Thermodynamic Uncertainty Relation for Biomolecular Processes

Abstract

Biomolecular systems like molecular motors or pumps, transcription and translation machinery, and other enzymatic reactions can be described as Markov processes on a suitable network. We show quite generally that in a steady state the dispersion of observables like the number of consumed/produced molecules or the number of steps of a motor is constrained by the thermodynamic cost of generating it. An uncertainty ǫ requires at least a cost of 2kB T /ǫ 2 independent of the time required to generate the output.PACS numbers: 05.70.Ln, Biomolecular processes are generally out of equilibrium and dissipative, with the associated free energy consumption coming most commonly from adenosine triphosphate (ATP) hydrolysis. The role of energy dissipation in a variety of processes related to biological information processing has received much attention recently [1][2][3][4][5][6][7][8][9][10][11][12][13][14], to give just one class of examples for which one tries to uncover fundamental limits involving energy dissipation in biomolecular systems.Chemical reactions catalyzed by enzymes are of central importance for many cellular processes. Prominent examples are molecular motors [15][16][17][18][19], which convert chemical free energy from ATP into mechanical work. In this case an observable of interest is the number of steps the motor made. Another commonly analyzed output in enzymatic kinetics is the number of product molecules generated by an enzymatic reaction, for which the Michaelis-Menten scheme provides a paradigmatic case [2].Quite generally, chemical reactions are well described by stochastic processes. An observable, like the rate of consumed substrate molecules or the number of steps of a motor on a track, is a random variable subjected to thermal fluctuations. Single molecule experiments [20][21][22][23][24] provide detailed quantitative data on such random quantities. Obtaining information about the underlying chemical reaction scheme through the measurement of fluctuations constitutes a field called statistical kinetics [25][26][27][28]. A central result in this field is the fact that the Fano factor quantifying fluctuations provides a lower bound on the number of states involved in an enzymatic cycle [15,28].For a non-zero mean output, the chemical potential difference (or affinity) driving an enzymatic reaction must also be non-zero, leading to a free energy cost. Is there a fundamental relation between the relative uncertainty associated with the observable quantifying the output and the free energy cost of sustaining the biomolecular process generating it?In this Letter, we show that such a general bound does indeed exist. Specifically, for any process running for a time t, we show that the product of the average dissipated heat and the square of the relative uncertainty of a generic observable is independent of t and bounded by 2k B T . This uncertainty relation is valid for general networks and can be proved within linear response theory. Beyond linear response theory, we show it analytically for unicyclic...

Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials

“…The product of the two quantities, Q, is, in fact, independent of t [1, 11], and it was further argued that Q is always greater than 2k B T for any process that can be described as Markov jump process on a suitable network. This notion is concisely written asThe validity of this inequality was claimed for general Markovian networks [1,2], and was partly proved at near equilibrium, linear response regime [1]. This effort has recently been followed by a general proof employing the large deviation theory [12][13][14].…”

“…Trade-offs between energetic cost and information processing in biochemical and biomolecular processes have been highlighted for the last decades [1][2][3][4][5]. Among others, Barato and Seifert [1] have recently conjectured a fundamental bound in the minimal heat dissipation (q) to generate an output with relative uncertainty ( ). To be specific, when a molecular motor moves along cytoskeletal filament [6][7][8], the chemical free energy transduced into the motor movement, which results in a travel distance of the motor X(t), is eventually dissipated as heat into the surrounding media [9,10], the amount of which increases with the time ( q ∼ t).…”

“…Trade-offs between energetic cost and information processing in biochemical and biomolecular processes have been highlighted for the last decades [1][2][3][4][5]. Among others, Barato and Seifert [1] have recently conjectured a fundamental bound in the minimal heat dissipation (q) to generate an output with relative uncertainty ( ).…”

“…First, the heat dissipated from this process in NESS is a housekeeping heat, which can be evaluated for Langevin systems as [1,19,20].…”

“…For a given thermodynamic affinity per cycle A, which itself is a deterministic quantity defined as the log-ratio between the forward and backward flux (or microscopic rate constants of chemical networks) [1,10,26], the mean and variance of the housekeeping heat can be related with those of another stochastic observable, such as reaction cycle step n(t) or travel distance X(t): q = A × n = A/L × X and (δq) 2 = A 2 × (δn) 2 = (A/L) 2 × (δX) 2 . Then, the squared uncertainty 2 ξ = σ 2 ξ /µ 2 ξ are identical for ξ = q(t), n(t), X(t).…”