# Information Thermodynamics for Deterministic Chemical Reaction Networks

Emanuele Penocchio<EMAIL_ADDRESS>Complex Systems and Statistical
Mechanics, Department of Physics and Materials Science, University of
Luxembourg, L-1511 Luxembourg  Francesco Avanzini<EMAIL_ADDRESS>Complex Systems and Statistical Mechanics, Department of Physics and Materials
Science, University of Luxembourg, L-1511 Luxembourg  Massimiliano Esposito
<EMAIL_ADDRESS>Complex Systems and Statistical Mechanics,
Department of Physics and Materials Science, University of Luxembourg, L-1511
Luxembourg

###### Abstract

Information thermodynamics relates the rate of change of mutual information
between two interacting subsystems to their thermodynamics when the joined
system is described by a bipartite stochastic dynamics satisfying local
detailed balance. Here, we expand the scope of information thermodynamics to
deterministic bipartite chemical reaction networks, namely, composed of two
coupled subnetworks sharing species, but not reactions. We do so by
introducing a meaningful notion of mutual information between different
molecular features, that we express in terms of deterministic concentrations.
This allows us to formulate separate second laws for each subnetwork, which
account for their energy and information exchanges, in complete analogy with
stochastic systems. We then use our framework to investigate the working
mechanisms of a model of chemically-driven self-assembly and an experimental
light-driven bimolecular motor. We show that both systems are constituted by
two coupled subnetworks of chemical reactions. One subnetwork is maintained
out of equilibrium by external reservoirs (chemostats or light sources) and
powers the other via energy and information flows. In doing so, we clarify
that the information flow is precisely the thermodynamic counterpart of an
information ratchet mechanism only when no energy flow is involved.

##  I Introduction

Engines commonly operate such that some components (e.g., pistons) directly
interact with the power source to harvest energy, whereas some other
components (e.g., wheels) produce the functionality the engine is designed
for. Chemical engines Amano  _et al._ (2021a), such as molecular motors, are
no exception Astumian and Bier (1994); Magnasco (1994); Keller and Bustamante
(2000). Indeed, they can be rationally described Bustamante  _et al._ (2001);
Kolomeisky and Fisher (2007); Lipowsky and Liepelt (2008); Mugnai  _et al._
(2020); Brown and Sivak (2020) and designed Kay  _et al._ (2007); Astumian
(2007); Mattia and Otto (2015); Amano  _et al._ (2021a); Das  _et al._ (2021)
as chemical reaction networks (CRNs) where energy-harvesting chemical Schliwa
and Woehlke (2003); Boekhoven  _et al._ (2010); Cheng  _et al._ (2015); Wilson
_et al._ (2016); Erbas-Cakmak  _et al._ (2017); Robert-Paganin  _et al._
(2020); Amano  _et al._ (2021b), photochemical Koumura  _et al._ (1999);
Serreli  _et al._ (2007); Greb and Lehn (2014); Ragazzon  _et al._ (2015); Li
_et al._ (2015); Foy  _et al._ (2017); Canton  _et al._ (2021) or
electrochemical Vives  _et al._ (2009); Pezzato  _et al._ (2018); Canton  _et
al._ (2021) processes are coupled with large-amplitude intramolecular motions
or self-assembly reactions. These CRNs are often bipartite Hartich  _et al._
(2014); Horowitz and Esposito (2014), as the energy-harvesting processes act
on some molecular properties (e.g., phosphorylation state, photo-isomerization
state, oxidation/reduction state) while the processes realizing the engine’s
functionality act on different properties (e.g., position in space, assembly
state). When this is the case, the chemical species involved in the
functioning of a chemical engine can be described with a double state ($x,y$),
and the whole network can be split into two coupled subnetworks characterizing
the interconversion of $x$-states by one kind of processes (e.g., self
assembly steps converting free states into assembled ones) possibly driven
out-of-equilibrium by processes of another kind interconverting $y$-states
(e.g., reaction with a chemical fuel acting as a substrate).

Assessing the thermodynamics of bipartite systems requires quantifying energy
and information exchanges between the subnetworks by applying the tools of
information thermodynamics Sagawa and Ueda (2010); Hartich  _et al._ (2014);
Horowitz and Esposito (2014); Parrondo  _et al._ (2015). Linear stochastic
models of biochemical machines comprising unimolecular or pseudo-unimolecular
reactions have been extensively studied from this perspective, revealing the
fundamental role of information flows between different degrees of freedom in
powering such single molecule machines Horowitz  _et al._ (2013); Barato and
Seifert (2014); Loutchko  _et al._ (2017); Takaki  _et al._ ; McGrath  _et
al._ (2017); Lathouwers and Sivak (2022). However, chemical engines may in
general comprise nonlinear processes such as bimolecular reactions and are
sometimes (almost always in case of synthetic ones) better described by
deterministic dynamics expressed in terms of kinetic equations evolving
experimentally measurable concentrations rather than probabilities. At
present, a theoretical framework able to systematically address the
information thermodynamics of nonlinear deterministic CRNs is missing.

Very recently, we applied information thermodynamics to an autonomous
synthetic molecular motor working at the deterministic level Amano  _et al._
(2022). Such analysis shows that information thermodynamics is in principle
not limited to stochastic setups, but it strongly relies on the fact that the
CRN describing the motor is only composed of unimolecular reactions. Indeed,
when this is the case, kinetic equations can be mapped into a Markov jump
process on a linear network of states by normalizing the concentrations by the
total concentration, which is a conserved quantity in linear networks. Once
this correspondence is in place, standard tools from information
thermodynamics can be applied Horowitz and Esposito (2014). In particular, the
mutual information Cover and Thomas (2012), a central quantity in information
thermodynamics, can be defined at the level of normalized concentrations Amano
_et al._ (2022).

In this paper, we go further by extending the framework of information
thermodynamics to nonlinear deterministic CRNs, where multimolecular reactions
cause the total concentration to vary in time. We start by introducing the
setup of deterministic CRNs in Section II, where the crucial notion of
bipartite networks is formally defined and used to identify the two
subnetworks, which influence each other despite the fact that each reaction
pertains unambiguously to only one subnetwork. In Section III, thermodynamic
quantities are introduced for both chemically-driven and light-driven CRNs,
and the second law for nonequilibrium regimes is formulated. The main result
of this paper is obtained in Section IV, where the notion of mutual
information for non-normalized concentration distributions is defined and used
to formulate a second law for each subnetwork. Crucially, these second laws
show that the two subnetworks exchange free-energy trough information and
energy flows. Our result is analogous to the one obtained in the framework of
stochastic thermodynamics of bipartite systems Hartich  _et al._ (2014);
Horowitz and Esposito (2014), but importantly information flow terms account
here for the non-normalized concentration distributions. Furthermore, the
resulting expressions of the subnetworks’ entropy production also consider the
variation of the subnetworks entropy due to the standard molar entropy carried
by the chemical species and the contribution of non-bipartite species which
may be present. To exemplify the use of our new framework, in Section V we
apply it to two paradigmatic examples of out of equilibrium chemistry with
nonlinear dynamics: a model for chemically-driven self-assembly Ragazzon and
Prins (2018); Das  _et al._ (2021) and a light-driven bimolecular motor
Ragazzon  _et al._ (2015); Sabatino  _et al._ (2019); Canton  _et al._ (2021);
Asnicar  _et al._ (2022); Corrà  _et al._ (2022). We briefly recap the basis
of their functioning while focusing on the insights brought by the information
thermodynamic analysis. Using numerical simulations, we also comment on their
efficiency and we quantitatively investigate the correspondence between the
concept of Brownian information ratchet Kay  _et al._ (2007); Astumian (2019)
and (thermodynamic) information flows.

Our approach can be applied to any CRN with a bipartite structure and coupled
to any kind of reservoir. We restricted the presentation to the case of
autonomous and homogeneous ideal dilute solutions, but extensions to
nonautonomous Forastiere  _et al._ (2020), non-homogeneous Falasco  _et al._
(2018); Avanzini  _et al._ (2019) and nonideal Avanzini  _et al._ (2021) CRNs
are possible, as well as to cases where external species are continuously
injected Avanzini and Esposito (2022). Based on the relevance of information
thermodynamics as a tool to elucidate free-energy processing in the stochastic
realm, we envision that our new framework will bring significant insights in
the understanding of deterministic chemical processes, from synthetic
molecular machines to complex biochemical networks.

Finally, we note that many forms of information processing are possible in
CRNs Xiang  _et al._ (2018); Grozinger  _et al._ (2019): e.g., logic gates
Green  _et al._ (2017); Wang  _et al._ (2020), machine learning Qian  _et al._
(2011); Deutsch (2018); Poole  _et al._ , sensing Malaguti and ten Wolde
(2021), copying and proofreading Andrieux and Gaspard (2008); Sartori and
Pigolotti (2015); Ouldridge  _et al._ (2017); Poulton  _et al._ (2019),
memories Padirac  _et al._ (2012); Schnitter  _et al._ (2022). At this stage,
how the present framework may be suitable to study all of them is left for
future inquiries.

##  II Setup

CRNs are treated here as ideal dilute solutions composed of chemical species,
identified by the label $\alpha\in Z$, which undergo elementary G. (1993) or
coarse-grained Wachtel  _et al._ (2018); Avanzini  _et al._ (2020); Penocchio
_et al._ (2021) chemical reactions, identified by the index
$\rho\in\mathcal{R}$:

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\boldsymbol{\alpha}\cdot\boldsymbol{\nu}_{+\rho}\ch{->[][\rho]}\boldsymbol{\alpha}\cdot\boldsymbol{\nu}_{-\rho}\,,$
(1)

with $\boldsymbol{\alpha}=(\dots,\alpha,\dots)^{\intercal}$ denoting the
vector of chemical species, while $\boldsymbol{\nu}_{+\rho}$ and
$\boldsymbol{\nu}_{-\rho}$ denote the vectors of stoichiometric coefficients
of reagents and products of reaction $\rho$. Throughout this paper, we will
use single arrows like in Eq. (1) to denote reactions that are actually
reversible: for every reaction $\rho\in\mathcal{R}$, the forward (resp.
backward) reaction $+\rho$ (resp. $-\rho$) interconverts
$\boldsymbol{\alpha}\cdot\boldsymbol{\nu}_{+\rho}$ (resp.
$\boldsymbol{\alpha}\cdot\boldsymbol{\nu}_{-\rho}$) into
$\boldsymbol{\alpha}\cdot\boldsymbol{\nu}_{-\rho}$ (resp.
$\boldsymbol{\alpha}\cdot\boldsymbol{\nu}_{+\rho}$). This choice will simplify
the hypergraph representation of the two CRNs examined in Sec. V (see Figs. 2
and 3).

In deterministic CRNs, the abundance of the chemical species is specified by
the concentration vector $\boldsymbol{z}=(\dots,[\alpha],\dots)^{\intercal}$,
which follows the rate equation

$\mathrm{d}_{t}\boldsymbol{z}=\mathbb{S}\boldsymbol{j}\,,$ (2)

where we introduced the stoichiometric matrix $\mathbb{S}$ and the current
vector $\boldsymbol{j}$. Each $\rho$ column ${\mathbb{S}_{\rho}}$ of the
stoichiometric matrix $\mathbb{S}$ specifies the net variation of the number
of molecules for each species undergoing the $\rho$ reaction (1),
${\mathbb{S}_{\rho}}=\boldsymbol{\nu}_{-\rho}-\boldsymbol{\nu}_{+\rho}$. The
current vector $\boldsymbol{j}=(\dots,j^{\rho},\dots)^{\intercal}$ specifies
the net reaction current for every $\rho$ reaction (1) as the difference
between the forward and backward flux, i.e., $j^{\rho}=j^{+\rho}-j^{-\rho}$.
In closed CRNs, where the chemical reactions involve only the chemical species
$Z$, the fluxes $j^{\pm\rho}$ depend only on the concentrations
$\boldsymbol{z}$. For elementary reactions satisfying mass action kinetics de
Groot and Mazur (1984); Laidler (1987); Pekař (2005),
$j^{\pm\rho}=k_{\pm\rho}\boldsymbol{z}^{\boldsymbol{\nu}_{\pm\rho}}$, with
$k_{\pm\rho}$ the kinetic constants (hereafter, for every vectors
${\boldsymbol{v}}$ and ${\boldsymbol{w}}$,
${\boldsymbol{v}}^{\boldsymbol{w}}=\prod_{i}v_{i}^{w_{i}}$). In open CRNs,
some chemical reactions exchange for instance matter and/or photons with
external reservoirs. Thus, the corresponding fluxes $j^{\pm\rho}$ depend on
the concentrations $\boldsymbol{z}$ as well as on the coupling mechanisms with
the reservoirs Avanzini and Esposito (2022). Note that, throughout the
manuscript, we will omit for compactness of notation the functions’ variables,
e.g., $\boldsymbol{z}=\boldsymbol{z}(t)$ and
$j^{\pm\rho}=j^{\pm\rho}(\boldsymbol{z})$ (if reaction $\rho$ is not coupled
to any reservoir).

We now define the formal conditions  for the chemical species and reactions
under which CRNs have a bipartite structure. An exemplification of the concept
is provided in Figure 1. First, the chemical species must split as $Z=a\cup
b$. The species $b$ – the bipartite species – are univocally identified with a
double state $(x,y)=\alpha\in b$. The species $a$ – the ancillary species –
are all the other species, when present.  Second, the set of chemical
reactions must split as $\mathcal{R}=A\cup X\cup Y$. The reactions $X$ (resp.
$Y$) interconvert the species $b$ in such a way that only their $x$ (resp.
$y$) state changes:

$\displaystyle\dots(x,y)\dots$ $\displaystyle\ch{->[][$\rho\in
X$]}\dots(x^{\prime},y)\dots\,,$ (3a) $\displaystyle\dots(x,y)\dots$
$\displaystyle\ch{->[][$\rho\in Y$]}\dots(x,y^{\prime})\dots\,.$ (3b)

In the above chemical reactions, we neglected the stoichiometric coefficients
as well as the other species involved for illustrative reasons. The reactions
$A$ are those that do not interconvert the species $b$ ($S^{\alpha}_{\rho}=0$
$\forall\alpha\in b$ and $\forall\rho\in A$):

$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\dots(x,y)\dots\ch{->[][\rho\in
A]}\dots(x,y)\dots\,.$ (4)

Notice that, when these conditions for the chemical species and reactions are
satisfied, the species $a$ can undergo every reaction
$\rho\in\mathcal{R}=A\cup X\cup Y$ and no reactions can change both the state
$x$ and the state $y$ of a $b$ species. When these conditions are not
satisfied, CRNs do not have a bipartite structure and their splitting into
coupled subnetworks (discussed in the next paragraphs of this section) would
not be possible. Finally, if the chemical reactions are coarse-grained, we
further assume that the sets $A$, $X$ and $Y$ are independent sets of coarse-
grained reactions. The reason for this will become clear in Subs. III.2.

Figure 1:  An example of bipartite network. To illustrate our notion of
bipartite structure in a CRN, we consider a generic molecule (blue “Pac-Man”)
able to bind two substrates. One substrate (orange triangle, denoted $\ch{a}$)
is generated from a precursor (denoted $\ch{a2}$) via reaction $A$:
$\ch{a2}\rightarrow 2\ch{a}$ (notice the use of hypergraph notation where the
production of two molecules of $\ch{a}$ is represented by a bifurcating
arrow). The substrate $\ch{a}$ can then bind the molecule via reactions
$X_{1}$ or $X_{2}$, which both change its $x$ state from $x_{1}$ (free) to
$x_{2}$ (bound). The other substrate (red sphere) is exchanged between the
molecule and compounds $\mathrm{F}$ and $\mathrm{W}$ through reactions $Y_{1}$
and $Y_{2}$, which both change the $y$ state of the molecule from $y_{1}$
(free) to $y_{2}$ (bound). As a result, the molecule can be found in four
double states $(x,y)$: $(x_{1},y_{1})$, $(x_{2},y_{1})$, $(x_{2},y_{2})$,
$(x_{1},y_{2})$, which form the set of bipartite species (set $b$). All other
species in the network ($\ch{a2}$, $\ch{a}$, $\mathrm{F}$ and $\mathrm{W}$)
can be treated either as ancillary species (set $a$) or, if their
concentration is controlled by external reservoirs, as chemostatted species
(i.e., controlled parameters). Crucially, as none of the transitions between
the double states can change both $x$ and $y$ at the same time, the resulting
CRN has a bipartite structure. To fix ideas, the Pac-Man shaped molecule can
be thought of as an enzyme binding (state $x$) the monomeric form of a
substrate and phosphorylated (state $y$) by adenosine triphosphate
($\mathrm{F}$) to adenosine diphosphate ($\mathrm{W}$) hydrolysis. A concrete
example with a simpler yet similar bipartite structure is found in a recently
reported single-molecule molecular motor, where the two states are the axial
chirality (which can be in R state or S state) and the chemical state (which
can be in diacid state or anhydride state) of the motor Borsley  _et al._
(2022). The latter results in a linear bipartite network to which the
treatment in Ref. Amano  _et al._ (2022) straightforwardly applies. On the
contrary, systems like the present network or the applications analyzed in
Section V require a generalization of the previous framework due to the
presence of nonlinear reactions and ancillary species.

In bipartite CRNs, we can apply the same splittings of the chemical species
and reactions to the stoichiometric matrix

$\mathbb{S}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt\crcr\kern 0.86108pt\cr
5.0pt\hfil\@arstrut$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\mathbb{S}^{a}$\hfil\kern 5.0pt\cr
5.0pt\hfil\@arstrut$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\mathbb{S}^{b}$\hfil\kern 5.0pt\cr}}}}\right)$}}\,,$ (5)

with

$\displaystyle\mathbb{S}^{a}$ $\displaystyle=\hbox{}\vbox{\kern
0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt\crcr\kern 0.86108pt\cr
5.0pt\hfil\@arstrut$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\mathbb{S}^{a}_{A}$\hfil\kern
5.0pt&5.0pt\hfil$\mathbb{S}^{a}_{X}$\hfil\kern
5.0pt&5.0pt\hfil$\mathbb{S}^{a}_{Y}$\hfil\kern 5.0pt\cr}}}}\right)$}}\,,$ (6c)
$\displaystyle\mathbb{S}^{b}$ $\displaystyle=\hbox{}\vbox{\kern
0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt\crcr\kern 0.86108pt\cr
5.0pt\hfil\@arstrut$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\mathbb{0}$\hfil\kern
5.0pt&5.0pt\hfil$\mathbb{S}^{b}_{X}$\hfil\kern
5.0pt&5.0pt\hfil$\mathbb{S}^{b}_{Y}$\hfil\kern 5.0pt\cr}}}}\right)$}}\,,$ (6f)

to the current vector

$\boldsymbol{j}=\big{(}\boldsymbol{j}^{A},\boldsymbol{j}^{X},\boldsymbol{j}^{Y}\big{)}\,,$
(7)

and to the concentration vector

$\boldsymbol{z}=(\boldsymbol{a},\boldsymbol{b})\,.$ (8)

The rate equation (2) thus becomes

$\displaystyle\mathrm{d}_{t}\boldsymbol{a}=\mathbb{S}^{a}\boldsymbol{j}=\mathbb{S}^{a}_{A}\boldsymbol{j}^{A}+\mathbb{S}^{a}_{X}\boldsymbol{j}^{X}+\mathbb{S}^{a}_{Y}\boldsymbol{j}^{Y}\,,$
(9a)
$\displaystyle\mathrm{d}_{t}\boldsymbol{b}=\mathbb{S}^{b}\boldsymbol{j}=\mathbb{S}^{b}_{X}\boldsymbol{j}^{X}+\mathbb{S}^{b}_{Y}\boldsymbol{j}^{Y}\,.$
(9b)

To account for the total concentration of the bipartite species in the same
state $x$ or $y$, we define the following marginal concentrations

$\displaystyle[x]$ $\displaystyle=\sum_{y}[(x,y)]\,,$ (10a) $\displaystyle[y]$
$\displaystyle=\sum_{x}[(x,y)]\,,$ (10b)

which are collected in the following vectors

$\displaystyle\boldsymbol{x}$ $\displaystyle=(\dots,[x],\dots)^{\intercal}\,,$
(11a) $\displaystyle\boldsymbol{y}$
$\displaystyle=(\dots,[y],\dots)^{\intercal}\,.$ (11b)

Here, $[(x,y)]$ is the concentration of the specific bipartite species
$\alpha\in b$ characterized by the double state $(x,y)$, i.e.,
$[(x,y)]=[\alpha]_{\alpha=(x,y)}$.

The bipartite structure of a CRN allows us to decompose it into two coupled
subnetworks $\mathcal{X}$ and $\mathcal{Y}$. The subnetwork $\mathcal{X}$
represents the interconversion of the states $\\{x\\}$ of the bipartite
species driven by the $X$ reactions. Analogously, the subnetwork $\mathcal{Y}$
represents the interconversion of the states $\\{y\\}$ driven by the $Y$
reactions. Notice that this does not imply that the concentrations $[x]$
(resp. $[y]$) change only because of the reactions $X$ (resp. $Y$) as we will
see for the example in Subs. V.1.

##  III Thermodynamics

Building on the setup of Sec. II, we introduce the thermodynamic theory of
deterministic CRNs Rao and Esposito (2016); Avanzini  _et al._ (2021);
Penocchio  _et al._ (2021) and we specialize it to bipartite CRNs.

###  III.1 Chemical Potentials

The free energy contributions carried by the chemical species ${Z}$ in ideal
dilute solutions are given by the vector of chemical potentials Holyst and
Poniewierski (2012):

$\boldsymbol{\mu}_{Z}=\boldsymbol{\mu}^{\circ}_{Z}+RT\ln\boldsymbol{z}\,,$
(12)

where $\ln\boldsymbol{z}=(\dots,\ln[\alpha],\dots)^{\intercal}$ (hereafter,
for every vector $\boldsymbol{v}=(\dots,v_{i},\dots)^{\intercal}$,
$\log\boldsymbol{v}=(\dots,\log v_{i},\dots)^{\intercal}$), $T$ is the
temperature of the thermal bath, $R$ is the gas constant, and
$\boldsymbol{\mu}^{\circ}_{Z}$ is the vector of the standard chemical
potentials, which in turn are given by the sum of a standard enthalpic
$\boldsymbol{h}^{\circ}_{Z}$ and entropic $\boldsymbol{s}^{\circ}_{Z}$
contributions according to

$\boldsymbol{\mu}^{\circ}_{Z}=\boldsymbol{h}^{\circ}_{Z}-T\boldsymbol{s}^{\circ}_{Z}\,.$
(13)

From a thermodynamic standpoint, CRNs are said to be open when they exchange
free energy with some external reservoirs besides the thermal bath at
temperature $T$. For instance, chemicals can be exchanged with chemostats,
i.e., particle reservoirs, and/or photons can be exchanged with radiation
sources. Each reservoir is in turn characterized by a chemical potential.

For chemostats, their chemical potentials are given by the vector

$\boldsymbol{\mu}_{\mathrm{ch}}=\boldsymbol{\mu}^{\circ}_{\mathrm{ch}}+RT\ln\boldsymbol{c}\,,$
(14)

where $\boldsymbol{\mu}^{\circ}_{\mathrm{ch}}$ and $\boldsymbol{c}$ are the
vectors of standard chemical potentials and of the concentrations of the
exchanged chemicals, respectively.

For radiation sources, the chemical potential of photons $\gamma_{\nu}$ at
frequency $\nu$ and concentration distribution $n_{\nu}$ reads Penocchio  _et
al._ (2021)

$\mu_{\nu}=u_{\nu}-RT\ln\left(\frac{f_{\nu}+n_{\nu}}{n_{\nu}}\right)\,,$ (15)

where $u_{\nu}=N_{\mathrm{A}}\hbar\omega_{\nu}$ is the energy carried by a
mole of photons at frequency $\nu$ and
$f_{\nu}={2\omega_{\nu}^{2}}/N_{\mathrm{A}}{\pi c^{3}}$ is the density of
photon states at frequency $\nu$ (with $N_{\mathrm{A}}$ the Avogadro’s number,
$\hbar$ the reduced Planck constant, $c$ the speed of light and
$\omega_{\nu}=2\pi\nu$ the angular frequency). When the radiation source is
thermal, photons are distributed according to the black body distribution at a
certain temperature $T_{\mathrm{r}}$:

$n_{\nu}^{T_{\mathrm{r}}}=\frac{f_{\nu}}{\exp(\hbar\omega_{\nu}/k_{\mathrm{B}}T_{\mathrm{r}})-1}\,,$
(16)

with $k_{\mathrm{B}}$ the Boltzmann constant. As a consequence, their chemical
potential becomes Ries and McEvoy (1991); Gräber and Milazzo (1997)

$\mu_{\nu}=u_{\nu}-RT\ln\left(\frac{f_{\nu}+n_{\nu}^{T_{\mathrm{r}}}}{n_{\nu}^{T_{\mathrm{r}}}}\right)=u_{\nu}\left(1-\frac{T}{T_{\mathrm{r}}}\right)\,.$
(17)

Since a radiation source is in equilibrium with a CRN only when the
corresponding chemical potential vanishes Penocchio  _et al._ (2021), i.e.,
$\mu_{\nu}=0$, Eq. (17) implies that this happens for a thermal radiation
source only when its temperature is the same as the one of the thermal bath,
i.e., $T_{\mathrm{r}}=T$.

In this work, we focus for simplicity on autonomous CRNs, namely, the chemical
potentials of the reservoirs ($\boldsymbol{\mu}_{\mathrm{ch}}$ in Eq. (14) for
chemostats and $\mu_{\nu}$ in Eq. (15) for radiation sources), and thus also
the quantities $\boldsymbol{c}$ and $n_{\nu}$, are constant in time.

###  III.2 Thermodynamic Forces

Chemical reactions are driven by thermodynamic forces named affinities
Prigogine (1961); Dilip Kondepudi (2015),

$\mathcal{A}_{\rho}=-\frac{\boldsymbol{\mu}_{Z}\cdot\mathbb{S}_{\rho}}{T}+{F}_{\rho}\,,$
(18)

which have two contributions. The first contribution, i.e.,
$-\boldsymbol{\mu}_{Z}\cdot\mathbb{S}_{\rho}/T$, accounts for the variation of
the free energy due to the interconversion of the $Z$ species in solution via
reaction $\rho$. The second, i.e., ${F}_{\rho}$, accounts for the external
force due to the coupling with the reservoirs.

For elementary reactions, affinities (18) satisfy the local detailed balance
condition,

$\mathcal{A}_{\rho}=R\ln\frac{j^{+\rho}}{j^{-\rho}}\,,$ (19)

which implies that they always have the same sign of the corresponding
reaction currents: $\mathcal{A}_{\rho}j^{\rho}\geq 0$. Coarse-grained
reactions do not in general satisfy Eq. (19), and hence their affinities may
not have the same sign as the currents (unless they are tightly coupled
Wachtel  _et al._ (2018)). However, every independent subset of coarse-grained
reactions $\mathcal{C}\subseteq\mathcal{R}$, namely, every subset whose
underlying elementary reactions involve unique intermediate (coarse-grained)
species which are not shared with other subsets, satisfies
$\sum_{\rho\in\mathcal{C}}\mathcal{A}_{\rho}j^{\rho}\geq 0$ Wachtel  _et al._
(2018); Avanzini  _et al._ (2020).

#### Example 1.

Consider the following reaction (assumed as elementary for illustrative
reasons) where the catalyst $\mathrm{M}$ interconverts the substrate (fuel)
$\mathrm{F}$ into the product (waste) $\mathrm{W}$ which are exchanged with
chemostats: In this case, the affinity coincides with the external force,

$\mathcal{A}={F}=\frac{\mu_{\mathrm{F}}-\mu_{\mathrm{W}}}{T}\,,$ (20)

while the reaction fluxes satisfy mass action kinetics,

$\displaystyle j^{+}=k_{+}[\mathrm{F}][\ch{M}]\,,$ $\displaystyle
j^{-}=k_{-}[\mathrm{W}][\ch{M}]\,,$ (21)

where $[\mathrm{F}]$, $[\mathrm{W}]$ and $[\ch{M}]$ are the concentrations of
the fuel, waste and catalyst, respectively. Since reaction
(LABEL:scheme:F2Wcg) is elementary, it satisfies the local detailed balance
condition (19), which in turn implies that the kinetic constants
$\\{k_{+},k_{-}\\}$ and the standard chemical potentials
$\\{\mu_{\mathrm{F}}^{\circ},\mu_{\mathrm{W}}^{\circ}\\}$ are not independent:

$\frac{\mu_{\mathrm{F}}^{\circ}-\mu_{\mathrm{W}}^{\circ}}{T}=R\ln\frac{k_{+}}{k_{-}}\,,$
(22)

where we used Eq. (14) to express the chemical potential of the chemostats.

#### Example 2.

Consider the coarse-grained reactions

$\displaystyle\begin{split}&\schemestart
E\arrow{-U>[$\gamma_\nu$]}Z\schemestop\,,\chemmove{\node[below]at(Uarrow@arctangent){$1$};}\quad\schemestart
Z\arrow{-U>[$\gamma_{\nu'}$]}E\schemestop\,,\chemmove{\node[below]at(Uarrow@arctangent){$2$};}\\\
\quad\\\ &\schemestart
E\arrow{-U>[$\gamma_\nu$]}E\schemestop\,,\chemmove{\node[below]at(Uarrow@arctangent){$3$};}\quad\schemestart
Z\arrow{-U>[$\gamma_{\nu'}$]}Z\schemestop\,,\chemmove{\node[below]at(Uarrow@arctangent){$4$};}\\\
\quad\\\ &\qquad\qquad\schemestart
E\arrow{-U>[]}Z\schemestop\,,\chemmove{\node[below]at(Uarrow@arctangent){$5$};}\end{split}$
(23)

describing a typical photoisomerization process occurring via the so-called
diabatic mechanism: upon the absorption of a photon $\gamma_{\nu}$, a species
in the $\ch{E}$ conformation is either converted into the $\ch{Z}$
conformation (reaction $1$) or it decays back to the same conformation by
dissipating all the absorbed energy (reaction $3$); analogously, another
photon $\gamma_{\nu^{\prime}}$, with possibly a different frequency
$\nu^{\prime}$, can trigger the conversion of $\ch{Z}$ into $\ch{E}$ (reaction
$2$) or be dissipated (reaction $4$); finally, $\ch{E}$ and $\ch{Z}$ can also
thermally interconvert (reaction $5$). We assume for simplicity that only
photons at the exact frequencies $\nu$ and $\nu^{\prime}$ can trigger the
corresponding transitions. In such a case, the affinities read

$\displaystyle\small\mathcal{A}_{1}=\frac{\mu_{\mathrm{E}}-\mu_{\mathrm{Z}}}{T}+\frac{\mu_{\nu}}{T}\,,\quad\mathcal{A}_{2}=\frac{\mu_{\mathrm{Z}}-\mu_{\mathrm{E}}}{T}+\frac{\mu_{\nu^{\prime}}}{T}\,,$
$\displaystyle\mathcal{A}_{3}=\frac{\mu_{\nu}}{T}\,,\quad\mathcal{A}_{4}=\frac{\mu_{\nu^{\prime}}}{T}\,,\quad\mathcal{A}_{5}=\frac{\mu_{\mathrm{E}}-\mu_{\mathrm{Z}}}{T}\,,$
(24)

with $F_{1}=F_{3}={\mu_{\nu}}/{T}$, $F_{2}=F_{4}={\mu_{\nu^{\prime}}}/{T}$,
and $F_{5}=0$, while expressions for the reaction currents
$\\{j^{1},j^{2},j^{3},j^{4},j^{5}\\}$ can be found in Ref. Penocchio  _et al._
(2021). The reactions (23) are not tightly coupled and, consequently,
affinities and currents do not have, in general, the same sign. However, as
they constitute an independent subset of coarse-grained reactions, they
satisfy

$\mathcal{A}_{1}j^{1}+\mathcal{A}_{2}j^{2}+\mathcal{A}_{3}j^{3}+\mathcal{A}_{4}j^{4}+\mathcal{A}_{5}j^{5}\geq
0\,.$ (25)

###  III.3 Entropy

The thermodynamic entropy per unit volume of open CRNs,

$S=S_{Z}+S_{\text{ch}}+S_{\text{ph}}\,,$ (26)

is given in general by the contribution of the chemical species $S_{Z}$, the
chemostats $S_{\text{ch}}$, and the photons $S_{\text{ph}}$. The contribution
carried by the chemical species reads Rao and Esposito (2016)

$\begin{split}S_{Z}=(\boldsymbol{s}^{\circ}_{Z}-R\ln\boldsymbol{z})\cdot\boldsymbol{z}+R\lVert\boldsymbol{z}\rVert=\boldsymbol{s}^{\circ}_{Z}\cdot\boldsymbol{z}+S^{\text{H}}_{Z}\,,\end{split}$
(27)

where $\lVert\boldsymbol{z}\rVert=\sum_{\alpha}[\alpha]$ and we introduced a
Shannon-like entropy for macroscopic non-normalized concentration
distributions,
$S^{\text{H}}_{Z}:=-R\ln\boldsymbol{z}\cdot\boldsymbol{z}+R\lVert\boldsymbol{z}\rVert$.
The latter will play a crucial role in recovering the information
thermodynamic framework for deterministic CRNs in Sec. IV.

Similarly, the contribution of the chemostats is given by

$S_{\text{ch}}=(\boldsymbol{s}^{\circ}_{\mathrm{ch}}-R\ln\boldsymbol{c})\cdot\boldsymbol{c}+R\lVert\boldsymbol{c}\rVert\,,$
(28)

with
$\boldsymbol{s}^{\circ}_{\mathrm{ch}}=-\partial\boldsymbol{\mu}^{\circ}_{\mathrm{ch}}/\partial
T$ the vector of the standard molar entropies of the chemostats. On the other
hand, the contribution due to the photons is expressed as the entropy of an
ideal Bose gas Dilip Kondepudi (2015)

$\small
S_{\text{ph}}=R\int\Big{[}\left(f_{\nu}+n_{\nu}\right)\ln\left(f_{\nu}+n_{\nu}\right)-n_{\nu}\ln
n_{\nu}-f_{\nu}\ln f_{\nu}\Big{]}\mathrm{d}\nu\,,$ (29)

where the integral runs in general over the whole spectrum.

The time derivative of the total entropy (26), i.e.,
${\mathrm{d}_{t}S=\boldsymbol{s}_{Z}\cdot\mathbb{S}\boldsymbol{j}}$ (with
$\boldsymbol{s}_{Z}=\boldsymbol{s}^{\circ}_{Z}-R\ln\boldsymbol{z}$), is given
by the sum of the entropy production rate $\dot{\Sigma}\geq 0$, accounting for
the dissipation, and of the entropy flow $\dot{S}_{e}$, accounting for the
reversible exchange of entropy with the reservoirs. It provides the
nonequilibrium formulation of the second law, which can be written as

$\displaystyle\dot{\Sigma}=\mathrm{d}_{t}S-\dot{S}_{e}\geq 0$ (30)

where, in autonomous CRNs (i.e., with $\boldsymbol{c}$ and $n_{\nu}$ being
constant in time),

$\displaystyle\dot{\Sigma}$
$\displaystyle=\boldsymbol{\mathcal{A}}\cdot\boldsymbol{j}\geq 0\,,$ (31)
$\displaystyle\dot{S}_{e}$
$\displaystyle=\frac{\boldsymbol{h}^{\circ}_{Z}}{T}\cdot\mathbb{S}\boldsymbol{j}-\boldsymbol{{F}}\cdot\boldsymbol{j}\,,$
(32)

with $\boldsymbol{\mathcal{A}}=(\dots,\mathcal{A}_{\rho},\dots)^{\intercal}$
and $\boldsymbol{\boldsymbol{{F}}}=(\dots,{F}_{\rho},\dots)^{\intercal}$.

In bipartite CRNs, we can apply the same splittings of the chemical species
$Z=a\cup b$ to the total entropy carried by the chemical species $S_{Z}$
(given in Eq. (27)) which leads to

$S={S_{a}+S_{b}}+S_{\text{ch}}+S_{\text{ph}}\,,$ (33)

where, given a set of species $v=\\{a,b\\}$,
$S_{v}=(\boldsymbol{s}^{\circ}_{v}-R\ln\boldsymbol{v})\cdot\boldsymbol{v}+R\lVert\boldsymbol{v}\rVert$
with $\boldsymbol{v}=(\dots,[\alpha],\dots)_{\alpha\in v}$ and
$\boldsymbol{s}^{\circ}_{v}=(\dots,s^{\circ}_{\alpha},\dots)_{\alpha\in v}$.
Furthermore, we can apply the splitting of the chemical reactions
$\mathcal{R}=A\cup X\cup Y$ to the entropy production rate and the entropy
flow:

$\displaystyle\dot{\Sigma}$
$\displaystyle=\dot{\Sigma}^{A}+\dot{\Sigma}^{X}+\dot{\Sigma}^{Y}\,,$ (34)
$\displaystyle\dot{S}_{e}$
$\displaystyle=\dot{S}_{e}^{A}+\dot{S}_{e}^{X}+\dot{S}_{e}^{Y}\,,$ (35)

where

$\displaystyle\dot{\Sigma}^{i}$ $\displaystyle=\sum_{\rho\in
i}{\mathcal{A}}_{\rho}j^{\rho}\geq 0\,,$ (36) $\displaystyle\dot{S}_{e}^{i}$
$\displaystyle=\sum_{\rho\in
i}\left(\frac{\boldsymbol{h}^{\circ}_{Z}}{T}\cdot\mathbb{S}_{\rho}-{F}_{\rho}\right)j^{\rho}\,,$
(37)

with $i=\\{A,X,Y\\}$. Note that the inequality in Eq. (36) holds because the
reactions in the sets $\\{A,X,Y\\}$ either satisfy the local detailed balance
condition (19), or are independent sets of coarse-grained reactions (see Subs.
II and IV.2).

##  IV Information Thermodynamics

We now formulate a second law for each subnetwork (defined in Sec. II) which,
compared to the second law (30) of the whole CRN, is (mainly) modified by an
information flow term accounting for the information exchange between the two
subnetworks. This information flow arises due to the fact that the two
subnetworks share some chemical species which are involved in both $X$ and $Y$
reactions.

###  IV.1 Mutual Information

We start by introducing the probability that bipartite species $\alpha\in b$
are in a specific double state $(x,y)$ as

$p_{x,y}=\frac{[{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(}x,y{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})}]}{\lVert\boldsymbol{b}\rVert}\,,$
(38)

with $\lVert\boldsymbol{b}\rVert=\sum_{\alpha\in b}[\alpha]$ and the marginal
probabilities

$\displaystyle p_{x}$
$\displaystyle=\sum_{y}p_{x,y}=\frac{[x]}{\lVert\boldsymbol{b}\rVert}\,,$
(39a) $\displaystyle p_{y}$
$\displaystyle=\sum_{x}p_{x,y}=\frac{[y]}{\lVert\boldsymbol{b}\rVert}\,,$
(39b)

that bipartite species are in state $x$ (resp. $y$) irrespectively of their
state $y$ (resp. $x$). In analogy to information theory when dealing with the
joint probability of a pair of random variables Cover and Thomas (2012), this
allows us to introduce the following Shannon entropies for the bipartite
species

$\displaystyle H_{x,y}=-\sum_{x,y}p_{x,y}\ln p_{x,y}\,,$ (40a) $\displaystyle
H_{x}=-\sum_{x}p_{x}\ln p_{x}\,,$ (40b) $\displaystyle H_{y}=-\sum_{y}p_{y}\ln
p_{y}\,,$ (40c)

and the mutual information

$\begin{split}\mathcal{I}_{b}=\sum_{x,y}p_{x,y}\ln\frac{p_{x,y}}{p_{x}p_{y}}\,,\end{split}$
(41)

which by construction satisfies

$H_{x,y}=H_{x}+H_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}-\mathcal{I}_{b}\,.$
(42)

Crucially, by applying the definition of probabilities in Eqs. (38) and (39),
the mutual information (41) can be directly expressed in terms of the
macroscopic non-normalized concentration distributions

$\mathcal{I}_{b}=\sum_{x,y}\frac{[{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(}x,y{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})}]}{\lVert\boldsymbol{b}\rVert}\ln\frac{[{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(}x,y{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})}]\lVert\boldsymbol{b}\rVert}{[x][y]}\,,$
(43)

which still measures, as in information theory Parrondo  _et al._ (2015);
Cover and Thomas (2012), the reduction in uncertainty about the states
$\\{x\\}$ when the states $\\{y\\}$ are known, and vice versa. This physically
means that $\mathcal{I}_{b}$ quantifies, in the framework of this paper, to
which extent being in a specific state $x$ correlates with being in a specific
state $y$ for the bipartite species and it can be determined by measuring the
concentrations $[(x,y)]$.

#### Example.

Consider the network in Fig. 1 with arbitrary concentrations $[(x,y)]$. On the
one hand, if
$[(x_{1},y_{1})]/[(x_{2},y_{1})]=[(x_{1},y_{2})]/[(x_{2},y_{2})]$, the $y$
state is not correlated with the $x$ state, as the relative concentration
between species in states $x_{1}$ and $x_{2}$ is independent of state $y$.
Analogously, if
$[(x_{1},y_{1})]/[(x_{1},y_{2})]=[(x_{2},y_{1})]/[(x_{2},y_{2})]$, the $x$
state is not correlated with the $y$ state, as the relative concentration
between species in states $y_{1}$ and $y_{2}$ is independent of state $x$. In
such a case, one cannot gain information about one state (e.g., the
concentration of molecules binding the triangular substrate) by measuring the
other (e.g., the concentration of molecules binding the small circular
substrate). This reflects into a vanishing mutual information (43). On the
other hand, if
$[(x_{1},y_{1})]/[(x_{2},y_{1})]\neq[(x_{1},y_{2})]/[(x_{2},y_{2})]$, the $y$
state correlates with the $x$ state, as the relative concentration between
species in states $x_{1}$ and $x_{2}$ now depends state $y$. For analogous
reasons, if
$[(x_{1},y_{1})]/[(x_{1},y_{2})]\neq[(x_{2},y_{1})]/[(x_{2},y_{2})]$, the $x$
state correlates with the $y$ state. In such a case, one can thus gain
information on one state by measuring the other. For instance, in the presence
of positive correlations between $x_{1}$ and $y_{1}$ and between $x_{2}$ and
$y_{2}$ (namely, $[(x_{1},y_{1})]>[(x_{1},y_{2})]$ and
$[(x_{2},y_{2})]>[(x_{2},y_{1})]$), knowing that the marginal concentration
$[y]$ is shifted towards state $y_{2}$ (most of the molecules bind the small
circular substrate) informs about the marginal concentration $[x]$ being
shifted towards state $x_{2}$ (most of the molecules bind the triangular
substrate). This reflects into a non-null mutual information (43).

###  IV.2 Entropy Decomposition

In standard information thermodynamics, the thermodynamic entropy of a system
coincides with the Shannon entropy. Thus, Eq. (42) splits the  former into
contributions from subsystems and their mutual information.  On the other
hand, as shown in Subs. III.3, the entropy (27) of deterministic CRNs is not
given by the Shannon entropy.  Nevertheless, the bipartite structure of the
network allows us to split the Shannon-like entropy of the bipartite species
$S^{\text{H}}_{b}=-R(\ln\boldsymbol{b})\cdot\boldsymbol{b}+R\lVert\boldsymbol{b}\rVert$
by following the same logic employed for the standard derivation of Eq. (42)
Cover and Thomas (2012), but exploiting the expression in Eq. (43) for the
mutual information. In this way, we get

$S^{\text{H}}_{b}=S^{\text{H}}_{x}+S^{\text{H}}_{y}-R\lVert\boldsymbol{b}\rVert\mathcal{I}_{b}+R\lVert\boldsymbol{b}\rVert(\ln\lVert\boldsymbol{b}\rVert-1)\,,$
(44)

where

$\displaystyle S^{\text{H}}_{x}$
$\displaystyle=-R(\ln\boldsymbol{x})\cdot\boldsymbol{x}+R\lVert\boldsymbol{x}\rVert\,,$
(45a) $\displaystyle S^{\text{H}}_{y}$
$\displaystyle=-R(\ln\boldsymbol{y})\cdot\boldsymbol{y}+R\lVert\boldsymbol{y}\rVert\,,$
(45b)

are the Shannon-like entropy of states $\\{x\\}$ and $\\{y\\}$, respectively,
and
$\lVert\boldsymbol{x}\rVert=\lVert\boldsymbol{y}\rVert=\lVert\boldsymbol{b}\rVert$.
Note that the last term in Eq. (44), i.e.,
$\lVert\boldsymbol{b}\rVert(\ln\lVert\boldsymbol{b}\rVert-1)$, emerges because
of the non-normalized concentration distributions.

By using Eq. (44) to express the entropy of the bipartite species
$S_{b}=S^{\text{H}}_{b}+\boldsymbol{s}^{\circ}_{b}\cdot\boldsymbol{b}$, the
total entropy (26) becomes

$\begin{split}S=\text{
}&S^{\text{H}}_{x}+S^{\text{H}}_{y}-R\lVert\boldsymbol{b}\rVert\mathcal{I}_{b}+R\lVert\boldsymbol{b}\rVert(\ln\lVert\boldsymbol{b}\rVert-1)+\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\boldsymbol{b}\\\
&+S_{a}+S_{\text{ch}}+S_{\text{ph}}\,,\end{split}$ (46)

where $\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\boldsymbol{b}$ accounts
for the standard molar entropy of the bipartite species and cannot be split
into a contribution due to the species in state $x$ and a contribution due to
the species in state $y$. This term is absent in the decomposition of the
entropy of bipartite Markov jump processes Horowitz and Esposito (2014) as
long as the states do not have an internal entropy.

###  IV.3 Entropy Production Decomposition

We now decompose the entropy production rate in such a way as to formulate the
second law for each subnetwork $\mathcal{X}$ and $\mathcal{Y}$. We start by
rewriting the second law (30) specifying the entropy flow according to Eq.
(35):

$\dot{\Sigma}=\mathrm{d}_{t}S-\dot{S}_{e}^{A}-\dot{S}_{e}^{X}-\dot{S}_{e}^{Y}\,.$
(47)

From Eq. (46), we further derive

$\begin{split}\mathrm{d}_{t}S=&\mathrm{d}_{t}S^{\text{H}}_{x}+\mathrm{d}_{t}S^{\text{H}}_{y}+\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\mathrm{d}_{t}\boldsymbol{b}-R\mathrm{d}_{t}\big{(}\lVert\boldsymbol{b}\rVert\mathcal{I}_{b}\big{)}+\\\
&+R\ln\lVert\boldsymbol{b}\rVert\mathrm{d}_{t}\lVert\boldsymbol{b}\rVert+\boldsymbol{s}_{a}\cdot\mathrm{d}_{t}\boldsymbol{a}\,,\end{split}$
(48)

where $\boldsymbol{s}_{a}=\boldsymbol{s}_{a}^{\circ}-R\ln\boldsymbol{a}$ and
the time derivative of the information term (43) can be specified as

$\mathrm{d}_{t}\mathcal{I}_{b}=\dot{\mathcal{I}}_{x}+\dot{\mathcal{I}}_{y}+\big{(}\ln\lVert\boldsymbol{b}\rVert-\mathcal{I}_{b}\big{)}{\mathrm{d}_{t}\ln\lVert\boldsymbol{b}\rVert}$
(49)

with

$\displaystyle\dot{\mathcal{I}}_{x}:=\frac{1}{\lVert\boldsymbol{b}\rVert}\Big{\\{}(\ln\boldsymbol{b})\cdot\mathbb{S}^{b}_{X}\boldsymbol{j}^{X}-(\ln\boldsymbol{x})\cdot\mathrm{d}_{t}\boldsymbol{x}\Big{\\}}\,,$
(50a)
$\displaystyle\dot{\mathcal{I}}_{y}:=\frac{1}{\lVert\boldsymbol{b}\rVert}\Big{\\{}(\ln\boldsymbol{b})\cdot\mathbb{S}^{b}_{Y}\boldsymbol{j}^{Y}-(\ln\boldsymbol{y})\cdot\mathrm{d}_{t}\boldsymbol{y}\Big{\\}}\,.$
(50b)

Crucially, the information flows $\dot{\mathcal{I}}_{x}$ and
$\dot{\mathcal{I}}_{y}$ quantify the changes in the mutual information (43)
per units of concentration, namely, the changes in the relative uncertainty
(correlations) between the $\\{x\\}$ and the $\\{y\\}$ states, due to the
dynamics of subnetworks $\mathcal{X}$ and $\mathcal{Y}$, respectively. The
third term in Eq. (49) accounts for the non-normalized nature of the
concentration distributions, vanishes at steady state, and does not contribute
to the entropy production decomposition. Indeed, by using Eq. (48) and Eq.
(49) into Eq. (47) and by splitting $\mathrm{d}_{t}\boldsymbol{a}$ and
$\mathrm{d}_{t}\boldsymbol{b}$ into the contributions due to the different
reactions $\mathcal{R}=A\cup X\cup Y$ as in Eqs. (9a) and (9b), we obtain

$\begin{split}\dot{\Sigma}(t)=&{\mathrm{d}_{t}S^{\text{H}}_{x}-\dot{S}_{e}^{X}-R\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}_{x}+\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\mathbb{S}^{b}_{X}\boldsymbol{j}^{X}+\boldsymbol{s}_{a}\cdot\mathbb{S}^{a}_{X}\boldsymbol{j}^{X}}+\\\
&+{\mathrm{d}_{t}S^{\text{H}}_{y}-\dot{S}_{e}^{Y}-R\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}_{y}+\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\mathbb{S}^{b}_{Y}\boldsymbol{j}^{Y}+\boldsymbol{s}_{a}\cdot\mathbb{S}^{a}_{Y}\boldsymbol{j}^{Y}}+\\\
&-{\dot{S}_{e}^{A}+\boldsymbol{s}_{a}\cdot\mathbb{S}^{a}_{A}\boldsymbol{j}^{A}}\end{split}$
(51)

which, by using Eq. (36), allows us to identify

$\displaystyle\dot{\Sigma}^{X}$
$\displaystyle=\mathrm{d}_{t}S^{\text{H}}_{x}-\dot{S}_{e}^{X}-R\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}_{x}+\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\mathbb{S}^{b}_{X}\boldsymbol{j}^{X}+\boldsymbol{s}_{a}\cdot\mathbb{S}^{a}_{X}\boldsymbol{j}^{X}\geq
0\,,$ (52a) $\displaystyle\dot{\Sigma}^{Y}$
$\displaystyle=\mathrm{d}_{t}S^{\text{H}}_{y}-\dot{S}_{e}^{Y}-R\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}_{y}+\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\mathbb{S}^{b}_{Y}\boldsymbol{j}^{Y}+\boldsymbol{s}_{a}\cdot\mathbb{S}^{a}_{Y}\boldsymbol{j}^{Y}\geq
0\,,$ (52b) $\displaystyle\dot{\Sigma}^{A}$
$\displaystyle=-\dot{S}_{e}^{A}+\boldsymbol{s}_{a}\cdot\mathbb{S}^{a}_{A}\boldsymbol{j}^{A}\geq
0\,.$ (52c)

Equation (52c) has the same form as the second law (30): it explicitly
expresses the dissipation of the $A$ reactions (which do not involve the
bipartite species) in terms of the entropy flow with the corresponding
reservoirs $\dot{S}_{e}^{A}$ and the variation of the total entropy due to the
interconversion of the $a$ species
$\boldsymbol{s}_{a}\cdot\mathbb{S}^{a}_{A}\boldsymbol{j}^{A}$.

Equations (52a) and (52b) are the key results of our work. They provide a
formulation of the second law for the subnetworks $\mathcal{X}$ and
$\mathcal{Y}$, respectively. Let us focus in the following on the subnetwork
$\mathcal{X}$, but analogous comments also hold for the subnetwork
$\mathcal{Y}$. Equation (52a) shows that the dissipation $\dot{\Sigma}^{X}$ is
balanced by different mechanisms: i) the variation of the subnetwork Shannon-
like entropy, $\mathrm{d}_{t}S^{\text{H}}_{x}$; ii) the entropy flow with the
reservoirs coupled to the $X$ reactions, $\dot{S}_{e}^{X}$; iii) the
information flow accounting for the information exchange with the
$\mathcal{Y}$ subnetwork, $\dot{\mathcal{I}}_{x}$; iv) the variation to the
subnetwork entropy due to the standard molar entropy carried by the chemical
species,
$\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\mathbb{S}^{b}_{X}\boldsymbol{j}^{X}$;
v) the variation to the subnetwork entropy due to the consumption/production
of the $a$ species,
$\boldsymbol{s}_{a}\cdot\mathbb{S}^{a}_{X}\boldsymbol{j}^{X}$. Note that the
first three mechanisms in Eq. (52a) also appear in an analogous decomposition
of entropy production for bipartite Markov jump processes Horowitz and
Esposito (2014), while the last two only emerge in bipartite deterministic
CRNs.

Each mechanism can act as a source (of entropy), when it contributes
positively to the entropy production, or as an output (of entropy), when it
contributes negatively. The sources must always dominate the outputs to
maintain the subnetwork out of equilibrium because of the unavoidable
dissipation. For instance, when $\dot{\mathcal{I}}_{x}<0$, the subnetwork
$\mathcal{X}$ is decreasing correlation with the subnetwork $\mathcal{Y}$: the
mutual information (43) decreases (see Eq. (49)). This consumption of mutual
information corresponds to a source of entropy
($-R\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}_{x}>0$) that can be used to
sustain the output mechanisms and/or to balance the dissipation. On the other
hand, when $\dot{\mathcal{I}}_{x}>0$ the subnetwork is increasing correlation
with the subnetwork $\mathcal{Y}$. This corresponds to an output of entropy
($-R\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}_{x}<0$) which has to be
sustained by an entropy source to balance the dissipation
($\dot{\Sigma}^{X}\geq 0$).

##  V Applications

We now apply our framework to two paradigmatic examples of chemical engines
with nonlinear dynamics: a model for chemically-driven self-assembly Ragazzon
and Prins (2018); Das  _et al._ (2021) and a light-driven bimolecular motor
Ragazzon  _et al._ (2015); Sabatino  _et al._ (2019); Canton  _et al._ (2021);
Asnicar  _et al._ (2022). We characterize their functioning at steady state
for various operating regimes.

###  V.1 Driven self-assembly

Figure 2:  Chemically-driven self-assembly of monomers into dimers. a)
Hypergraph representation of the bipartite chemical reaction network. Arrows
are used to indicate the conventional direction of the reactions, which are
all reversible. Reactions are labelled according to the CRN (53); b-d)
Numerical simulations of the model with three different sets of parameters.
Information flow ($RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$), energy
flow ($\dot{\mathcal{E}}$), and dissipation of the self-assembly subnetwork
($T\dot{\Sigma}^{X}=\dot{\mathcal{E}}+RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$)
at steady state are plotted against the net thermodynamic force acting on the
system ($F=({\mu_{\mathrm{F}}-\mu_{\mathrm{W}}})/T$, with $T=298$K,
$\mu^{\circ}_{\mathrm{F}}=11$ kJ/mol and
$\mu^{\circ}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{W}}=-11$
kJ/mol, up to an arbitrary constant) in a range corresponding to
$[\mathrm{F}]$ from $1\cdot 10^{-4}$M to $4\cdot 10^{2}$M, with
$[\mathrm{W}]=1$M. The efficiency of the internal free-energy transduction
($\eta$) is computed according to Eq. (56). The standard chemical potentials
ruling the relative thermodynamic stability of the four species are, up to an
arbitrary constant: $\mu^{\circ}_{\mathrm{M}_{1}}=-2$ kJ/mol,
$\mu^{\circ}_{\mathrm{L}_{1}}=-3$ kJ/mol, $\mu^{\circ}_{\mathrm{L}_{2}}=-4$
kJ/mol, and $\mu^{\circ}_{\mathrm{M}_{2}}=9$ kJ/mol for case b);
$\mu^{\circ}_{\mathrm{M}_{1}}=2$ kJ/mol, $\mu^{\circ}_{\mathrm{L}_{1}}=-3$
kJ/mol, $\mu^{\circ}_{\mathrm{L}_{2}}=-6$ kJ/mol, and
$\mu^{\circ}_{\mathrm{M}_{2}}=4$ kJ/mol for case c); and
$\mu^{\circ}_{\mathrm{M}_{1}}=2$ kJ/mol, $\mu^{\circ}_{\mathrm{L}_{1}}=-3$
kJ/mol, $\mu^{\circ}_{\mathrm{L}_{2}}=-2$ kJ/mol, and
$\mu^{\circ}_{\mathrm{M}_{2}}=4$ kJ/mol for case d). The independent kinetic
parameters are the same as in Ref. Penocchio  _et al._ (2019). Colored spheres
indicate which is the most populated species at steady state for low,
intermediate and high forces regimes, while dotted vertical lines mark the
value of the force maximizing the concentration of the target species
$\mathrm{M}_{2}$. Note that, in panel c), the information flow
($RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$, blue solid line) coincides
exactly with the dissipation of the self-assembly subnetwork
($T\dot{\Sigma}^{X}$, green solid line).

As a first application, we analyze a minimalist model (see Fig. 2a)
epitomizing the basic working principles of chemically-driven self-assembly
Ragazzon and Prins (2018); Falasco  _et al._ (2019); Penocchio  _et al._
(2019); Das  _et al._ (2021): an external force is exploited to increase the
concentration of a target species with respect to the equilibrium one.
Prominent examples of this kind of mechanism are the formation of microtubules
out of tubulin dimers fueled by guanosine 5’-triphosphate (GTP) Desai and
Mitchison (1997); Hess and Ross (2017) and the ATP-driven self-assembly of
actin ﬁlaments Howard (2001). Driven self-assembly has also been exploited in
experiments such as the controlled gelation of dibenzoyl-L-cysteine to form
nanoﬁbers Boekhoven  _et al._ (2010) and the chemically fueled transient self-
assembly of ﬁbrous hydrogel materials Boekhoven  _et al._ (2015).

In the model depicted in Fig. 2a, the direct aggregation of two monomers
$\mathrm{M}_{1}$ to form the dimer $\mathrm{M}_{2}$ is coupled with the
exergonic conversion of a high energetic species F into a low energetic one W.
In particular, both the monomer $\mathrm{M}_{1}$ and the dimer
$\mathrm{M}_{2}$ can catalyze the F-to-W conversion via their activated
species $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ (e.g.,
$\mathrm{F}+\mathrm{M}_{1}\ch{->}\mathrm{L}_{1}\ch{->}\mathrm{M}_{1}+\mathrm{W}$).
This leads, by properly fixing the concentrations of F and W, to a
nonequilibrium steady state enriched in the dimer $\mathrm{M}_{2}$. Unlike
conventional equilibrium self-assembly, the efficacy of this synthetic
procedure is not determined by the relative thermodynamic stability of the
species, but rather by a kinetic asymmetry Astumian and Bier (1996) in how the
monomers $\mathrm{M}_{1}$ and the dimer $\mathrm{M}_{2}$ react with
$\mathrm{F}$ and $\mathrm{W}$ Ragazzon and Prins (2018); Das  _et al._ (2021).
This built-in kinetic asymmetry is often referred to as an information ratchet
mechanism  because the rate at which $\mathrm{F}$ and $\mathrm{W}$ react with
the system is somehow dependent on information about the reacting species:
that is, a species will react more quickly with fuel if doing so enables
forward cycling or prevents backward cycling Ragazzon and Prins (2018);
Astumian (2019).

The model depicted in Fig. 2a corresponds to the open CRN

$\begin{split}2\mathrm{\mathrm{M}_{1}}\ch{->[$\mathrm{
}$][$1$]}\mathrm{\mathrm{M}_{2}}\quad\quad\quad\quad
2\mathrm{\mathrm{L}_{1}}\ch{->[$\mathrm{ }$][$2$]}\mathrm{\mathrm{L}_{2}}&\\\
\quad\\\
\schemestart$\mathrm{M}_1$\arrow{-U>[$\mathrm{F}$][]}$\mathrm{L}_1$\schemestop\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$3{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}}$}};}\,\quad\schemestart$\mathrm{M}_1$\arrow{-U>[$\mathrm{W}$][]}$\mathrm{L}_1$\schemestop\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$3{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{w}}$}};}&\\\
\quad\\\
\schemestart$\mathrm{M}_2$\arrow{-U>[$2\mathrm{F}$][]}$\mathrm{L}_2$\schemestop\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}}$}};}\,\quad\schemestart$\mathrm{M}_2$\arrow{-U>[$2\mathrm{W}$][]}$\mathrm{L}_2$\schemestop\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{w}}$}};}&\end{split}$
(53)

where there are four bipartite species
$b=\\{\mathrm{M}_{1},\mathrm{M}_{2},\mathrm{L}_{1},\mathrm{L}_{2}\\}$ (with
monomeric/dimeric states $x\in\\{1,2\\}$ and non-activated/activated states
$y\in\\{\mathrm{M},\mathrm{L}\\}$) and two chemostats $\mathrm{F}$ and
$\mathrm{W}$. No $a$ species are present.  The marginal concentrations read

$\displaystyle[\mathrm{M}]$
$\displaystyle=[\mathrm{M}_{1}]+[\mathrm{M}_{2}]\,,$ (54a)
$\displaystyle[\mathrm{L}]$
$\displaystyle=[\mathrm{L}_{1}]+[\mathrm{L}_{2}]\,,$ (54b) $\displaystyle[1]$
$\displaystyle=[\mathrm{M}_{1}]+[\mathrm{L}_{1}]\,,$ (54c) $\displaystyle[2]$
$\displaystyle=[\mathrm{M}_{2}]+[\mathrm{L}_{2}]\,,$ (54d)

Here, all the reactions satisfy mass action kinetics. The self-assembly
reactions $\\{1,2\\}$ are nonlinear with respect to the chemical species
$\mathrm{M}_{1}$ and $\mathrm{L}_{1}$, and only change their $x$ state, i.e.,
$X=\\{1,2\\}$. They thus define the self-assembly subnetwork $\mathcal{X}$.
Conversely, the fueling and waste-forming reactions
$\\{3{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}},3{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{w}},4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}},4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{w}}\\}$,
accounting for the coupling with the chemostats, only change the $y$ state of
the species, i.e.,
$Y=\\{3{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}},3{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{w}},4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}},4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{w}}\\}$.
They thus define the fueling subnetwork $\mathcal{Y}$. We stress that here,
due to the nonlinearity of the $X$ reactions,
$\lVert\boldsymbol{b}\rVert=[\mathrm{M}]+[\mathrm{L}]=[1]+[2]=[\mathrm{M}_{1}]+[\mathrm{M}_{2}]+[\mathrm{L}_{1}]+[\mathrm{L}_{2}]$
is not a conserved quantity of the dynamics, which prevents direct application
of previous formulations of information thermodynamics for bipartite networks
Horowitz and Esposito (2014); Amano  _et al._ (2022). In addition, the CRN in
Eq. 53 provides an example where the marginal concentrations of $y$ states
($[\mathrm{M}]$ and $[\mathrm{L}]$) change due to $X$ reactions ($\\{1,2\\}$).

We now characterize the energetics of the nonequilibrium steady state which
will eventually be reached by the dynamics in the long time limit. We do so by
computing the various contributions to the total dissipation (51), thus
specializing Eqs. (52a) and (52b) for the case at study (notice that Eq. (52c)
as well as all the terms involving $a$ species do not play any role here). At
steady state, the time derivatives of Shannon-like entropies of states
$\\{x\\}$ and $\\{y\\}$ vanish
($\mathrm{d}_{t}S^{\text{H}}_{x}=\mathrm{d}_{t}S^{\text{H}}_{y}=0$) and the
dynamics is fully characterized by the net current
$j=-j^{1}=j^{2}=(j^{3{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}}}+j^{3{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{w}}})/2=-(j^{4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}}}+j^{4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{w}}})$
, which is positive when flowing anticlockwise across the hyper-graph in Fig.
2a according to the sign convention in Eq. (53). Furthermore, the rate at
which the fuel species $\mathrm{F}$ is injected into the system at steady
state to keep its concentration constant
($I_{\mathrm{F}}=j^{3{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}}}+2j^{4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathrm{f}}}$)
corresponds to the rate at which the waste species $\mathrm{W}$ is extracted
($I_{\mathrm{F}}=-I_{\mathrm{W}}=-j^{3\mathrm{w}}-2j^{4\mathrm{w}}$). As a
consequence, Eqs. (52a) and (52b) boil down to:

$\displaystyle T\dot{\Sigma}^{X}$
$\displaystyle=\underbrace{j(\mu^{\circ}_{\mathrm{\mathrm{M}_{2}}}-\mu^{\circ}_{\mathrm{\mathrm{L}_{2}}}+2\mu^{\circ}_{\mathrm{\mathrm{L}_{1}}}-2\mu^{\circ}_{\mathrm{\mathrm{M}_{1}}})}_{=:\dot{\mathcal{E}}}+\underbrace{RTj\ln\frac{[\mathrm{\mathrm{L}_{1}}]^{2}[\mathrm{\mathrm{M}_{2}}]}{[\mathrm{\mathrm{M}_{1}}]^{2}[\mathrm{\mathrm{L}_{2}}]}}_{=:RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}}\,,$
(55a) $\displaystyle T\dot{\Sigma}^{Y}$
$\displaystyle=I_{\mathrm{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})-\dot{\mathcal{E}}-RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}\,,$
(55b)

where $\dot{\mathcal{I}}=-\dot{\mathcal{I}}_{x}=\dot{\mathcal{I}}_{y}$ follows
directly from Eqs. (50a), (50b) and (54), and we introduced the energy flow
$\dot{\mathcal{E}}/T=-\dot{S}_{e}^{X}+\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\mathbb{S}^{b}_{X}\boldsymbol{j}^{X}$
accounting for the standard contribution to the variation of the CRN free
energy due to $X$ reactions, with
$I_{\mathrm{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})-\dot{\mathcal{E}}=-\dot{S}_{e}^{Y}+\boldsymbol{s}^{\circ}_{\boldsymbol{b}}\cdot\mathbb{S}^{b}_{Y}\boldsymbol{j}^{Y}$
following directly from Eq. (37) (after identifying
${F}_{\mathrm{3f}}=\mu_{\mathrm{F}}/T$,
${F}_{\mathrm{3w}}=\mu_{\mathrm{W}}/T$,
${F}_{\mathrm{4f}}=2\mu_{\mathrm{F}}/T$, and
${F}_{\mathrm{4w}}=2\mu_{\mathrm{W}}/T$).

Equations (55a) and (55b) fully characterize the free-energy exchanges between
the CRN and the chemostats, on the one hand, and between the subnetworks
$\mathcal{X}$ and $\mathcal{Y}$, on the other hand. The former is accounted
for by $I_{\mathrm{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})$, quantifying the
fueling work performed by the external force
$F=(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})/T$ Penocchio  _et al._ (2019). This
work is entirely delivered to the subnetwork $\mathcal{Y}$ since
$I_{\mathrm{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})$ appears only in Eq. (55b).
The free-energy exchange between $\mathcal{Y}$ and $\mathcal{X}$ is accounted
for by the energy and information flows, which together
($\dot{\mathcal{E}}+RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$) constitute
the only possible source of free energy for the subnetwork $\mathcal{X}$ (see
Eq. (55a)). This implies that the subnetwork $\mathcal{X}$ can be kept out of
equilibrium only when part of the work
$I_{\mathrm{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})$ is not dissipated by the
subnetwork $\mathcal{Y}$, but output to the subnetwork $\mathcal{X}$ via
$\dot{\mathcal{E}}$ and $\dot{\mathcal{I}}$. We view this process as an
internal free-energy transduction. Based on this understanding, one can define
the efficiency of the internal free-energy transduction as the ratio between
the free energy transferred to the subnetwork $\mathcal{X}$ and the work
performed on the subnetwork $\mathcal{Y}$

$\displaystyle
0\leq\eta=\frac{RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}+\dot{\mathcal{E}}}{{I_{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})}=1-\frac{T\dot{\Sigma}^{Y}}{{I_{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})}\leq
1\,,$ (56)

which is bounded by zero and one due to the non-negativity of
$\dot{\Sigma}^{Y}$. Physically, $\eta$ measures the fraction of the fueling
work devoted to keep the subnetwork $\mathcal{X}$ out of equilibrium, which is
exactly the goal of driven self-assembly.

Such a level of resolution on how free energy is dissipated leads to a
refinement of a previous analysis of this model Penocchio  _et al._ (2019).
First, Eq. (56) characterizes the steady state performance of the self-
assembly (what is called “maintenance phase” in Ref. Penocchio  _et al._
(2019)). Second, if the subnetwork $\mathcal{X}$ performed work
$\mathcal{W}_{ext}<0$ against the environment (as in the driven synthesis
setup of Ref. Penocchio  _et al._ (2019)), Eq. (56) would allow us to split
the overall efficiency
$\eta_{\mathrm{ds}}=-\dot{\mathcal{W}}_{ext}/\mathrm{I_{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})$
into two contributions:

$\displaystyle\eta_{\mathrm{ds}}=\frac{RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}+\dot{\mathcal{E}}}{{I_{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})}\cdot\frac{-\dot{\mathcal{W}}_{ext}}{\dot{\mathcal{E}}+RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}}=\eta\cdot\eta_{ext}\leq
1\,,$ (57)

where $\eta_{ext}$ measures the fraction of the free-energy transduced towards
the subnetwork $\mathcal{X}$
($\dot{\mathcal{E}}+RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$) which is
fruitfully converted into useful power delivered to the environment
($-\dot{\mathcal{W}}_{ext}$).

As a further comment, we note that $\eta$ can vanish because of two reasons: a
thermodynamic and a kinetic one. The former occurs when the external force is
null ($F=(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})/T=0$), which directly implies
that both subnetworks reach equilibrium
($\dot{\Sigma}^{Y}=\dot{\Sigma}^{X}=0$). The latter occurs when the external
force is not null ($F=(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})/T\neq 0$), but the
sum of $\dot{\mathcal{E}}$ and $RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$
vanishes, thus implying $\dot{\Sigma}^{X}=0$, but $\dot{\Sigma}^{Y}>0$. From a
kinetic standpoint, this second case is often referred as a scenario where the
system does not display kinetic asymmetry Ragazzon and Prins (2018); Astumian
(2019); Das  _et al._ (2021). From an information thermodynamic standpoint,
this happens when there is no internal free-energy transduction between the
two subnetworks and, consequently, the subnetwork $\mathcal{X}$ remains at
equilibrium even though $\mathcal{Y}$ is maintained in a nonequilibrium steady
state by the chemostats. This also implies that all the dissipation happens at
the level of the subnetwork $\mathcal{Y}$ ($\dot{\Sigma}=\dot{\Sigma}^{Y}$).

To better illustrate our results, we examine three different scenarios. The
first one (Fig. 2b) is the same as in Refs. Penocchio  _et al._ (2019) and
Falasco  _et al._ (2019), where the dimeric state is energetically highly
unfavored in the non-activated state
($\mu^{\circ}_{\mathrm{M}_{2}}-2\mu^{\circ}_{\mathrm{M}_{1}}\gg 0$), but
favored in the activated state
($\mu^{\circ}_{\mathrm{L}_{2}}-2\mu^{\circ}_{\mathrm{L}_{1}}<0$). Under these
conditions, the energy flow is positive for any value of the external force
and has a maximum at $F=63$ J K-1mol-1 (which also corresponds to the value of
the force maximizing the concentration of $\mathrm{M}_{2}$). This happens
because $\dot{\mathcal{E}}$ given in Eq. (55) is proportional to i) the steady
state current $j>0$ and ii)
$(\mu^{\circ}_{\mathrm{\mathrm{M}_{2}}}-\mu^{\circ}_{\mathrm{\mathrm{L}_{2}}}+2\mu^{\circ}_{\mathrm{\mathrm{L}_{1}}}-2\mu^{\circ}_{\mathrm{\mathrm{M}_{1}}})>0$.
Thus, the energy flow always contributes positively to the internal free-
energy transduction from the subnetwork $\mathcal{Y}$ to the subnetwork
$\mathcal{X}$. In contrast, the information flow $\dot{\mathcal{I}}$ is
negative for small values of the external force ($F\lesssim$ 35 J K-1mol-1),
thus representing a cost for the free-energy transduction. Physically, this
means that the subnetwork $\mathcal{X}$ is spending a fraction of the free
energy provided by the energy flow to create correlations with the subnetwork
$\mathcal{Y}$. For intermediate values of the external force (35 J K-1mol-1
$\lesssim F\lesssim$ 100 J K-1mol-1), the information flow has a finite
positive value meaning that correlations are generated by the subnetwork
$\mathcal{Y}$ and used as a source by the subnetwork $\mathcal{X}$ to stay out
of equilibrium.

In the second scenario (Fig. 2c), there is no energetic preference between
monomeric and dimeric states:
$\mu^{\circ}_{\mathrm{M}_{2}}-2\mu^{\circ}_{\mathrm{M}_{1}}=0$ and
$2\mu^{\circ}_{\mathrm{L}_{2}}-\mu^{\circ}_{\mathrm{L}_{1}}=0$. As a
consequence, the energy flow vanishes at any value of the force, and the only
source of free energy for the subnetwork $\mathcal{X}$ is the (always
positive) information flow. In such a case, the subnetwork $\mathcal{Y}$ acts
as a Maxwell Demon powered by the chemostats and transducing free-energy
towards the subnetwork $\mathcal{X}$ purely in form of information. The value
of the force $F$ maximizing the concentration of the target species
$\mathrm{M}_{2}$ turns out to be $67$ J K-1mol-1.

In the third scenario (Fig. 2d), there still is no energetic preference
between the non-activated monomer and non-activated dimer
($\mu^{\circ}_{\mathrm{M}_{2}}-2\mu^{\circ}_{\mathrm{M}_{1}}=0$), but the
activated dimer is unfavored with respect to the activated monomer
($\mu^{\circ}_{\mathrm{L}_{2}}-2\mu^{\circ}_{\mathrm{L}_{1}}>0$). As a
consequence, the energy flow is always negative, and the term
$RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$ must always be positive and
larger than $|\dot{\mathcal{E}}|$ in order to keep the subnetwork
$\mathcal{X}$ out of equilibrium. The value of the force $F$ maximizing the
concentration of the target species $\mathrm{M}_{2}$ turns out to be $68$ J
K-1mol-1.

In all the three scenarios, the internal free-energy transduction mechanism
gets stalled for large values of the external force ($F\gtrsim$ 100 J
K-1mol-1): $\dot{\mathcal{E}}$ and
$RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$ go to zero. Thus, the
subnetwork $\mathcal{X}$ approaches equilibrium ($\dot{\Sigma}^{X}\approx 0$),
despite the subnetwork $\mathcal{Y}$ being far from equilibrium. This can also
be understood in terms of the so-called negative differential response of the
steady state current Falasco  _et al._ (2019): $j$ decreases when increasing
$F$.

In all the three scenarios, the efficiency $\eta$ is finite for small values
of the external force, presents a maximum for intermediate values and then
vanishes at large values. As shown in Appendix B, $\eta$ is constant in the
linear regime for very small forces. The fact that the efficiency is maximized
for finite values of the force is therefore a truly far-from-equilibrium
feature, which in this specific case can be ascribed to the negative
differential response of the current $j$ with respect to the force.
Interestingly, in the third scenario (Fig. 2d), the value of the force
maximizing the efficiency gets close to the one maximizing the concentration
of the target species $\mathrm{M}_{2}$, indicating the possibility to fine-
tune the parameters in order to optimize both the kinetic and the
thermodynamic performance.

As a final comment, we notice that in the second and third scenarios the
information flow constitutes the only free-energy source keeping the
subnetwork $\mathcal{X}$ out of equilibrium. This is in line with the fact
that the functioning of this model is often described in terms of an
information ratchet mechanism, hinting that the generation of information is
the key working principle. However, the first scenario shows that there may be
regimes where the information flow is negative or null, meaning that the
information flow does not power the assembling reactions, but rather
represents a cost. We therefore conclude that the thermodynamic concept of
information flow is not synonymous with the kinetic concept of information
ratcheting, but they are related. In particular, the connection appears to be
vague only in those conditions where a positive energy flow (i.e., an energy
ratchet like contribution) is present.

###  V.2 Light driven directional bimolecular motor

Figure 3:  Light-driven bimolecular motor. a) Hypergraph representation of the
bipartite chemical reaction network and chemical structures of the involved
species. Arrows are used to indicate the conventional direction of the
reactions, which are all reversible (photochemical reactions are theoretically
reversible Penocchio  _et al._ (2021), but practically irreversible as
explained in A.1). Reactions are labelled according to equations (58), (59),
and (60); b,c) Numerical simulations of the motor operated at 365 nm (b) and
436 nm (c). Information flow
($RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$), energy flow
($\dot{\mathcal{E}}$), and dissipation of the self-assembly reactions
($T\dot{\Sigma}^{X}=\dot{\mathcal{E}}+RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}$)
at steady state are plotted against the net thermodynamic force acting on the
system ($F=\mu_{\nu}/T$, with $T=298$K) in a range corresponding to a photon
flow from $10^{-15}$ to 10-7 moles of photons per second. Experimentally
probed conditions are marked by the vertical dotted lines. The efficiency of
the internal free-energy transduction ($\eta$) is computed according to Eq.
(63). Experimental parameters in both the regimes are taken from the original
experimental data in Ref. Ragazzon  _et al._ (2015) and Refs. Sabatino  _et
al._ (2019); Asnicar  _et al._ (2022). At steady state, on-average directed
motion of the macrocycles $\ch{M}$ with respect to the axles is achieved, as a
clockwise current arises with respect to the hypergraph and reactions $1r$ and
$2l$ are kinetically unfavorable compared to reactions $1l$ and $2r$.

As a second application, we consider a class of synthetic bimolecular motors
powered by light Ragazzon  _et al._ (2015); Canton  _et al._ (2021); Corrà
_et al._ (2022). Their functioning consists in the autonomous directional
threading and dethreating of a crown-ether macrocycle through an asymmetric
molecular axle existing in two different conformations Sabatino  _et al._
(2019); Asnicar  _et al._ (2022).

This class of systems can be represented by the CRN in Fig. 3a. The threading
of the macrocycle (or ring) $\mathrm{M}$ through the free axle, either in the
conformation $\mathrm{E_{f}}$ or $\mathrm{Z_{f}}$, can be described by four
different self-assembly reactions forming the ring-axle complex, either in the
conformation $\mathrm{E_{c}}$ or $\mathrm{Z_{c}}$:

$\begin{split}&\schemestart$\mathrm{M}$+$\mathrm{E_f}$\arrow{->[][$1l$]}$\mathrm{E_c}$\schemestop\,,\quad\schemestart$\mathrm{M}$+$\mathrm{E_f}$\arrow{->[][$1r$]}$\mathrm{E_c}$\schemestop\\\
&\schemestart$\mathrm{M}$+$\mathrm{Z_f}$\arrow{->[][$2l$]}$\mathrm{Z_c}$\schemestop\,,\quad\schemestart$\mathrm{M}$+$\mathrm{Z_f}$\arrow{->[][$2r$]}$\mathrm{Z_c}$\schemestop\,,\end{split}$
(58)

where labels $l$ and $r$ discriminate over which side of the axle the
threading/de-threading of the ring occurs (see Fig. 3a). The reactions (58)
follow mass action kinetics. The switching of the axle between its
conformations in the free state as well as in the complex state can happen via
four (in practice irreversible) coarse-grained photochemical reactions (see
Appendix A.1 for the expression of their currents), all triggered by photons
$\gamma_{\nu}$ at frequency $\nu$:

$\begin{split}&\schemestart$\mathrm{E_f}$\arrow{-U>[$\gamma_\nu$][]}$\mathrm{Z_f}$\schemestop\,,\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$3\nu$}};}\,\quad\schemestart$\mathrm{Z_f}$\arrow{-U>[$\gamma_\nu$][]}$\mathrm{E_f}$\schemestop\,,\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$4\nu$}};}\\\
\quad\\\
&\schemestart$\mathrm{E_c}$\arrow{-U>[$\gamma_\nu$][]}$\mathrm{Z_c}$\schemestop\,,\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$5\nu$}};}\,\quad\schemestart$\mathrm{Z_c}$\arrow{-U>[$\gamma_\nu$][]}$\mathrm{E_c}$\schemestop\,.\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$6\nu$}};}\end{split}$
(59)

Changes of the axle’s conformation are also possible via thermal reactions
following mass-action kinetics:

$\displaystyle\schemestart$E_c$\arrow{->[][$8$]}$Z_c$\schemestop\,,\quad\schemestart$E_f$\arrow{->[][$7$]}$Z_f$\schemestop\,.$
(60)

Finally, four “futile” photochemical reactions are also possible, where
photons are absorbed without leading to a conformation change of the axle:

$\begin{split}&\schemestart$\mathrm{E_f}$\arrow{-U>[$\gamma_\nu$][]}$\mathrm{E_f}$\schemestop\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$9\nu$}};}\,\quad\schemestart$\mathrm{Z_f}$\arrow{-U>[$\gamma_\nu$][]}$\mathrm{Z_f}$\schemestop\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$10\nu$}};}\\\
&\schemestart$\mathrm{E_c}$\arrow{-U>[$\gamma_\nu$][]}$\mathrm{E_c}$\schemestop\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$11\nu$}};}\,\quad\schemestart$\mathrm{Z_c}$\arrow{-U>[$\gamma_\nu$][]}$\mathrm{Z_c}$\schemestop\chemmove{\node[below]at(Uarrow@arctangent){\footnotesize{$12\nu$}};}\,.\end{split}$
(61)

For this CRN, we define the set of bipartite species as
$b=\\{\mathrm{E_{f}},\mathrm{Z_{f}},\mathrm{E_{c}},\mathrm{Z_{c}}\\}$ and the
set of ancillary species as $a=\\{\mathrm{M}\\}$. Accordingly, reactions can
be split into three sets. The futile photochemical reactions
$A=\\{9\nu,10\nu,11\nu,12\nu\\}$ do not lead to any change in the
concentrations, but they contribute to the dissipation of the CRN, as shown in
the following. The reactions $X=\\{1l,1r,2l,2r\\}$ interconvert the assembled
state of the bipartite species $x\in\\{\mathrm{f},\mathrm{c}\\}$ between the
free and the complex state. They thus define the self-assembly subnetwork
$\mathcal{X}$. The reactions $Y=\\{3\nu,4\nu,5\nu,6\nu,7,8\\}$ interconvert
the isomerization state of the bipartite species
$y\in\\{\mathrm{E},\mathrm{Z}\\}$. They thus define the isomerization
subnetwork $\mathcal{Y}$.

As done for the previous example, we study the network’s energetics at steady
state by specializing Eqs. (52a), (52b) and (52c)

$\displaystyle T\dot{\Sigma}^{X}$
$\displaystyle=\underbrace{j(\mu^{\circ}_{\mathrm{Z_{c}}}-\mu^{\circ}_{\mathrm{E_{c}}}+\mu^{\circ}_{\mathrm{E_{f}}}-\mu^{\circ}_{\mathrm{Z_{f}}})}_{=:\dot{\mathcal{E}}}+\underbrace{RTj\ln\frac{[{\mathrm{E_{f}}}][{\mathrm{Z_{c}}}]}{[{\mathrm{Z_{f}}}][\mathrm{E_{c}}]}}_{=:RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}}\,,$
(62a) $\displaystyle T\dot{\Sigma}^{Y}$
$\displaystyle=\dot{W}^{Y}_{\nu}-\dot{\mathcal{E}}-RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}\,,$
(62b) $\displaystyle T\dot{\Sigma}^{A}$ $\displaystyle=\dot{W}^{A}_{\nu}\,,$
(62c)

with
$j=j_{1l}+j_{1r}=-(j_{2l}+j_{2r})=-j_{3\nu}-j_{7}+j_{4\nu}=j_{5\nu}+j_{7}-j_{4\nu}$
the steady state net current (considered as positive when flowing clockwise
across the hyper-graph in Fig. 3a),
$\dot{\mathcal{I}}=-\dot{\mathcal{I}}_{x}=\dot{\mathcal{I}}_{y}$, and
$\dot{\mathcal{E}}$ the energy flow. Furthermore, we introduced the two work
sources $\dot{W}^{A}_{\nu}$ and $\dot{W}^{Y}_{\nu}$ quantifying the rates at
which free-energy is provided by the radiation to futile and
photoisomerization reactions, respectively (see Eq. (74) and (73) for their
expressions). The former is fully dissipated by the futile reactions (see Eq.
(62c)) and, consequently, it is useless to the motor’s functioning. The latter
accounts for the fraction of the free-energy absorbed from the radiation which
can actually be used to sustain the motor’s functioning Corrà  _et al._
(2022). Similarly to the previous example, the $\mathcal{Y}$ subnetwork
(involving the photoisomerization reactions) is the only one coupled to the
external force $F=\mu_{\nu}/T$, but part of the free-energy exchanged with the
radiation can be transduced towards subnetwork $\mathcal{X}$ (involving self-
assembly reactions) through the energy $\dot{\mathcal{E}}$ and information
$\dot{\mathcal{I}}$ flows. These two mechanisms are the only possible free-
energy sources for subnetwork $\mathcal{X}$ which can be used at steady state
to sustain the self-assembly reactions, i.e., the motor’s functioning.

Based on the above understanding, the efficiency at which the free-energy
harvested by subnetwork $\mathcal{Y}$ from the radiation is transduced into
free-energy available to subnetwork $\mathcal{X}$ can be defined as:

$\displaystyle
0\leq\eta=\frac{RT\lVert\boldsymbol{b}\rVert\dot{\mathcal{I}}+\dot{\mathcal{E}}}{\dot{W}^{Y}_{\nu}}=1-\frac{T\dot{\Sigma}^{Y}}{\dot{W}^{Y}_{\nu}}\leq
1\,.$ (63)

which is analogous to Eq. (56) except that the source work $\dot{W}^{Y}_{\nu}$
is performed by the radiation instead of the chemostats. Similar comments on
the properties of the efficiency $\eta$ as those for the case of driven self-
assembly apply here as well.

To illustrate our results, we simulated the CRN in Fig. 3a under two
previously characterized experimental regimes which just differ by the
wavelength ($\lambda$, assumed as monochromatic) of the radiation Ragazzon
_et al._ (2015). Based on experimental measures, the threading of the ring
through the axles is always energetically favored independently of the
isomerization state, but it is more favored in the $\ch{E}$ state, namely
$\mu_{\mathrm{E_{c}}}^{\circ}-\mu_{\mathrm{E_{f}}}^{\circ}-\mu_{\mathrm{M}}^{\circ}{<}\mu_{\mathrm{Z_{c}}}^{\circ}-\mu_{\mathrm{Z_{f}}}^{\circ}-\mu_{\mathrm{M}}^{\circ}{<}0$.
Furthermore, because of steric hindrance, threading in the $\ch{E}$ state
happens preferentially via reaction $1l$ in Eq. (58), and de-threadering in
the $\ch{Z}$ state happens preferentially via reaction $2r$. Such a kinetic
preference favors the net directed motion of the ring with respect to the
axles when the CRN is brought out of equilibrium. This is often recognized as
an energy ratchet mechanism in the literature of light-driven molecular
motors, and provides the kinetic asymmetry allowing the the motor’s
functioning Kay  _et al._ (2007); Ragazzon  _et al._ (2015); Astumian (2016);
Canton  _et al._ (2021); Corrà  _et al._ (2021). From a thermodynamic
standpoint, this corresponds to a positive energy flow $\dot{\mathcal{E}}$ for
any value of the external force since i)
$(\mu^{\circ}_{\mathrm{Z_{c}}}-\mu^{\circ}_{\mathrm{E_{c}}}+\mu^{\circ}_{\mathrm{E_{f}}}-\mu^{\circ}_{\mathrm{Z_{f}}})>0$
and ii) $j>0$ (see Eq. (62)).

In the first regime ($\lambda=365$nm, Fig. 3b), the complexes $\mathrm{E_{c}}$
and $\mathrm{Z_{c}}$ are more effective than their free counterparts
$\mathrm{E_{f}}$ and $\mathrm{Z_{f}}$ in absorbing light, thus causing
photochemical reactions $5\nu$ and $6\nu$ in Eq. (59) to occur faster than the
photochemical reactions $3\nu$ and $4\nu$ in Eq. (59). This is often
recognized as an information ratchet mechanism providing an additional kinetic
bias to the energy ratchet mechanism Kay  _et al._ (2007); Ragazzon  _et al._
(2015); Astumian (2016); Corrà  _et al._ (2021); Canton  _et al._ (2021). In
the second regime ($\lambda=436$nm, Fig. 3c), the photochemical properties of
the axles are independent of the assembled state and, therefore, the motor’s
functioning is achieved by virtue of the energy ratchet mechanism alone
Ragazzon  _et al._ (2015). In both cases, we find that relatively high values
of the force are needed to take the self-assembly subnetwork out of
equilibrium ($T\dot{\Sigma}^{X}>0$) and, in contrast to the previous example,
the dissipation increases monotonically up to a plateau, as well as the energy
flow. On the other hand, the information flow is initially negative and
reaches a plateau only after displaying a minimum. The plateau value is
positive when the irradiation wavelength is 365 nm, while the information flow
is always negative when the irradiation wavelength is 436 nm. As in the
previous example, connections can be drawn between information/energy flow and
information/energy ratcheting, but clearly the two concepts do not coincide.
In particular, when the mechanism is a pure energy ratchet ($\lambda$ = 436
nm, Fig. 3c), the only positive contribution to $T\dot{\Sigma}^{X}$ comes from
the energy flow. When information ratcheting is added ($\lambda$ = 365 nm,
Fig. 3b) regimes with positive information flow arise. However, when both
energy and information ratchet mechanisms are active, regimes where the
information flow is null or negative are observed, and when there is no
information ratchet mechanism, the information flow is always not null and
negative.

Finally, in both cases the efficiency stays finite in the linear regime (see
Appendix B) and drops to zero at high forces as $T\dot{\Sigma}^{X}$ saturates,
with values of $3\times 10^{-2}\%$ for $\lambda$ = 365 nm and $3\times
10^{-4}\%$ for $\lambda$ = 436 nm at the experimentally probed conditions
Ragazzon  _et al._ (2015) (marked by the vertical dotted lines in Fig. 3b,c).
Interestingly, while at $\lambda$ = 436 nm the efficiency drops to zero
monotonically, it presents a maximum at intermediate forces for $\lambda$ =
365 nm.

##  VI Conclusion

In this work, we formulated information thermodynamics for deterministic
bipartite CRNs with nonlinear dynamics. The resulting picture is that these
networks can be divided into two subnetworks made of different sets of
reactions, which interact (due to the presence of shared species) by
exchanging free-energy in the form of information and/or energy flows. To get
there, we defined the mutual information in terms of the concentrations of
bipartite species (Eq. (43)), showing that it can be expressed in terms of
Shannon-like contributions to the total entropy (Eq. (44)). Furthermore, we
specialized the second law for each subnetwork (Eq. (52)), in analogy with
information thermodynamics of Markov jump processes. Crucially, informational
terms appear and may play the role of free-energy sources sustaining reactions
which would not occur spontaneously. At odds with information thermodynamics
of stochastic processes, information flow terms in deterministic CRNs must
account for non-normalized concentration distributions. Also, variation of the
subnetworks entropy due to the standard molar entropy carried by the chemical
species and the contribution of non-bipartite species which may be present
have to be taken into account.

Our framework generalizes a previous formulation developed for linear dynamics
Amano  _et al._ (2022). It directly applies to most of the artificial
molecular machines and motors reported until now, and to many relevant models
in systems chemistry and biochemistry too (e.g., biomolecular motors, signal
transduction). Here, we analyzed two epitomes of nonequilibrium supramolecular
chemistry where the nonlinear self-assembly steps explicitly require
leveraging our new results, as standard methods of information thermodynamics
fail to treat non-normalized concentration distributions. Unprecedented
insights were provided on i) how free-energy is used to power the chemically-
driven self-assembly of monomers and an experimental light-driven bimolecular
motor, and ii) on the corresponding performance evaluated according to their
thermodynamic efficiencies that we introduced (Eqs. (56) and (63)).
Furthermore, the role of information as a physical quantity is clarified and
compared with the concept of information ratcheting. Albeit related, we found
counterexamples to a general one-to-one correspondence between the two, thus
establishing that the two concepts leverage two different notions of
information, thus pertaining to distinct levels of descriptions (a
thermodynamic level and a kinetic one).

Our results can be straightforwardly extended to cases where the external
parameters $\boldsymbol{c}$ and $n_{\nu}$ change in time according to a time
dependent protocol Forastiere  _et al._ (2020), as well as when fluxes between
the system and the reservoirs are controlled instead of the concentrations
Avanzini and Esposito (2022). Also, if nonideal solutions were considered
Avanzini  _et al._ (2021), additional terms would appear in the expression of
the subnetworks entropy production without qualitatively altering the results.
Furthermore, our approach should apply to multipartite CRNs too, namely when
species and reactions can be split into more than two subnetworks exchanging
information (e.g., a bipartite network where the species can also diffuse
between two compartments defining an additional label for both species and
reactions) Horowitz (2015); Wolpert (2020). However, additional work is
required to meaningfully explore this direction, especially concerning the
notions of thermodynamic efficiency.

It is now clear that information processing is a key feature of living systems
which is prone to quantitative analyses Tkačik and Bialek (2016); Mattingly
_et al._ (2021). With our work, we developed some of the theoretical tools
needed to extend the approach of information thermodynamics towards the
deterministic domain. As concentration distributions are often easier to
characterize than probability distributions, we foresee that the current
understanding of the role of information in chemical systems can benefit to
some extent from the results that we derived.

###### Acknowledgements.

This research was funded by the Luxembourg National Research Fund, grant
ChemComplex (C21/MS/16356329), and by the FQXi Foundation, grant ‘Information
as a fuel in colloids and superconducting quantum circuits’ (FQXi-IAF19-05).

##  Appendix A Photochemistry

In this appendix we derive the main quantities needed to reproduce the results
in Section V.2, namely, the expressions of photochemical reaction currents and
the photochemical work.

###  A.1 Photochemical currents

In a typical photochemical experiment of the kind considered in Section V.2,
the solution is contained in a square cuvette and irradiated from one side
with a fixed photon flow $I_{0}$ quantifying the amount of moles of photons
impinging on the system’s surface $\Omega$ per unit time Montalti  _et al._
(2006). The radiation frequency is selected through interference filters with
a bandwidth $\Delta\nu$ of approximately 10 nm in unit of wavelength Montalti
_et al._ (2006), and all the frequency-dependent properties of the system are
considered as constants over this interval. This approximation is equivalent
to assuming the radiation to be monochromatic at the chosen frequency.
According to the Beer-Lambert law, the intensity of the photon flow decreases
along the direction of propagation $r$ (with $r=0$ being the point where the
radiation starts interacting with the solution and $r=r_{f}$ being the optical
path) as:

$I(r)=I_{0}\cdot 10^{-r\boldsymbol{\epsilon}\cdot\boldsymbol{b}}\,,$ (64)

where ${\boldsymbol{\epsilon}}$ is the vector collecting the molar extinction
coefficients of each species. Note that the concentrations $\boldsymbol{b}$
are considered uniform in the sample (i.e., we consider a well-mixed
solution). To express the current of the photoisomerization reaction
$\mathrm{E}\ce{->[\text{light}]}\mathrm{Z}$, we compute the number of photons
absorbed per unit of time and volume by the species $\mathrm{E}$, and we
multiply it by the quantum yield $\phi_{\mathrm{E\rightarrow Z}}$ of the
process, accounting for the probability that the photoisomerization happens
once $\mathrm{E}$ absorbed a photon Mauser and Gauglitz (1998):

$j^{\mathrm{E\rightarrow
Z}}=\frac{I_{0}}{V}(1-10^{-r_{f}\boldsymbol{\epsilon}\cdot\boldsymbol{b}})\frac{\epsilon_{\mathrm{E}}[\mathrm{E}]}{\boldsymbol{\epsilon}\cdot\boldsymbol{b}}\phi_{\mathrm{E\rightarrow
Z}}\,.$ (65)

The above expression simplifies under the the working conditions of interest
Ragazzon  _et al._ (2015), i.e., the total absorbance of the solution at
steady state is small ($r_{f}\boldsymbol{\epsilon}\cdot\boldsymbol{b}\ll 1$),
to

$j^{\mathrm{E\rightarrow
Z}}=\frac{\ln(10)r_{f}\epsilon_{\mathrm{E}}\phi_{\mathrm{E\rightarrow
Z}}}{V}I_{0}[\mathrm{E}]=k_{\mathrm{E\rightarrow Z}}I_{0}[\mathrm{E}]\,,$ (66)

which is the expression we used in our simulations in Sec. V.2. The currents
of other photoisomerization reactions are derived in the same way, as well as
the currents of futile photochemical reactions, like for
$\mathrm{E}\ce{->[\text{light}]}\mathrm{E}$

$j^{\mathrm{E\rightarrow
E}}=\frac{\ln(10)r_{f}\epsilon_{\mathrm{E}}\phi_{\mathrm{E\rightarrow
E}}}{V}I_{0}[\mathrm{E}]=k_{\mathrm{E\rightarrow E}}I_{0}[\mathrm{E}]\,,$ (67)

with $\phi_{\mathrm{E\rightarrow E}}=1-\phi_{\mathrm{E\rightarrow Z}}$.
Equations (66) and (67) show that photochemical reactions in low-absorbance
regimes under controlled constant monochromatic irradiation can be treated as
irreversible unimolecular mass action reactions Mauser and Gauglitz (1998);
Montalti  _et al._ (2006) by introducing effective kinetic constants, e.g.,
$k_{\mathrm{E\rightarrow Z}}$ and $k_{\mathrm{E\rightarrow E}}$. We note that
a more detailed analysis taking into account the underlying excited state
dynamics (photoisomerization reactions usually follow the so-called diabatic
mechanism Balzani  _et al._ (2014)) would show that this kind of reactions are
in principle reversible, but the backward pathways (e.g., $\mathrm{Z}$ gets
thermally excited and decays towards $\mathrm{E}$ by emitting a photon) are so
unlikely that they can be safely neglected for any practical purpose, thus
further justifying the above derivation Penocchio  _et al._ (2021).

###  A.2 Photochemical work

The concentration distribution of photons in the system can be expressed as
$n_{\nu}(r)=n_{\nu}^{T}+n_{0}10^{-r\boldsymbol{\epsilon}\cdot\boldsymbol{b}}$,
where the black body distribution $n_{\nu}^{T}$ at the temperature $T$ and
$n_{0}$ account for the contribution due to the thermal bath and the external
radiation, respectively. Within the experimental approximations, the latter
can still be expressed in terms of a black body distribution
($n_{0}=n_{\nu}^{T_{\mathrm{r}}}$) by assigning to the external radiation the
temperature $T_{\mathrm{r}}$ at which a black body would emit the same number
of photons in the interval selected by the interference filter. To find
$T_{\mathrm{r}}$, we therefore impose

$I_{0}=\Omega
c\int_{\nu-\Delta\nu/2}^{\nu+\Delta\nu/2}n_{\nu^{\prime}}^{T_{\mathrm{r}}}\mathrm{d}\nu^{\prime}\approx\Omega
cn_{\nu}^{T_{\mathrm{r}}}\Delta\nu\,,$ (68)

with $\Omega$ the irradiated area and $c$ the speed of light. The
approximation of treating the the black body distribution as almost constant
over the integration interval $\Delta\nu$ is not necessary, but we mention it
because it simplifies the identification of $T_{\mathrm{r}}$ without
qualitatively changing the results. For the plots shown in Fig. 3, Eq. (68)
has been solved numerically without employing the mentioned approximation. In
the low absorbance regime of interest in Ref. Ragazzon  _et al._ (2015), we
have that:

$\displaystyle n_{\nu}(r)=$ $\displaystyle
n_{\nu}^{T}+n_{\nu}^{T_{\mathrm{r}}}10^{-r\boldsymbol{\epsilon}\cdot\boldsymbol{b}}$
$\displaystyle\approx$ $\displaystyle
n_{\nu}^{T}+n_{\nu}^{T_{\mathrm{r}}}(1-\ln(10)r\boldsymbol{\epsilon}\cdot\boldsymbol{b})\,.$
(69)

As a consequence, the chemical potential of the radiation (15) will be a
function of $r$ too. By considering $\mu_{\nu}(r)$ to be constant in the
region of space from $r$ to $r+\mathrm{d}r$, the free energy absorbed per unit
of time from the radiation in the same region and in the low absorbance regime
can be expressed as

$\displaystyle\mu_{\nu}(r)(I_{0}(10^{-r\boldsymbol{\epsilon}\cdot\boldsymbol{b}}-10^{-(r+\mathrm{d}r)\boldsymbol{\epsilon}\cdot\boldsymbol{b}}))$
$\displaystyle\approx\mu_{\nu}(r)I_{0}\ln(10)\boldsymbol{\epsilon}\cdot\boldsymbol{b}\,\mathrm{d}r\,.$
(70)

The free energy harvested from the radiation by the whole system per unit time
and volume is thus given by

$\dot{W}_{\nu}=\frac{1}{V}\int_{0}^{r_{f}}\mu_{\nu}(r)I_{0}\ln(10)\boldsymbol{\epsilon}\cdot\boldsymbol{b}\,\mathrm{d}r\,,$
(71)

and it can be split into the contributions of $Y$ and $A$ reactions as

$\dot{W}_{\nu}=\dot{W}_{\nu}^{Y}+\dot{W}_{\nu}^{A}\,,$ (72)

with

$\dot{W}_{\nu}^{Y}=\sum_{\rho\in
Y_{\mathrm{ph}}}\frac{j^{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rho}}{r_{f}}\int_{0}^{r_{f}}\mu_{\nu}(r)\mathrm{d}r=\sum_{\rho\in
Y_{\mathrm{ph}}}j^{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rho}\langle\mu_{\nu}(r)\rangle\,,$
(73)

and

$\dot{W}_{\nu}^{A}=\sum_{\rho\in
A}\frac{j^{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rho}}{r_{f}}\int_{0}^{r_{f}}\mu_{\nu}(r)\mathrm{d}r=\sum_{\rho\in
A}j^{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rho}\langle\mu_{\nu}(r)\rangle\,,$
(74)

with $Y_{\mathrm{ph}}=\\{3\nu,4\nu,5\nu,6\nu\\}\subset Y$ the set of
photoisomerization reactions in Eq. (59). Equations (73) and (74) allow us to
identify the average chemical potential $\langle\mu_{\nu}(r)\rangle$ as the
photochemical force associated to the radiation, i.e.,
$F=\langle\mu_{\nu}(r)\rangle/T$, which is the expression used for the results
reported in Fig. 3. We stress that this is a consequence of the low absorbance
approximation, which makes the photochemical currents $j_{\rho}$ independent
of $r$. If this approximation did not hold, the currents $j_{\rho}$ would
enter the integrals in Eqs. (73) and (74) and the average chemical potential
would not play the role of the net thermodynamic force Corrà  _et al._ (2022).

##  Appendix B Linear regime

In this appendix, we compute the efficiencies (56) and (63) in the linear
regime, i.e., when the external force acting on the CRN is small enough ($F\ll
R$) that the concentrations at steady state are well approximated by linear
shifts from the equilibrium ones proportional to the external force according
to $[\alpha]_{\mathrm{ss}}=[\alpha]_{\mathrm{eq}}(1+m_{\alpha}F/R)$ with
$m_{\alpha}$ the coefficient of proportionality. In these conditions, the
chemical potentials (12) read
$\boldsymbol{\mu}_{Z}=\boldsymbol{\mu}_{Z}^{\mathrm{eq}}+\boldsymbol{m}TF$
with $\boldsymbol{m}=(\dots,m_{\alpha},\dots)^{\intercal}$, while the current
of a generic chemical reaction $\rho$ following mass action kinetics can be
expressed at the first order in $F$ as
$j^{\rho}=-j_{\mathrm{eq}}^{+\rho}(\mathbb{S}_{\rho}\cdot\boldsymbol{m})F/R$
(where we used $j_{\mathrm{eq}}^{+\rho}=j_{\mathrm{eq}}^{-\rho}$).

###  B.1 Driven self-assembly

Without loss of generality, we consider
$[\mathrm{F}]=[\mathrm{F}]_{\mathrm{eq}}(1+F/R)$ and
$[\mathrm{W}]=[\mathrm{W}]_{\mathrm{eq}}$, which is consistent with
$(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})/T=F$ in the first order in $F$. Under
these conditions, $\eta$ as computed with Eq. (56) is constant up to the
second order in $F$. Indeed, for $T\dot{\Sigma}^{X}$ we have

$\displaystyle T\dot{\Sigma}^{X}=$ $\displaystyle
j(\mu_{\mathrm{\mathrm{M}_{2}}}-\mu_{\mathrm{\mathrm{L}_{2}}}+2\mu_{\mathrm{\mathrm{L}_{1}}}-2\mu_{\mathrm{\mathrm{M}_{1}}})=$
$\displaystyle=$
$\displaystyle\frac{TF^{2}}{R}j^{+2}_{\mathrm{eq}}(2m_{\mathrm{L}_{1}}-m_{\mathrm{L}_{2}})(m_{\mathrm{\mathrm{M}_{2}}}-m_{\mathrm{\mathrm{L}_{2}}}+2m_{\mathrm{\mathrm{L}_{1}}}-2m_{\mathrm{\mathrm{M}_{1}}})\,,$
(75)

where the steady state current has been computed from reaction $2$. For
$I_{\ch{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})$, we have

$\displaystyle I_{\ch{F}}(\mu_{\mathrm{F}}-\mu_{\mathrm{W}})=$ $\displaystyle
TF(j^{3\mathrm{F}}+2j^{4\mathrm{F}})=$ $\displaystyle=$
$\displaystyle\frac{TF^{2}}{R}(j^{+3\mathrm{F}}_{\mathrm{eq}}(m_{\mathrm{M}_{1}}+1-m_{\mathrm{L}_{1}})+2j^{+4\mathrm{F}}_{\mathrm{eq}}(m_{\mathrm{M}_{2}}+2-m_{\mathrm{L}_{2}}))\,.$
(76)

Therefore:

$\displaystyle\eta=\frac{j^{+2}_{\mathrm{eq}}(2m_{\mathrm{L}_{1}}-m_{\mathrm{L}_{2}})(m_{\mathrm{\mathrm{M}_{2}}}-m_{\mathrm{\mathrm{L}_{2}}}+2m_{\mathrm{\mathrm{L}_{1}}}-2m_{\mathrm{\mathrm{M}_{1}}})}{j^{+3\mathrm{F}}_{\mathrm{eq}}(m_{\mathrm{M}_{1}}+1-m_{\mathrm{L}_{1}})+2j^{+4\mathrm{F}}_{\mathrm{eq}}(m_{\mathrm{M}_{2}}+2-m_{\mathrm{L}_{2}})}\,.$
(77)

After analytically computing the coefficients $\boldsymbol{m}$ by solving the
steady state of the linearized dynamics, Eq. (77) yields efficiencies of
$7.5\%$, $3.7\%$, and $3.2\%$ for the linear regimes of scenarios b, c and d,
respectively.

###  B.2 Light driven bimolecular motor

In the case of the light driven bimolecular motor, we take
$n_{\nu}(r)=n_{\nu}^{T}(1+F10^{-r\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}}/R)$.
By doing so, the chemical potential (15) at first order in $F$ reads
$\mu_{\nu}(r)=TF10^{-r\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}}$
provided that $f_{\nu}\gg n_{\nu}^{\mathrm{T}}$, as it is the case at both the
frequencies considered in Ref. Ragazzon  _et al._ (2015). For the equilibrium
concentrations $\boldsymbol{b}_{\mathrm{eq}}$, the low absorbance
approximation does not hold, and therefore the free energy absorbed per unit
of time from the radiation in the region of space between $r$ and
$r+\mathrm{d}r$ reads

$\displaystyle\mu_{\nu}(r)\left(I_{0}^{\mathrm{T}}\frac{F}{R}10^{-r\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}}-I_{0}^{\mathrm{T}}\frac{F}{R}10^{-(r+\mathrm{d}r)\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}}\right)$
$\displaystyle=$ $\displaystyle
I_{0}^{\mathrm{T}}\frac{TF^{2}}{R}10^{-2r\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}}\ln(10)\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}\,\mathrm{d}r\,,$
(78)

with $I_{0}^{T}=\Omega
c\int_{\nu-\Delta\nu/2}^{\nu+\Delta\nu/2}n_{\nu^{\prime}}^{T}\mathrm{d}\nu^{\prime}$.
The free energy harvested from the radiation by the whole system per unit time
and volume is obtained by integrating Eq. (78) over $r_{f}$,

$\displaystyle\dot{W}_{\nu}=$
$\displaystyle\frac{I_{0}^{\mathrm{T}}TF^{2}\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}\ln(10)}{VR}\int_{0}^{r_{f}}10^{-2r\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}}\,\mathrm{d}r$
$\displaystyle=$
$\displaystyle\frac{I_{0}^{\mathrm{T}}TF^{2}}{2VR}(1-10^{-2r_{f}\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}})\,,$
(79)

and it can be decomposed in the contributions of $Y$ and $A$ reactions as

$\dot{W}_{\nu}=\dot{W}_{\nu}^{Y}+\dot{W}_{\nu}^{A}\,,$ (80)

with

$\begin{split}\dot{W}_{\nu}^{Y}=&\dot{W}_{\nu}\Gamma^{Y}_{\mathrm{eq}}=\\\
=&\dot{W}_{\nu}\Big{(}\epsilon_{\mathrm{E_{f}}}[\mathrm{E_{f}}]_{\mathrm{eq}}\phi_{\mathrm{\mathrm{E_{f}}\rightarrow\mathrm{Z_{f}}}}+\epsilon_{\mathrm{Z_{f}}}[\mathrm{Z_{f}}]_{\mathrm{eq}}\phi_{\mathrm{\mathrm{Z_{f}}\rightarrow\mathrm{E_{f}}}}+\\\
&+\epsilon_{\mathrm{E_{c}}}[\mathrm{E_{c}}]_{\mathrm{eq}}\phi_{\mathrm{\mathrm{E_{c}}\rightarrow\mathrm{Z_{c}}}}+\epsilon_{\mathrm{Z_{c}}}[\mathrm{Z_{c}}]_{\mathrm{eq}}\phi_{\mathrm{\mathrm{Z_{c}}\rightarrow\mathrm{E_{c}}}}\Big{)}/\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}\end{split}$
(81)

and

$\begin{split}\dot{W}_{\nu}^{A}=&\dot{W}_{\nu}\Gamma^{A}_{\mathrm{eq}}=\\\
=&\dot{W}_{\nu}\Big{(}\epsilon_{\mathrm{E_{f}}}[\mathrm{E_{f}}]_{\mathrm{eq}}\phi_{\mathrm{\mathrm{E_{f}}\rightarrow\mathrm{E_{f}}}}+\epsilon_{\mathrm{Z_{f}}}[\mathrm{Z_{f}}]_{\mathrm{eq}}\phi_{\mathrm{\mathrm{Z_{f}}\rightarrow\mathrm{Z_{f}}}}+\\\
&+\epsilon_{\mathrm{E_{c}}}[\mathrm{E_{c}}]_{\mathrm{eq}}\phi_{\mathrm{\mathrm{E_{c}}\rightarrow\mathrm{E_{c}}}}+\epsilon_{\mathrm{Z_{c}}}[\mathrm{Z_{c}}]_{\mathrm{eq}}\phi_{\mathrm{\mathrm{Z_{c}}\rightarrow\mathrm{Z_{c}}}}\Big{)}/\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}\,,\end{split}$
(82)

where $\Gamma^{Y}_{\mathrm{eq}}$ and $\Gamma^{A}_{\mathrm{eq}}$ quantify the
fractions of the power which are provided by the radiation to
photoisomerizations and futile reactions, respectively (with
$\Gamma^{Y}_{\mathrm{eq}}$ \+ $\Gamma^{A}_{\mathrm{eq}}=1$). Finally, for the
sum of the energy and information flows we have:

$\displaystyle
T\dot{\Sigma}^{X}=j(\mu_{\mathrm{Z_{c}}}-\mu_{\mathrm{E_{c}}}+\mu_{\mathrm{E_{f}}}-\mu_{\mathrm{Z_{f}}})=$
$\displaystyle=$
$\displaystyle\frac{TF^{2}}{R}(j_{\mathrm{eq}}^{+2l}+j_{\mathrm{eq}}^{+2r})(m_{\mathrm{Z_{c}}}-m_{\mathrm{Z_{f}}}-m_{\mathrm{M}})(m_{\mathrm{Z_{c}}}-m_{\mathrm{E_{c}}}+m_{\mathrm{E_{f}}}-m_{\mathrm{Z_{f}}})\,.$
(83)

Therefore, Eq. (63) in the linear regime boils down to

$\displaystyle\footnotesize\eta=\frac{2V(j_{\mathrm{eq}}^{+2l}+j_{\mathrm{eq}}^{+2r})(m_{\mathrm{Z_{c}}}-m_{\mathrm{Z_{f}}}-m_{\mathrm{M}})(m_{\mathrm{Z_{c}}}-m_{\mathrm{E_{c}}}+m_{\mathrm{E_{f}}}-m_{\mathrm{Z_{f}}})}{I^{T}_{0}(1-10^{-2r_{f}\boldsymbol{\epsilon}\cdot\boldsymbol{b}_{\mathrm{eq}}})\Gamma^{Y}_{\mathrm{eq}}}\,,$
(84)

which verifies that the efficiency is constant up to the second order in $F$.
Here, solving the steady state of the linearized dynamics to compute the
coefficients $\boldsymbol{m}$ and then find the actual values of $\eta$ in the
two experimental regimes would require to know the rates of the backward
photochemical reactions, which cease to be negligible at very small forces.

## References

  * Amano  _et al._ (2021a) Shuntaro Amano, Stefan Borsley, David A. Leigh, and Zhanhu Sun, “Chemical engines: driving systems away from equilibrium through catalyst reaction cycles,” Nature Nanotechnology 16, 1057–1067 (2021a).
  * Astumian and Bier (1994) R. Dean Astumian and Martin Bier, “Fluctuation driven ratchets: Molecular motors,” Phys. Rev. Lett. 72, 1766–1769 (1994).
  * Magnasco (1994) Marcelo O. Magnasco, “Molecular combustion motors,” Phys. Rev. Lett. 72, 2656–2659 (1994).
  * Keller and Bustamante (2000) David Keller and Carlos Bustamante, “The mechanochemistry of molecular motors,” Biophys. J. 78, 541–556 (2000).
  * Bustamante  _et al._ (2001) Carlos Bustamante, David Keller, and George Oster, “The physics of molecular motors,” Acc. Chem. Res. 34, 412–420 (2001).
  * Kolomeisky and Fisher (2007) Anatoly B. Kolomeisky and Michael E. Fisher, “Molecular motors: A theorist’s perspective,” Annu. Rev. Phys. Chem. 58, 675–695 (2007).
  * Lipowsky and Liepelt (2008) Reinhard Lipowsky and Steffen Liepelt, “Chemomechanical coupling of molecular motors: Thermodynamics, network representations, and balance conditions,” J. Stat. Phys. 130, 39–67 (2008).
  * Mugnai  _et al._ (2020) Mauro L. Mugnai, Changbong Hyeon, Michael Hinczewski, and D. Thirumalai, “Theoretical perspectives on biological machines,” Rev. Mod. Phys. 92, 025001 (2020).
  * Brown and Sivak (2020) Aidan I. Brown and David A. Sivak, “Theory of nonequilibrium free energy transduction by molecular machines,” Chemical Reviews 120, 434–459 (2020).
  * Kay  _et al._ (2007) Euan R. Kay, David A. Leigh, and Francesco Zerbetto, “Synthetic molecular motors and mechanical machines,” Angew. Chem. Int. Ed. 46, 72–191 (2007).
  * Astumian (2007) R. Dean Astumian, “Design principles for brownian molecular machines: how to swim in molasses and walk in a hurricane,” Phys. Chem. Chem. Phys. 9, 5067–5083 (2007).
  * Mattia and Otto (2015) Elio Mattia and Sijbren Otto, “Supramolecular systems chemistry,” Nat. Nanotechnol. 10, 111–119 (2015).
  * Das  _et al._ (2021) Krishnendu Das, Luca Gabrielli, and Leonard J. Prins, “Chemically fueled self-assembly in biology and chemistry,” Angew. Chem. Int. Ed. 60, 20120–20143 (2021).
  * Schliwa and Woehlke (2003) Manfred Schliwa and Günther Woehlke, “Molecular motors,” Nature 422, 759–765 (2003).
  * Boekhoven  _et al._ (2010) Job Boekhoven, Aurelie M. Brizard, Krishna N. K. Kowlgi, Ger J. M. Koper, Rienk Eelkema, and Jan H. van Esch, “Dissipative self-assembly of a molecular gelator by using a chemical fuel,” Angew. Chem. Int. Ed. 49, 4825–4828 (2010).
  * Cheng  _et al._ (2015) Chuyang Cheng, Paul R. McGonigal, Severin T. Schneebeli, Hao Li, Nicolaas A. Vermeulen, Chenfeng Ke, and J. Fraser Stoddart, “An artificial molecular pump,” Nat. Nanotechnol. 10, 547–553 (2015).
  * Wilson  _et al._ (2016) Miriam R. Wilson, Jordi Solà, Armando Carlone, Stephen M. Goldup, Nathalie Lebrasseur, and David A. Leigh, “An autonomous chemically fuelled small-molecule motor,” Nature 534, 235–240 (2016).
  * Erbas-Cakmak  _et al._ (2017) Sundus Erbas-Cakmak, Stephen D. P. Fielden, Ulvi Karaca, David A. Leigh, Charlie T. McTernan, Daniel J. Tetlow, and Miriam R. Wilson, “Rotary and linear molecular motors driven by pulses of a chemical fuel,” Science 358, 340–343 (2017).
  * Robert-Paganin  _et al._ (2020) Julien Robert-Paganin, Olena Pylypenko, Carlos Kikuti, H. Lee Sweeney, and Anne Houdusse, “Force generation by myosin motors: A structural perspective,” Chem. Rev. 120, 5–35 (2020).
  * Amano  _et al._ (2021b) Shuntaro Amano, Stephen D. P. Fielden, and David A. Leigh, “A catalysis-driven artificial molecular pump,” Nature 594, 529–534 (2021b).
  * Koumura  _et al._ (1999) Nagatoshi Koumura, Robert W. J. Zijlstra, Richard A. van Delden, Nobuyuki Harada, and Ben L. Feringa, “Light-driven monodirectional molecular rotor,” Nature 401, 152–155 (1999).
  * Serreli  _et al._ (2007) Viviana Serreli, Chin-Fa Lee, Euan R. Kay, and David A. Leigh, “A molecular information ratchet,” Nature 445, 523–527 (2007).
  * Greb and Lehn (2014) Lutz Greb and Jean-Marie Lehn, “Light-driven molecular motors: Imines as four-step or two-step unidirectional rotors,” J. Am. Chem. Soc. 136, 13114–13117 (2014).
  * Ragazzon  _et al._ (2015) Giulio Ragazzon, Massimo Baroncini, Serena Silvi, Margherita Venturi, and Alberto Credi, “Light-powered autonomous and directional molecular motion of a dissipative self-assembling system,” Nat. Nanotechnol. 10, 70–75 (2015).
  * Li  _et al._ (2015) Quan Li, Gad Fuks, Emilie Moulin, Mounir Maaloum, Michel Rawiso, Igor Kulic, Justin T. Foy, and Nicolas Giuseppone, “Macroscopic contraction of a gel induced by the integrated motion of light-driven molecular motors,” Nat. Nanotechnol. 10, 161–165 (2015).
  * Foy  _et al._ (2017) Justin T. Foy, Quan Li, Antoine Goujon, Jean-Rémy Colard-Itté, Gad Fuks, Emilie Moulin, Olivier Schiffmann, Damien Dattler, Daniel P. Funeriu, and Nicolas Giuseppone, “Dual-light control of nanomachines that integrate motor and modulator subunits,” Nat. Nanotechnol. 12, 540–545 (2017).
  * Canton  _et al._ (2021) Martina Canton, Jessica Groppi, Lorenzo Casimiro, Stefano Corra, Massimo Baroncini, Serena Silvi, and Alberto Credi, “Second-generation light-fueled supramolecular pump,” J. Am. Chem. Soc. 143, 10890–10894 (2021).
  * Vives  _et al._ (2009) Guillaume Vives, Henri-Pierre Jacquot de Rouville, Alexandre Carella, Jean-Pierre Launay, and Gwénaël Rapenne, “Prototypes of molecular motors based on star-shaped organometallic ruthenium complexes,” Chem. Soc. Rev. 38, 1551–1561 (2009).
  * Pezzato  _et al._ (2018) Cristian Pezzato, Minh T. Nguyen, Dong Jun Kim, Ommid Anamimoghadam, Lorenzo Mosca, and J. Fraser Stoddart, “Controlling dual molecular pumps electrochemically,” Angew. Chem. Int. Ed. 57, 9325–9329 (2018).
  * Hartich  _et al._ (2014) D Hartich, A C Barato, and U Seifert, “Stochastic thermodynamics of bipartite systems: transfer entropy inequalities and a maxwell’s demon interpretation,” J. Stat. Mech.: Theory Exp. 2014, P02016 (2014).
  * Horowitz and Esposito (2014) Jordan M. Horowitz and Massimiliano Esposito, “Thermodynamics with continuous information flow,” Phys. Rev. X 4, 031015 (2014).
  * Sagawa and Ueda (2010) Takahiro Sagawa and Masahito Ueda, “Generalized jarzynski equality under nonequilibrium feedback control,” Phys. Rev. Lett. 104, 090602 (2010).
  * Parrondo  _et al._ (2015) Juan M. R. Parrondo, Jordan M. Horowitz, and Takahiro Sagawa, “Thermodynamics of information,” Nat. Phys. 11, 131 (2015).
  * Horowitz  _et al._ (2013) Jordan M. Horowitz, Takahiro Sagawa, and Juan M. R. Parrondo, “Imitating chemical motors with optimal information motors,” Phys. Rev. Lett. 111, 010602 (2013).
  * Barato and Seifert (2014) Andre C. Barato and Udo Seifert, “Stochastic thermodynamics with information reservoirs,” Phys. Rev. E 90, 042150 (2014).
  * Loutchko  _et al._ (2017) Dimitri Loutchko, Maximilian Eisbach, and Alexander S. Mikhailov, “Stochastic thermodynamics of a chemical nanomachine: The channeling enzyme tryptophan synthase,” J. Chem. Phys. 146, 025101 (2017).
  * (37) Ryota Takaki, Mauro L. Mugnai, and D. Thirumalai, “Information flow, gating, and energetics in dimeric molecular motors,” BioRxiv:2021.12.30.474541 (2021).
  * McGrath  _et al._ (2017) Thomas McGrath, Nick S. Jones, Pieter Rein ten Wolde, and Thomas E. Ouldridge, “Biochemical machines for the interconversion of mutual information and work,” Phys. Rev. Lett. 118, 028101 (2017).
  * Lathouwers and Sivak (2022) Emma Lathouwers and David A. Sivak, “Internal energy and information flows mediate input and output power in bipartite molecular machines,” Phys. Rev. E 105, 024136 (2022).
  * Amano  _et al._ (2022) Shuntaro Amano, Massimiliano Esposito, Elisabeth Kreidt, David A. Leigh, Emanuele Penocchio, and Benjamin M. W. Roberts, “Insights from an information thermodynamics analysis of a synthetic molecular motor,” Nat. Chem. (2022), 10.1038/s41557-022-00899-z.
  * Cover and Thomas (2012) Thomas M. Cover and Joy A. Thomas,  _Elements of Information Theory_ (John Wiley & Sons, Inc., Hoboken, 2012).
  * Ragazzon and Prins (2018) G. Ragazzon and L. J. Prins, “Energy consumption in chemical fuel-driven self-assembly,” Nat. Nanotechnol. 13, 882–889 (2018).
  * Sabatino  _et al._ (2019) Andrea Sabatino, Emanuele Penocchio, Giulio Ragazzon, Alberto Credi, and Diego Frezzato, “Individual-molecule perspective analysis of chemical reaction networks: The case of a light-driven supramolecular pump,” Angew. Chem. Int. Ed. 58, 14341–14348 (2019).
  * Asnicar  _et al._ (2022) Daniele Asnicar, Emanuele Penocchio, and Diego Frezzato, “Sample size dependence of tagged molecule dynamics in steady-state networks with bimolecular reactions: Cycle times of a light-driven pump,” The Journal of Chemical Physics 156, 184116 (2022).
  * Corrà  _et al._ (2022) Stefano Corrà, Marina Tranfić Bakić, Jessica Groppi, Massimo Baroncini, Serena Silvi, Emanuele Penocchio, Massimiliano Esposito, and Alberto Credi, “Kinetic and energetic insights into the dissipative non-equilibrium operation of an autonomous light-powered supramolecular pump,” Nat. Nanotechnol. 17, 746–751 (2022).
  * Astumian (2019) R. Dean Astumian, “Kinetic asymmetry allows macromolecular catalysts to drive an information ratchet,” Nat. Commun. 10, 3837 (2019).
  * Forastiere  _et al._ (2020) Danilo Forastiere, Gianmaria Falasco, and Massimiliano Esposito, “Strong current response to slow modulation: A metabolic case-study,” J. Chem. Phys. 152, 134101 (2020).
  * Falasco  _et al._ (2018) Gianmaria Falasco, Riccardo Rao, and Massimiliano Esposito, “Information thermodynamics of turing patterns,” Phys. Rev. Lett. 121, 108301 (2018).
  * Avanzini  _et al._ (2019) Francesco Avanzini, Gianmaria Falasco, and Massimiliano Esposito, “Thermodynamics of chemical waves,” J. Chem. Phys. 151, 234103 (2019).
  * Avanzini  _et al._ (2021) Francesco Avanzini, Emanuele Penocchio, Gianmaria Falasco, and Massimiliano Esposito, “Nonequilibrium thermodynamics of non-ideal chemical reaction networks,” J. Chem. Phys. 154, 094114 (2021).
  * Avanzini and Esposito (2022) Francesco Avanzini and Massimiliano Esposito, “Thermodynamics of concentration vs flux control in chemical reaction networks,” J. Chem. Phys. 156, 014116 (2022).
  * Xiang  _et al._ (2018) Yiyu Xiang, Neil Dalchau, and Baojun Wang, “Scaling up genetic circuit design for cellular computing: advances and prospects,” Nat. Comput. 17, 833–853 (2018).
  * Grozinger  _et al._ (2019) Lewis Grozinger, Martyn Amos, Thomas E. Gorochowski, Pablo Carbonell, Diego A. Oyarzún, Ruud Stoof, Harold Fellermann, Paolo Zuliani, Huseyin Tas, and Angel Goñi-Moreno, “Pathways to cellular supremacy in biocomputing,” Nat. Commun. 10, 5250 (2019).
  * Green  _et al._ (2017) Alexander A. Green, Jongmin Kim, Duo Ma, Pamela A. Silver, James J. Collins, and Peng Yin, “Complex cellular logic computation using ribocomputing devices,” Nature 548, 117–121 (2017).
  * Wang  _et al._ (2020) Fei Wang, Hui Lv, Qian Li, Jiang Li, Xueli Zhang, Jiye Shi, Lihua Wang, and Chunhai Fan, “Implementing digital computing with dna-based switching circuits,” Nat. Commun. 11, 1–8 (2020).
  * Qian  _et al._ (2011) Lulu Qian, Erik Winfree, and Jehoshua Bruck, “Neural network computation with dna strand displacement cascades,” Nature 475, 368–372 (2011).
  * Deutsch (2018) J.M. Deutsch, “Computational mechanisms in genetic regulation by rna,” J. Theor. Biol. 458, 156–168 (2018).
  * (58) William Poole, Thomas Ouldridge, Manoj Gopalkrishnan, and Erik Winfree, “Detailed balanced chemical reaction networks as generalized boltzmann machines,”  Preprint at https://arxiv.org/abs/2205.06313 (2022).
  * Malaguti and ten Wolde (2021) Giulia Malaguti and Pieter Rein ten Wolde, “Theory for the optimal detection of time-varying signals in cellular sensing systems,” eLife 10, e62574 (2021).
  * Andrieux and Gaspard (2008) David Andrieux and Pierre Gaspard, “Nonequilibrium generation of information in copolymerization processes,” Proc. Natl. Acad. Sci. 105, 9516–9521 (2008).
  * Sartori and Pigolotti (2015) Pablo Sartori and Simone Pigolotti, “Thermodynamics of error correction,” Phys. Rev. X 5, 041039 (2015).
  * Ouldridge  _et al._ (2017) Thomas E. Ouldridge, Christopher C. Govern, and Pieter Rein ten Wolde, “Thermodynamics of computational copying in biochemical systems,” Phys. Rev. X 7, 021004 (2017).
  * Poulton  _et al._ (2019) Jenny M. Poulton, Pieter Rein ten Wolde, and Thomas E. Ouldridge, “Nonequilibrium correlations in minimal dynamical models of polymer copying,” Proc. Natl. Acad. Sci. 116, 1946–1951 (2019).
  * Padirac  _et al._ (2012) Adrien Padirac, Teruo Fujii, and Yannick Rondelez, “Bottom-up construction of in vitro switchable memories,” Proc. Natl. Acad. Sci. 109, E3212–E3220 (2012).
  * Schnitter  _et al._ (2022) Fabian Schnitter, Benedikt Rieß, Christian Jandl, and Job Boekhoven, “Memory, switches, and an or-port through bistability in chemically fueled crystals,” Nat. Commun. 13, 2816 (2022).
  * G. (1993) Svehla G., “Nomenclature of kinetic methods of analysis (iupac recommendations 1993),”  _Pure and Applied Chemistry_ ,  65, 2291 (1993).
  * Wachtel  _et al._ (2018) Artur Wachtel, Riccardo Rao, and Massimiliano Esposito, “Thermodynamically consistent coarse graining of biocatalysts beyond michaelis–menten,” New J. Phys. 20, 042002 (2018).
  * Avanzini  _et al._ (2020) Francesco Avanzini, Gianmaria Falasco, and Massimiliano Esposito, “Thermodynamics of non-elementary chemical reaction networks,” New J. Phys. 22, 093040 (2020).
  * Penocchio  _et al._ (2021) Emanuele Penocchio, Riccardo Rao, and Massimiliano Esposito, “Nonequilibrium thermodynamics of light-induced reactions,” J. Chem. Phys. 155, 114101 (2021).
  * de Groot and Mazur (1984) S. R. de Groot and P. Mazur,  _Non-Equilibrium Thermodynamics_ (Dover, 1984).
  * Laidler (1987) K. J. Laidler,  _Chemical Kinetics_ (Harper Collins Publishers, 1987).
  * Pekař (2005) Miloslav Pekař, “Thermodynamics and foundations of mass-action kinetics,” Prog. React. Kinet. Mech. 30, 3–113 (2005).
  * Borsley  _et al._ (2022) Stefan Borsley, Elisabeth Kreidt, David A. Leigh, and Benjamin M. W. Roberts, “Autonomous fuelled directional rotation about a covalent single bond,” Nature 604, 80–85 (2022).
  * Rao and Esposito (2016) Riccardo Rao and Massimiliano Esposito, “Nonequilibrium thermodynamics of chemical reaction networks: Wisdom from stochastic thermodynamics,” Phys. Rev. X 6, 041064 (2016).
  * Holyst and Poniewierski (2012) Robert Holyst and Andrzej Poniewierski,  _Thermodynamics for Chemists, Physicists and Engineers_ (Springer, Dordrecht (NL), 2012).
  * Ries and McEvoy (1991) Harald Ries and A.J. McEvoy, “Chemical potential and temperature of light,” J. Photochem. Photobiol. A 59, 11–18 (1991).
  * Gräber and Milazzo (1997) Peter Gräber and Giulio Milazzo,  _Bioenergetics_ (Birkhäuser Basel, 1997).
  * Prigogine (1961) I. Prigogine,  _Introduction to Thermodynamics of Irreversible Processes_ , 2nd ed. (Interscience Publishers, New York, 1961).
  * Dilip Kondepudi (2015) Ilya Prigogine Dilip Kondepudi,  _Modern Thermodynamics. From heat engines to dissipative structures_ , 2nd ed. (Wiley, 2015).
  * Penocchio  _et al._ (2019) Emanuele Penocchio, Riccardo Rao, and Massimiliano Esposito, “Thermodynamic efficiency in dissipative chemistry,” Nat. Commun. 10, 3865 (2019).
  * Falasco  _et al._ (2019) Gianmaria Falasco, Tommaso Cossetto, Emanuele Penocchio, and Massimiliano Esposito, “Negative differential response in chemical reactions,” New J. Phys. 21, 073005 (2019).
  * Desai and Mitchison (1997) Arshad Desai and Timothy J. Mitchison, “Microtubule polymerization dynamics,” Annu. Rev. Cell Dev. Biol. 13, 83–117 (1997).
  * Hess and Ross (2017) H. Hess and Jennifer L. Ross, “Non-equilibrium assembly of microtubules: from molecules to autonomous chemical robots,” Chem. Soc. Rev. 46, 5570–5587 (2017).
  * Howard (2001) J. Howard,  _Mechanics of Motor Proteins and the Cytoskeleton_ (Sunderland, MA: Sinauer Associates, 2001).
  * Boekhoven  _et al._ (2015) Job Boekhoven, Wouter E. Hendriksen, Ger J. M. Koper, Rienk Eelkema, and Jan H. van Esch, “Transient assembly of active materials fueled by a chemical reaction,” Science 349, 1075–1079 (2015).
  * Astumian and Bier (1996) R.D. Astumian and M. Bier, “Mechanochemical coupling of the motion of molecular motors to atp hydrolysis,” Biophys. J. 70, 637–653 (1996).
  * Astumian (2016) R. D. Astumian, “Optical vs. chemical driving for molecular machines,” Faraday Discuss. 195, 583–597 (2016).
  * Corrà  _et al._ (2021) Stefano Corrà, Lorenzo Casimiro, Massimo Baroncini, Jessica Groppi, Marcello La Rosa, Marina Tranfić Bakić, Serena Silvi, and Alberto Credi, “Artificial supramolecular pumps powered by light,” Chem. Eur. J. 27, 11076–11083 (2021).
  * Horowitz (2015) Jordan M Horowitz, “Multipartite information flow for multiple maxwell demons,” J. Stat. Mech. 2015, P03006 (2015).
  * Wolpert (2020) David H Wolpert, “Minimal entropy production rate of interacting systems,” New Journal of Physics 22, 113013 (2020).
  * Tkačik and Bialek (2016) Gašper Tkačik and William Bialek, “Information processing in living systems,” Annu. Rev. Condens. Matter Phys. 7, 89–117 (2016).
  * Mattingly  _et al._ (2021) H. H. Mattingly, K. Kamino, B. B. Machta, and T. Emonet, “Escherichia coli chemotaxis is information limited,” Nature Physics 17, 1426–1431 (2021).
  * Montalti  _et al._ (2006) Marco Montalti, Alberto Credi, Luca Prodi, and Teresa Gandolfi,  _Handbook of Photochemistry_ , 3rd ed. (CRC Press, 2006).
  * Mauser and Gauglitz (1998) H. Mauser and G. Gauglitz,  _Photokinetics : theoretical fundamentals and applications_ (Elsevier, 1998).
  * Balzani  _et al._ (2014) Vincenzo Balzani, Paola Ceroni, and Alberto Juris,  _Photochemistry and photophysics : concepts, research, applications_, 1st ed. (Wiley-VCH Verlag, 2014).