Thermodynamics of Animal Locomotion

Muscles are biological actuators extensively studied in the frame of Hill’s classic empirical model as isolated biomechanical entities, which hardly applies to a living organism subjected to physiological and environmental constraints. Here we elucidate the overarching principle of a living muscle action for locomotion, considering it from the thermodynamic viewpoint as an assembly of actuators (muscle units) connected in parallel, operating via chemical-to-mechanical energy conversion under mixed (potential and flux) boundary conditions. Introducing the energy cost of effort as the generalization of the well-known oxygen cost of transport in the frame of our compact locally linear nonequilibrium thermodynamics model, we analyze oxygen consumption measurement data from a documented experiment on energy cost management and optimization by horses moving at three different gaits. Horses adapt to a particular gait by mobilizing a nearly constant number of muscle units minimizing waste production per unit distance covered; this number significantly changes during transition between gaits. The mechanical function of the animal is therefore determined both by its own thermodynamic characteristics and by the metabolic operating point of the locomotor system.

Figure 1

Four-quadrant plot of ${\mathrm{COE}}_{-}$ (north direction), $\eta$ (west), $I_M$ (east), and $P$ (south): (a) ${\mathrm{COE}}_{-}$ vs $\eta$, (b) ${\mathrm{COE}}_{-}$ vs $I_M$, (c) $P_M$ vs $\eta$, (d) $P_M$ vs $I_M$, from Eqs. (1)–(4), using ${\mathrm{\Phi }}_{+}=({\mu }_{+}-{\mu }_{M+}/R_{+})$, with $R_M=10$, $R_{+}=10$, $R_{-}=0$, $R_E=100$, $\alpha =0.5$, ${\mu }_{-}={\mu }_{M-}={10}^{-4}$, and ${\mu }_{+}=100$ (all in arbitrary units). The arrows show the direction along which $I_M$ increases. The red star symbol (respectively, blue squares and green dots) indicates the position of $P_{\max }$ (respectively, ${\mathrm{COE}}_{-}^{*}\equiv \min ({\mathrm{COE}}_{-})$ and ${\eta }_{\max }$).

Figure 2

(a)–(c) Experimental data from [26], horse B of mass $M=140\mathrm{kg}$: oxygen flux ${\mathrm{\Phi }}_{{\mathrm{O}}_2}$ and $\mathrm{COT}\equiv {\mathrm{\Phi }}_{O_2}/v$, plotted against the speed $v$ for walk (red stars), trot (blue dots), and gallop (green squares), and their fits with our modeling, Eq. (5), and fitting parameters in Table 1. Note the COT dramatic increase for the high-speed walk shown in the inset: this slope change marks the change of muscular effort regime in this region (see Supplemental Material [27]). (c) For each gait, the histogram of the observed range of velocities naturally used by the horse let free to run on the ground. (d),(e) $N_{H_w}{\phi }_{-}/M$ and the specific $\mathrm{COT}/M$ are plotted against the scaled speed $v/v^{*}\propto i_M$. In both cases, the thick dark line is a two-parameter fit of the aggregated data. All gaits are considered as a collection of a different number of activated muscle units; the ratio $N_{H_w}/N$ vs $v^{*}$ is shown in the inset. The ratio $v_w^{*}/v^{*}$ is represented with blue dots, $B_{v_w}/B_v$ with red $+$, and $R_v/R_{v_w}$ with dark stars. The red dotted line $1/v^{*}$ serves as a guide for the eyes for comparison purposes.