

Based on the relationships developed so far, there are three equations that can be used for initial velocity.

The first equation already mentioned is that the initial velocity is equal to the concentration of ES times k2.

The second equation is that the initial velocity is equal to the fractional saturation of the enzyme (Y) times the maximum velocity or Vmax.

• The fractional saturation is equal to the concentration of ES divided by the total amount of E.

The third equation, which can be derived from substitutions and algebra is initial velocity is equal to the maximum velocity or Vmax times the substrate concentration divided by the concentration of substrate plus Km.

• This equation is called the Michaelis Menten equation.

Michaelis-Menten Equation and Graph

The Michaelis-Menten Equation describes the reaction velocity vs substrate concentration graph shown here.

The maximum value of initial velocity is Vmax and is the value that is approached by the curve at high concentrations of substrate.

The Km is the substrate concentration when the reaction reaches half Vmax.

Summary of Michaelis-Menten Kinetics

To summarize, here's a list of features that the Michaelis-Menten Kinetics model to describe enzymes.

There are three key assumptions that allow the model to be developed.

Everything is based on the concentration of the enzyme substrate complex.

The rate constant K2 or Kcap is the rate of catalysis or conversion from substrate to product.

V0 is the initial velocity of product formation.

The Vmax is the maximum possible initial velocity of product formation for a given concentration of enzyme and is reached when all E is bound to S the

Km which is analogous to Kd is the substrate concentration at the initial velocity that is one half of Vmax