How does subdiffusion arise in cells and why is it important?

In this month's Physics Today there is an interesting article entitled "Strange kinetics of single molecules in living cells" that discusses recent interpretations of single-molecule tracking experiments. In these experiments, fluorescent molecules or microbead probes are attached to some organelle or other piece of cellular material in a live cell and then tracked using video tracking microscopy. The paths taken by the molecules or beads are then analyzed and their motion is interpreted through random walk models. The goal is to learn something about the intracellular environment from the complicated motions of these probes.

The diffusion of these probes is almost always subdiffusive, which means that their mean squared displacements (the second moment of their position vs. time) grows slower than linearly with time, or

where r(t) is the position of the probe at time t and 0 < a < 1. The brackets denote an ensemble average, or a second moment calculated over a large number of probe trajectories. In these experiments, there does not exist a large enough number of probes for sufficient averaging, so instead the time average of a few probes is calculated. However, this produces wildly different results from particle to particle. The reason is that ergodicity may not apply to cellular transport. Ergodicity is a well-known property from statistical mechanics of systems whose ensemble averages are equivalent to time averages as time tends to infinity.

This article presented two possible models for why the probes behave in this way. One model, the continuous time random walk (CTRW), is nonergodic and subdiffusive for heavy-tailed probability distributions of particle waiting times. The other interpretation is that the cellular environment is spatially inhomogeneous so that "the environment sampled by the molecule during its motion through the cell differs from one trajectory to another."

If my understanding of their reasoning is correct, then I don't think that these two possibilities are logically equivalent. The random environment of the cell is a real, physical thing. The CTRW model is just that: a model. I feel that presenting these as two possibilities to explain the motion of the probes is like saying the earth revolves around the sun because either there is an attractive gravitational force between the two or the orbit is roughly elliptical with the sun at one foci. The first is a statement about the physics of the phenomenon and the other is a mathematical model. Perhaps the random cellular environment is the cause for the CTRW model to be valid. This line of reasoning I can accept.

The article concludes with very interesting remarks on why subdiffusion of proteins and biomolecules should occur at all. Subdiffusion is a way to make certain reactions more efficient by preventing the reactants from diffusing too far apart from one another. Considering normal diffusion (a = 1 in the expression above) as the most efficient manner of passive transport for cellular materials, subdiffusion may be understood as an evolutionary compromise between fast transport and efficient use of cargo in a crowded environment. Cells should not be viewed as "small, well-mixed reaction flasks," since their order actually enables crucial cellular processes.

Other notes:

1. Advances in improving the experiments' temporal resolution and finding smaller and brighter light emitters are the primary challenges to optics from single-molecule tracking experiments.
2. Fractional Brownian motion (first developed by B. B. Mandelbrot) is another random walk model that leads to subdiffusion but does not break ergodicity. It may model single particles in many-body systems, such as a monomeric unit in a polymer chain.
3. A fundamental question in cell biology concerns how the chromosomes are packed inside the nucleus. Are they separated by unseen walls or does their connectedness and limited volume keep them effectively disentangled.
4. While reading this article the following thoughts came to mind: superdiffusive transport, like transport of vacuoles by molecular motors, is a characteristic of nonequilibrium systems. Subdiffusion does not require a nonequilibrium system since the cells' physical constrains are the likely limiting factor to transport. Does this make subdiffusion and superdiffusion fundamentally different things?