Tuesday, 4 July 2017

Becoming more certain about uncertainty in molecular systems

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).
This motor is used to create the chemical fuel source of the cell (ATP) from its components (ADP and inorganic phosphate P). In order to drive this process, a current of hydrogen ions flows through the top half of the motor, causing it to systematically rotate in one direction with respect to the bottom half. This rotation is physically linked to the reaction ADP + P -> ATP, and so ATP is created. This one-directional rotational motion only arises because the current of hydrogen ions continuously supplies more energy (more technically, free energy) to the system than is needed to create the ATP. We say that the current of ions drives the system.

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 factor.

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.

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