356 8.4  Reaction, Diffusion, and Flow

where ν is the universal rate constant for a transition state, also given by kBT/​h from Eyring

theory, and has a value of ~6 × 10¹² s⁻¹ at room temperature, where h is Planck’s constant. These

rate equations form the basis of the Arrhenius equation that is well known to biochemists.

Processes exhibiting detailed balance are such that each elemental component of that

process is equilibrated with its reverse process, that is, the implicit assumption is one of

microscopic reversibility. In the case of single biomolecules undergoing some reaction and/​

or molecular conformational change, this means the transition would go in reverse if the time

coordinate of the reaction were inverted. A useful way to think about kinetic transitions from

state 1 to state 2 is to consider what happens to the free energy of the system as a function of

some reaction coordinate (e.g., the time elapsed since the start of the reaction/​transition was

observed). If the rate constant for 1 → 2 is k12 and for the reverse transition 2 → 1 is k21, then

the principle of detailed balance predicts that the ratio of these rate constants is given by the

Boltzmann factor:

(8.78)

k

k

G

k T

B

12

21

=

−



^



^

exp

∆

where ΔG is the free energy difference between states 1 and 2. For example, this process

was encountered previously in Chapter 5 in probing the proportional of spin-​up and spin-​

down states of magnetic atomic nuclei in NMR. This expression is generally true whether

the system is in thermal equilibrium or not. Historically, rate constants were derived from

ensemble average measurements of chemical flux from each reaction, and the ratio is k12/​

k21, which is the same as the ratio of the ensemble average concentrations of molecules in

each state at equilibrium. However, using the ergodic hypothesis (see Chapter 6), a single-​

molecule experiment can similarly be used, such that each rate constant is given by the

reciprocal of the average dwell time that the molecule spends in a given state. Therefore, by

sampling the distribution of a lifetime of a single-​molecule state experimentally, for example,

the lifetime of a folded or unfolded state of a domain, these rate constants can in principle

be determined, and thus the shape of the free energy landscape can be mapped out for the

molecular transitions.

Note also that the implicit assumption in using the Boltzmann factor for the ratio of the

forward and reverse rate constants is that the molecules involved in the rate processes are all

in contact with a large thermal bath. This is usually the surrounding solvent water molecules,

and their relatively high number is justification for using the Boltzmann distribution to

approximate the distribution in their accessible free energy microstates. However, if there

is no direct coupling to a thermal bath, then the system is below the thermodynamic limit,

for example, if a component of a molecular machine interacts directly with other nearby

components with no direct contact with water molecules or, in the case of gene regulation

(see Chapter 9), where the number of repressor molecules and promoters can be relatively

small, which potentially again do not involve direct coupling to a thermal bath. In these cases,

the actual exact number of different microstates needs to be calculated directly, for example,

using a binomial distribution, which can complicate the analysis somewhat. However, the

core of the result is the same in the ratio of forward to reverse rate constants, which is given

by the relative number of accessible microstates for the forward and reverse processes,

respectively.

As we saw from Chapter 6, single-​molecule force spectroscopy in the form of optical or

magnetic tweezers or AFM can probe the mechanical properties of individual molecules. If

these molecular stretch experiments are performed relatively slowly in quasi-​equilibrium

(e.g., to impose a sudden molecular stretch and then take measurements over several seconds

or more before changing the molecular force again), then there is in effect microscopic

reversibility at each force, meaning that the mechanical work done on stretching the mol­

ecule from one state to another is equal to the total free energy change in the system, and thus

the detailed balance analysis earlier can be applied to explore the molecular kinetic process

using slow-​stretch molecular data.