General technique of calculating the drift velocity and diffusion coefficient in arbitrary periodic systems

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...

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“…where we used the fact that C ′ 1 = 0 for a unicyclic network [35]. We now show that C 2 Γ − /C 2 1 reaches its maximum when k…”

Section: Calculations For Unicyclic Networkmentioning

confidence: 56%

Thermodynamic Uncertainty Relation for Biomolecular Processes

2015

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“…The chemomechanical dynamics of a molecular motor can be mapped on an appropriate kinetic network, which allows us to express the transport properties of the motor, V and D, in terms of a set of rate constants {k ij }, where k ij is associated with the transition from the i-th to j-th state [5,13,14] (see SI text). The formal expressions of V and D in terms of {k ij (f, [ATP])} are used to fit simultaneously the experimental data of V and D obtained under varying conditions of ATP and f [12].…”

Section: Resultsmentioning

confidence: 99%

“…To obtain the expression of V and D for multicyclic kinetic network model in terms of a set of rate constants {k ij }, we have generalized the technique by Koza [13] (Alternatively, technique based on the large deviation theory can be used. See Ref.…”

Section: Calculation Of V and D Of Kinesin-1 In 6-state Multi-cyclic mentioning

confidence: 99%

Energetic costs, precision, and efficiency of a biological motor in cargo transport

Hwang

Hyeon

2017

Preprint

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Molecular motors play key roles in organizing the interior of cells. An efficient motor in cargo transport would travel with a high speed and a minimal error in transport time (or distance) while consuming minimal amount of energy. The travel distance and its variance of motor are, however, physically constrained by energy consumption, the principle of which has recently been formulated into the thermodynamic uncertainty relation. Here, we reinterpret the uncertainty measure (Q) defined in the thermodynamic uncertainty relation such that a motor efficient in cargo transport is characterized with a small Q. Analyses on the motility data from several types of molecular motors show that Q is a nonmonotic function of ATP concentration and load (f ). For kinesin-1, Q is locally minimized at [ATP] ≈ 200 µM and f ≈ 4 pN. Remarkably, for the mutant with a longer neck-linker this local minimum vanishes, and the energetic cost to achieve the same precision as the wild-type increases significantly, which underscores the importance of molecular structure in transport properties. For the biological motors studied here, their value of Q is semi-optimized under the cellular condition ([ATP] ≈ 1 mM, f = 0 − 1 pN). We find that among the motors, kinesin-1 at single molecule level is the most efficient in cargo transport.Biological systems are in nonequilibrium steady states (NESS) in which the energy and material currents flow constantly in and out of the system. Subjected to incessant thermal and nonequilibrium fluctuations, cellular processes are inherently stochastic and error-prone. To minimize detrimentaion effects, cells are equipped with a plethora of energy-consuming error-correction mechanisms. Trade-off relations between the energetic cost and information processing are ubiquitous in cellular processes, and have been a recurring theme in biology for many decades [1? ? -4].A recent study by Barato and Seifert [5] has formulated a concise inequality known as the thermodynamic uncertainty relation, the trade-off between energy consumption and precision of an observable from dissipative processes in NESS. To be more explicit, the uncertainty measure Q, defined as a product between the energy consumption (heat dissipation, Q(t)) from an energy-driven process in the steady state and the squared relative error of an output observable X(t), 2 X (t) = δX 2 / X 2 , is constant. It has been further conjectured that for an arbitrary chemical network formulated by Markov jump processes Q cannot be smaller than 2k B T ,The measure Q quantifies the uncertainty of a dynamic process. The smaller the value of Q, the more regular and predictable is the trajectory generated from the process, rendering the output observable more precise. The proof and physical significance of this inequality have been discussed [5][6][7][8][9][10]. In the presence of large fluctuations inherent to cellular processes, harnessing energy into precise motion is critical for accuracy in cellular computation.The uncertainty measure Q can be used to assess the e...

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“…2B), no purely kinetic two-state model can fit the data (2,11,19,22). Four transitions are expected biochemically: (0 3 1) ATP binding; (1 3 2) hydrolysis; (2 3 3) release of P i or, from the forward head (20,23), ADP; (3 3 4 ϵ 0) release of ADP, or P i from the original rear head.…”

Section: Randomnessmentioning

confidence: 99%

Kinesin crouches to sprint but resists pushing

Fisher

Kim

2005

Proc. Natl. Acad. Sci. U.S.A.

104

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Recent optical trap experiments have applied resisting, assisting, and sideways loads to conventional kinesin moving on microtubules at fixed [ATP]. To gain insight into intermediate motions when the motor protein takes its 8.2-nm steps, the velocity and randomness data have been analyzed by using discrete-state stochastic models with a three-dimensional ''energy landscape.'' The bead size and tether angle play a crucial role. The analysis implies that on binding ATP the motor ''crouches,'' the point of attachment of the tether at the necklinker junction moving downward toward the microtubule by 0.5-0.7 nm, while inching forward by only 0.1-0.2 nm, before completing the step from a transition state by a unitary ''sprint'' of ϳ7.8 nm. These inferences accord with high-resolution observations that exclude a previously predicted substep of 1.8 -2.1 nm. Assisting and leftward loads are opposed in that the perpendicular component of the tension in the tether is enhanced by ϳ2 pN, which reduces the velocity, but sideways lurching is not supported.motor protein ͉ three-dimensional landscape ͉ vectorial loading

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General technique of calculating the drift velocity and diffusion coefficient in arbitrary periodic systems

Cited by 69 publications

References 22 publications

Thermodynamic Uncertainty Relation for Biomolecular Processes

Thermodynamic Uncertainty Relation for Biomolecular Processes

Energetic costs, precision, and efficiency of a biological motor in cargo transport

Kinesin crouches to sprint but resists pushing

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