Before going on, I want to emphasize the precise connection between the mathematics of diffusion - which is the statistical description of random-walking - and flexible polymers - whose statistical description is also in terms of random walks.

When we discussed diffusion we saw that the probability of finding a particle at (r,t) if it starts out from (0,0) is given by the `spreading Gaussian' distribution

This probability distribution is for the three-dimensional case, and is normalized to one particle, i.e. its integral over all of space is 1.

In fact the microscopic model of diffusion we previously introduced is the same as the random-flight model of a flexible polymer:

The ensemble of trajectories of a particle undergoing Brownian motion are random walks, and therefore is the ensemble of conformations of random-flight polymers.

All statistical properties of these two problems are the same, given that we properly identify time with polymer length.

In the limit of such a long polymer (or random walk of a particle) that there is a strong separation of the size of the conformation and the segment size b, we can use solutions of the diffusion equation to calculate statistical properties of polymers.

In particular the spreading Gaussian P(r,t) represents the probability distribution for the end-to-end vector of a long random-flight polymer. All we need to do is assert the connection between total time and polymer length, for the case where the steps taken in our microscopic diffusion model and in our polymer model are the same:

So, we should replace Dt ® Nb²/6 in P(r,t), to get:

We now have the probability distribution for the end-to-end vector, for a long, flexible random-walk polymer. Note that we still have òd³ r P(r,N) = 1, which is the correct normalization for the end-to-end vector (if we sum up all possibilities, we must get a net probability of 1).

This probability distribution is a simple Gaussian distribution in space, with a characteristic width of » N^1/2 b, which makes sense since the two ends should be typically R_ideal = N^1/2 b apart.

However, note that the maximum of the distribution is at |r| = 0. This means that the most likely end-to-end vector is 0. This expresses something important - that random walks tend to revisit points in space. We'll make this more precise below.

The end-to-end vector distribution is a quantity associated with the equilibrium static statistics of a flexible polymer. This means that we should be able to find a connection to thermodynamics. To do this we need to remember that if we know the probability distribution of some quantity x in thermal equilibrium, P(x), then we know the free energy when x is fixed to some value:

In the case of the end-to-end vector, we can find the free energy of the polymer when its ends are constrained to be r apart:

For very strong stretching where |r| ® N b there is no longer an exact connection between solutions of the diffusion equation and polymer conformations, and more complicated calculations are necessary.

The distribution P(r,N) can be used to find the probability that a flexible polymer of N segments happens to form a closed loop.

This is just the probability for r to be zero. Now, P(r,N) has dimensions of inverse volume, since it is a normalized probability distribution. We need to integrate over some volume to get a probability. So what we really need to ask is what the probability is that the two ends are within some distance d of each other. This tells us that we should compute the probability

What is the corresponding probability for a chain of 600 amino acids?