In-vivo entropy production of A. subaru

Abstract

Entropy production is often used as a proxy for energy consumption of a non-equilibrium system. Lower bounds can be estimated from coarse-grained observations, and this has been done for various biological systems. Here, we apply these tools to a more macroscopic system whose true energy consumption is also known. We find that while entropy production does give a lower bound, it is some 25 orders of magnitude away from being saturated. To be certain of this result, we survey different methods of estimating irreversibility, and write down a novel kNN estimator.

Figures (6)

• Figure 1: (A) Automobilus subaru (left) is a large quadrirotus creature, originally from Japan. Shown with H. sapiens (centre) and S. demersus (right) for scale. (B) Raw data time-trace, showing speed, engine RPM, and coolant temperature. These three numbers are recorded about every 140 ms, using a device huang00freematics which plugs into the OBD-II port epa1993obdii. We regard the temperature as a constant, and use the other two as $x(t) \in \mathbb{R}^2$ for our EPR calculations (figures \ref{['fig:k-means']}, \ref{['fig:many-methods']}).
• Figure 2: Entropy production provides a bound on the rate of energy consumption, at a given temperature. Besides our work (black star), we plot some competing papers; only the one on L. catesbeianus hair cell bundles roldan2021quantifying provides figures for both axes (red star). For all other points, we have estimated energy consumption (as described in Methods) and plot irreversibility / entropy production rates taken from the following papers: Bos taurusskinner2021estimating, H. sapienslynn2021broken, Sturnus vulgarisferretti2022signatures, microtubules and HEK cells skinner2021improved, neurons pietzonka2024thermodynamic, and the Turin public transit system biazzo2020city. The two red points are sub-cellular scale, while the green points are single cells.
• Figure 3: Clustering data using k-means to compute irreversibility. Same data as figure \ref{['fig:raw-data']}, with $k=13$ clusters. Discretising on a grid instead would result in many empty squares. (We observe that this subaru has a CVT speedkar99subaru, as fixed gear ratios would show up as radial lines on this plot.) Right, width of arrows is $J_{ij}-J_{ji}$, showing non-equilibrium fluxes for 1st-order Markov formula (\ref{['eq:histogramSdot']}).
• Figure 4: The effect of hyper-parameters and bias on several methods of estimating entropy production, equations (\ref{['eq:gaussSdot']}) to (\ref{['eq:turSdot']}). All estimates from data (blue points) are deterministic, except for (\ref{['eq:histogramSdot']}) where k-means clustering has a random start, hence we plot 20 repetitions of the clustering algorithm, and the maximum. The shuffling of data, and the noise, are also random (grey points). On all plots, orange points are estimates corrected for bias; these are closer to being independent of the hyper-parameter. The dotted orange line at $\,\Sigma=0.5\,$bits/s is the value in \ref{['eq:onenumber']}, and used in figure \ref{['fig:Comparison']}. For the TUR estimate (\ref{['eq:turSdot']}), we compute winding about many points $x$, on a grid, and highlight the maximum. All plots use the same data, with $d=2$ and $N=134,600$ points over $t_\mathrm{max}=324$ minutes, a longer time-series like that shown in figure \ref{['fig:raw-data']}B.
• Figure 5: Sketch of three kNN irreversibility estimators, using a toy dataset with $N=500$ points in $d=2$ dimensions. We perform a 10th order Markov estimate, hence $z_{t}=(x_{t},x_{t+1},\ldots,x_{t+\ell})\in\mathbb{R}^{20}$; the plot shows two principal components. For a given point $z_{t}$, the radii of the circles containing $k$ forward points (blue), or $k$ reverse points $\tilde{z}_{t'}$ (orange), are the input to the plug-in estimator (\ref{['eq:wangSdot']}). For the KSG-like estimator $\,\Sigma_\text{count}$ from (\ref{['eq:digammaSdot']}), instead the number of orange points within the blue circle is what matters. Finally, $\,\Sigma_\text{union}$ from (\ref{['eq:noshadSdot']}) finds distance to the $k$-th nearest point among the union of all $z_{t}$ and $\tilde{z}_{t}$, and then counts both blue and orange points within this green circle.