Chapter 8 – Theoretical Biophysics  353

The key physical process is one of a thermal ratchet. In essence, a molecular motor is sub­

ject to Brownian motions due to thermal fluctuations and so can translocate in principle both

forward and backward along its molecular track. However, the action of binding to the track

coupled to an external energy input of some kind (e.g., ATP hydrolysis in the case of many

molecular motors such as myosin molecules translocating on F-​actin) results in molecular

conformation changes, which biases the random motion in one particular direction, thus

allowing it to ratchet along the track once averaged over long times in one favored direction

as opposed to the other. But this does not exclude the possibility for having “backward” steps;

simply these occur with a lower frequency than the “forward” steps due to the directional

biasing of the thermal ratchet. The minimal model is of such importance that we discuss it

formally in Worked Case Example 8.2.

Reaction–​diffusion processes are also very important for pattern formation in biology

(interested students should read Alan Turing’s classic paper, Turing, 1952). In essence, if there

is reaction–​diffusion of two of more components with feedback and some external energy

input, this results in disequilibrium, that is, stable out of thermal equilibrium behavior,

embodied by the Turing model. With boundary conditions, this then results in oscilla­

tory pattern formation. This is the physical basis of morphogenesis, namely, the diffusion

and reaction of chemicals called morphogens in tissues resulting in distinctly patterned cell

architectures.

Experimentally, it is difficult to study this behavior since there are often far more bio­

molecule components than just two involved, which make experimental observations diffi­

cult to interpret. However, a good bacterial model system exists in Escherichia coli bacteria.

Here, the position of cell division of a growing cell is determined by the actions of just three

proteins MinC, MinD, and MinE. Min is named as such because deleting a Min component

results in the asymmetrical cell division in the formation of a small mini cell from one of the

ends of a dividing cell. ATP hydrolysis is the energy input in this case.

A key feature in the Min system, which is contrasted with other different types of bio­

logical oscillatory patterns such as those purely in solution, is that one of the components

is integrated into the cell membrane. This is important since the mobility in the cell mem­

brane is up to three orders of magnitude lower than in the cytoplasmic solution phase, and

so this component acts as a time scale adapter, which allows the period of the oscillatory

behavior to be tuned to a time scale, which is much longer than the time taken to diffuse

the length of a typical bacterial cell. For example, a typical protein in the cytoplasm has a

diffusion coefficient of ~5 μm² s⁻¹ so will diffuse a 1D length of few microns in ~1 s (see

Equation 2.12). However, the cell division time of E. coli is typically a few tens of minutes,

depending on the environmental conditions. This system is reduced enough in complexity to

also allow synthetic Min systems to be utilized in vitro to demonstrate this pattern formation

behavior (see Chapter 9). Genetic oscillations are another class of reaction–​diffusion time-​

resolved molecular pattern formation. Similar time scale adapters potentially operate here,

for example, in slow off-​rates of transcription factors from promoters of genes extending the

cycle time to hundreds of seconds.

8.4.2  REACTION-​LIMITED REGIMES

In instances of comparatively rapid diffusion, the bottleneck in processes may be the reaction

kinetics, and these processes are said to be reaction limited. A good example of this behavior

is the turnover of subunits in a molecular complex. As discussed in Chapter 3, a popular

experimental tool to probe molecular turnover is fluorescence recovery after photobleaching

(FRAP).

With FRAP, an intense confocal laser excitation volume is used to photobleach a region

of a fluorescently labeled biological sample, either a tissue or a single cell, and any diffusion

and subsequent turnover and reincorporation of subunits into the molecular complex in

that bleached zone will be manifested as a recovery in fluorescence intensity in the original

bleached zone with time following the photobleach. This system can often be satisfactorily

modeled as a closed reaction–​diffusion environment confined to a finite volume, for example,