# Patterns that persist: Heritable information in stochastic dynamics

Peter M. Tzelios  Department of Chemical Engineering
Columbia University, New York, NY 10027  Kyle J. M. Bishop***Address
correspondence to<EMAIL_ADDRESS>Department of Chemical Engineering
Columbia University, New York, NY 10027

###### Abstract

Life on earth is distinguished by long-lived correlations in time. The
patterns of material organization that characterize the operation of living
organisms today are contingent on events that took place billions of years
ago. We and other living organisms are patterns that persist. This contingency
is a necessary component of Darwinian evolution: patterns in the present
inherit some of their features from those in the past. Despite its central
role in biology, heritable information is difficult to recognize in prebiotic
systems described in the language of chemistry or physics. Here, we consider
one such description based on continuous time Markov processes and investigate
the persistence of heritable information within large sets of dynamical
systems. While the microscopic state of each system fluctuates incessantly,
there exist few systems that relax slowly to their stationary distribution
over much longer times. These systems, selected for their persistence, are
further distinguished by their mesoscopic organization, which allows for
accurate course grained descriptions of their dynamics at long times. The slow
relaxation of these stable patterns is made possible by dissipative currents
fueled by thermodynamic gradients present in the surrounding reservoirs. We
show how the rate of entropy production within a system sets an upper bound on
the lifetime of its persistent patterns. The observation that dissipation
enables persistence suggests that other features of living systems such as
homeostasis and autocatalysis may emerge as consequences of a more general
principle based on heritable information. We also consider the probability of
finding persistence within large sets of dynamical systems. We show that
physical constraints based on continuity and locality can strongly influence
the probability of persistence and its dependence on system size. Finally, we
describe how heritable information can be quantified in practice using
universal compression algorithms. We demonstrate this approach on an
experimental system of magnetically-driven, colloidal rollers and discuss the
application of these methods to origins of life research.

“Wherever matter is so poised, arranged, and adjusted, as to continue in
perpetual motion, and yet preserve a constancy in the forms, its situation
must, of necessity, have all the same appearance of art and contrivance which
we observe at present.”

—David Hume, Dialogs Concerning Natural Religion, Part VIII

## Introduction

What kind of explanation can we expect to find for the origins of life[1, 2,
3, 4, 5, 6, 7] as we know it? We would appreciate a detailed historical
narrative addressing what, when, where, and how our most distant ancestors
came to be. What were the material components of pre-life? When and where on
Earth or elsewhere were they found? How did their transient fluctuations give
way to a branching sequence of correlated events spanning billions of years?
Sadly, even the most important features of this history may never be known.
Recognizing these limitations, we might instead welcome synthetic
demonstrations whereby key features of living systems are reproduced from
simple abiotic ingredients. Synthetic models of life should adapt their
organization to harness energy from the surroundings and fuel dissipative
functions.[8, 9, 10] These models should maintain local conditions favorable
to their operation to provide autonomy from an ever-changing environment.[11,
12] As even the most reliable machines deteriorate, life-like systems should
make copies of themselves to preserve dynamic patterns across generations.[13,
14] Such synthetic demonstrations of dissipative adaptation, homeostasis, and
autocatalysis promise to provide useful insights into the operation of early
life-like systems. However, the _a priori_ emphasis on key features of life
risks obscuring the historical processes by which these features emerged. To
better understand and perhaps recreate that process, we require an appropriate
conceptual framework with which to approach the physical history of life’s
origins.

To illustrate the important role of such a framework, consider another
origin—not of life—but of bacterial mobility (Figure 1a). How did bacteria
“learn” to swim? While we lack a full historical narrative of this transition,
artifacts like the fossil record can provide clues. At one point in time,
there were organisms recognizable as bacteria that lacked the biomolecular
machinery needed to swim. Today, we observe bacteria that can swim. Moreover,
we know how they do it—for example, how the flagellar motor converts proton
gradients into rotary motion of helical filaments to propel bacteria through
viscous fluids. Building on this knowledge, we’ve created synthetic
demonstrations of primitive, micron-scale motors that swim like bacteria and
unlike bacteria.[15, 16] However, these historical artifacts and synthetic
demonstrations cannot explain the origin of bacterial mobility without the
conceptual framework of Darwinian evolution by natural selection. Even without
key details of this evolutionary transition, we are nevertheless confident
that it can be explained by the differential survival and reproduction of
bacteria exhibiting heritable variation. We reason that swimming must be good
for the survival of bacteria descendent from those that chanced upon this
winning innovation.

Figure 1: Examples of persistent patterns containing features correlated to
past events _ca._ one billion years ago. (a) Schematic illustration of a
bacterium highlighting the actuated appendage used to swim (flagellum) and its
genetic material (DNA). Image adapted from work by Ali Zifan. (b) Photograph
of a Lake Superior agate. Image by Lech Darski.

The central hypothesis of this Chapter is that the conceptual framework for
understanding life’s origins is a suitable generalization of natural selection
that describes the evolution of inanimate physical systems.[17, 18] The
critical ingredient for such a framework is _heritable information_.[1] In
biology, the replication of the genome provides the most important (though not
the only) mechanism by which heritable information is passed from one
generation to the next. An organism’s genome contains _patterns_[19] that
persist over time—for example, the highly conserved genes shared by all living
organisms. The persistence of these patterns over billions of years is
remarkable considering the comparatively short lifetime of DNA—the physical
medium in which genes are encoded. Genetic patterns are not just molecular
structures but also dynamic processes that harness matter and energy in
service of their persistence. Those patterns that survive the march of time
capture our attention with “the appearance of art and contrivance”; those that
fade away and the many that never were do not.[5] Outside the context of
biology, we will refer to this blind selection mechanism as the _persistence
filter_ whereby time reveals those patterns that persist in a given
environment.

The qualitative notion of patterns that persist can be quantified using
dynamical models that describe the time evolution of physical systems. Here,
we consider systems described by a finite number of discrete
states—corresponding, for example, to the different possible configurations of
system components. The system state changes in time due to stochastic
transitions, which reflect uncertainties introduced by the fluctuating
environment and by hidden variables neglected in the model dynamics. On
sufficiently long times, these dynamics lead the system to explore all
possible states with finite probability as determined by the system components
and the surrounding environment. On shorter times, however, the state of the
system is determined also by its history: knowledge of the present state (and
of the system dynamics) teaches us something about the not-so-distant
past.[20] This heritable information can be quantified in terms of the mutual
information between the current state of the system and its state at a
previous time. All physical systems exhibit this simple form of heredity;
however, only precious few exhibit life-like persistence.

Living systems are remarkable as their patterns persist on time scales far
exceeding those of the microscopic processes from which they are made. By
contrast, consider the persistent patterns of the Lake Superior agate—mineral
formations created by geological events more than a billion years ago (Figure
1b). The banded deposits of iron-rich minerals are highly correlated across
time for the simple reason that their atomic components are themselves
unchanged. Meanwhile, some 98 percent of the atoms in the human body are
replaced each year. The patterns that distinguish us as unique individuals—and
those shared with other living organisms—exhibit a type of dynamical
persistence exceedingly rare among physical systems. But just how improbable
is life-like persistence within a prescribed class of physical systems? What
are the mechanisms by which their physical histories are preserved? Under what
conditions is dynamical persistence most likely to be found, and how will we
recognize it? This Chapter seeks answers—however tentative and incomplete—to
these important questions.

We investigate the persistent patterns found within a set of dynamical systems
based on continuous time Markov processes on a finite state space. To prohibit
trivial forms of persistence—like that of the agate—we fix the lifetime of
each state to unity. While the microscopic state of each system fluctuates
incessantly, there exist few systems that relax slowly to their stationary
distribution over much longer times. These systems, selected for their
persistence, are further distinguished by their mesoscopic organization, which
allows for accurate course grained descriptions of their dynamics at long
times. The slow relaxation of these stable patterns is made possible by
dissipative currents fueled by thermodynamic gradients present in the
surrounding reservoirs. We show how the rate of entropy production within a
system sets an upper bound on the lifetime of its persistent patterns. The
observation that dissipation enables persistence suggests that other features
of living systems such as homeostasis and autocatalysis may emerge as
consequences of a more general principle based on heritable information. We
also consider the probability of finding persistence within large sets of
dynamical systems. We show that physical constraints based on continuity and
locality can strongly influence the probability of persistence and its
dependence on system size. Finally, we describe how heritable information can
be quantified in practice using universal compression algorithms. We
demonstrate this approach on an experimental system of magnetically-driven,
colloidal rollers and discuss the application of these methods to origins of
life research.

Overall, this Chapter advocates for a physical perspective of life and its
origins that places heredity at the center. Importantly, the form of heredity
we describe is not based on specific mechanisms such as self-replicating
molecules but rather on the stable features of dynamical systems. This
generalization of heritable information allows for the application of
evolutionary concepts such as genotype and phenotype within even simple, non-
biological systems. The cost of this generality, however, is the challenge of
connecting abstract representations in terms of microstates and mesostates
with the physical world of our experience. We therefore begin with a tutorial
introduction to Markov state models covering the basic concepts needed for the
present investigation. Building on insights from these abstract models, the
final section aims to apply the ideas of this Chapter to an experimental model
of driven colloids. It is our hope that this work will inspire others to go
searching for heritable information in other experimental systems from active
matter[21] to systems chemistry[22] and beyond.

## Markov Processes

To investigate the origins of dynamical persistence in physical systems, we
consider a class of continuous time Markov processes on systems containing $N$
discrete states.[23] The Markov property implies that the transition to state
$i$ from state $j$ occurs with a rate $w_{ij}(t)$ independent of the system’s
history.[24] This framework has been widely applied in describing the dynamics
of physical systems across many scales including molecular dynamics,[25, 26]
chemical kinetics,[27] and biomolecular machines.[28, 29] The success of these
models relies on their ability to average over fast microscopic processes and
obtain coarse grained descriptions of slow mesoscopic processes of interest.
For example, in describing the kinetics of chemical reactions, a coarse
grained state may specify the numbers of different reacting species but not
their microscopic positions and momenta nor those of the surrounding solvent
molecules. Transitions from one state to the next are inherently stochastic
due to fluctuations in the fast microscopic processes neglected under coarse
graining. Additionally, these transitions are coupled to one or more external
reservoirs that specify the temperature and chemical potential of the
environment(s). The presence of multiple reservoirs out of equilibrium with
one another allows for describing open thermodynamic systems characterized by
currents of energy and species within and across the system.[30, 31]

#### Stochastic Dynamics.

The dynamics can be visualized as a random-walk on a graph comprised of $N$
states connected by directed edges representing stochastic transitions (Figure
2a).[23] Given that the system is in state $j$ at time $t$, it will jump to a
different state $i$ with probability $w_{ij}(t)dt$ during a differential time
interval $dt$. In general, the transition rates $w_{ij}$ depend on time—for
example, due to changes in the surrounding environment; however, we limit our
discussion to homogeneous processes with constant rates. Upon arriving in
state $i$, the system waits for some time before jumping to another state $k$.
The waiting time is exponentially distributed with a mean value of
$(\sum_{k}w_{ki})^{-1}$ for state $i$. We focus on systems in which all states
have a common lifetime by which other times are measured—that is,
$\sum_{k}w_{ki}=1$ for all $i$. As shown below, the ability of such systems to
preserve features of their past over long times ($t\gg 1$) derives—not from
local kinetic traps—but from mesoscopic organization and dissipative currents.

Figure 2: A 12-state system with persistent dynamics. (a) Graphical
representation of the system. The numbered states are connected by directed
edges representing stochastic transitions. An edge to state $i$ from state $j$
is characterized by a transition rate $w_{ij}=d_{j}^{-1}$, where $d_{j}$ is
the out-degree of state $j$. Each state $j$ has an average lifetime of one by
which other times are scaled—in other words, $\sum_{i}w_{ij}=1$. At long times
($t\gg 1$), the dynamics of the system is well approximated by a coarse
grained description involving two mesostates (shaded circles). (b) Stochastic
dynamics of the system. Starting from the initial condition $X(0)=2$, the plot
shows one realization of the fluctuating state $X(t)$ as a function of time
(red curve). The colormap shows the transient probability distribution
$p_{i}(t)$ obtained by solving the master equation (1). (c) Mutual information
between the past and the present. On average, knowledge of the present state
$X(t)$ provides $I(\tau)$ bits of information about the past state
$X(t-\tau)$. The plot shows how this information decays with increasing time
lag $\tau$ from its maximum value, corresponding to the entropy of the
stationary distribution $p_{i}(t\rightarrow\infty)=\pi_{i}$, to its asymptotic
value of zero. The red curve shows the time-dependent information content
associated with the coarse grained description.

We consider a discrete set of possible dynamics parameterized by a directed
adjacency matrix $A$. This binary matrix has elements $A_{ij}=1$ when there
exists a transition to state $i$ from state $j$ and $A_{ij}=0$ otherwise. The
associated transition rate is specified as $w_{ij}=A_{ij}/d_{j}$ where
$d_{j}=\sum_{i}A_{ij}$ is the out degree of state $i$. In other words, all
transitions exiting state $i$ proceed at a common rate such that its mean
lifetime is unity. Of the $2^{N(N-1)}$ possible adjacency matrices, we
consider only those that encode strongly connected graphs, for which there
exists a directed path from $i$ to $j$ and from $j$ to $i$ for every pair of
states $(i,j)$. Despite the simplifying assumptions of equal rates and unit
lifetimes, this space of possible dynamics is sufficiently large to include
some of the complex dynamical behaviors expected in real systems. To
facilitate a comprehensive exploration of these dynamics, we limit our initial
investigation to the ca. $10^{114}$ possible systems with less than or equal
to $N=20$ states.

#### Master Equation.

The time-dependent probability distribution over the $N$ states is governed by
the so-called master equation

$\frac{dp_{i}}{dt}=\sum_{j}W_{ij}p_{j}$ (1)

where $p_{i}(t)\equiv\Pr(X(t)=i)$ denotes the probability that the system is
in state $i$ at time $t$, and $W_{ij}=w_{ij}-\delta_{ij}\sum_{k}w_{ki}$ are
the elements of the transition rate matrix $W$. This linear equation describes
the conservation of probability as it flows and spreads among the states of
the system.

To solve the master equation, we first decompose the rate matrix into its
eigenvalues and eigenvectors as $W=-Q\Lambda Q^{-1}$, where $Q$ is the matrix
of eigenvectors and $\Lambda$ is the diagonal matrix of eigenvalues with
elements $\Lambda_{nn}=\lambda_{n}$. The transient probability distribution
can then be expressed as a series of eigenmodes that decay independently as

$p_{i}(t)=\sum_{j}T_{ij}(t)p_{j}(0)\quad\text{with}\quad
T_{ij}(t)=\sum_{n}Q_{in}e^{-\lambda_{n}t}Q_{nj}^{-1}$ (2)

where $p_{j}(0)$ is the initial distribution at time $t=0$ (Figure 2b). The
matrix $T(t)$ describes a linear transformation by which the initial
probability vector $\bm{\mathbf{p}}(0)$ is squeezed, rotated, and sheared to
produce the distribution $\bm{\mathbf{p}}(t)$ at time $t$. For a strongly
connected graph with constant transition rates, exactly one of the eigenvalues
is equal to zero (denoted $\lambda_{1}$). The associated eigenmode corresponds
to the unique steady-state solution,
$p_{i}(t\rightarrow\infty)=\pi_{i}=Q_{i1}/\sum_{i}Q_{i1}$.[23] The other
eigenvalues have positive real parts, $\mathrm{Re}(\lambda_{n})>0$, indicating
that the stationary solution is stable. By convention, we number the
eigenvalues in order of increasing real part such that
$0=\lambda_{1}<\text{Re}(\lambda_{2})\leq\text{Re}(\lambda_{3})\leq\dots\leq\text{Re}(\lambda_{N})$.

#### Dynamical Persistence.

Once the system relaxes to its steady-state distribution $\pi_{i}$, it retains
no memory of its initial condition $p_{i}(0)$. We seek to identify persistent
dynamics that retain features of their past for as long as possible (Figure
2c). This type of dynamical persistence can be quantified in terms of the
mutual information $I(\tau)$ between the current state of the system $X(t)$
and its past state $X(t-\tau)$

$I(\tau)=H(X(t-\tau))-H(X(t-\tau)\mid X(t))$ (3)

Here, $H(X(t-\tau))$ is the marginal entropy representing our uncertainty in
the state of the system at time $t-\tau$. Upon observing the system’s state at
time $t$, we expect to learn something about its past thereby reducing our
uncertainty. The conditional entropy $H(X(t-\tau)\mid X(t))$ represents the
expected uncertainty about the system’s past state given knowledge of its
present state. Using equation (2) for the distribution $p_{i}(t)$, the mutual
information $I(\tau)$ can be written more explicitly as

$I(\tau)=-\sum_{i}\pi_{i}\log_{2}\pi_{i}+\sum_{i,j}T_{ij}(\tau)\pi_{j}\log_{2}(T_{ij}(\tau))$
(4)

where the two terms in equation (4) correspond to the entropies in equation
(3). The use of binary logarithms implies that mutual information is measured
in bits.

Figure 2c shows how the amount of heritable information $I(\tau)$ decays with
increasing lag time $\tau$ for a particular system with persistent dynamics.
For short lags ($\tau\ll 1$), knowledge of the present unambiguously
determines the recent past such that $H(X(t-\tau)\mid X(t))\approx 0$; the
mutual information equals the entropy of the steady-state distribution. For
long lags, when even the slowest eigenmode has relaxed
($\text{Re}(\lambda_{2})\tau\gg 1$), knowledge of the present provides no
information about the past, and the mutual information approaches zero. At
intermediate times, $I(\tau)$ decreases monotonically with $\tau$ as the
different eigenmodes relax in succession from fastest to slowest.

In light of the relationship between mutual information and lag time, we can
approach the question of persistence from two complementary perspectives. In
one, persistent systems are those that preserve large amounts of information
about their past for a specified lag time, $I(\tau)\geq 1$. Alternatively, we
can invert this relationship and define persistent systems as those that
preserve a specified amount of information for long times, $\tau(I)\gg 1$. For
systems of given size $N$, there exists an inherent trade-off between the
amount of heritable information $I$ and the time $\tau$ for which it is
preserved. We make use of both perspectives on persistence in exploring this
trade-off for the set of dynamical systems introduced above.

#### Coarse Graining.

Systems with persistent dynamics preserve information about their past for
long times, but it remains unclear how this information is encoded. What
features of the past persist to the present? To answer this question, we first
note that systems with dynamical persistence are characterized by “slow”
eigenmodes that relax more slowly than the remaining “fast” modes—often by
orders of magnitude. At intermediate time scales, the fast modes are fully
relaxed while the slow modes remain largely unchanged and determined by the
system’s history. Importantly, the fast modes act to redistribute probability
_within_ specific clusters of states—so-called mesostates—while the slow modes
describe transitions _between_ mesostates. Consequently, the separation of
time scales implies the existence of coarse grained descriptions that
characterize the slow dynamics of the system among the different mesostates.
When the fast and slow times scales are strongly separated, transitions
between the different mesostates are well approximated by Markov processes
like those describing the original microscopic dynamics.

Figure 2 shows a system with $N=12$ states organized to form $M=2$ mesostates.
While the average eigenvalue is equal to one, there exist one slow mode that
relaxes more slowly with rate $\lambda_{2}\approx 0.001$. This mode describes
the relaxation of the system between the two mesostates. On the slow time
scale, the system is accurately described by a coarse grained description
based on mesostate probabilities, $P_{k}(t)=\sum_{i}\Pi_{ki}p_{i}(t)$, where
$P_{k}(t)\equiv\Pr(Y(t)=k)$ denotes the probability that the system is in
mesostate $k$ at time $t$. The conditional probability,
$\Pi_{ki}\equiv\Pr(Y(t)=k\mid X(t)=i)$, serves to define the mesostate by
associating each state $i$ to a mesostate $k$. To maximize the information
content of this lossy description, we assign $\Pi_{ki}=1$ when state $i$ is in
mesostate $k$ and $\Pi_{ki}=0$ otherwise (see Methods). In this way, the
mutual information $I(\tau)$ computed from the mesostate probabilities
accurately captures the persistent features of the system’s dynamics (Figure
2c).

#### Entropy Production.

The dynamical systems we investigate are, in general, out-of-equilibrium.
Consequently, they do not satisfy the condition of detailed balance, which
requires the equality of forward and reverse rates at equilibrium,
$w_{ij}\pi_{j}=w_{ji}\pi_{i}$. Instead, the steady-state distribution
$\pi_{i}$ is characterized by dissipative currents that flow through the
network of states due to thermodynamic forces imposed by the contacting
reservoirs. The rate of entropy production associated with these processes can
be evaluated as

$\sigma=\sum_{i,j}(w_{ij}\pi_{j}-w_{ji}\pi_{i})\ln\left(\frac{w_{ij}\pi_{j}}{w_{ji}\pi_{i}}\right)\geq
0$ (5)

where equality holds at equilibrium. As we will show, the rate of entropy
production sets bounds on persistence within our set of possible dynamics. In
short, dissipation enables persistence. To explore this relationship, it is
necessary to modify the dynamics to ensure that each nonzero transition rate,
$w_{ij}\neq 0$, is accompanied by another for the reverse process, $w_{ji}\neq
0$.

We introduce a parameter $\gamma\in[0,1)$ that controls the extent of
disequilibrium and allows for moving continuously between equilibrium and
nonequilibrium dynamics. Under conditions of maximum disequilibrium
($\gamma\rightarrow 1$), transition rates are related to the adjacency matrix
as $w^{\text{neq}}_{ij}=A_{ij}/d_{j}$ as described above. In the opposite
limit ($\gamma=0$), the equilibrium transition rates are defined as
$w^{\text{eq}}_{ij}=A^{\text{eq}}_{ij}/d^{\text{eq}}_{j}$ where
$A^{\text{eq}}_{ij}$ is the symmetric adjacency matrix formed by the logical
disjunction of the directed adjacency matrix and its inverse—that is,
$A^{\text{eq}}_{ij}=A_{ij}\lor A_{ji}$. For intermediate values of $\gamma$,
the transition rates are equal to a weighted average of these limiting
scenarios

$w_{ij}=w^{\text{eq}}_{ij}+\gamma(w^{\text{neq}}_{ij}-w^{\text{neq}}_{ij})$
(6)

When $\gamma=0$, the system satisfies detailed balance. The equilibrium
distribution is $\pi^{\text{eq}}_{j}\propto d^{\text{eq}}_{j}$, where
$d_{j}^{\text{eq}}=\sum_{i}A^{\text{eq}}_{ij}$ is the degree of state $j$
within the equilibrium graph. In the limit as $\gamma\rightarrow 1$,
transitions between states become irreversible; the rate of entropy production
diverges even as the dynamics remains well-behaved. Unless otherwise noted,
the results below correspond to systems under conditions of maximum
disequilibrium, $\gamma\rightarrow 1$.

## Results

### The Persistence Filter

We consider a large number of systems with different rate matrices $W$, each
initialized in a particular state drawn from the stationary distribution
$\bm{\mathbf{\pi}}$. These systems evolve independently of one another for a
specified time $\tau$, longer than the microscopic time scale associated with
each state (here, $\tau\gg 1$). Upon observing the current states of the
different systems, what do we learn about their respective histories? For the
vast majority, we learn nothing. The mutual information $I(\tau)$ between the
past and the present has relaxed to zero. The state of such systems is fully
explained by their components, their environment, and the whim of chance.
However, there exist few exceptional systems—distinguished by their dynamical
persistence—for which the present state provides a window into the past. These
systems are further characterized by their mesoscopic organization, which
reveals “clever” strategies that enable their persistence. This purposeful
organization is designed by a blind selection mechanism—here, termed the
_persistence filter_ —whereby time erases all but the most persistent patterns
of dynamical organization.

To demonstrate the persistence filter in action, we sample the space of
possible dynamics for $N=12$ states to compute the distribution of information
$I(\tau)$ for a specified lag time $\tau$ (Figure 3a). To efficiently sample
rare systems with comparatively large $I(\tau)$, we use the Wang-Landau
algorithm[32] originally developed for computing the density of states.
Initially at $\tau=0$, the amount of information averaged over the possible
dynamics is $\langle I(0)\rangle=3.5$ bits, which corresponds to the average
entropy of the respective stationary distributions. As time progresses, the
average information decays exponentially as the systems relax. The fraction of
systems that encode one or more bits about their past also decreases
exponentially. By $\tau=15$, only one in ca. $10^{20}$ systems is sufficiently
persistent to preserve 0.7 bits of information.

Figure 3: The persistence filter for systems with $N=12$ states. (a)
Distribution of mutual information $I(\tau)$ across all possible systems at
different lag times $\tau$. Bins of the histograms are uniformly spaced on a
logarithmic scale; the area under each curve is normalized to one. The inset
shows the tail of the distribution corresponding to those systems with
persistent dynamics—defined here as $I(\tau)\geq 0.7$ bits. (b) Four systems
sampled at random from those satisfying the constraint $I(\tau)\geq I_{\min}$
for $I_{\min}=0.7$ bits for different $\tau$. At long times, the sampled
networks exhibit mesoscopic organization that enable their dynamical
persistence. (c) Mutual information for those systems in (b) as a function of
lag time $\tau$.

Figure 3b shows four 12-state systems with persistent dynamics sampled from
the tail of the information distribution at different lag times $\tau$. These
systems are selected to preserve $0.7$ or more bits of information about their
past state—that is, $I(\tau)\geq 0.7$. As time progresses, systems satisfying
this constraint become increasingly rare within the set of ca. $10^{39}$
possibilities. These exceptional systems are further distinguished by
increasing levels of mesoscopic organization as evidenced by the modular
structure of their network representations. This organization is a consequence
of the slowly decaying eigenmodes selected by the persistence filter. By
looking for information about the past, we are led to discover highly
organized systems designed for the purpose of preserving that information.

The persistence filter is analogous to Darwinian evolution by natural
selection, in which the “survival of the fittest” is replaced by the less
memorable aphorism “persistence of the most persistent”. Those patterns of
dynamical organization determined by their history—not by system components
and their environment alone—must persist in time if we are ever to recognize
them. Moreover, the longer these patterns persist in spite of fast microscopic
dynamics, the more organized the system must be to preserve them. Here, by
“organized” we mean “improbable”; only an exceptional few systems armed with
clever strategies can effectively preserve heritable information. Below, we
discuss some of these strategies and their respective frequencies within our
set of possible dynamics.

### Mechanisms of Persistence

To better understand the mechanisms of dynamical persistence, it is helpful to
consider the limiting cases—namely, those systems with $M$ modes that decay as
slowly as possible. Specifically, we identify systems that minimize the
$M^{\mathrm{th}}$ eigenvalue $\lambda_{M}$ of their respective rate matrices
$W$. As detailed in the Methods, we approach this combinatorial optimization
problem using Markov Chain Monte Carlo (MCMC) sampling with simulated
annealing to explore the space of possible dynamics in search of those with
the smallest $\lambda_{M}$.[33] For the dynamics considered here, the average
eigenvalue is constant and equal to one. Thus, by minimizing $\lambda_{M}$, we
identify systems with strong time-scale separation,
$\lambda_{M}\ll\lambda_{M+1}$, that allow for an accurate coarse grained
description in terms of $M$ mesostates.

The network in Figure 2a shows the optimal $12$-state system that minimizes
the second eigenvalue $\lambda_{2}$. The system is divided into two
mesostates, which are organized to prohibit transitions from one to the other
(Figure 2a). Transitions between mesostates occur at a rate
$\lambda_{2}=0.0011$, which is orders of magnitude slower than transitions
within each mesostate (e.g., $\lambda_{3}=0.65$). Inspection of the network
organization reveals the mechanism underlying this separation of time-scales.
Within the blue mesostate, there exists a single linear path from state 1 to
state 6, which is the sole gateway to the red mesostate. At each step along
this path, competing transitions lead back to earlier states. Only through a
series of increasingly improbable events can the system escape these sisyphean
cycles and transit to the opposite mesostate.

In general, minimizing the $M^{\mathrm{th}}$ eigenvalue $\lambda_{M}$ leads to
the discovery of persistent systems with $M$ mesostates, which preserve ca.
$\log_{2}M$ bits of information for as long as possible. Figure 4 shows the
result of this analysis for networks with $N=4M$ states for $M=2,3,4,5$. We
observe that the optimal networks are organized into $M$ mesostates of equal
size (here, four states). As above, these clusters of states are organized
internally to maximize their persistence and prohibit transitions to
neighboring mesostates. As a result, the eigenvalues of the $M$ slow modes,
$\lambda_{n}$ for $n=2,\dots,M$, have similar values comparable to the
lifetime of each mesostate. By contrast, the remaining eigenvalues—those of
the fast modes—are of order unity.

Figure 4: (a) Networks with $N=4M$ states that minimize the eigenvalue
$\lambda_{M}$ for $M=2,3,4,5$. Different colors are used to distinguish the
$M$ mesostates; darker (lighter) shades denote higher (lower) probability at
steady-state. (b) Eigenvalues for the networks in (a) highlighting time-scale
separation between the slow modes and fast modes.

For systems with three or more mesostates ($M\geq 3$), transitions _between_
mesostates can also influence persistence; however, those _within_ mesostates
are significantly more important. In Figure 4, all mesostates within these
maximally persistent systems share a common internal organization analogous to
that discussed above for $M=2$ (cf. Figure 2a). By contrast, the transitions
between mesostates are loosely organized into sparse loops with no other
discernible patterns. Small changes in the network connectivity among
mesostates serve to increase the relaxation rate $\lambda_{M}$ by ca. 10%;
such perturbations reduce persistence but only slightly. As the number of
states grows, it becomes increasingly challenging to identify the global
optimum in $\lambda_{M}$ among the ca. $2^{N(N-1)}$ possibilities.
Fortunately, such technical details do not influence our central conclusion
that dynamical persistence and network organization are inextricably linked.
Any form of selection for persistence will select also for mesoscopic
organization similar to that shown in Figure 4.

Such coarse grained descriptions represent a type of _pattern_ whereby certain
microscopic details are no longer required to describe or predict the system’s
mesoscopic behavior. At intermediate time scales ($\lambda_{M+1}^{-1}\ll
t\ll\lambda_{M}^{-1}$), the system remains localized in one of many possible
mesostates, which uniquely determines the quasi-steady distribution over its
associated states. This persistent pattern is analogous to the _genotype_ of a
living organism, which persists from generation to generation with little
change. Different genotypes specify different dynamical processes or
_phenotypes_ in the service of maintaining their persistence. These processes
include metabolism, motility, and reproduction among life’s many remarkable
innovations. Though comparatively primitive, the fluctuating currents within
an occupied mesostate describe similar dissipative processes organized to
foster its persistence.

### Effects of Size $N$ and Disequilibrium $\gamma$

The lifetime of a mesostate optimized to persist increases faster than
exponentially with the number states it contains. Figure 5a illustrates this
result for networks containing two mesostates that minimize $\lambda_{2}$. The
minimum eigenvalue $\lambda^{\min}_{2}$ characterizing the transition rate
between mesostates decreases with the number of states as approximated by the
relation, $\lambda^{\min}_{2}\approx 1/\Gamma(N/2+1)$. This result follows
from the network architecture (Figure 2a), which requires a specific series of
transitions to escape the mesostate—namely, $i\rightarrow i+1$ with
probability $1/i$ for $i=1,\dots,N/2$. With sufficient organization, as few as
48 states with “microscopic” lifetimes of order $10^{-42}$ s (the Planck time)
can form a cluster with “mesoscopic” lifetimes of order $10^{17}$ s (the age
of the universe). Such persistence, however, has a thermodynamic cost that
cannot be paid at equilibrium.

Figure 5: (a) Minimum eigenvalue $\lambda^{\min}_{2}$ as a function of network
size $N$ for different degrees of disequilibrium $\gamma$. The solid markers
correspond to specific networks identified by Monte Carlo optimization; the
red dashed curve denotes the approximate relation for equilibrium networks,
$\lambda_{2}^{\min}\approx 47N^{-3}$; the purple dashed curve denotes the
approximate relation for non-equilibrium networks with $\gamma\rightarrow 1$,
$\lambda_{2}^{\min}\approx 1/\Gamma(N/2+1)$. (b) Network representations of
systems with $N=12$ states that minimize $\lambda_{2}^{\min}$ for different
$\gamma$. The colored labels indicate the range of $\gamma$ for which each
system is optimal; colors are matched to the boxed markers in (a). (c)
Eigenvalues $\lambda_{2}$ and entropy production $\sigma$ for the systems in
(b) as parameterized by the disequilibrium parameter $\gamma$.

Equilibrium networks ($\gamma=0$) can also be organized to maximize
persistence; however, the resulting relaxation rates scale algebraically (not
exponentially) with the number of states. Figure 5a shows that the minimum
eigenvalue $\lambda_{2}^{\min}$ scales with network size $N$ as approximated
by $\lambda_{2}^{\min}\approx 47N^{-3}$ for equilibrium networks with
$\gamma=0$ and $N\gg 1$ (see Appendix A). These equilibrium networks are
organized into two highly-connected clusters connected by a linear chain of
states (Figure 5b). For large $N$, each cluster contains one third of the $N$
states (as opposed to one half for the non-equilibrium networks). To achieve
the same $10^{60}$ fold enhancement in persistence achieved by $N=48$ states
for $\gamma\rightarrow 1$, an equilibrium network ($\gamma=0$) would require
more than $N=10^{20}$ states. We emphasize that these results are influenced
by our starting assumption that all microstates have a common lifetime. Real
systems—like that of the agate in Figure 1b—may contain low-energy states in
which the system can be trapped for long times.

As $\gamma$ increases from zero to one, the system that maximizes persistence
steps through a series of intermediate architectures between the limiting
cases of equilibrium and maximum disequilibrium (Figure 5b). The different
architectures correspond to different strategies for achieving persistence.
The relative success of these strategies depends on the availability of
thermodynamic resources needed to power dissipative currents. For
nonequilibrium systems, these cyclic currents act to concentrate the occupancy
probability deep within the interior of each mesostate, thereby preventing
transitions between mesostates. For networks of a given size, the rate of
these transitions—as measured by the eigenvalue $\lambda_{2}$—decreases
monotonically with the rate of entropy production $\sigma$ as parameterized by
$\gamma$ (Figure 5c). The observed relationship between these quantities
suggests a thermodynamic cost[10] of persistence whereby the rate of entropy
production bounds the maximum lifetime of each mesostate. Similar bounds on
the precision of stochastic processes have been established[34, 35] and
provide a basis for thermodynamic inference[36] of unobserved properties in
nonequilibrium systems.

By searching for systems with persistent dynamics, we are led to discover
nonequilibrium systems that make use of available resources to power their
persistence. However, dissipative currents can also be used to erase
information about the past and accelerate relaxation to the steady-state.
Thus, finite rates of entropy production are necessary but not sufficient to
achieve persistence. This distinction is important in the context of life
where features such as entropy production, compartmentalization, and
replication among others are given privileged status in distinguishing life
from non-life. Here, we see that entropy production is a corollary of
dynamical persistence; dissipative processes enable useful strategies for
achieving persistence. Here, by “useful” we mean “probable”; there are many
more ways to preserve heritable information by harnessing available energy
sources. We hypothesize that other features of life such as
compartmentalization and replication can be understood in similar fashion as
consequences of persistence within appropriate classes of dynamical systems.

### Probability of Persistence

Over time, the persistence filter sifts through many dynamical systems to
select those few that preserve information about their respective histories.
The filter does not create these systems; instead, it focuses our attention on
their unlikely existence among myriad alternatives with little or no history
at all. To understand the existence of dynamical persistence—most notably that
of life on Earth—it is instructive to ask how unlikely it is within a
particular class of dynamical systems? Moreover, which types of systems are
more likely to foster persistence? To address these questions, we search our
space of possible dynamics to estimate the fraction of systems that encode one
bit of information relaxing slower than a prescribed rate $\nu$. Specifically,
we use the Wang-Landau algorithm[32] to compute the cumulative distribution
$\Pr(\lambda_{2}<\nu)$ for the eigenvalue $\lambda_{2}$. For now, each of the
ca. $2^{N(N-1)}$ possible dynamics is assumed to be equally likely. As
discussed below, this prior distribution over the possible dynamics has a
significant impact on the probability of finding persistent systems.

Figure 6: Probability of persistence. The plot shows the cumulative
probability $\Pr(\lambda_{2}<\nu)$ that the eigenvalue $\lambda_{2}$ is slower
than a prescribed rate $\nu$. Colored markers correspond to results of the
Wang-Landau calculations for systems of different sizes $N$. The black curve
shows the approximate relation for systems with the maximal
persistence—namely, $\lambda_{2}=\lambda_{2}^{\min}\approx 1/\Gamma(N/2+1)$
and $\Pr(\lambda_{2}=\lambda_{2}^{\min})\approx N!/2^{N(N-1)}$.

Figure 6a shows the monotonic relationship between probability and
persistence: systems with smaller eigenvalues $\lambda_{2}$ are increasingly
hard to find. For a given size $N$, the eigenvalue $\lambda_{2}$ is bounded by
its minimum value shown in Figure 5a. These maximally persistent systems
correspond to $N!$ degenerate networks that differ only in the labeling of
their states. The probability of selecting one such system at random can
therefore be approximated as $\Pr(\lambda_{2}=\lambda_{2}^{\min})\approx
N!/2^{N(N-1)}$. This estimate represents a lower bound as it includes all
possible adjacency matrices—even those that are not strongly connected. The
black curve in Figure 6 shows that these approximate results for maximally
persistent systems are representative of the general trends.

For a given relaxation rate $\nu$, the probability of finding a persistent
system decreases with increasing system size $N$ (Figure 6). The number of
non-persistent systems grows faster with $N$ than does the number of
persistent systems. This observation suggests that persistence is most likely
to be achieved using the smallest—that is, the simplest—system possible. The
chances of finding dynamical persistence in macroscopic systems becomes
vanishingly small, and yet, here we are. To reconcile this apparent
contradiction, we must recognize that our prior distribution over the space of
possible systems is a poor representation of physical reality. In particular,
the assumption that all adjacency matrices are equally likely lacks the
features of _continuity_ and _locality_ characteristic of physical dynamics.

Figure 7: Constraints of continuity and locality alter the probability of
persistence. (a) For $N=2^{S}$ states, we can associate each state with a
sequence of $S$ binary variables or “spins”. In the absence of constraints,
transitions are allowed between any two states (left). With the continuity
constraint, only transitions that flip one spin are allowed (center). Further
addition of the locality constraint implies that the flipping of a spin
depends only on its nearest neighbors (right). (b) Adjacency matrices $A$
illustrate the possible transitions allowed (white) for $N=16$ ($S=4$). For
the locality constraint (right), possible transitions with the same color must
take the same value in the adjacency matrix. (c) Cumulative probability
$\Pr(\lambda_{2}<\nu)$ that the eigenvalue $\lambda_{2}$ is slower than a
prescribed rate $\nu$. The data from Figure 6 is reproduced (left) to
facilitate comparison with the results of the constrained dynamics (center,
right). Similarly, data for $N=16$ ($S=4$) is reproduced on successive plots
to highlight the influence of the constraints.

#### Continuity Constraint.

The principle of continuity implies that a microscopic transition to state $i$
from state $j$ is more likely to occur if those states are “close” to one
another in phase space. As a specific example, we consider a digital phase
space of $S$ binary variables (or “spins”) where each of the $N=2^{S}$ total
states corresponds to a binary sequence of length $S$ (e.g., $01001\dots$).
Microscopic transitions from one state to another are more likely to occur
when those states differ by only a few spins. Here, we assume that each state
can transition to $S$ possible neighboring states by flipping only one spin;
transitions that involve the flipping of two or more spins are prohibited. As
a result, the number of possible dynamics (adjacency matrices) is reduced from
ca. $2^{N(N-1)}$ down to ca. $2^{N\log_{2}N}$ (Figure 7b, center). Owing to
the continuity constraint, large changes in phase space can occur only through
sequences of many small changes.

Within the reduced space of possible dynamics, the probability of finding
persistent systems increases by orders of magnitude (Figure 7, center). For
systems of $N=16$ states ($S=4$ spins), the fraction with two mesostates that
relax slower than $\lambda_{2}<0.003$ increases from ca. $10^{-34}$ to
$10^{-12}$ on applying the continuity constraint. This increase is due to the
significant reduction in the total number of systems, most of which are highly
connected and non-persistent. The continuity constraint also excludes
persistent systems but to lesser extent. For $N=16$, the minimum eigenvalue
$\lambda_{2}^{\min}$ increases from $2.2\times 10^{-5}$ to $1.5\times 10^{-3}$
when excluding those systems that violate the continuity principle. As with
any optimization, the addition of constraints necessarily increases the
objective function at the minimum. More importantly, these results show how
physical constraints on the space of possible dynamics can serve to make
persistence more probable.

#### Locality Constraint.

The principle of locality further implies that transitions between neighboring
states in phase space are largely independent of system features that are
“far” away in physical space. Continuing our example, the rate of flipping a
particular spin may depend on neighboring spins but not on the detailed
configurations of those far away in physical space. The exclusion of systems
with non-local dynamics further constrains the space of possible dynamics and
can therefore impact conclusions regarding the probability of persistence.
Here, we arrange the $S$ spins within a linear array and limit the range of
interactions to nearest neighbors (Figure 7, right). Thus, the rate of
flipping each spin depends only on the configuration of two neighboring spins
and is independent of all other spins. This constraint further reduces the
number of possible dynamics from ca. $2^{N\log_{2}N}$ down to ca.
$2^{8\log_{2}N}$ for nearest neighbor interactions in 1D.

The rightmost plot of Figure 7c shows the probability of persistence for
systems satisfying the continuity and locality constraints described above.
Owing to the significant reduction in the total number of systems, those
satisfying these constraints require more states to achieve high levels of
persistence. On applying the locality constraint, the probability of
persistence increases as compared to systems satisfying the continuity
constraint alone—but not by much. More important, however, is the dependence
on system size $N$. For systems satisfying both continuity and locality, the
probability of persistence appears to increase with the number of states $N$.
These results are rather anecdotal—applying only to numerical results of a
particular model for small systems—but are nevertheless suggestive of an
important hypothesis. For certain distributions over possible dynamics within
physical systems, persistent dynamics—of one form or another—are all but
guaranteed for sufficiently large systems. For other distributions, however,
persistence is all but impossible as $N$ grows to the astronomical values
typical of macroscopic systems. This hypothesis has implications for the
search for abiotic systems that exhibit characteristics of life.

#### New Strategies for Persistence.

With the application of the continuity and locality constraints, the newly
optimal systems adopt different strategies to maximize their persistence.
Figure 8a shows three such systems with 16 states and different constraints
that minimize the second eigenvalue $\lambda_{2}$, thereby preserving one bit
of information for as long as possible. To facilitate comparison between these
systems, the positions of the $2^{4}$ states are fixed at the corners of an
$4$-dimensional hypercube; the networks differ only in their directed
transitions.

Figure 8: New strategies for persistence. (a) Network representations of
systems with $S=4$ spins that minimize $\lambda_{2}$ for the physical
constraints of Figure 7. Each spin corresponds to a particular dimension of
the 4D hypercube. (b) State-time images showing the system state as a function
of time for the networks in (a). Each image is obtained from a single
stochastic trajectory from $t=0$ to $t=500$ with a common initial condition.

With no constraints, the maximally persistent system is analogous to that
shown in Figure 2a; however, the network organization is obscured by the
hypercube representation. There is no meaningful relationship between the
$S=4$ spins and the $N=2^{S}$ states. As a result, the organization of the the
system into two mesostates is not at all apparent when visualized through the
dynamics of the four spins (Figure 8b, left), which fail to provide a useful
description of the persistent pattern. This pattern is nevertheless present:
the system remains localized in the blue mesostate throughout the duration of
the simulation pictured here.

By contrast, for systems that satisfy the continuity constraint, the identity
of the mesostate is clearly encoded by a single variable in the phase space
(Figure 8a, center). In this way, the persistent dynamics of the system is
immediately visible from the dynamics of the four spins (Figure 8b, center).
The continuity constraint requires that transitions between states occur along
one dimension of the hypercube at a time. Subject to this constraint, the
strategy for achieving dynamical persistence is qualitatively similar to that
of the unconstrained system. Within each mesostate, there exists a single
linear path to escape to the other mesostate (e.g., $0000\rightarrow
0001\rightarrow 0011\rightarrow 0010\rightarrow 0110\rightarrow
0100\rightarrow 0101\rightarrow 0111\rightarrow 1111$). At each step along
this path, competing transitions lead back to earlier states, when permitted
by the physical constraints. Further addition of the locality constraint
results in similar networks that encode the mesostate within a single physical
spin (Figure 8b, right). In this way, the search for persistent dynamics leads
to error correction mechanisms that preserve the configuration of one spin.
More generally, this result suggests how physical constraints serve to
concentrate mesoscopic information within few variables of a larger phase
space.

### Measuring Persistence in Practice

For the idealized models described above, we quantify dynamical persistence
using the mutual information between the system state at two different times.
This approach requires knowledge of the system’s microscopic dynamics in order
to distinguish patterns associated with its history from those determined by
its components and environment alone. In practice, however, we don’t know the
underlying dynamics, and our measurements of the system state are incomplete.
How then can we expect to recognize and quantify persistent dynamics in
experimental models of life-like systems? Here, we described one approach
based on computable information density (CID)[37] and apply it to an
experimental model of driven colloidal particles.

#### Computable Information Density (CID).

The mutual information $I(\tau)$ between two system states separated by a time
lag $\tau$ is the difference of two entropies: that of the stationary
distribution $H(X(t))$ and that of the future state conditioned on the past
$H(X(t)\mid X(t-\tau))$. By observing the system state at regular time
intervals $\tau$, we can estimate these quantities without explicit knowledge
of the system dynamics. The observed sequence,
$x=\\{X_{1},X_{2},\dots,X_{n}\\}$, is analogous to a text string where each
letter denotes a possible state of the system (Figure 9a). The desired
entropies determine our ability to compress this string to form a more concise
but nevertheless exact representation of the state sequence. Consequently, we
can estimate the entropy of a string from the length of its binary
representation after lossless compression. The computable information density
(CID) is defined as the ratio between the binary code length $\mathcal{L}(x)$
of the compressed string and the length $n$ of the original string

$\text{CID}(x)=\frac{\mathcal{L}(x)}{n}$ (7)

This quantity has been used previously as an order parameter to describe phase
transitions[37] and correlation lengths[38] in nonequilibrium systems. Here,
we use the CID to measure time correlations and quantify dynamical persistence
in Markov processes.

We perform lossless compression of state sequences using the tree-structured
Lempel–Ziv algorithm (LZ78)[39, 40] as illustrated by the example in Figure
9a. Given a string of length $n$ from a finite alphabet of size $N$, the
algorithm parses the string into a sequence of “phrases” where the next phrase
is the shortest phrase not seen in the past. In the compressed representation,
each phrase can be described by a pointer to the corresponding prefix phrase
and by the terminal letter that completes the phrase. If there are $c(n)$
phrases in a particular string of length $n$, then the specification of the
pointer requires at most $\log_{2}c(n)$ bits and the terminal letter at most
$\log_{2}N$ bits. Therefore, the binary length of the compressed string is
bounded as

$\mathcal{L}(x)\leq c(n)(\log_{2}c(n)+\log_{2}N)$ (8)

We use this bound to approximate the binary code length in evaluating the CID.

Figure 9: (a) Example of the tree-structured Lempel-Ziv compression algorithm
(LZ78).[40] A string of length $n=12$ with an alphabet of size $N=3$ is parsed
into a sequence of $c(n)=7$ unique phrases. The compressed output contains the
sequence of phrases, each specified by the index of their prefix phrase and
their terminal letter. (b) The mutual information $I(\tau)$ for the 12-state
system in Figure 5 (solid curve) is approximated by $\hat{I}(\tau)$ (markers)
using sequences of $n=10^{4}$ states generated with a prescribed time step
$\tau$. All realizations of the stochastic process are initialized from state
1.

The LZ78 code—like LZ77 used in prior work[37, 38]—is “universal” in that it
compresses any sequence without _a priori_ knowledge of the process by which
it was generated. Moreover, the code is “optimal” in that the
$\mathrm{CID}(x)$ converges to the entropy rate for a stationary ergodic
process[40]

$\lim_{n\rightarrow\infty}\mathrm{CID}(x)=\lim_{n\rightarrow\infty}\frac{1}{n}H(X_{1},X_{2},\dots,X_{n})$
(9)

For Markov processes in particular, the entropy rate is identical to the
conditional entropy $H(X(t)\mid X(t-\tau))$ introduced above.[40] In this way,
the CID provides a convenient estimate for the conditional entropy, which is
exact in the limit of long sequences. We note, however, that the difference
between the expected CID and the entropy rate—the so-called redundancy—decays
slowly as $(\log_{2}n)^{-1}$.[41]

To estimate the marginal entropy $H(X(t))$, we shuffle the state sequence $x$
and evaluate the CID of the randomized sequence $x_{\text{sh}}$.[42] The
process of shuffling acts to destroy time correlations between successive
states. For long times ($n\tau\gg\lambda_{1}^{-1}$), the shuffled sequence
becomes indistinguishable from a set of $n$ independent samples drawn from the
stationary distribution. With these preliminaries, the mutual information
$I(\tau)$ can be estimated from state sequence $x$ with sample time $\tau$ as

$I(\tau)\approx\hat{I}(\tau)=\langle\mathrm{CID(x_{\text{sh}})}\rangle-\mathrm{CID(x)}$
(10)

where $\langle~{}\rangle$ denotes an average over the shuffled sequences.
Figure 9b compares this computable approximation based on observed states to
the exact result based on knowledge of the dynamics for the 12-state system of
Figure 2. Deviations between the two derive from the finite number of sampled
states (here, $n=10^{4}$) and the finite sample times for short lags
$\tau<(n\lambda_{1})^{-1}\approx 0.1$.

#### Quantifying Persistence in Dynamic Assemblies of Colloidal Rollers.

CID-based estimates of mutual information can be used to quantify dynamical
persistence in experimental models of nonequilibrium systems. Here, we
consider one such model based on colloidal particles driven to roll along a
solid surface by a time-varying magnetic field (Figure 10a; see Methods for
details).[43, 44] The particles are confined within a circular track and
driven to roll along circular orbits by fields of the form

$\bm{\mathbf{B}}(t)=B_{0}\left(\sin(\Omega t)\sin(\omega
t)\bm{\mathbf{e}}_{x}+\cos(\Omega t)\sin(\omega
t)\bm{\mathbf{e}}_{y}+\cos(\omega t)\bm{\mathbf{e}}_{z}\right)$ (11)

where $B_{0}$ is the constant magnitude of the field, $\omega$ is the rolling
frequency, and $\Omega$ is the comparatively slow frequency with which the
rolling direction changes. This rotating field drives ferromagnetic spheres to
roll across the surface with speed $U$ around circular orbits of radius
$U/\Omega$.

As they move along the track, the particles assemble to form dynamic clusters
mediated by hydrodynamic, magnetic dipole-dipole, and excluded volume
interactions. The clusters of different shapes and sizes move at different
speeds within the common driving field. Fast-moving clusters ($U>R\Omega$)
roll into the outer wall of the track thereby disrupting their organization
and slowing their motion. These clusters form a moving front or “stampede” of
rollers that orbit the track at a frequency $\Omega$ specified by the driving
field (Figure 10). By contrast, slow-moving clusters ($U<R\Omega$) that fail
to keep up with the stampede move erratically in place until the moving front
of rollers returns. As the stampede passes over the slow-moving clusters, the
latter may be incorporated into the stampede or otherwise reconfigured by
interactions with other clusters. The motivating hypothesis of this system is
that over time the particles will organize to form persistent clusters
distinguished by their stability in the driving field.

Figure 10: (a) Magnetic Janus particles (MJPs; 4 $\mu$m radius) are driven to
roll around a circular track of radius $R=400~{}\mu$m, width $100~{}\mu$m, and
height $50~{}\mu$m by a time-varying magnetic field $\bm{\mathbf{B}}(t)$.
Particles and clusters thereof appear as bright spots on fluorescent
microscopy images—here, annotated to show the circular track. The inset shows
the downsampling and thresholding of the raw image to create the coarse
grained representation used in our analysis. (b) Time-lapse images show
cluster dynamics within the moving front. Here, we adopt a rotating frame of
reference in which the average position of the front is stationary. (c) Mutual
information estimate $\hat{I}(\tau)$ as a function of lag time $\tau$ for the
system shown in (a). Here, the driving field is specified by equation (11)
with magnitude $B_{0}=3$ mT, rolling frequency $\omega=62.8$ rad/s, and
rotation frequency $\Omega=0.0140$ rad/s; the video is captured over 70
minutes with a frame rate of 0.67 fps. The markers denote the estimates
computed from the videos; the solid curve is a least-squares fit of the form
$C\tau^{-\alpha}e^{-\beta\tau}$ with $\alpha=0.49$ and $\beta=0.029$ s-1. The
inset shows the same data on a log-log plot.

To test this hypothesis, we analyze microscopy videos of particle motion to
estimate the mutual information $\hat{I}(\tau)$ as a function of lag time
(Figure 10c). For each time interval $\tau$, the videos are coarse grained to
create a sequence of $n$ binary images containing $h\times w$ pixels
comparable in size to the individual particles. Following Cavagna _et al._
,[42] we use $z$-order (Morton space-filling) curves to transform the three-
dimensional ($n\times h\times w$) bit array into a one-dimensional bit string.
These orderings preserve locality of the bits in space and time and are
therefore better in capturing time correlations in cluster dynamics. We use
the LZ78 algorithm to compress the bit string and estimate its binary code
length $\mathcal{L}$ using the bound in equation (8). Dividing by the number
of images $n$, we obtain the CID measuring the number of bits of information
per frame in the coarse grained video. To estimate the amount of heritable
information $\hat{I}(\tau)$, we repeat this analysis on shuffled image
sequences and evaluate the difference in the respective CIDs following
equation (10).

As expected, the heritable information $\hat{I}(\tau)$ decreases monotonically
with increasing lag time as physical memories of the system’s past fade.
Interestingly, however, these memories decay slowly as approximated by a
truncated power-law with an exponent of $\alpha\approx 0.5$ over the lag times
investigated. At longer times, the heritable information $\hat{I}(\tau)$
decays exponentially with a rate comparable to the slow rotation frequency
$\Omega$ of the driving field. Longer experiments are required to determine
whether or not any information is preserved for longer times, $\Omega\tau>1$.
The presence of such long-lived correlations—if they exist—would be suggestive
of the type of persistent patterns from which the physicochemical precursors
of life evolved. Despite the lack of evidence for such patterns here, the
proposed methodology should be useful for detecting dynamical persistence in
this and other experimental models of active matter.[21, 16]

CID-based measures of heritable information can be applied in practice using
relatively small numbers of sampled states. In the present example, the space
of binary image sequences contains an astronomical $2^{hwn}$ possible states;
however, meaningful estimates of heritable information are obtained from a
single image sequence of only $n\sim 100$ frames. Due to the physical
principles of continuity and locality, the dynamics on this large state space
admit a far more concise description based on moving particles and their local
interactions. Consequently, the use of the z-order curve during compression is
essential to capturing the decomposition of the system into local subsystems
exhibiting similar dynamics. Almost no heritable information is detected when
the image sequences are encoded differently prior to compression. Presumably,
the dependence of $\hat{I}$ on the details of state encoding should vanish in
the limit as $n\rightarrow\infty$; however, this asymptotic behavior has
limited relevance in practice. Care is therefore required in interpreting
$\hat{I}$ for finite $n$. Our present analysis captures local forms of
heredity (e.g., the persistence of a particular cluster geometry) but is
unlikely to capture more complex forms based on communication and cooperation
among clusters, which require more data to resolve.

We emphasize that the use of universal compression algorithms allows for the
detection of persistent patterns with little to no knowledge of the patterns
themselves or the mechanisms by which they persist. Such patterns are
distinguished by surprising amounts of heritable information that persist over
time scales much longer than those of internal relaxation or external driving.
Once discovered, the relevant patterns that enable compression can be
interrogated more explicitly to identify the persistent features of the system
and understand their dynamical stability. By searching for information about
the past, one aims to discover new modes of organization that enable that
information to survive the passage of time. We hypothesize that this
perspective—one that elevates heritable information above other features of
life—will help in identifying experimental models that exhibit primitive forms
of heredity relevant to life’s origins.

## Conclusions

So what kind of explanation can we expect to find for the origins of life? The
answer advocated here is a conceptual framework based on heritable information
in dynamical systems. Within this framework, we consider an ensemble of
possible dynamical systems. The prior probability assigned to each possibility
is chosen to reflect our knowledge of the physical world and the constraints
it imposes (e.g., locality, continuity). We imagine that many of these
hypothesized systems occur somewhere in the world like so many primordial
soups with different components and environments. As the systems change in
time, we look for information about their past within their present
configurations. For the majority of such systems, the present state is
explained by their components and environment alone. Others, however, contain
additional information about their history. In rare instances, these histories
are not frozen relics of the past but are actively maintained by dissipative
processes that appear organized for the purpose of preserving heritable
information. Once identified, these persistent patterns capture our attention
and demand explanation of the mechanisms—the so-called “free floating
rationales” [5]—that underlie their existence. The origins of life is informed
by this persistence filter whereby patterns of organization are selected by
the passage of time on the basis of their persistence alone. In these
concluding remarks, we highlight some of the (many) remaining questions about
this proposed framework.

In our numerical investigation, we imposed the artificial constraint that all
microstates have a common lifetime. As a result, the persistent systems we
identify are necessarily dissipative and require purposeful organization to
preserve information for times longer than this microscopic time scale. By
contrast, real systems have a spectrum of microstate lifetimes, some of which
may be extremely long as evidenced by the stability of the agate (Figure 1a).
Our measure of persistence based on mutual information between the past and
present does not discriminate between these different types of persistence.
Further work is needed to more clearly distinguish dynamic and static
persistence—that of the bacterium and the agate, respectively (Figure 1). In
the present context, we use the term dynamical persistence to describe systems
that preserve heritable information on timescales longer than even the longest
microstate lifetime. Such persistence is not a property of the microstates
themselves but rather their mesoscopic organization. The connection between
material organization and heritable information is a central theme of this
Chapter and demands further investigation. As Darwinian evolution informs our
understanding of biological function, heritable information may provide new
principles for understanding material organization in nonequilibrium systems.

We further impose the simplifying assumption that each dynamical system is
isolated within its own unchanging environment. In reality, these systems are
embedded within larger systems and interact with one another to create
complex, fluctuating environments. The carving of the world into system and
environment, microstates and mesostates, fast and slow processes is messy
business filled with ambiguities. The simple models described here are useful
for building intuition precisely because they are free from such ambiguities.
The study of experimental systems—even those as simple as the colloidal
rollers discussed above—requires difficult choices to best “carve nature at
its joints”. In navigating these choices, the concept of heritable information
can provide a useful guide. We seek the most concise representations that
capture the system’s heritable information on a prescribed time scale. For
this reason, we coarse grain the images of the rollers as to omit features
smaller than the particles themselves. Such _ad hoc_ coarse graining
procedures can surely be improved using machine learning (ML) techniques such
as variational autoencoders, which create coarse grained representations that
maximize a specified objective function (e.g., the amount of heritable
information). The pattern-finding capabilities of ML algorithms offer many
exciting opportunities to identity persistent patterns of heritable
information in diverse material systems.

One question that we have not attempted to address is the semantic meaning of
heritable information. We instead adopt Shannon’s perspective and analyze
dynamical systems as communications channels that transmit information between
the past and the present without regard to its meaning. Such channels are a
critical prerequisite for the hereditary processes underlying life as we know
it. In this Chapter, we have sought to explain how inanimate material systems
are capable of transmitting information through time. For example, the
12-state system of Figure 2 transmits 1 bit of information about its distant
past to the present; however, we have not discussed how this bit was written
or what it might mean. Returning to the analogy with Natural Selection, it
would appear that persistent patterns succeed in preserving heritable
information but lack the capacity for mutation and thereby evolutionary
change. This view is incorrect: change is ubiquitous, persistence in spite of
such change is rare and the defining characteristic of life. The distinction
between change and persistence is often a matter of time scales. What persists
on one time scale changes over longer times. Our genome is stable over our
lifetime but changes in part over generations even as the vast majority of
distinctly human genetic patterns persist longer still. By exploring the
mechanisms of dynamical persistence in material systems and estimating their
respective probabilities under different conditions, we move closer to
understanding the origins of heredity and thereby life itself.

## Methods

### Coarse Graining

For a system with rate matrix $W$ characterized by $M$ slow eigenvalues, the
$N$ states can be clustered into $M$ mesostates to provide a coarse grained
description of the system dynamics. We use a spectral clustering approach to
associate each state $i$ with a mesostate $k$ to determine the elements
$\Pi_{ki}$ of the conditional probability matrix introduced in the main text.
Two states $i$ and $j$ are considered to be in the same mesostate if they
evolve in time to neighboring points in probability space after the relaxation
of the fast modes. The space of possible distributions $p_{i}$ is constraine
by the requirement that $0\leq p_{i}\leq 1$ and by the normalization condition
$\sum_{i}p_{i}=1$. As illustrated in Figure 11 for $N=3$, this space can be
viewed as an $(N-1)$-dimensional simplex in $N$-dimensional Euclidean space,
where each vertex corresponds to a state of the system. Under the dynamics of
the master equation (1), the simplex is squeezed, sheared, and rotated by the
linear transformation $T(t)$, ultimately collapsing to a single point
describing the stationary distribution. At all times, the collapsing simplex
must remain within its initial bounds, thereby restricting the possible
eigenvalues and eigenvectors.

Figure 11: Geometric representation of the 2D probability space (light blue)
of a 3-state system. In time, the simplex of possible distributions (dark
blue) contracts toward the stationary state (red dot). This system is
characterized by time scale separation between its fast eigenmode
($\lambda_{3}=1.5$) and its slow eigenmode ($\lambda_{2}=0.013$).
Consequently, the system admits a coarse grained representation—here, one in
which states 1 and 2 are combined to form a mesostate.

For systems with timescale separation, the relaxation of the fast modes causes
the $N$ vertices of the simplex to cluster into $M$ mesostates, which
approximate a new simplex of reduced dimensionality. The matrix element
$T_{ij}(t)$ describes the $i^{\mathrm{th}}$ coordinate of the
$j^{\mathrm{th}}$ vertex at time $t$. At intermediate times—much longer than
the fast time $\lambda^{-1}_{\text{fast}}$ but much shorter than the slow time
$\lambda^{-1}_{\text{slow}}$—these vertex coordinates are well approximated as

$T_{ij}(t)\approx\sum_{m}Q_{im}Q^{-1}_{mj}$ (12)

where the sum includes only the slow modes. We use the $k$-means clustering
algorithm to group these points into $M$ mesostates. Alternatively, one may
cluster points in the ($M-1$)-dimensional eigenspace by applying the $k$-means
algorithm to the $N$ coordinates, $Q^{-1}_{mj}$.

### Monte Carlo Optimization

We use the Metropolis–Hastings algorithm to sample the space of possible
dynamics in search of those that minimize some property $U$—for example, the
relaxation rate $\text{Re}(\lambda_{2})$. This procedure is analogous to
finding the minimum energy configuration of an equilibrium system. The space
of possible dynamics corresponds to that of possible adjacency matrices for
strongly connected graphs. Starting from one such strongly connected graph, we
consider two possible Monte Carlo moves—bit flips and bit swaps—at each step
of the Markov chain. During a bit flip, we select one of the $N(N-1)$ possible
edges at random and flip the corresponding value of the adjacency matrix
$A_{ij}$ to add or remove the edge. During a bit swap, we select at random two
elements of the adjacency matrix such that $A_{ij}=1$ and $A_{kl}=0$ and swap
their values. If an attempted move breaks the connectivity constraint, the
attempt is rejected. Otherwise, the move from the current dynamics $n$ to the
candidate dynamics $m$ is accepted with probability
$\min(1,e^{-\beta(U_{m}-U_{n})})$. This algorithm samples the dynamics $n$ in
accordance with the Boltzmann distribution $p_{n}\propto e^{-\beta U_{n}}$,
where $\beta\in[0,\infty)$ is a tunable parameter. To identify the dynamics
that minimize $U$, we use simulated annealing[33] to gradually increase
$\beta$ thereby biasing the Markov chain towards dynamics with smaller $U$.

### Experiments on Ferromagnetic Rollers

We prepared magnetic Janus particles (MJPs) by e-beam deposition of successive
layers of metal—5 nm Ti adhesive layer, 25 nm Ni magnetic layer—onto
monolayers of 4 $\mu$m fluorescent sulfonated polystyrene (PS) particles.[45]
The nickel layer gave the particles a permanent magnetic moment $m\approx
3\times 10^{-14}$ A m2 directed parallel to the Janus equator.[46] To mitigate
their agglomeration in water, the Janus particles were coated with a thin
layer of silicon dioxide by sputter deposition. The circular track was
fabricated in polydimethyl siloxane (PDMS) by soft lithography and treated
with a surfactant solution (0.2 w/v% Pluronic F127 in water) to further
prevent particle adhesion. The aqueous dispersion of MJPs were pipetted into
the PDMS track and sealed with a glass cover slide using UV-curable adhesive.

During the rolling experiments, the particles settled under gravity to the
glass substrate where they were imaged from below by an inverted microscope.
The time-varying magnetic field $\bm{\mathbf{B}}(t)$ was generated using a
custom triaxial electromagnet [44] with a waveform given by equation (11). The
magnitude of the field was held constant with $B_{0}=3$ mT as measured by a
gaussmeter. The rolling frequency $\omega$ was chosen to be sufficiently slow
such that magnetic particles could role in sync with the applied field—that
is, $\omega<mB_{0}/\lambda$ where $\lambda\approx 8\pi\eta a^{3}$ is the
hydrodynamic resistance to rotation for a sphere in a fluid of viscosity
$\eta$. The rotation frequency $\Omega$ on which the direction of the rotating
field changes was chosen empirically to permit “fast” clusters to follow the
field around track but not “slow” clusters and individual particles. Here, we
use a rolling frequency of $\omega=62.8$ rad/s and a rotation frequency of
$\Omega=0.0140$ rad/s. The frame rate of image acquisition was $0.67$ fps.

The video analyzed in Figure 10 contained 2801 grayscale images with
$4908\times 3264$ pixels each. Each image was cropped, downsampled, and
thresholded to create a $320\times 320$ binary image, where each square pixel
($5.8\times 5.8~{}\mu$m2) denotes the presence (1) or absence (0) of a
particle at that location. Those pixels located outside of the circular track
were always zero and therefore omitted from the analysis. In estimating the
mutual information $\hat{I}(\tau)$, we considered 33 time lags $\tau$, each an
integer multiple of the time step for image acquisition, $\Delta t=1.5$ s. For
each lag, frames were removed from the binary image sequence such that the
remaining frames were spaced by $\tau$. Following Cavagna _et al._ ,[42] we
ordered the 3D bit arrays into six 1D bit strings using as many permutations
of the $z$-order curve (e.g., $(x,y,t)$, $(y,t,x)$, etc.). The resulting CIDs
were averaged over these six strings to improve the isotropy of the analysis
and avoid privileging some dimensions over others. Each bit string was
compressed using the LZ78 algorithm (Figure 9), and the CID computed using
equation (7) with the binary code length $\mathcal{L}$ estimated by equation
(8) and the number of frames $n$ determined by the lag as $n=\lfloor
2801\Delta t/\tau\rfloor$. Finally, the binary image sequence was randomly
shuffled in time and its CID computed. In accordance with equation (10), the
mutual information estimate $\hat{I}(\tau)$ was evaluated as the difference
between the average CID of eight shuffled sequences and that of the original
sequence.

## References

  * [1] Erwin Schrödinger  “What is Life?”  Cambridge University Press, 1944
  * [2] Freeman Dyson  “Origins of life”  Cambridge University Press, 1999
  * [3] Manfred Eigen  “From strange simplicity to complex familiarity: a treatise on matter, information, life and thought”  Oxford University Press, 2013
  * [4] Addy Pross  “What is Life?: How chemistry becomes biology”  Oxford University Press, 2016
  * [5] Daniel C Dennett  “From bacteria to Bach and back: the evolution of minds”  WW Norton & Company, 2017
  * [6] Paul Davies  “The demon in the machine: how hidden webs of information are solving the mystery of life”  University of Chicago Press, 2019
  * [7] Stuart A Kauffman  “A world beyond physics: the emergence and evolution of life”  Oxford University Press, 2019
  * [8] Jeremy L England  “Dissipative adaptation in driven self-assembly”  In _Nature nanotechnology_ 10.11 Nature Publishing Group, 2015, pp. 919–923
  * [9] Esra Te Brinke, Joost Groen, Andreas Herrmann, Hans A Heus, Germán Rivas, Evan Spruijt and Wilhelm TS Huck  “Dissipative adaptation in driven self-assembly leading to self-dividing fibrils”  In _Nat. Nanotech._ 13.9 Nature Publishing Group, 2018, pp. 849–855
  * [10] Yong Dou, K. Dhatt-Gauthier and Kyle J.M. Bishop  “Thermodynamic costs of dynamic function in active soft matter”  In _Curr. Opin. Solid State Mater. Sci._ 23.1 Elsevier, 2019, pp. 28–40  DOI: 10.1016/j.cossms.2018.11.002
  * [11] Ximin He, Michael Aizenberg, Olga Kuksenok, Lauren D Zarzar, Ankita Shastri, Anna C Balazs and Joanna Aizenberg  “Synthetic homeostatic materials with chemo-mechano-chemical self-regulation”  In _Nature_ 487.7406 Nature Publishing Group, 2012, pp. 214–218
  * [12] Michael M Lerch, Alison Grinthal and Joanna Aizenberg  “Homeostasis as inspiration—Toward interactive materials”  In _Adv. Mater._ 32.20 Wiley Online Library, 2020, pp. 1905554
  * [13] Mathieu Colomb-Delsuc, Elio Mattia, Jan W Sadownik and Sijbren Otto  “Exponential self-replication enabled through a fibre elongation/breakage mechanism”  In _Nat. Commun._ 6.1 Nature Publishing Group, 2015, pp. 1–7
  * [14] Xiaojin He, Ruojie Sha, Rebecca Zhuo, Yongli Mi, Paul M Chaikin and Nadrian C Seeman  “Exponential growth and selection in self-replicating materials from DNA origami rafts”  In _Nat. Mater._ 16.10 Nature Publishing Group, 2017, pp. 993–997
  * [15] Pierre Illien, Ramin Golestanian and Ayusman Sen  “‘Fuelled’motion: Phoretic motility and collective behaviour of active colloids”  In _Chem. Soc. Rev._ 46.18 Royal Society of Chemistry, 2017, pp. 5508–5518
  * [16] Clemens Bechinger, Roberto Di Leonardo, Hartmut Löwen, Charles Reichhardt, Giorgio Volpe and Giovanni Volpe  “Active particles in complex and crowded environments”  In _Rev. Mod. Phys._ 88.4 APS, 2016, pp. 045006
  * [17] Addy Pross  “Toward a general theory of evolution: extending Darwinian theory to inanimate matter”  In _J. Sys. Chem._ 2.1 Springer, 2011, pp. 1
  * [18] Nigel Goldenfeld and Carl Woese  “Life is Physics: Evolution as a Collective Phenomenon Far From Equilibrium”  In _Annu. Rev. Condens. Matter Phys._ 2.1, 2011, pp. 375–399  DOI: 10.1146/annurev-conmatphys-062910-140509
  * [19] Daniel C Dennett  “Real patterns”  In _The journal of Philosophy_ 88.1 JSTOR, 1991, pp. 27–51
  * [20] Nathan C Keim, Joseph D Paulsen, Zorana Zeravcic, Srikanth Sastry and Sidney R Nagel  “Memory formation in matter”  In _Rev. Mod. Phys._ 91.3 APS, 2019, pp. 035002
  * [21] M Cristina Marchetti, Jean-François Joanny, Sriram Ramaswamy, Tanniemola B Liverpool, Jacques Prost, Madan Rao and R Aditi Simha  “Hydrodynamics of soft active matter”  In _Rev. Mod. Phys._ 85.3 APS, 2013, pp. 1143
  * [22] Gonen Ashkenasy, Thomas M Hermans, Sijbren Otto and Annette F Taylor  “Systems chemistry”  In _Chem. Soc. Rev._ 46.9 Royal Society of Chemistry, 2017, pp. 2543–2554
  * [23] Jürgen Schnakenberg  “Network theory of microscopic and macroscopic behavior of master equation systems”  In _Rev. Mod. Phys._ 48 APS, 1976, pp. 571–585
  * [24] Nicolaas Godfried Van Kampen  “Stochastic processes in physics and chemistry”  Elsevier, 1992
  * [25] John D. Chodera and Frank Noé  “Markov state models of biomolecular conformational dynamics”  In _Curr. Opin. Struct. Biol._ 25, 2014, pp. 135–144  DOI: 10.1016/j.sbi.2014.04.002
  * [26] Brooke E. Husic and Vijay S. Pande  “Markov state models: from an art to a science”  In _J. Am. Chem. Soc._ 140.7, 2018, pp. 2386–2396  DOI: 10.1021/jacs.7b12191
  * [27] Daniel T Gillespie  “Stochastic simulation of chemical kinetics.”  In _Annu. Rev. Phys. Chem._ 58, 2007, pp. 35–55  DOI: 10.1146/annurev.physchem.58.032806.104637
  * [28] Anatoly B. Kolomeisky and Michael E. Fisher  “Molecular Motors: A Theorist’s Perspective”  In _Annu. Rev. Phys. Chem._ 58.1, 2007, pp. 675–695  DOI: 10.1146/annurev.physchem.58.032806.104532
  * [29] Jason A. Wagoner and Ken A. Dill  “Molecular Motors: Power Strokes Outperform Brownian Ratchets”  In _J. Phys. Chem. B_ 120.26, 2016, pp. 6327–6336  DOI: 10.1021/acs.jpcb.6b02776
  * [30] Massimiliano Esposito  “Stochastic thermodynamics under coarse graining”  In _Phys. Rev. E_ 85.4, 2012, pp. 041125  DOI: 10.1103/PhysRevE.85.041125
  * [31] Udo Seifert  “Stochastic thermodynamics: From principles to the cost of precision”  In _Physica A_ 504 Elsevier, 2018, pp. 176–191
  * [32] Fugao Wang and D. P. Landau  “Efficient, multiple-range random walk algorithm to calculate the density of states”  In _Phys. Rev. Lett._ 86 American Physical Society, 2001, pp. 2050–2053  DOI: 10.1103/PhysRevLett.86.2050
  * [33] Scott Kirkpatrick, C Daniel Gelatt and Mario P Vecchi  “Optimization by simulated annealing”  In _Science_ 220.4598 American association for the advancement of science, 1983, pp. 671–680
  * [34] Andre C. Barato and Udo Seifert  “Thermodynamic uncertainty relation for biomolecular processes”  In _Phys. Rev. Lett._ 114.15, 2015, pp. 158101  DOI: 10.1103/PhysRevLett.114.158101
  * [35] Todd R. Gingrich, Jordan M. Horowitz, Nikolay Perunov and Jeremy L. England  “Dissipation bounds all steady-state current fluctuations”  In _Phys. Rev. Lett._ 116.12, 2016, pp. 120601  DOI: 10.1103/PhysRevLett.116.120601
  * [36] Jordan M. Horowitz and Todd R. Gingrich  “Thermodynamic uncertainty relations constrain non-equilibrium fluctuations”  In _Nat. Phys._ 16.1 Springer US, 2020, pp. 15–20  DOI: 10.1038/s41567-019-0702-6
  * [37] Stefano Martiniani, Paul M. Chaikin and Dov Levine  “Quantifying hidden order out of equilibrium”  In _Phys. Rev. X_ 9.1 American Physical Society, 2019, pp. 11031  DOI: 10.1103/PhysRevX.9.011031
  * [38] Stefano Martiniani, Yuval Lemberg, Paul M. Chaikin and Dov Levine  “Correlation lengths in the language of computable information”  In _Phys. Rev. Lett._ 125.17 American Physical Society, 2020, pp. 170601  DOI: 10.1103/PhysRevLett.125.170601
  * [39] Jacob Ziv and Abraham Lempel  “Compression of individual sequences via variable-rate coding”  In _IEEE Trans. Inf. Theory_ 24.5, 1978, pp. 530–536  DOI: 10.1109/TIT.1978.1055934
  * [40] Thomas M Cover  “Elements of information theory”  John Wiley & Sons, 1999
  * [41] Serap A. Savari  “Redundancy of the Lempel-Ziv incremental parsing rule”  In _IEEE Trans. Inf. Theory_ 43.1, 1997, pp. 9–21  DOI: 10.1109/18.567642
  * [42] Andrea Cavagna, Paul M Chaikin, Dov Levine, Stefano Martiniani, Andrea Puglisi and Massimiliano Viale  “Vicsek model by time-interlaced compression: a dynamical computable information density”  In _arXiv:2007.11322_ , 2020
  * [43] Michelle Driscoll, Blaise Delmotte, Mena Youssef, Stefano Sacanna, Aleksandar Donev and Paul Chaikin  “Unstable fronts and motile structures formed by microrollers”  In _Nat. Phys._ 13.4 Nature Publishing Group, 2017, pp. 375–379
  * [44] Yong Dou, Peter M Tzelios, Dimitri Livitz and Kyle J M Bishop  “Programmable topotaxis of magnetic rollers in time-varying fields”  In _Soft Matter_ Royal Society of Chemistry, 2020
  * [45] Wenjie Fei, Peter M Tzelios and Kyle J M Bishop  “Magneto-capillary particle dynamics at curved interfaces: Time-varying fields and drop mixing”  In _Langmuir_ 36 ACS Publications, 2020, pp. 6977–6983
  * [46] Wenjie Fei, Michelle M Driscoll, Paul M Chaikin and Kyle J M Bishop  “Magneto-capillary dynamics of amphiphilic Janus particles at curved liquid interfaces”  In _Soft Matter_ 14.23 Royal Society of Chemistry, 2018, pp. 4661–4665

## Acknowledgements

This work was supported in part by the Center for Bio-Inspired Energy Science,
an Energy Frontier Research Center funded by the U.S. Department of Energy,
Office of Science, Basic Energy Sciences under Award DE-SC0000989. K.J.M.B. is
grateful to L. Bishop for useful discussions of this work.

##  Appendix A Persistence in Equilibrium Systems

Equilibrium systems with two mesostates that minimize the second eigenvalue
$\lambda_{2}$ are organized into two clusters connected by a linear chain of
states (Figure 5b). Each cluster contains $N_{c}$ states, which are fully
connected to one another by reversible transitions; the linear chain contains
$N_{l}$ states. Here, we aim to estimate the minimum eigenvalue
$\lambda_{2}^{\min}$ and the allocation of the $N=2N_{c}+N_{l}$ states between
the clusters and the chain.

To simplify the analysis, we coarse grain each cluster as a single state with
an escape rate $k$. The probability of being in any state within the cluster
relaxes quickly to $N_{c}^{-1}$ on time scales of order one. The rate of
exiting the cluster from the single outlet is also $N_{c}^{-1}$ owing to our
rules relating transition rates to the number of outgoing transitions.
Therefore, we can approximate the escape rate for the coarse grained cluster
as $k=N_{c}^{-2}$.

Within the linear chain of $N_{l}$ states, the probability of being in state
$i$ evolves as

$\frac{d{p}_{i}}{dt}=\tfrac{1}{2}p_{i-1}-p_{i}+\tfrac{1}{2}p_{i+1}$ (13)

for the internal states $i=2,\dots,N_{l}-1$. At the ends of the chain, the
state probabilities evolve as

$\displaystyle\frac{dp_{1}}{dt}$
$\displaystyle=kp_{L}-p_{1}+\tfrac{1}{2}p_{2}$ (14)
$\displaystyle\frac{dp_{N_{l}}}{dt}$
$\displaystyle=\tfrac{1}{2}p_{N_{l}-1}-p_{N_{l}}+kp_{R}$ (15)

where $p_{L}$ and $p_{R}$ are the probabilities that the system is in the
coarse grained clusters on the left and right, respectively. These
probabilities evolve in time as

$\displaystyle\frac{dp_{L}}{dt}$ $\displaystyle=-kp_{L}+\tfrac{1}{2}p_{1}$
(16) $\displaystyle\frac{dp_{R}}{dt}$
$\displaystyle=\tfrac{1}{2}p_{N_{l}}-kp_{R}$ (17)

Together, equations (13)–(17) specify the dynamics of the system neglecting
the internal dynamics within the two clusters, which are assumed to be fast.

For large numbers of states, we can approximate the discrete states
$i=1,2,\dots,N_{l}$ by a continuum $x\in[0,N_{l}]$ where the probability
density $p(x,t)$ is governed by the diffusion equation

$\frac{\partial p}{\partial t}=\frac{1}{2}\frac{\partial^{2}p}{\partial
x^{2}}$ (18)

At the boundaries ($x=0,N_{l}$), conservation of probability requires that

$\displaystyle\frac{dp_{L}}{dt}$
$\displaystyle=\left.\frac{1}{2}\frac{\partial p}{\partial
x}\right\rvert_{0}=-kp_{L}+\frac{1}{2}p(0,t)$ (19)
$\displaystyle\frac{dp_{R}}{dt}$
$\displaystyle=\left.-\frac{1}{2}\frac{\partial p}{\partial
x}\right\rvert_{N_{l}}=\frac{1}{2}p(N_{l},t)-kp_{R}$ (20)

The discrete equations above correspond to a finite difference approximation
of this continuum description. We seek the slowest decaying eigenmode of the
form $p(x,t)=\hat{p}(x)e^{-\lambda t}$. Substituting this expression into the
diffusion equation, we obtain harmonic solutions $\hat{p}(x)$ of the form

$\hat{p}(x)=A\cos(\sqrt{2\lambda}x)+B\cos(\sqrt{2\lambda}x)$ (21)

Substituting this result into equations (19) and (20) we obtain the following
characteristic equation for $\lambda$

$\sqrt{2\lambda}\left(2(k-\lambda)^{2}-\lambda\right)\sin(\sqrt{2\lambda}N_{l})-4\lambda(\lambda-k)\cos(\sqrt{2\lambda}N_{l})=0$
(22)

Anticipating that $\lambda\propto N^{-3}$, we can expand this equation in a
Taylor series about $\lambda=0$ to obtain a quadratic approximation for the
first two roots valid for large numbers of states $N\gg 1$. Substituting
$k=N_{c}^{-2}$ and introducing $\alpha=N_{l}/N$, the second root $\lambda_{2}$
can then be approximated as

$\lambda_{2}=\frac{8}{\alpha(\alpha-1)^{2}N^{3}}+O(N^{-4})$ (23)

which reaches it minimum value of $\lambda^{\min}_{2}=54/N^{3}$ for
$\alpha=1/3$. Consistent with our numerical results (Figure 5b), the states
are partitioned equally in three groups corresponding to the two clusters and
the linear chain. The minimum eigenvalue $\lambda_{2}^{\min}$ scales as
$N^{-3}$ as illustrated in Figure 5a. The difference in the prefactor (54 here
vs. 47 in Figure 5a) is due to the coarse graining of the connected clusters
in our present analysis.