Stochastic Aspects of Diffusion:
Single Molecules

A single molecule in solution: In class we dicussed the stochastic nature of diffusion. Usually it is very hard to directly 'see' simple molecules because of their small size (simple molecules are usually a few angstroms (10-10 m) in diameter). Some biomolecules, such as double stranded DNA, can be extremely long (lengths > 10's of microns (10-6m)) and can be directly visualized in solution using fluorescence microscopy.

Click here to view a single lambda-phage DNA molecule (length = 20 microns) in an aqueous solution (viscosity of solvent = 1 cp) . The image size is 12 microns x 12 microns and the counter is showing time in seconds. What you hopefully noticed are the large fluctuations in both the configuration of the molecule and the center of mass.

Aside: You also might have noticed that the apparent size of the DNA is much smaller than 20 microns! DNA is a polymer and due to entropic reasons will stay in a coiled configuration (this is beyond the scope of 10.50 and is discussed in detail in 10.537 and 10.668).

Molecular trajectories: We can quantify the motion of the molecules by tracking the center of mass of the DNA as a function of time. The trajectories of 4 molecules during a 1 second time interval are shown in the figure to the right. Since this is a stochastic process we see that the trajectories of individual molecules are quite different!
To analyze the system further we will calculate ensemble averaged quantities. The vectoral position of a molecule at time=0 is defined as ri(0). We monitor the displacement from this postion, ri(t)-ri(0). On taking the ensemble average over 100 molecules we arrive at < ri(t)-ri(0)> =0 which merely reflects the fact that there is no bulk flow in the experiment. However, the mean squared displacement (<[ri(t)-ri(0)]2>) is nonzero and grows linearly in time (cf. figure to the right). This is consistent with our discussion in class (section 2.9 of Deen).
We add one technical note. In the experiment we are only able to track the in-plane motion of the molecule which is denoted here as the x-y plane. To extract the diffusion coefficent of the DNA we use the 2-dimensional analog of equation 2.9-21 in Deen: <[ri(t)-ri(0)]2> = 4 D t.