Chemical kinetics

Relationships between product (P) formed in a unit of time (ΔP/ Δt)

Velocity (v) of the reaction

Rate of equation ΔPΔt

= V = k[S]

S Pk1

k-1

Single-substrate mechanism for an enzyme reaction. k1, k-1 and k2 are the rate constants for the individual steps.

Enzyme kineticsEnzyme binds supstrate in enzyme-substrate form.

Progress curve for an enzyme-catalyzed reaction

Initial slope = v0 = ΔtΔ[P]

Δ[P]Δ[P]

Δt ΔtProgress curve at two different enzyme concentration in the presence of the high initial concentrations of substrate:[S] >> [E]

In this case = the rate product formation depends on enzyme concentration and not on the substrate concentration.

The Michaelis –Menten Equation

Michaelis-Menten kinetics describes the kinetics of many enzymes. It is named after Leonor Michaelis and Maud Menten.

This kinetic model is relevant to situations where the concentration of enzyme is much lower than the concentration of substrate (i.e. where enzyme concentration is the limiting factor), and when the enzyme is not allosteric.

Determination of constants

Saturation curve for an enzyme showing the relation between the concentration of substrate and rate.

To determine the maximum rate of an enzyme mediated reaction, the substrate concentration ([S]) is increased until a constant rate of product formation is achieved.

This is the maximum velocity (Vmax) of the enzyme.

In this state enzyme active sites are saturated with substrate.

Note that at the maximum velocity, the other factors that affect the rate of reaction (ie. pH, temperature, etc) are at optimal values.

Reaction rate/velocity V

The speed V means the number of reactions per second that are catalyzed by an enzyme.

With increasing substrate concentration [S], the enzyme is asymptotically approaching its maximum speed Vmax, but never actually reaching it.

Because of that, no [S] for Vmax can be given.

Instead, the characteristic value for the enzyme is defined by the substrate concentration at its half-maximum speed (Vmax/2).

This KM value is also called the Michaelis-Menten constant.

Rate (or kinetic assay):

Measure concentration of analyte or product in initial stages only of the reaction (usually < 5 mins.)

Determine INITIAL RATE (= SLOPE of the line as close as possible to start of reaction).

Measurement and Interpretation of Rate Automated instrument may make readings at two set times,

say 1 min and 5 mins after initiating reaction.

Computes the rate between these two times

C5 - C1

5 - 1(but must be sure the concentration - time graph is close to linear over this time)i e

Michaelis-Menten constant 'KM'

Since Vmax cannot be reached at any substrate concentration (because of its asymptotic behaviour, V keeps growing at any [S], albeit ever more slowly), enzymes are usually characterized by the substrate concentration at which the rate of reaction is half its maximum.

This substrate concentration is called the Michaelis-Menten

constant (KM).

This represents (for enzyme reactions exhibiting simple Michaelis-Menten kinetics) the dissociation constant (affinity for substrate) of the enzyme-substrate (ES) complex.

Low values indicate that the ES complex is held together very tightly and rarely dissociates without the substrate first reacting to form product.

KM can only be used to describe an enzyme's affinity for

substrate when product formation is rate-limiting, i.e., when

k2 << k-1 and KM becomes k-1/k1.

Often, k2 >> k-1, or k2 and k-1 are comparable.

Derivation of the Michaelis-Menten Equation

This derivation of "Michaelis-Menten" was actually described by Briggs and Haldane. It is obtained as follows:

The enzymatic reaction is supposed to be irreversible, and the product does not rebind the enzyme.                               

Because we follow the steady state approximation, The concentrations of the intermediates are assumed to equillibrate much faster than those of the product and substrate, i.e. their time derivatives are zero:                                                                           

Let's define the Michaelis constant:                    

This simplifies the form of the equation:

                 The total (added) concentration of enzyme is a sum of that which is free in the solution and that which is bound to the substrate, and the free enzyme concentration is derived from this:[E0] = [E] + [ES][E] = [E0] − [ES] (2)

Using this concentration (2), the bound enzyme concentration (1) can now be written:                               

(1)

                                                                                   (3)

                  

(4)

The rate (or velocity) of the reaction is:

Rearranging gives:

Substituting (3) in (4) and multiplying numerator and denominator by [S]:                                    

This equation may be analyzed experimentally with a Lineweaver-Burk diagram or a Hanes-Woolf Plot.

This equation may be analyzed experimentally with a Lineweaver-Burk diagram or a Hanes-Woolf Plot.The plot provides a useful graphical method for analysis of the Michaelis-Menten equation:

Taking the reciprocal gives:

V = reaction velocity (the reaction rate), Km = Michaelis-Menten constant, Vmax = maximum reaction velocity [S] is the substrate concentration.

Effect of enzyme concentration on reaction rate

Effect of substrate concentration on reaction rate (hyperbolic)

An increase substrate concentration initially leads to a linear increase in reaction rate

This trend continues as long as the initial substrate concentration does not saturate or occupy all available active sites.

As the concentration of substrate reaches levels where the active sites are saturated, the initial reaction rate starts to decrease

Eventually the substrate concentration is so high that it continuously keeps the active sites occupied and saturated, reaching a maximum initial velocity

Km on the graph indicates where half Vmax is reached. This type of kinetics is termed hyperbolic and is usually

shown by simple, monomeric enzymes.

Units for expressing enzyme activity

  Reaction rate implies substrate utilised per unit time or

product formed per unit time.   The katal is the SI unit but is not often used in ordinary

conversation. It is defined as the transformation of mole of substrate per second.

Enzyme activity is defined as the amount of enzyme converting 1 μm of substrate per second.

  Turnover number is another common term i.e. the number

of substrate molecules converted by one enzyme molecule under specified conditions.

  Specific activity refers to enzyme activity per mass of

protein i.e. all the protein in a sample may not be enzyme. This unit also gives an indication of enzyme purity i.e. an impure enzyme will give low activity per unit mass.

Meaninig of Km

Michaelis constants have been determined for many of the commonly used enzymes. The size of Km tells us several things about a particular enzyme:

1. A small Km indicates that the enzyme requires only a small amount of substrate to become saturated. Hence, the maximum velocity is reached at relatively low substrate concentrations.

2. A large Km indicates the need for high substrate concentrations to achieve maximum reaction velocity.

The substrate with the lowest Km upon which the enzyme acts as a catalyst is frequently assumed to be enzyme's natural substrate, though this is not true for all enzymes.

A Km of 10-7 M indicates that the substrate has a greater affinity for the enzyme than if the Km is 10-5 M.

The Catalytic Constant kcat

At high substrate concentration the overall velocity of the reaction is Vmax and the rate is determined by the enzyme concentration.

The rate constant observed under these conditions is called the catalytic constant, kcat, defined as:

kcat indicates the maximum number of substrate molecules converted to product each second by each active site. This is called turnover number.

The catalytic constant measures how fast a given enzyme can catalyze a specific reaction (describing the effectiveness of an enzyme)

The unit for kcat is s-1 (for the most enzymes, kcat is 102 to 103 s-1)