# Scale-dependent irreversibility in living matter

Tzer Han Tan1,2†, Garrett A. Watson1†, Yu-Chen Chao1†, Junang Li1, Todd R.
Gingrich3,
Jordan M. Horowitz4,5,6∗, Nikta Fakhri1∗

1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA
02139, USA
2Center for Systems Biology Dresden, Max Planck Institute of Molecular Cell
Biology and Genetics,
Dresden 01307, Germany (_current address_)
3Department of Chemistry, Northwestern University, Evanston, IL 60208, USA
4Department of Biophysics, University of Michigan, Ann Arbor, Michigan, 48109,
USA
5Center for the Study of Complex Systems, University of Michigan, Ann Arbor,
Michigan, 48109, USA
6Department of Physics, University of Michigan, Ann Arbor, Michigan, 48109,
USA

∗Corresponding authors.
†These authors contributed equally

## Abstract

A defining feature of living matter is the ability to harness energy to self-
organize multiscale structures whose functions are facilitated by irreversible
nonequilibrium dynamics [1]. While progress has been made in elucidating the
underlying principles, what remains unclear is the role that thermodynamics
plays in shaping these structures and their ensuing functions. Here, we
unravel how a fundamental thermodynamic connection between the physical energy
dissipation sustaining a nonequilibrium system and a measure of statistical
irreversibility (arrow of time), can provide quantitative insight into the
mechanisms of nonequilibrium activity across scales [2]. Specifically, we
introduce a multiscale irreversibility metric and demonstrate how it can be
used to extract model-independent estimates of dissipative timescales. Using
this metric, we measure the dissipation timescale of a multiscale cellular
structure – the actomyosin cortex – and further observe that the
irreversibility metric maintains a monotonic relationship with the underlying
biological nonequilibrium activity. Additionally, the irreversibility metric
can detect shifts in the dissipative timescales when we induce spatiotemporal
patterns of biochemical signaling proteins upstream of actomyosin activation.
Our experimental measurements are complemented by a theoretical analysis of a
generic class of nonequilibrium dynamics, elucidating how dissipative
timescales manifest in multiscale irreversibility.

## Main text

Living systems operate far from equilibrium, constantly consuming energy. This
energy is harnessed at the molecular-scale through ATP hydrolysis and
dissipated on larger spatiotemporal scales spanning the entire cell. Canonical
examples of these cellular structures are the actomyosin cortex [3, 4] and
mitotic spindle [5, 6], which are required for vital functions such as cell
division. These living systems are, in fact, examples of the broader class of
active matter, where large scale rearrangements and functions are driven by
energy-consuming microscopic processes, such as mechanochemical reactions [1].
The inherent multiscale nature of such systems makes it challenging to detect
the nonequilibrium activity, let alone unravel its characteristic features [7,
8].

While various methods have been developed to detect the existence of
nonequilibrium activity, they often fail to reveal any mechanistic insight.
For instance, a clear signature of a nonequilibrium process is directed motion
or flux. This observation has led to a handful of methods to identify the
existence of such fluxes and to extract from their fluctuations estimates (or
bounds) on the nonequilibrium dissipation, typically in individual mesoscopic
systems [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. However, there is no clear
method available to extract characteristic features of the nonequilibrium
activity from the estimates of these fluxes. On the other hand, departures
from the fluctuation-dissipation theorem (FDT) are inherently connected to
nonequilibrium activity [19, 20, 21] in a way that could reveal more detailed
information about the timescales of nonequilibrium fluctuations; however, such
analyses are typically focused on identifying structure in a time-dependent
(or frequency-dependent) effective temperature whose interpretation usually
requires prior information about the mechanisms [22, 23, 24, 25, 26, 27, 28].
Specifically for mesoscopic diffusive systems, the deviations of the velocity
FDT can be used to estimate the dissipation using the Harada-Sasa equality
[19], an approach that has been successfully utilized to estimate the
energetic efficiency of molecular motors [29, 30] and to probe the statistics
of active forces [31]. Even still, the FDT approach requires active
perturbations in order to measure response properties, which in practice can
be challenging. Thus, the development of new frameworks that can identify the
scales of energy dissipation is crucial for a mechanistic understanding of
nonequilibrium processes in complex biological and active matter systems.

### Irreversibility measures the arrow of time

Any nonequilibrium process is accompanied by the irrecoverable loss of energy
to its environment, often in the form of heat, which commensurately produces
entropy in the environment. The unidirectionality of this flow implies a
direction to time’s arrow. In other words, the process is inherently
irreversible [32, 33, 34, 35, 2]. Here, we show that by measuring
irreversibility one can reveal the temporal structure of nonequilibrium
activity from experimental time series data collected in a living system – the
actomyosin cortex of a starfish oocyte. Thereby, we discover that recent
developments in nonequilibrium thermodynamics that were previously thought
only able to confirm or deny the existence of nonequilibrium [36, 37, 38] can,
in fact, be adapted to unravel timescale information about dissipative
processes. Nonequilibrium thermodynamics offers a quantitative connection
between dissipation and the statistical irreversibility (time-reversal
asymmetry) of the dynamics. For a nonequilibrium steady state, the relative
entropy or Kullback-Leibler divergence (KLD) between the probability
${\mathcal{P}}(x_{1},\dots,x_{n})$ to observe a stationary series of
measurements $(x_{1},\dots,x_{n})$ of a time-symmetric observable and the
probability to observe the reverse sequence ${\mathcal{P}}(x_{n},\dots,x_{1})$
bounds the dissipation (entropy production) as [33, 34, 2, 39, 40]

$\Sigma\geq\sigma=\lim_{n\to\infty}\sigma_{n}\equiv\lim_{n\to\infty}\frac{k_{\rm
B}}{n\tau}D[{\mathcal{P}}(x_{1},\dots,x_{n})||{\mathcal{P}}(x_{n},\dots,x_{1})],$
(1)

where $\tau$ is the sampling interval and $k_{\rm B}$ is Boltzmann’s constant.
Generically, the entropy production $\Sigma$ can be related to the energy
dissipation once one identifies the nature of the thermodynamic reservoirs in
the environment with which the system exchanges energy: For example, chemical
reactions occurring at rate $r$ driven by a chemical potential difference
$\Delta\mu$ produce entropy equal to the chemical work $\Sigma=r\Delta\mu$.
The KLD, $D(p||q)=\int dx\ p(x)\ln p(x)/q(x)$, is an information-theoretic
measure of distinguishability that generically decreases under coarse graining
[41], with equality when every nonequilibrium degree of freedom is included.
As such, $\sigma$ measures the observed statistical distinguishability between
the forward and reverse senses of time, making it a quantitative measure of
irreversibility. With this in mind, we call $\sigma$ the (observed)
irreversibility. In the absence of detailed knowledge of the system, Eq. (1)
has been developed into a robust experimental and computational tool to detect
nonequilibrium activity even in the absence of observable flows [35, 2, 42,
43, 44, 36, 37, 45, 46]. Here, we exploit the flexibility in the choice of
observable in Eq. (1) to probe multiple timescales of irreversibility by
repeatedly estimating $\sigma$ for progressively more coarse-grained time-
series data. Coarse graining in time allows us to smear over dynamic processes
with fast characteristic timescales, thereby removing their influence on the
observed irreversibility. Then by tracking the variation of irreversibility
with the coarseness of the data, we can unravel timescales important to
dissipation. Such a coarse-graining approach to time series analysis has
previously been successfully employed as a basis for the multiscale sample
entropy method for complexity assessment [47].

### Actomyosin cortex: a multiscale dissipative structure

The actomyosin cortex is a thin, cross-linked network of filamentous actin and
myosin motors that is anchored to the cell membrane [3]. The cortex of an
immature starfish oocyte is in a nonequilibrium steady state that persists
over very long times ($>24\ hours$ post extraction). The nonequilibrium
processes in the cortex, in particular the force fluctuations generated by the
activity of myosin motors, are powered by constant ATP consumption at the
molecular scale. We varied the amount of the nonequilibrium activity in the
actomyosin cortex via ATP depletion treatments (low activity steady state) or
over-expression of myosin motors (high activity steady state) [48]. To probe
the local force fluctuations, we took advantage of endogenous fluorescent
cortical granules ($\sim 1\ \mu m$ in size) that are embedded in the cortex
(Figs. 1a-c ). With high photostability and modest photobleaching for up to an
hour, they enabled long duration ($>1\ hour$), high-speed microscopy ($>10\
Hz$). To verify that the cortical granules are embedded in the actin cortex,
we depolymerized actin using cytochalasin D and observed that these granules
dissociated from the cortex. We used a single molecule tracking algorithm [49]
to obtain the centroid positions $r(t)$ of the cortical granules (Fig. 1b) as
a function of time $t$, sampled every $\tau$ seconds. We verified that the
actomyosin network is spatially homogeneous over the course of our
observations (Fig. S1) and stitched particle trajectories from individual
granules together into one long trajectory of length $T$ seconds for
subsequent analysis.

Figure 1: Positional fluctuations and position-increment autocorrelation
function of endogenous cortical granules probe local force fluctuations of the
actomyosin cortex. (a) Cortical cross-section of a flattened starfish oocyte
imaged using widefield near-infrared fluorescence microscopy. Puncta represent
individual granules. Schematic shows the cortical granules (yellow) embedded
in the actin network (red) activated by myosin motor minifilaments (blue). (b)
Example position trajectories of embedded cortical granules color coded with
time. (c) 2D coarse-grained trajectories at different lag time $\vartheta$
color coded by time. (d-f) Comparison of position-increment autocorrelation
function $C^{\vartheta}(\Delta t)$ with various lag times
$\vartheta=1,2.5,5,10,18\ s$ as a function of correlation time $\Delta t$ for
an oocyte with (d) low activity (ATP depletion), (e) medium activity (wild
type), and (f) high activity (myosin over expression) displays an activity-
dependent cross-over with coarse-graining lag time. Notice how the observed
short-time elastic response characterized by a sudden and rapid drop
disappears at lower lag times for higher activity treatments.

To estimate the irreversibility $\sigma$, we require data obtained from a
stationary steady-state process. Here, the underlying dynamics of the
actomyosin network is stationary, which leads to stationary local forces
fluctuations acting on the cortical granules. Considering that the diffusive
dynamics of the cortical granules are overdamped [50], we can interpret
changes in position as readouts of these stationary local force fluctuations
of the underlying nonequilibrium actomyosin network. We estimated these
position increments $\Delta r^{\vartheta}(t)=r(t+\vartheta)-r(t)$ using an
adjustable lag time $\vartheta$, with larger $\vartheta$ corresponding to more
coarse-grained data (Fig. 1c and S2). (Stationarity of the time-series was
confirmed by a unit root test.) Thus by varying $\vartheta$, we generate a
collection of observables whose estimated KLD $\sigma(\vartheta)$ captures the
irreversibility on scales larger than $\vartheta$, scales that have not been
smeared out by coarse graining. It is important to note that such a measure is
not sensitive to the frequency of sampling (as long as the sampling is faster
than the rate of dissipation), as less-frequent sampling manifests in the
formalism as simply an additional form of coarse graining.

As a first step in understanding the underlying dynamical processes in the
actomyosin cortex, we analyzed the normalized time-lagged position-increment
autocorrelation function $C^{\vartheta}(\Delta t)=\langle\Delta
r^{\vartheta}(t+\Delta t)\Delta r^{\vartheta}(t)\rangle/\langle\Delta
r^{\vartheta}(t)^{2}\rangle$ (Fig. 1d-f). In the low activity steady state
(Fig. 1d), the position increments are delta-correlated at $\Delta t=0$
followed by a negative correlation for small $\Delta t$ with diminishing
magnitude for longer lag times $\vartheta$. This is consistent with the
dynamics of the cortical granules being driven primarily by thermal
fluctuations in a locally elastic network on short times, as evidenced by the
mean squared displacement curve (Fig. S3) [7, 8]: small displacements are
immediately followed by a displacement that returns the particle to its
original position, resulting in oppositely oriented increments. On longer
times, nonequilibrium activity overcomes the elasticity of the network. This
effect is stronger in medium and high activity steady states (Figs. 1e,f),
where the short time equilibrium elastic response (negative dip in the
autocorrelation function) is smeared out at smaller lag times:
$\vartheta^{*}\approx 2.5-10\ s$ for the medium activity steady state and
$\vartheta^{*}\approx 1-2.5\ s$ for the high activity steady state. One might
suspect that these times roughly correlate with the timescale of a
nonequilibrium mechanism, since they are sensitive to the level of activity.
In the following, we corroborate this conclusion by demonstrating the
emergence of this timescale in an analysis of the multiscale irreversibility,
which by Eq. (1) offers us a model-independent verification that these
timescales are indeed related to the dissipation.

### Irreversibility reveals nonequilibrium timescales

Using the position-increment time-series data, we interrogate the
irreversibility in the network’s force fluctuations (which we take to be even
under time reversal) by estimating the KLD $\sigma(\vartheta)$ between the
position increment trajectory $\Delta r^{\vartheta}(t)$ and its reverse
$\Delta r^{\vartheta}(T-t)$ for various values of lag time $\vartheta$. We
used a $k$-nearest-neighbor method to estimate the KLD [51] as detailed in the
SI Sec. 1.6 (Fig. S2, S4-5). For lag times $\vartheta\geq 8\ s$, we verified
that our KLD estimates plateaued with number of samples prior to reaching our
sample size of $9.1\times 10^{5}$ data points. For lag times $\vartheta<8\ s$,
thermal fluctuations are dominant, and despite increasing the sample size to
$3.1\times 10^{6}$ data points, the convergence is still not satisfactory.

Figure 2: Time irreversibility reveals the intrinsic timescale of
nonequilibrium activity and correctly rank orders nonequilibrium activity
above the dissipation timescale. (a) Estimated irreversibility $\sigma$ for
the high ($N\approx 10\times 10^{6}$ for $\vartheta<5\ s$, $N\approx 1\times
10^{6}$ for $\vartheta>5\ s$) and medium ($N\approx 5\times 10^{6}$ for
$\vartheta<5\ s$, $N\approx 1\times 10^{6}$ for $\vartheta>5\ s$) activity
treatment as a function of lag time $\vartheta$. The shaded region for low lag
time ($\vartheta<5\ s$) of the high and medium activity curves shows the range
of 5 bootstraps statistics with $N\approx 2.5\times 10^{6}$ and $N\approx
1.5\times 10^{6}$ data points respectively. The shaded region for high lag
time ($\vartheta>5\ s$) shows the range of 5 bootstraps statistics with
$N\approx 0.3\times 10^{6}$ data points. (b) Estimated irreversibility using
lower order KLD ($\sigma_{2},\sigma_{3},\sigma_{4}$) for the high activity
treatment ($N\approx 3\times 10^{6}$). The shaded region shows the range of 5
bootstraps statistics with $N\approx 1\times 10^{6}$. (c) Estimated
irreversibility $\sigma$ as a function of high lag time $\vartheta$ for high,
medium and low activity. Each line corresponds to a distinct oocyte, with
$N\approx 1\times 10^{6}$. The error bars show the range of 5 bootstraps
statistics with $N\approx 0.3\times 10^{6}$. All estimated irreversibility
$\sigma$ is calculated using the $k$-nearest neighbor method with $k=1$ on
steady state oocytes.

Our main result, depicted in Fig. 2a, is an estimate of the irreversibility
$\sigma(\vartheta)$ as a function of lag time $\vartheta$ for the high and
medium activity steady states, where the effects are most pronounced.
Strikingly, the estimated irreversibility is a nonmonotonic function of lag
time peaked between $\vartheta^{*}\approx 1-3\ s$ for the high activity state
and $\vartheta^{*}\approx 8-10\ s$ for the medium activity state, when using
the most converged estimate with $k=1$, where $k$ denotes the number of
nearest neighbors used in our estimation of the KLD (SI Sec. 1.5, Fig. S5). We
also verified that the results for the medium activity state are qualitatively
similar for $k=2$ and $5$, but smaller in magnitude and noisier (Fig. S6).

We can understand this nonmonotonic behavior as follows. For large lag times
($\vartheta\gtrsim 10\ s$), we have coarse grained out too much information,
which generically leads to a smaller KLD with smaller irreversibility
estimates. For shorter times ($\vartheta\lesssim 1\ s$), thermal noise
dominates our estimates, leading to poorer convergence and a smaller estimated
irreversibility. This is consistent with the statistics of displacements being
dominated by the high frequency fluctuations coming from the background
reversible thermal noise [23, 7, 8]. The irreversible noise acting on our
probes originates in the jostling of the myosin motors whose effects on the
probes requires a longer time to manifest [7]. Thus, the nonmonotonic behavior
of the irreversibility (KLD) tells a story that with the smallest lag times,
which include all the timescales, the largest dominant contribution is the
reversible thermal motion. Then, as we progressively remove timescales (by
increasing the time lag), the thermal fluctuations are smeared over, leading
to the emergence of the irreversible active noise, which is eventually coarse
grained away resulting in smaller observed irreversibility. The peak
represents a trade-off between the timescale at which irreversibility is most
prominent and the suppression due to coarse graining. The consequence is that
the peak reveals the timescale of nonequilibrium irreversibility originating
from the myosin on/off rate, which remarkably is consistent with the position-
increment autocorrelation cross-over time estimated from Fig. 1. However here,
we inferred this timescale using a model-independent irreversibility metric,
allowing us to characterize the dissipation.

Remarkably, we found that even the observed irreversibility estimated from
short time correlations using low-order KLDs $\sigma_{n}$ for $n=2,3,4$ are
non-monotonic with a peak that reveals the correct order of magnitude for the
timescale of non-equilibrium activity (Fig. 2b). Generically, such low-order
KLDs provide a looser bound on the actual dissipation (entropy production) of
a process. However, for biological experiments where statistics could be
limited, this result implies that the irreversibility metric estimated using
low-order KLDs $\sigma_{n}$ can still be useful in providing an estimate of
the nonequilibrium timescale in the system.

Beyond the dissipation timescale $\vartheta\gtrsim 8\ s$, we can be confident
our estimator is well converged and consistently capturing the detectable
irreversibility of the process. This allows us to infer relative
irreversibility of systems with different amounts of nonequilibrium activity.
This is verified in Fig. 2c, where we have plotted the estimated
irreversibility as a function of time lag for three steady states – high,
medium, and low activity – each for multiple oocytes. Though the estimates all
decrease with lag time, they clearly appear clustered into three zones
corresponding to the three activity levels. The finite-length trajectories
results in small statistical error in our estimates (shaded lines) and the
dominate variability comes from two biological replicates (Fig. 2c.) Our
observations that irreversibility can distinguish levels of nonequilibrium
activity aligns with the observed variation in irreversibility measured in
mechanosensory hair cells for different activity treatments [38]. Despite the
KLD only offering a bound to the physical dissipation, there is a monotonic
relationship between the detectable irreversibility and the expected
nonequilibrium activity. This supports the hypothesis that cellular activity
and irreversibility are linked and that irreversibility can be considered a
readout of cellular activity.

Figure 3: Time irreversibility reveals the dissipative timescales and levels
of non-equilibrium activity in patterned cortex (a) Patterns of Rho signaling
waves in Ect2 overexpressed oocytes (green) overlaid on cortical granules
(white) before (Stage 0) and after (Stages 1-4) ATP depletion drug is added.
Scale bar: $25\ \mu m$ (b) Time series of local Rho-GTP concentration
oscillations from Stage 0 to Stage 4. (c) Power spectral density of local Rho
oscillations show a prominent peak at the oscillation frequency. The standard
error of the mean is smaller than the linewidths. (d) Estimated
irreversibility $\sigma$ for the 5 different stages as a function of lag time
$\vartheta$. Inset shows the timescale of peak irreversibility as a function
of Rho oscillation period. The standard error of the mean for Rho oscillation
periods in the inset is smaller than the marker. ($N\approx 1.2\times 10^{5}$
for Stage 0, $N\approx 3.5\times 10^{4}$ for Stage 1, $N\approx 1.6\times
10^{5}$ for Stage 2, $N\approx 4.0\times 10^{4}$ for Stage 3, $N\approx
1.0\times 10^{5}$ for Stage 4.)

Actomyosin networks are orchestrated in space and time via biochemical
signaling proteins. In particular, Rho-GTP is a highly conserved signaling
protein pivotal in regulating cellular mechanics. The question we next ask is
whether Rho-GTP patterns can tune dissipation timescales? We induced steady
state patterns of Rho-GTP by increasing the GEF (guanine exchange factor)
responsible for activating Rho-GTP. Rho-GTP patterns maintained constant
oscillatory periods within the observed non-equilibrium steady states [52, 53]
(Fig. 3, see Method). The locally oscillating Rho-GTP pattern (Fig. 3a)
activates myosin molecular motors and modulates actin network organization
[54], which results in an effective active drive on the cortical granules. The
precise activity timescales induced by the Rho-GTP signaling is difficult to
ascertain, due to the nonlinear effects in the actomyosin network organization
[54], but we can vary the Rho-GTP oscillation frequency by modulating the
intracellular ATP concentrations via sodium azide treatment (Fig. 3b-c).
Remarkably, the estimated irreversibility calculated from the dynamics of the
granules under this perturbation exhibits a shift in the peak timescales in a
manner that correlate with Rho-GTP oscillation frequency (Fig. 3d). Moreover,
the magnitude of the irreversibility peak also reflect the changes in
intracellular ATP levels. This further corroborates the observations in Fig.
2c that the observed variation in irreversibility corresponds to physical
variations in dissipation.

### Analytic validation of irreversibility metric

Having observed structure in the observed irreversibility of an active
biological system, we now turn to validating this approach by analyzing a
simple model for a diffusing particle experiencing active noise, such as the
cortical granules. Consider a two-dimensional overdamped Brownian particle in
a viscous fluid with viscosity $\gamma$ at temperature $T$ perturbed by an
active noise $\boldsymbol{\mathcal{A}}(t)$. The particle’s position ${\bf
r}_{t}=(x_{t},y_{t})$ at time $t$, then evolves according to the Langevin
equation

$\gamma{\dot{\bf r}}(t)=\boldsymbol{\mathcal{A}}(t)+\sqrt{2\gamma k_{\rm
B}T}{\boldsymbol{\eta}}(t).$ (2)

where $\boldsymbol{\eta}(t)$ is zero-mean Gaussian white noise
$\langle\boldsymbol{\eta}(t)\boldsymbol{\eta}(s)\rangle={\hat{I}}\delta(t-s)$.
To make the model tractable, the active noise is taken as a nonequilibrium
Ornstein-Uhlenbeck process,

$\dot{\boldsymbol{\mathcal{A}}}(t)=-{\hat{K}}\boldsymbol{\mathcal{A}}(t)+\sqrt{2k_{\rm
B}T_{A}}{\boldsymbol{\xi}}(t),\quad{\hat{K}}=\left(\begin{array}[]{cc}\kappa&\alpha\\\
-\alpha&\kappa\end{array}\right),$ (3)

with zero-mean Gaussian white noise $\boldsymbol{\xi}(t)$ at active-noise
temperature $T_{A}$. If ${\hat{K}}$ were symmetric
(${\hat{K}}={\hat{K}}^{T}$), then these dynamics would describe an equilibrium
system, as that force on the active nose (${\hat{K}}\boldsymbol{\mathcal{A}}$)
would be derivable from a potential. Thus, the activity arises solely due to
the asymmetric, off-diagonal elements of the force matrix $\alpha$, which set
the timescale of this nonequilibrium mechanism.

As for the granules, we probe the dissipative timescale $\alpha$ by
calculating the irreversibility $\sigma(\vartheta)$ of the position increments
$\Delta{\bf r}^{\vartheta}(t)={\bf r}(t+\vartheta)-{\bf r}(t)$. The results of
the semi-analytic calculation are presented in Fig. 4a for three
representative values of the (dimensionless) dissipative timescale
$\alpha/\kappa=2,5,10$ (low, medium, and high activity). We observe that there
is a pronounced drop-off in the irreversibility controlled by the
characteristic scale $\vartheta\approx\kappa/\alpha$, consistent with the
experimental observations. Noticeably, when the coarse-graining exceeds the
dissipative timescale ($\vartheta>\alpha$), the irreversibility cannot detect
any activity, as expected.

Figure 4: Time irreversibility reveals the dissipative timescale of a 2D
Brownian particle driven by active Ornstein-Uhlenbeck (OU) noise (a) The
analytic solution (shaded line) and normalized irreversibility
$\sigma/\max(\sigma)$ (solid line) as calculated using our method for
$\alpha=2,5,10$ with thermal temperature $T=0$. The inset shows the magnitude
of $\sigma$ versus the actual entropy production rate $\Sigma$. (b)
Irreversibility $\sigma$ as a function of lag time for different thermal
temperature $T$ as calculated by our method. Inset shows the corresponding
semi-analytic result.

To verify our estimator can detect the dissipative timescale, we simulated
trajectories of our model and calculated the irreversibility using the
$k$-nearest-neighbor estimator. In the absence of thermal noise ($T=0$), the
observed irreversibility of the simulated trajectories at different coarse-
graining times $\sigma(\vartheta)$ exhibit a pronounced drop-off at the
characteristic timescale $\kappa/\alpha$, in good agreement with the semi-
analytic result (Fig. 4a). Importantly, the maximum value of the observed
irreversibility scales with the entropy production of the process (Fig. 4a,
inset), indicating that the magnitude of the observed irreversibility contains
information about the relative magnitude of the dissipation, similar to what
was observed in the experimental data in Fig. 2b and 3d.

For non-zero thermal temperature ($T>0$), we find that the observed
irreversibility at lower lag time drops to smaller values, revealing a peak
near the characteristic nonequilibrium timescale (Fig. 4b). At timescales
shorter than the characteristic timescale, the semi-analytic calculation
predicts a progressively smaller (but constant) irreversibility when thermal
noise is increased (Fig. 4b, inset). This numerical result corroborates with
the interpretation of our experimental results in Fig. 2a, whereby at short
lag time, the observed irreversibility is dominated by reversible thermal
noise, leading to poor convergence and thus small irreversibility.

Nonequilibrium processes are inherently asymmetric under time reversal. Our
results demonstrate how a fundamental thermodynamic connection between the
physical energy dissipation sustaining a nonequilibrium system and a measure
of statistical irreversibility, can provide quantitative insight into the
mechanisms of nonequilibrium activity. The dependence of irreversibility on
observation frequency (lag time) reveals the striking signature of the
timescale of the nonequilibrium activity in a peak of our model-independent
irreversibility metric. The basic idea of estimating irreversibility for
various levels of coarse graining can lead to important inferences whenever
there is a well-defined notion of dissipative scales. Analogous procedures
could be used to identify spatial scales in living and non-living active
matter systems, allowing us to probe the nonequilibrium coupling of structure
with function [55, 56, 57, 58].

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## Acknowledgments

This research was supported by a Gordon and Betty Moore Foundation (to N.F.),
Sloan Research Fellowship (to N.F.), and the Human Frontier Science Program
Career Development Award (to N.F.). We gratefully acknowledge the Gordon and
Betty Moore Foundation for supporting T.R.G. and J.M.H. as Physics of Living
Systems Fellows through Grant GBMF4513. This research was supported in part
through the computational resources and staff contributions provided for the
Quest high performance computing facility at Northwestern University which is
jointly supported by the Office of the Provost, the Office for Research, and
Northwestern University Information Technology. This research was supported in
part by the National Science Foundation under Grant No. NSF PHY-1748958. N.F.
and J.H. acknowledge discussions during KITP program ‘ACTIVE20: Symmetry,
Thermodynamics and Topology in Active Matter’.

## Author contributions

N.F. and J.M.H. designed and supervised the research. T.H.T. and Y.C.C.
performed the experiments. T.H.T., G.A.W., Y.C.C., J.L. and T.R.G. analyzed
the data. All authors discussed the results and wrote the paper.

## Competing interests

The authors declare no competing interests.