Microscale velocity-dependent unbinding generates a macroscale performance-efficiency tradeoff in actomyosin systems

Figures

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  Fig. 1. A nonlinear model accounting for 2-state myosin binding dynamics and actuation of a mass. a Schematic of the 2-state model of myosin binding dynamics in muscle, created using BioRender98. Myosin cycles between detached and attached states. b, c Time series of actuation, represented in force-velocity (b) and power-velocity (c) spaces compared with the predictions of Hill’s model in Eq. (1) (black-dashed line). Color corresponds to dimensionless time T (color bar, right). d Time series of model attachment (teal line), velocity (magenta line), and force (gray line) for α = 2, with T = kat dimensionless time. e Time series of myosin attachment for α = 0, 1, 2, 5 (black, blue, orange, and green lines, respectively); a larger fraction of myosin unbind with increasing nonlinearities.

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  Fig. 2. A performance-efficiency tradeoff emerges. a As the rate of unbinding (α) increases, mechanical power output (solid lines) and power consumption (dashed lines) decreases, however, because power consumption drops at a faster rate than power output, (b) we find increasing efficiency with the nonlinearity. c, d Performance-efficiency tradeoff emerges in 2-state model (black-dash lines) and in-vivo data taken from the literature (scatter points, N = 26 measurements with associated error bars as reported in their respective studies); as the unbinding rate α increases, the actomyosin system is less powerful (lower PC) yet it gains higher instantaneous efficiency η values.

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  Fig. 3. Introducing nonlinearity in HillBot demonstrates output and efficiency tradeoff. a Photograph of HillBot. b Schematic of electromechanical model of HillBot. c Schematic of HillBot lifting a weight against gravity. d Time series of data collected from HillBot; normalized, measured force (gray open circles), velocity (magenta open circles), and voltage (teal open circles) as a function of time overlayed on the model’s predictions (solid lines). e FV and (f) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{W}V$$\end{document}W°V curves for α = 0, 4 given by Hill’s equation (blue and green lines, respectively), experimental measurements with HillBot (open circles), and electromechanical model (black dashed lines). g The PID controller dynamically reduces voltage for increasing α to match Hill-type outputs (similar to cross-bridge unbinding in the 2-state model). h, i HillBot’s electromechanical model (black dashed lines) and N = 169 trials in HillBot (scatter points) show an exchange of power…

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  Fig. 4. Nonlinear coupling generates a tradeoff in performance and efficiency not found in linear actuation. a, b Comparison of linear C = 0.72 and nonlinear α = 4 force-power-velocity relations for PC = 0.25. c Linear relationship between applied voltage and motor handicap; experimentally, we decrease the voltage supply to decrease the motor constant to lower the PC in Series-L. d Motor constant C and nonlinearity α values that generate FV-relations with an equal PC; for example, PC(C = 0.72) = PC(α = 4) = 0.25 as in (a, b). e Efficiency decreases with PC in Series-L, whereas efficiency increases with nonlinear actuation, disentangling PC′s effect on η.

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  Fig. 5. α meta-analysis results. a Hill’s force-velocity (FV) and (b) mechanical power-velocity (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{W}V$$\end{document}W°V) relations for α = 0, 4, 15. c List of muscle types found in the meta-analysis. Muscles are split into four groups: those where α has a broad range (yellow), low values (blue), high values (red), and near α* (green). Numbered label corresponds to (d); those labeled NH indicate non-human muscles. d Muscles found in the human body from the meta-analysis. Muscles are colored according to their label in (c). e Distribution of α in measured biological muscle tissue for N = 136 muscle samples, including those from mammals, birds, mollusks, fish, amphibians, insects, reptiles, and single mouse myoblast cells (red dashed line). The red dashed line indicates a measurement of the FV-relation from a myoblast cell. The nonlinearity α ranges between 0 and 15, centered at α* = 3.85 ± 2.32 (median, gray line ± IQR, shaded gray).

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  Fig. 6. The performance-efficiency tradeoff quantified. a Objective function applied to in-vivo meta-analysis data (purple scatter points, N = 26 measurements with associated error bars as reported in their respective studies) and the simple 2-state actomyosin model (dashed black line). ϕHill is minimized at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha }_{\,{\mbox{2-st}}\,}^{* }=5.72$$\end{document}α2-st*=5.72, agreeing with α* = 3.85 ± 2.32 (gray shaded region). Inset: Efficiency η (red symbols) and power characteristic PC (blue symbols) as functions of the nonlinearity parameter α. b Objective function ϕHill as a function of the nonlinearity parameter α in HillBot (purple circles, N = 169 trials) with electromechanical model (black dashed line). The gray region corresponds to the characteristic nonlinearity α*. Sub-panel shows an exchange of power characteristic (blue circles) and efficiency (red circles) with α; ϕHill is minimized at \documentclass[12pt]{minimal} \usepackage{amsmath}…