Efficiency of molecular motors at maximum power

Molecular motors transduce chemical energy obtained from hydrolizing ATP into mechanical work exerted against an external force. We calculate their efficiency at maximum power output for two simple generic models and show that the qualitative behaviour depends crucially on the position of the transition state or, equivalently, on the load distribution factor. Specifically, we find a transition state near the initial state (sometimes characterized as a “power stroke”) to be most favorable with respect to both high power output and high efficiency at maximum power. In this regime, driving the motor further out of equilibrium by applying higher chemical potential differences can even, counter-intuitively, increase the efficiency. Introduction. – Molecular motors are essential for directed transport within the cell [1]. They typically operate under nonequilibrium conditions due to the unbalanced chemical potentials of molecules like ATP or ADP involved in the chemical reactions accompanying the motor steps. In contrast to macroscopic engines, fluctuation effects are important thus allowing for backward steps even in directed motion. The stochastic dynamics of these motors under an applied load force can be probed experimentally by single molecule assays (see, e. g., for kinesin [2], myosin [3–5] or ATPase [6]). Generically, such biomotors are modelled either in terms of continuous “flashing ratchets” [7–10] or by a (chemical) master equation on a discrete state space [11–17]. For macroscopic engines working between two heat baths at temperatures T2 > T1, efficiency is bounded by the Carnot limit ηC = 1−T1/T2. Since this limit can only be achieved by driving the engine infinitesimally slowly, thus leading to an infinitesimally small power output, it is arguably more meaningful to characterize engines by their efficiency at maximum power [18, 19]. This quantity has been studied for more than 30 years under the label of “finite-time thermodynamics” [18–22]. Recently, this concept has been transferred to microscopic (Brownian) heat engines in a variety of different model systems [23–25]. In contrast to heat engines, biomotors are driven by chemical potential differences. The efficiency of such motors is bounded by ηmax = 1 [26]. This bound can only be reached in an equilibrium situation corresponding to a vanishing power output of the motor. In analogy with heat engines, we here propose to investigate such motors under the condition of maximum power output. We start with a simple model system for a chemically driven biomotor [11] and show that the qualitative results also apply to a more realistic motor model involving a second cycle. In this second model, a futile cycle leads to ATP consumption even at the stall force. Mechanical and chemical cycles are no longer tightly coupled. In both cases, the efficiency at maximum power crucially depends on the position of the transition state or, equivalently, on the load distribution factor. In fact, a transition state near the initial position is most favorable with respect to a maximal motor power output. For the efficiency at maximum power, we obtain two counter-intuitive results : (i) it increases when the transition state position is changed in such a way that the power output rises and (ii) it can increase when the system is driven further out of equilibrium by a higher chemical potential difference. Model I. – We first consider a linear molecular motor with equivalent discrete states (sites) Xn (n = 0,±1,±2, . . . ) with distance l between the sites and nextneighbour transitions between these states subject to a force F in backward direction, see Fig 1. Forward reactions are assumed to be driven by ATP molecules with chemical potential μATP and backward transitions by ADP and P molecules with chemical potentials μADP and μP, respectively, ATP +Xn w ⇋ w Xn+1 +ADP + P. (1)