Chemical kinetics is the study of chemical (or biological) systems whose composition changes with time. In biological systems, for example, the cellular machinery involves a very large number of chemical reactions catalyzed by proteins, association (and dissociation) of proteins with other proteins and with DNA molecules, and conformational changes in the stucture of the biological molecules. Each of these processes can be described at some level as a simple chemical reaction between 2 or more states of the system. In this section we will work out how the various states evolve as a function of time from some initial population.

Unimolecular reactions

Unimolecular reactions, as the name implies, concerns a single molecule which can exist in a number of different states. The spontaneous fluctuation of the structure of a protein molecule from one conformation to another is an example of a unimolecular reaction. For example, the protein molecule can fluctuate between an unfolded random coil state to a folded native state, or between an ‘open’ conformation and a ‘closed’ conformation within the native state. The simplest unimolecular reaction in which a molecule fluctuates between two states or two conformations is written as

where k is the rate constant for the forward reaction and k’ is the rate constant for the reverse reaction . k and k’ in a unimolecular reaction have units of s^-1.

The rate equations that describe the change in populations of A and B as a function of time are:

Eventually the system reaches equilibrium when the rate at which A(B) is depleted equals the rate at which it is replenished and, on average, the relative populations of A and B do not change with time. Therefore at equilibrium

which implies that or where B_eq and A_eq refer to the populations of A and B at equilibrium and K is the equilibrium constant which we have encountered before. Recall that K was defined as where  is the free energy difference between states A and B. Therefore, the equilibrium constant and hence the free energy difference between the two states also determine the ratio of the forward and reverse rate constants.

For unimolecular reactions, the populations at equilibrium are independent of the initial concentration of molecules. We can normalize all concentrations to unity so that and A and B in the previous equations refer to the fractional populations in the two states with the constraint that  at all times. At equilibrium we have

and 

If the forward and backward rate constants are equal we get the equilibrium populations to be ½ in each state and the equilibrium constant K = 1.

If  then  and at equilibrium the populations in B will be 10 times that in A.

Solution of the rate equations

If the system is initially in a non-equilibrium situation, the populations in A and B will change as a function of time until the system reaches equilibrium. The time-dependence of A and B are governed by the coupled differential equations, the rate equations written above.

To solve the equations, we define  and with  or 

x_A and x_B are the populations in A and B, respectively, that deviate from the equilibrium populations.

The differential equations can be rewritten as

Therefore

The deviations from the equilibrium decay to zero with a characteristic rate constant k_r that is the sum of the forward and reverse rate constants 

where A(0) and B(0) are the populations at some initial time chosen as t = 0.

If the initial conditions A(0) = 1 and B(0) = 0 we get

A(t) decreases exponentially to its equilibrium value and B(t) increases exponentially. You can also check that A(t)+B(t)=1

Alternative method

The rate equations can be written in matrix form as

where is the rate matrix.

The solution is of the form

where are the eigenvalues of the rate matrix and are given by

or

which gives two solutions for 

Therefore

Here a₁ and b₁ are equilibrium populations when the exponential terms have decayed to zero

and a₂ and b₂ are obtained from the initial conditions.

Using the same initial conditions as before A(0)=1 and B(0)=0 we get

To summarize, this example is a simple unimolecular reaction involving only 2 states and the time-dependent change in the population of either states is described by a single-exponential with a characteristic rate constant given by the sum of the forward and reverse rates.

Unimolecular reaction with one intermediate state

The next example involves an extra intermediate state in the reaction scheme

For example, in a folding reaction A could be the unfolded state of a protein, B the compact denatured state or ‘molten globule’ state and C the final native state of the protein.

To keep the equations simple we assume irreversible steps and .

In principle, there is a reverse rate which we assume is considerably smaller than the forward rates and hence can be ignored.

The rate equations for the kinetic scheme shown above can be written as

or in the matrix form as:

with the solution of the form

The three eigenvalues of the rate matrix are given by

with

Since there are no reverse rates in the scheme, the time-dependence of A is particularly simple with a single-exponential decay:

where we have assumed that the initial population is all in A.

The time-dependence of B has the form

To determine the coefficients b₁, b₂, and b₃ we apply the conditions

which gives b₃ = 0 and 

Therefore 

By substituting the solution of A(t) and B(t) into the second rate equation

we get 

and 

Finally 

If you plot these solutions as a function of time, you will find that A(t) decreases in a single-exponential phase, B(t) increases exponentially, reaches some maximum value determined by and then decays back to zero, and C(t) increases in a bi-exponential manner.

We can now work out two limiting cases.

Case I

First step is fast and the second step is slow

In this limit, the intermediate state B increases with a characterstic rate k₁ to the maximum value of 1, and then decays with a characterstic rate k₂ , and C increases at the slow rate k₂.

Case II

First step is slow and the second step is fast

In this limit, B increases rapidly with a characteristic rate k₂ to a low steady-state concentration of k₁ / k₂ << 1, and then both A and B decay with the slow rate k₁, which is also the rate at which C increases.

Both the concentration of the intermediate B(t) and the rate of change dB/dt are small under these conditions, and one can assume that. This approximation is called the steady-state approximation and helps simplify the solution of more complicated rate equations with reverse rates included in the kinetic scheme.

We will apply the steady-state approximation when we work through the bimolecular reactions.

Biomolecular reactions

Biomolecular reactions introduce complications not seen in unimolecular reactions.

First, the rate of the reaction depends on the concentration of the molecules.

Second, the reaction consists of two steps: the reacting molecules have to first come close together to make contact, and then the the molecules can either diffuse away or react to form the products.

The simplest form of a bimolecular reaction is

where k_d is a diffusion-limited, bimolecular rate constant for forming the encounter complex AB, is a unimolecular rate constant for the encounter complex to fall apart before the reaction occurs, and k_a is the rate constant for the reaction step.

Here and k_a have units of s^-1 whereas k_d has units of M^-1s^-1.

We can write down the rate equation for the encounter complex [AB] as

Steady-state approximation

Under conditions where either k_a >> k_d or >> k_d the concentration of the encounter complex [AB] is small at all times and one can use the steady-state approximation
 
 

Therefore, the rate at which the product is formed is given by

Therefore, under conditions where the steady-state approximation is valid, the bimolecular reaction can be written as a one step process with a characteristic rate constant k_obs

where

There are two limiting cases:

Case I: Dissociation rate of the encounter complex much smaller than the reaction rate

In this limit, also called the diffusion-controlled limit, 

The rate at which the product is formed is governed by the rate at which the molecules A and B diffuse and make contact.

Case II: Reaction rate is much smaller than the dissociation rate of the encounter complex

In this limit, also called the reaction-controlled limit, 

and the observed rate constant is the product of the equilibrium constant for forming the complex and the reaction rate constant k_a.

The simplest biomolecular reaction

We will now work out the solution for the simplest bimolecular reaction

where we have dropped the subscript from k_obs (=k). Note that k is a bimolecular rate constant and has units of M^-1s^-1.

For every molecule of P that is formed, one molecule of A and one molecule of B are consumed in the reaction. Therefore

A special case of the bimolecular reaction is when A and B are present in equal concentrations initially: 

Therefore, the differential equation simplifies to:

Therefore, for this special case, a plot of 1/[A] versus time is a straight line with slope given by k.

We can rearrange the above solution to give

Note that, unlike the unimolecular reaction, the rate at which the fractional population [A]/[A]₀ changes with time depends upon the initial concentration [A]₀.

Difference between bimolecular and exponential decay

How does this solution differ from an exponential decay of the form

where k_uni is a unimolecular rate constant?

For an exponential decay, a plot of ln[A] versus time gives a straight line with a constant slope which has a magnitude equal to k_uni

Let’s see how the slope changes with time in a bimolecular reaction.

For a bimolecular reaction, the effective rate constant [A]k decreases as the concentration of [A] depletes.

This makes intuitive sense, since the probability that two molecules would come together in order to react decreases as the concentration of reactants depletes.

In the next lecture notes, we will work out the solution to the bimolecular equation when we have .

Additional Reading

P. W. Atkins, Physical Chemistry, Oxford University Press