366 8.4  Reaction, Diffusion, and Flow

the cell will change its direction more frequently, whereas if food is abundant, the cell will

tumble less frequently, ultimately resulting in a biased random walk in the average direction

of a nutrient concentration gradient (i.e., in the direction of food). Remarkably, this system

does not require traditional gene regulation (see Chapter 7) but relies solely on interactions

between a network of several different chemotaxis proteins.

Worked Case Example 8.2: Reaction–​Diffusion Analysis of Molecular Motors

In the minimal model for motor translocation on a 1D molecular track parallel to the x-​

axis release of a bound motor that occurs only in the region x > 0 and motor binding to the

track that occurs along a small region x0 to x0 +​ Δx, with a mean spacing between binding

sites on the track of d, the bound motor translocates along the track with a constant speed

v, and the on-​rate for an unbound motor to bind to the track is kon, while the off-​rate for a

bound motor to unbind from the track is koff (see Figure 8.7a).

a Calculate an ordinary differential equation of the 1D Fokker–​Planck equation in steady

state for the probability density function P(x,t) for a bound motor.

b In reference to this equation, find expressions for the binding motor probability Pb(x,t)

in three different regions of x0 < x < x0 +​ Δx, x0 < x < 0, and x > 0, and sketch a plot of Pb

vs x for the case of a “fast” motor and a “slow” motor.

c If the force exerted by a crossbridge of a bound motor linked to the track is −κx, such

that it is a “forward” force for x0 < x < 0 and a “backward” force for x > 0, calculate an

expression for the mean work done W per binding site traversed and the average force

〈F〉 per binding site, and sketch a graph of the variation of 〈F〉 vs v.

Answers

a If P(x,t) is the probability density function for a bound molecular motor to its track

assumed to be a line along the x-​axis, then the 1D Fokker–​Planck equation can be

rewritten as

(8.101)

d

d

d

d

on

off

P

t

k

P

k

P

v ^P

x

=

−

(

) −

−

=

1

0

Assuming in steady state, this equation equates to zero.

FIGURE 8.7  Reaction–​diffusion analysis of molecular motor translocation. (a) Schematic

of translocating motor on a 1D track. (b) Sketch of variation of motor binding probability

on the track with respect to distance x for a “fast” and “slow” motor and (c) sketch of the

predicted variation of average force per bound motor crossbridge (F with motor trans­

location speed v).