358 8.4  Reaction, Diffusion, and Flow

so can be used as an estimator for the free energy difference, which in turn can be related to

actual rate constants for unfolding.

Note that a key potential issue with routine reaction-​limited regime analysis is the

assumption that a given biological system can be modeled as a well-​mixed system. In

practice, in many real biological processes, this assumption is not valid due to compart­

mentalization of cellular components. Confinement and compartmentalization are often

pervasive in biology, which needs to feature in more realistic, complex, mathematical

analysis.

A commonplace example of reaction-​limited processes is a chemical reaction that can

be modeled using Michaelis–​Menten kinetics. A useful application of this approach is to

model the behavior of the molecular machine F1Fo ATP synthase, which is responsible for

making ATP inside cells (see Chapter 2). This is an example of a cyclical model, with the pro­

cess having ~100% efficiency for the molecular action of the F1Fo, requiring cooperativity

between the three ATP binding sites:

1 There is an empty ATP-​waiting state for one site brought about by the orientation of

the rotor shaft of the F1 machine, while the other two sites are each bound to either

an ATP or ADP molecule.

2 ATP binding to this site drives an 80°–​90° rotation step of the rotor shaft, such that the

wait time before the step is dependent on ATP concentration with a single exponen­

tial distribution indicating that a single ATP binding event triggers the event.

3 This binding triggers ATP hydrolysis and phosphate release but leaving an ADP mol­

ecule still bound in this site.

4 Unbinding ADP from this site is then coupled to another rotor shaft rotation of 30°–​

40°, such that the wait time does not depend on ATP concentration but has a peaked

distribution, indicating that it must be due to at least two sequential steps of ~1 ms

duration each (correlated with the hydrolysis and product release).

5 The 1–​4 cycle repeats.

The process of F1-​mediated ATP hydrolysis can be represented by a relatively simple chem­

ical reaction in which ATP binds to F1 with rate constants k1 and k2 for forward and reverse

processes, respectively, to form a complex ATP·F1, which then re-​forms F1 with ADP and

inorganic phosphate as the products in an irreversible manner with rate constant k3. This

reaction scheme follows Michaelis–​Menten kinetics. Here, a general enzyme E binds revers­

ibly to its substrate to form a metastable complex ES, which then breaks effectively irrevers­

ibly to form product(s) P while regenerating the original enzyme E. In this case, F1 is the

enzyme and ATP is the substrate. If the substrate ATP is in significant excess over F1 (which

in general is true since only a small F1 amount is needed as it is regenerated at the end of each

reaction and can be used in subsequent reactions), this results in steady-​state values of the

complex F1·ATP, and it is easy to show that this results in a net rate of reaction k given by the

Michaelis–​Menten equation:

(8.81)

v

v

S

K

S

m

=

[ ]

+[ ]

max

where

[S] is the substrate concentration and Km is the Michaelis constant which is a measure of

the substrate-​binding affinity, so in the case of F1Fo [S] is ATP concentration

Km=​ (k1 +​ k2)/​k1

kmax is the maximum rate of reaction given by k3[F1·ATP] (square brackets here indicates

a respective concentration)

In the case of a nonzero load on the F1 complex (e.g., it is dragging a large fluorescent F-​actin

filament that can be used as lever to act as a marker for rotation angle of the F1 machine),

then the rate constants are weighted by the relevant Boltzmann factor exp(−ΔW/​kBT) where