Miniature work-to-work converter engine powered by motor protein

TL;DR

This work introduces a miniature information-based engine that converts motor-protein–driven motion into work output within an optical-trap setup, functioning as a work-to-work converter powered by a kinesin–microtubule system. The authors develop a 1D and 2D stochastic framework, derive analytical expressions for bead dynamics, run-time statistics, and the full thermodynamics of the engine, and validate predictions with extensive simulations. Notably, the average work per cycle can exceed multiple $k_B T$ and reach regimes where the work distribution becomes nontrivial, driven primarily by motor binding/unbinding stochasticity rather than bath fluctuations. The results provide quantitative guidance for designing high-performance microscale engines and highlight the role of information (state knowledge of motor binding) in enhancing thermodynamic performance, with feasible paths toward experimental realization.

Abstract

Designing a miniature microscale engine that can override the role of thermal fluctuations has remained elusive and is an important open challenge. Here we provide the design and theoretical framework for a unique information-based engine - a work-to-work converter - comprising a sub-micron size bead and motor protein-microtubule (MT) complex in an optical trap setup. We demonstrate how by implementing a simple motor protein state-dependent feedback protocol of the optical trap stiffness, this engine is able to harness and convert the movement of a motor protein into work output. Unlike other conventional microengines, the fidelity and performance of this engine is determined by the stochasticity of motor (un)binding characteristics. We obtain an analytical form of the work distribution function, average work output and average power output, providing quantitative predictions for engine performance which are validated by stochastic simulations. Remarkably, the average work output per cycle is at least an order of magnitude higher than the thermal fluctuations and supersedes the performance of other microscale engines realized so far.

Figures (11)

• Figure 1: Engine Cycle: (a) At A, motor attaches at optical trap center. For step AB, trap stiffness $k_t$ varies linearly such that $k_t(t) = k_o + \mu t$. At B, motor detaches from MT and $k_t$ is instantaneously reduced to $k_o$, corresponding to the step BC. For step CA, the detached motor relaxes to the center of optical trap and stays there until next motor attachment event occurs. This sequence of events completes one engine cycle. (b) Trap stiffness variation with time : $k_t$ vs $t$ (c) Variation of bead position $x$ with $t$, and its correspondence with $k_t$ vs $t$ for one complete engine cycle.
• Figure 2: Performance of kinesin-1 and kinesin-3 driven engine: $\langle W_c\rangle$ vs $v_o$ for (a) kinesin-1 and (b) kinesin-3. Contour plot of $\langle W_c\rangle$ in $(v_o- k_o)$ plane for (c) kinesin-1 and (d) kinesin-3. (e) Efficiency $\eta$ vs $v_o$ (f) Average power output: $P$ vs $v_o$. For kinesin-1, $\epsilon_o = 0.72 ~ s^{-1}$soppina2022kinesin, $f_s = 5.7$ pN brenner2020force, and $f_m = 4$ pN guo2019force, and for kinesin-3, $\epsilon_o = 0.23~ s^{-1}$soppina2022kinesin, $f_s = 3$ pN budaitis2021pathogenic, and $f_m = 2.7$ pN budaitis2021pathogenic. The trap stiffness $k_o = 0.005 ~ pN ~nm^{-1}$rai2013molecularbrenner2020force in (e) and (f).
• Figure 3: PDF of $W_c$ and its characteristics: (a) and (b) are plots of P($W_c$) vs $W_c$ for an engine driven by kinesin-1 and kinesin-3 motor respectively, when $k_o = 0.005~ pN ~nm^{-1}$. For (a), $v_o = 0.8 ~\mu m~s^{-1}$, and for (b), $v_o = 2.5 ~\mu m~s^{-1}$. Rest of the parameters are same as in Fig.\ref{['fig3']}. Analytical result of PDF in Eq.\ref{['Pw']} (blue curve) is compared with 1D stochastic simulations (performed with $10^6$ samples) with discrete motor step size of $8~ nm$ (green circles) and continuous step size (red squares). The maximum work output $W_{max} = - 169.42 ~ k_BT$ for (a), and $W_{max} = - 7.90 ~ k_BT$ for (b). (c) and (d) corresponds to plot of the characteristics of P($W_c$) for an engine driven by kinesin-1 and kinesin-3 motor respectively. In Region $(I)$ (green), $P(W_c)$ monotonically decreases to zero. In Region $(II)$ (blue) $P'(W_c) \rightarrow -\infty$ as $W_c \rightarrow W_{max}$. In Region $(III)$ (red) both $P'(W_c) \rightarrow \infty$ and $P(W_c) \rightarrow \infty$ as $W_c \rightarrow W_{max}$. The purple curve corresponds to $\alpha = \alpha_c/2$ while the black curve corresponds to $\alpha = \alpha_c$.
• Figure 4: Comparison with 2D stochastic simulations: (a) and (b) correspond to plots of $\langle W_c\rangle$ vs $v_o$ for engine driven by kinesin-1 and kinesin-3 motor respectively. (c) $P(W_c)$ vs $W_c$ for engine driven by kinesin-1 motor when $v_o=0.8~\mu m ~s^{-1}$howard1989movement. (d) $P(W_c)$ vs $W_c$ for engine driven by kinesin-3 when $v_o=2.5~\mu m~s^{-1}$soppina2022kinesin. In the panels, the blue curves in (a) and (b) correspond to analytical expression $\langle W_c\rangle$ in Eq.\ref{['eq:Wc1']} and for (c) and (d) it corresponds to analytical expression of $P(W_c)$ in Eq.\ref{['Pw']}. Comparison is done with 2D stochastic simulation with bead radius $R = 0.05 ~ \mu m$(red squares) and $R = 0.1 ~ \mu m$ (green circles). For simulations, motor rest length $l_o = 110 ~ nm$erickson2011molecularuccar2019force and trap stiffness along the MT and perpendicular to it are chosen as $k_o^x=k_o=0.005~pN~nm^{-1}$ and $k_o^y=k_o^x/3$ashkin1992biophysjbormuth2008opticsexp, respectively. All other parameters are same as Fig.\ref{['fig3']} for all panels. Stochastic simulation curves are obtained from $10^6$ independent runs.
• Figure S1: Mean run time $\langle \tau_1\rangle$ for (a) kinesin-1 (b) kinesin-3 as a function of $v_o$. Run time distribution Eq.\ref{['eq:P1']}$P(\tau_1)$ (c) for kinesin-1. Here, $k_o = 0.005 ~ pN~ nm^{-1}$, $f_s = 5.7~ pN$ , $f_m = 4.0 ~pN$ , $\epsilon_o = 0.72 ~ s^{-1}$ . (d) for kinesin-3. Here, $k_o = 0.005 ~ pN~ nm^{-1}$, $f_s=3.0 ~ pN$ , $f_m = 2.7 ~ pN$ , $\epsilon_o = 0.23 ~ s^{-1}$.