Bringing calorimetry (back) to life

TL;DR

This work develops nonequilibrium calorimetry as a quantitative tool for biology by defining excess heat and the nonequilibrium heat capacity $C_T$, and by proposing AC-calorimetry and quasipotential methods to estimate it. It analyzes minimal biophysical models of ciliary beating and molecular motors in both Langevin (diffusion) and Markov-jump formulations, revealing nonmonotonic and sometimes negative $C_T$ as functions of activity parameters like energy input, barrier height, and external load. The findings show $C_T$ can reach magnitudes near $k_B$ and exhibit rich behavior tied to nonequilibrium driving, suggesting calorimetry as a potential diagnostic for biological function—albeit with substantial experimental sensitivity requirements. Collectively, the paper provides a framework and concrete predictions linking nonequilibrium thermodynamics to microscale biological dynamics, motivating future experimental tests and extensions to more complex, coarse-grained networks.

Abstract

Micro-calorimetry offers significant potential as a quantitative method for studying the structure and function of biological systems, for instance, by probing the excess heat released by cellular or sub-cellular structures, isothermal or not, when external parameters change. We present the conceptual framework of nonequilibrium calorimetry, and as illustrations, we compute the heat capacity of biophysical models with few degrees of freedom related to ciliar motion (rowing model) and molecular motor motion (flashing ratchets). Our quantitative predictions reveal intriguing dependencies of the (nonequilibrium) heat capacity as a function of relevant biophysical parameters, which can even take negative values as a result of biological activity.

Figures (22)

• Figure 1: Nonequilibrium calorimetry at a glance. (a) Step excitation: temperature (top) and heat flow (bottom) as a function of time. Illustration of the released power (= heat flux) when the bath temperature is changed $T\to T+\text{d}T$, with $T=25^o$C and d$T=1^o$C in this example. The excess heat $\delta \cal Q^{\text{exc}}$ due to that change in temperature equals the shaded area, and the nonequilibrium heat capacity equals $C_T= -\delta \cal Q^{\text{exc}}/\text{d}T$. (b) Oscillatory (AC) excitation: temperature (top) and heat flow (bottom); see text for further details. The excess heat and nonequilibrium heat capacity are estimated respectively; see also Fig. \ref{['comp']} for a specific illustration in a stochastic model of ciliar motion.
• Figure 2: Diffusive rower model for a single cilium. (a) Sketch of the motion of a single cilium whose tip motion is probed by a bead (orange circle). (b) Sketch of the stochastic oscillation mechanism of the rower model described in \ref{['rowm']}: the tip of the cilium (orange circle) diffuses in a harmonic potential centered in $-A$ (green line) until it reaches $-a>-A$, after which the potential's minimum is immediatly switched to $A$ (red line) until the next passage through $a$ etc. (c) Numerical simulation for the cilium's trajectory over one second for bath temperature $\, T=300 \,\mathrm{K}$, friction coefficient $\gamma = 0.2\,\text{pN s}/\mu\text{m}$, stiffness $\kappa = 1.5\,\text{pN}/\mu\text{m}$, and length parameters $A = 1\,\mu\text{m}$ and $a = 0.25\,\mu\text{m}$; see also Ref. gupta2025.
• Figure 3: Nonequilibrium (AC-) calorimetry at work: rower model for ciliar motion.(a) Top: heat-bath temperature oscillations $T_b(t)=T[1-\epsilon_b\sin(\omega_b t)]$ used for AC-calorimetry. Bottom: AC and DC expected heat fluxes. (b) The differences in the expected heat fluxes $\dot{\mathcal{Q}}(t) - \dot{\mathcal{Q}}_T$ in Eq. \ref{['mama']} specifically for the stochastic nonequilibrium dynamics associated with the rower model of a single cilium motion; see Eq. \ref{['rowm']}. The data are shown for reference temperature $T = 295\,\text{K}$, temperature relative amplitude $\epsilon_b = 0.01$, temperature oscillation frequency $\omega_b= (\pi/5)$Hz. For the sake of visualization, the nonequilibrium heat capacity $C_T$ is taken in this plot to be ten times larger than its actual value. The in-phase Fourier component is $B_T\epsilon T \sin(\omega t)$ while the out-of-phase component is $C_T \epsilon_b T \omega_b \cos(\omega_b t)$. Here, $C_T = 0.15\,k_B$, friction coefficient $\gamma = 0.2\,\text{pN s}/\mu\text{m}$, stiffness $\kappa = 1.5\,\text{pN}/\mu\text{m}$, and length parameters $A = 1\,\mu\text{m}$ and $a = 0.25\,\mu\text{m}$.
• Figure 4: Expected excess heat flux as a function of time for the diffusive rower obtained from AC calorimetry (a) in equilibrium, and (b) out of equilibrium.Estimates from numerical simulations of the time-dependent (expected) excess heat flux, obtained as the difference between AC (total $\dot{\cal {Q}}(t)$) and DC (housekeeping $\dot{\cal Q}_T$) expected heat fluxes as a function of time for the rowing model for parameter values $\kappa = 1.5 \mathrm{pN/\mu m}$, $a = 0.25 \mu\mathrm{m}$, and temperature modulation $T_b(t)= (1-\epsilon_b\,\sin(\omega_b t))T$, with $T=300$K, $\epsilon_b=0.01$, $\omega_b= \pi/5$Hz. For the equilibrium case (a) $A=0$ and $\gamma= 0.1$ pN s/$\mu$m, while for the nonequilibrium cases (b) $A = 0.5 \mu\mathrm{m}$ and $\gamma$ as specified in the legend, where we also added the equilibrium case at $\gamma= 0.1$ pN s/$\mu$m for comparison. All results are averages over $10^5$ numerical simulations.
• Figure 5: Nonequilibrium heat capacity in the rower model as a function of biophysical parameters. Heat capacity at temperature $T=300K$ using AC-calorimetry with $T_b(t) = T \,(1 - \epsilon_b \sin(\omega_b t))$, with $\epsilon_b=0.01$ and $\omega_b=2\pi$Hz: (a) as a function of $A$, half the distance between the minima of the potentials with $r=6.25 \mu m$ (or equivalently $\gamma =0.1\,\mathrm{pN\cdot s}/\mu\mathrm{m}$), and (b) as a function of $r$, the bead radius of the tip for $A = 0.5\,\mu\mathrm{m}$. The symbols are results from numerical simulations and the solid lines fits to empirical functions. The simulation is performed for $10^4$ trajectories over $200\,\mathrm{s}$, corresponding to an average over $200$ oscillation periods, with $a = 0.25\,\mu\mathrm{m}$, $\kappa = 1.5\,\mathrm{pN}/\mu\mathrm{m}$. The expected heat flux is obtained from \ref{['powerrowing']} and the heat capacity using Eqs. \ref{['mama']} and \ref{['cac']}. Error bars are given by standard error of the mean. The fits are: (a) $C_T(A)/k_B = d_0 + d_1 A + d_2 A^2 + d_3 A^3$, with $d_0 = 0.49$, $d_1 = -0.14\,\mu\mathrm{m}^{-1}$, $d_2 = 0.013\,\mu\mathrm{m}^{-2}$, and $d_3 = -4.0 \times 10^{-4}\,\mu\mathrm{m}^{-3}$; and $C_T(r)/k_B = c_0 + c_1 r + c_2 r^2 + c_3 r^3$, (b) with $c_0 = 0.41$, $c_1 = -1.1 \times 10^{-2}\,\mu\mathrm{m}^{-1}$, $c_2 = 1.6 \times 10^{-4}\,\mu\mathrm{m}^{-2}$, and $c_3 = 1.7 \times 10^{-5}\,\mu\mathrm{m}^{-3}$.