By Jenny Poulton
Due to the unpredictability of motion at the microscopic scale, molecular
processes have randomness associated with them, exhibiting what we call thermodynamic
fluctuations. A group in Germany lead by Barato and Seifert have written a series of papers,
beginning with "Thermodynamic uncertainty relation for biomolecular processes" (preprint here), exploring how uncertainty in the number of reaction
steps taken by a molecular process is related to the degree to which the
system is constantly consuming energy.
To be more precise, Barato and Seifert consider the number of times a system
completes a cycle in a given time window. A good example of this kind
of setup is the rotary motor F0F1-ATPsynthase (below, image taken from Wikipedia).
In general, small driven systems have a bias towards stepping forward, but there
is still a non-zero probability of stepping backwards due to thermodynamic fluctuations.
We also cannot predict exactly how long the system will take to complete each step of the cycle, and so the time taken per step is variable. Thus the number of cycles completed
in a given time is uncertain. It is, however, possible to define an average
of the net number of cycles in a time window µ and a variance σ2, which is a
mathematical measure of the typical deviation from the average due to fluctuations.
The Fano factor F = σ2/µ gives a measure of the relative importance of
the random fluctuations about the average.
In the paper "Thermodynamic uncertainty relation for biomolecular processes", Barato and Seifert relate the energy consumption and the Fano
factor via F ≤ 2kT /E. Here E is the energy consumed per cycle, T is the
temperature and k is Boltzmann’s constant. This expression means that the
Fano factor is at least as big as the quantity 2kT /E. Thus a cycle which uses
a certain amount of fuel E has an upper limit to its precision, and there is an
evident trade-off between the amount of energy dissipated per cycle and the Fano
In the original paper, the authors only prove their relation for very simple
processes. However, it has since been generalised in this paper (preprint here). The result is actually based on very
deep statements about the types of fluctuating processes that are possible in
physical systems. One of the challenges now is to take this fundamental insight and apply it to gain a better understanding of practical systems. Fortunately, the F0F1-ATPsynthase rotary motor is not the only example of an interesting biological system that undergoes
driven cycles; the cell contains a huge variety of molecular motors that can also be understood in this way (preprint here). Molecular timekeepers that are vital to the cellular life cycle also depend on driven cycles. Understanding the trade-offs between unwanted variability and energy consumption will be vital in engineering such systems.