In this problem, we extend the one$-$dimensional model of diffusion in the presence of crowding molecules to account for the difference in size between a tracer particle $($considered to be present at low concentration$)$ and the crowders. This situation is relevant for the data shown in Figure $14\mathrm{.}24($A$).$ The tracer particles are assumed to be undergoing random walk motion on the larger tracer lattice with lattice constant b$,$ while the crowders are hopping between adjacent sites of the smaller lattice, with lattice constant a $($see Figure $14\mathrm{.}26).$ The two lattices are introduced to account for the difference in size between the two molecular species.

$($a$)$ Calculate the diffusion coefficient by considering the possible trajectories of the tracer particles and their probabilities. Note that the tracer can hop to an adjacent site of the tracer lattice only if there are no crowders present. Express your answer in terms of the diffusion coefficient D$0$ of the tracer particles in the absence of crowders, the volume fraction of the crowders $\backslash$phi $,$ and the ratio of the tracer and crowder sizes r $=$ b$/$a$.$

$($b$)$ Plot ln$($D$/$D$0)$ as a function of the volume fraction for different values of r$.$ How well does this model explain the data shown in Figure $14\mathrm{.}24($A$)?$ To make this comparison, you will need to estimate the sizes of the molecules used in the experiment from their molecular masses and a typical protein density that is $1\mathrm{.}3$ times that of water. The data are provided on the book$$s website.

Figure $14\mathrm{.}26$: Lattice model of tracer particles of size b diffusing in the presence of crowding molecules of size a$.$ The tracer particle can hop to the neighboring tracer site only if there are no crowding molecules present in the r $=$ b$/$a adjacent crowding molecule sites. The fraction of sites occupied by crowding molecules is $\backslash$phi $.$