Figure $16\mathrm{.}21$: Schematic of a

one$-$state motor. The motor can only be

in one state in each box. The rate

constants characterize the probability

that the motor will jump lef or right

per unit time.

Consider the simplest possible model of the motion of a motor. Assume that the

motor has no internal states and simply hops from one site to the next with forward

rate $k+(F)$ and backward rate $k-(F)$ under the action of the applied force $F.$ The

probability per unit time of the motor moving forward by one site is $k+(F),$ while the

probability per unit time of the motor moving backward by one site is $k-(F).$ We

specify the explicit dependence of these rates on the applied force, which we

assume to be applied in the backward direction.

For the simplicity of the mathematical derivation, we incorporate the effects of

energy utilization by allowing the forward rate constant and the backward rate

constant to be different, even when the applied force F is zere. So in reality, our

treatment here is of a "quasi$-$one$-$state" motor, which does have different rate

constants for taking a forward step versus a backward step, but we set aside for the

moment the internal complexities of the motor that permit these differences.

Here your aim is to determine the evolution equation for the probability distribution of

the motor state $p(n,t),$ considering that the motor can only be in one internal state.

$($A$)$ Derive and show that the probability distribution satisfies

$\frac{p(n,t+t)-p(n,t)}{t}=k_{+}[p(n-1,t)-p(n,t)]+k_{-}[p(n+1,t)-p(n,t)].$

$($B$)$ Show using Taylor series approximations that the probability distribution function satisfies the

convective$-$diffusion form given below

$\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{t}}=-V\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{x}}+D\frac{de{l}^{2}p}{del{x}^2}$

Identify the diffusivity and the drift velocity $($convective speed$)$ in the above equations and write

these down explicitly as a function of the rate constants. Explain how you went from a discrete

version to a continuous version of the equations and the assumptions underlying this

approximation.

$($C$)$ Next you will show that a change of variable allows you to convert this to a pure diffusion

equation. Use the change of variables below that allows you to write the equation in a moving

frame.

$\frac{?}{\text{b}\text{a}\text{r}}(t)=t,\stackrel{}{x}=x-Vt.$

Derive and write down the appropriate equation, In the case of initial conditions where the motor is

localized at the origin $($the transformed $x$ variable $=0)$ at $t=0,$ write down the solution based on

lecture notes. Moving back to original variables show that the solution is

please write clearly, show all steps, and label all the parts of the solution