The equation describes diffusion of a molecule in one dimension. In multiple dim

Question

The equation describes diffusion of a molecule in one dimension. In multiple dimensions, this becomes 2 = 2nDt where "n" is the number of dimensions and zis the magnitude of the displacement of the particle in all dimensions combined, not just in the horizontal direction. We won't work with multiple dimensions in this problem.) The goals of this problem is to understand where this equation comes from using the model of a random walk and to practice working with distributions and calculations on distributions. Instead of treating space and time as continuous variables, we'll describe motion in one dimension as on a discrete grid with discrete time steps. Suppose a molecule begins at x=0 and can take one step of 1 nm to either the left or right. Then after one step, it could be at -1 nm or 1 nm. There is one path leading to -1 nm and one path leading to +1 nm, so two paths over all. Each past is equally probably, so if we did many trials, the average position, )is zero, which we can calculate via (1 path) × (-linn) + (1 path) × 1nm 2paths z) = On the other hand, the average square of position, ), is equal to 1 nm2. The quick way to see this is that (-1 nm)2 - 1 nm2 and (Inm)2 - 1 nm2, so whether x=-1 nm or x=1 nm, xl_1 nm2, so (z, must be 1 nm 2 on average. We can do the longer calculation as: r = (1 path) × (-1nm)2 + (1 path) × (1nm)-= lu 2 paths

Explanation / Answer

Cells constantly regulate the expression levels of their

genes. A central motif in this regulatory process is the binding

of transcription factor proteins to specific sites along the

DNA. The precision of transcriptional regulation is limited,

ultimately, by randomness in the arrival of transcription factor

TF molecules at these sites.

However, because the target site dimensions are

as small as a nanometer, there is concern about whether diffusionlike

models are appropriate or not. We first point out

that we are by no means the first to apply the diffusion equation

down to the length scales characterizing individual molecules:

in the chemical physics literature there are numerous

examples where diffusion-limited association rates are computed

within this framework, and even small deviations from

the theory are taken seriously.

With this remark in mind we note that our aim is to compute

the equilibrium behavior of the occupancy of the binding

site on long-time scales. We do not track the evolution of

motion of a single molecule along the DNA, but actually

imagine a number of molecules diffusing in the bulk and

along the DNA. Our results will concern the noise in the

0 limit, or, more precisely, the noise averaged over times

which are much longer than the characteristic time that the

TF particle needs to diffuse over length scales of the receptor

or sliding length. Concretely, the noise in gene regulation is

generally averaged on the cell division or protein lifetime

scale, both of which are in the range of at least minutes,

while diffusion across a region 1 nm in size at D

1 m2 / s will take on the order of 1 s, and will be comparatively

short even if such region is bigger by 2 orders of

magnitude. This means that—over the relevant timescale—

we will be able to define an ensemble of particles that

samples the spatial neighborhood of the binding site extremely

well and will thus be justified in using the diffusion

equations. We expect that our continuous approach is inappropriate

for probing small distances at short times, but this

is not relevant here.

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