sectionentry [tocbeforeskip=1pt]section section subsection paragraph

# Characterizing the detailed balance property by means of measurements in
chemical networks

Eugenia Franco University of Bonn, Institute for Applied Mathematics
Endenicher Allee 60
D-53115 Bonn
GERMANY<EMAIL_ADDRESS>Bernhard Kepka University of Bonn, Institute
for Applied Mathematics
Endenicher Allee 60
D-53115 Bonn
GERMANY<EMAIL_ADDRESS>Juan J. L. Velázquez University of Bonn,
Institute for Applied Mathematics
Endenicher Allee 60
D-53115 Bonn
GERMANY<EMAIL_ADDRESS>
###### Abstract

In this paper we study how to determine if a linear biochemical network
satisfies the detailed balance condition, without knowing the details of all
the reactions taking place in the network. To this end, we use the formalism
of response functions $R_{ij}(t)$ that measure how the system reacts to the
injection of the substance $j$ at time $t=0$, by measuring the concentration
of the substance $i\neq j$ for $t>0$. In particular, we obtain a condition
involving two reciprocal measurements (i.e. $R_{ij}(t),\,R_{ji}(t)$) that is
necessary, but not sufficient for the detailed balance condition to hold in
the network. Moreover, we prove that this necessary condition is also
sufficient if a topological condition is satisfied, as well as a stability
property that guarantees that the chemical rates are not fine-tuned.

###### keywords:

Detailed balance; biochemical network; response function; cut vertex.

###### Contents

  1. 1 Introduction
    1. 1.1 Plan of the paper
    2. 1.2 Notations
  2. 2 Linear biochemical systems and detailed balance
    1. 2.1 Notions of graph theory
    2. 2.2 Linear biochemical systems
    3. 2.3 Detailed balance
    4. 2.4 Lack of detailed balance in biochemical systems
  3. 3 Pathwise detailed balance
    1. 3.1 Pathwise detailed balance and pathwise symmetry
    2. 3.2 Pathwise detailed balance and detailed balance
  4. 4 Stability classes of chemical networks
    1. 4.1 Stability classes
    2. 4.2 Stability/instability of a property in a stability class
    3. 4.3 Classes of graphs for which the pathwise detailed balance condition is stable
  5. 5 Stable pathwise detailed balance without detailed balance implies the existence of a cut vertex
    1. 5.1 Construction of a suitable path in a graph without cut vertices
    2. 5.2 Stable pathwise detailed balance implies the existence of a cut vertex
  6. 6 Approximating the response functions using the time series of the Markov process associated to the biochemical network
  7. 7 Extended detailed balance condition
  8. 8 Alternative choices of measurements
  9. 9 Concluding remarks

##  1 Introduction

A common feature of most of the subsystems that are parts of a biological
organism is that they operate out of equilibrium. Therefore they must exchange
energy with their surroundings in order to function.

The most common molecule used by cells to transport energy is Adenosine
triphosphate (ATP), although other molecules as Guanosine triphosphate (GTP),
Nicotinamide Adenine Dinucleotide Phosphate (NADPH) are sometimes also used.
The energy stored in ATP, or in any other aforementioned molecules, is
realised during a chemical reaction via the dephosphorilization of ATP in
Adenosine diphosphate (ADP).

We now mention few examples of biological systems for which it has been
conclusively shown that they work in out of equilibrium conditions. A well
known case are the kinetic proof-reading mechanisms of the immune system.
These mechanisms allow the immune system to obtain a much larger
discrimination between antigens, than the one that could be expected from the
difference of their affinities with the receptors of the immune cells. This
was studied by Hopfield [14] and Ninio [23]. Similar mechanisms take place in
the recognition and error correcting systems acting in processes like mRNA
transcription, DNA duplication and others, see [3] for a review on the topic.

Another example of biological system for which it has been experimentally
shown that the system works out of equilibrium is the actively beating
Chlamydomonasflagella (see [2]). In general all the systems involving
molecular motors, like kinesins, which are involved in the transport of
proteins inside the cells, or like the actin/myosin motors, which act as
propellers for eukaryotic cells [1], [20], or bacteria like Listeria [26],
must necessarily work out of equilibrium in order to transport molecules or to
produce mechanical work to yield cell displacement.

As another example of biological system working out of equilibrium we refer to
the pump and leak mechanism, that is used by cells to maintain a constant
size, despite the huge difference of osmotic pressure between the interior and
the exterior of the cell, see [27] as well as [17] and [22]. A related fact is
the way in which the family of ABC proteins, which are on the membrane and
acquire different configurations, allow molecules to traverse the membrane.
The concentrations of substance on both the sides of the membrane are out of
equilibrium, therefore, in order to be able to select the molecules allowed to
cross the membrane, the ABC proteins need to use energy in the form of ATP
(see [11]). We remark also that it has been suggested (see [18]) that the
values of the Voltage potential on the membrane of some red blood cells are
created by a passive mechanism, known as Donnan effect, without requiring
active sodium/potassium pumps.

As we will explain in detail later, in this paper we are interested in
understanding if a biochemical system satisfies the detailed balance condition
or not, by measuring the evolution of some suitably chosen substances in the
system. A biochemical system satisfies the detailed balance condition if all
the reactions are balanced at the stationary state. We refer to Section 2.3
for a precise definition of the detailed balance property for the type of
systems of interest for this paper.

An essential feature of all the biological systems introduced above, that
require the use of energy, is that they function due to the fact that they are
open systems, working in out of equilibrium conditions. More precisely, in all
the examples above, the system is able to work because there exists a
continuous influx of molecules able to store energy (for instance ATP) and a
continuous outflux of by-product (for instance ADP). However, we can
approximate the dynamics of these open systems by means of effective systems
that are closed and for which the detailed balance fails, see Section 2.4 for
more details on this approximation. This is the reason why we are interested
in studying systems that do not satisfy the detailed balance condition.

In this paper we study how to determine if a linear biochemical system
satisfies the detailed balance condition. To this end, we use the formalism of
the response functions, introduced in [25]. Some mathematical questions
referring to the response function formalism were studied in [12]. The main
idea behind this approach is to study the main properties of a linear
biochemical system analysing how the system reacts to certain inputs. More
precisely, we study a biochemical system by analysing the properties of the
response functions $R_{ij}(t)$. These response functions describe the
evolution in time of the concentration of the substance $i$, after the arrival
of a molecule of substance $j$ to the system. The advantage of studying a
biochemical system via the response functions $R_{ij}(t)$ is that these
functions can be, at least in principle, measured, without requiring a
detailed knowledge of all the rates of the reactions that take place in the
biochemical system.

We restrict the attention to the case of linear biochemical systems with a
finite number of states, say $L\in\mathbb{N}$. This simplifies the analysis
due to the fact that we have a clear correspondence between systems of ODEs
and graphs, that will allow us to deduce results using graph theory. The
relation between graphs and biochemical reaction networks is more involved in
the non-linear case, see [6, 9].

We stress that the analysis of biochemical systems, that do not satisfy the
detailed balance property, is a very active area in bio-physics, see for
instance [2, 16, 19, 21, 28]. The novelty of this paper is that we approach
the problem using the formalism of response functions. Although, as we explain
in Section 4.2, the results obtained in this paper can be understood also as
results for stochastic processes, making the connection with the approach
studied in [21].

In this paper we prove that, if a linear biochemical system satisfies the
detailed balance condition, then for every states $i,j$ there exists a
constant $c>0$ such that

$R_{ji}(t)=cR_{ij}(t)\text{ for every }t\geq 0.$ (1.1)

We remark that (1.1) provides a necessary condition for the detailed balance
property. In particular, consider a linear biochemical system, with a finite
number of states, such that there exists a couple of states, say $1,2$, and
two positive times $t_{1}\neq t_{2}$, such that
$R_{12}(t_{1})/R_{21}(t_{1})\neq R_{12}(t_{2})/R_{21}(t_{2})$. Then the system
does not satisfy the detailed balance condition.

It is then natural to ask the following question. Assume that equality (1.1)
holds for a particular choice of states $i$ and $j$. Can we infer that the
detailed balance condition holds? If not, can we infer the detailed balance
condition by performing more measurements?

We prove that, if the number of states in the system, $L$, is equal to $3$,
then the fact that two states in the system satisfy the property (1.1) is
equivalent to the detailed balance property of the system. However, this
result holds only for $L=3$ and we prove it to be false for $L>3$, via a
counterexample in $L=4$ (see Example 3.12).

Nevertheless, since we are dealing with biological applications, it is
important to understand if the fact that (1.1) holds for a biochemical
network, for some $i$ and $j$, is a robust property. Indeed, small mutations
can be expected to modify the chemical rates of a system. Therefore, chemical
systems, that do not satisfy the detailed balance property, and satisfy (1.1)
for some $i,j$ only for some fine-tuned rates, cannot be expected from the
biological point of view, unless one finds an underlying reason for the
network to have fine-tuning.

This motivates us to introduce a definition of stability classes of
biochemical systems. A stability class is just a set of biochemical systems
involving the same set of substances and whose reactions share the network
architecture. In particular, if a biochemical system in a certain class is
such that the substances $i$ and $j$ do not interact with each other (namely
an element with state $i$ cannot jump to state $j$ and vice versa), then the
same will be true for all the biochemical systems in the class. Moreover if at
stationary state a biochemical system is such that the reactions between the
substances $i$ and $j$ balance, then the same will be true for all the
biochemical systems in the class.

These features are properties that are not easily modified by evolution,
because they would require a very substantial mutation or a very large number
of mutations. In other words, small mutations, leading to changes in the
coefficients of a reaction network, will not modify the class of the network.
Therefore, we will say that a property is stable or robust in a certain class,
if small changes in the reactions rates do not destroy the property. We refer
to Section 4 for the precise definition of stability classes.

One could impose different concepts robustness and of classes of networks from
the ones that we introduce in this paper. However, we will see that the
definitions that we give not only have some biochemical rationale, but they
are also flexible enough to allow us to prove rigorous mathematical results.

In particular we prove that the fact that (1.1) holds for some pair $i,j$ is
an unstable property for many of the stability classes of biochemical systems
that do not satisfy the detailed balance condition. This means that if we
slightly modify the reactions rates of the biochemical network in one of these
stability classes, then we can obtain a biochemical network belonging to the
same class, for which (1.1) does not hold for any $i,j$. This means that to
have a biochemical system belonging to this class and satisfying (1.1) for
some pair $i,j$, without satisfying the detailed balance condition, requires a
fine-tuning of the parameters in the system.

Hence, from the practical point of view, if we assume that fine-tuning of the
chemical rates does not take place and if we know a priori that a biochemical
system belongs to a stability class with a suitable architecture and we have a
couple of reciprocal measurements $R_{ij},\,R_{ji}$ satisfying (1.1), then we
can infer that the detailed balance condition is satisfied. An example of
class of biochemical systems for which (1.1) is unstable is the class of
systems in which each of the substances interact with all the other substances
in the system and there is at least one reaction in which detailed balance
fails, i.e. one reaction is not balanced at the stationary state. See
Corollary 5.7 in Section 5.

We also prove that there exists a class of biochemical systems that do not
satisfy the detailed balance condition and that satisfy (1.1) for some states
$i,j$ in a stable way. This class of system is characterized by a topological
property of the graph induced by the reactions among the substances in the
system.

The specific property of the architecture of the network that allows us to
have (1.1) in a stable manner, without satisfying the detailed balance
condition, is the existence of a cut vertex in the graph corresponding to the
reaction network. A cut vertex is a vertex of the graph with the property that
removing it, as well as all the edges connected to it, results in a
disconnected graph. The consequence of this is that, if the property
$\eqref{eq:intro p(DB)}$ holds for one couple of vertices $(i,j)$, then the
detailed balance condition is satisfied, unless either the biochemical system
has a particular architecture or the reaction rates are fine-tuned.

The results of this paper can also be interpreted in terms of so-called hidden
Markov processes. Let us mention that the detailed balance condition is
referred to as reversibility in the theory of stochastic processes. The
equations under consideration in this paper describe also the evolution of the
probability densities of a set of states of a Markov process. Hidden Markov
processes are characterized by the fact that only some of the states are
visible. If both states $(i,j)=(1,2)$ are in the observable part of the Markov
process, then the condition (1.1) indicates an observable condition that
allows to determine (for stable systems in the sense explained above) if the
detailed balance condition holds. Finally we recall that hidden Markov
processes are a particular type of semi-Markov processes, for which the
transition probabilities are given by suitable integral operators, see [10].

The results in this paper have some analogies with the theory of inverse
problems, in the sense that we try to determine if a property (reversibility,
i.e. detailed balance) holds for a system, by performing a set of
measurements. The type of equations considered in this paper can be thought as
parabolic equations in a discrete setting. The theory of inverse problems for
elliptic and parabolic equations is very well developed. We refer to the book
[15] for examples of problems of determining the coefficients of an elliptic
or parabolic equation from a small set of measurements. However, we are not
aware of results determining the reversibility of the underlying stochastic
process from some measurements. It would be natural to consider the type of
problems studied in this paper in the continuous setting (i.e. to determine
the reversibility of some stochastic process from some particular
measurements).

In this paper, we are mainly interested in the study of biochemical systems
that are closed, i.e. we assume that the number of substances in the system
does not change in time. In Section 7 we explain how to generalize some of the
concepts and the results of the paper for nonconservative systems, i.e. some
substances are degraded or added to the system. More precisely, we introduce
the notion of extended detailed balance and we prove that (1.1) is a necessary
condition to have that the extended detailed balance condition holds.

Finally, in Section 8 we illustrate why reverse measurements, as in (1.1), are
the natural types of measurements to study if a system satisfies or not the
detailed balance condition. Indeed if these two measurements are such that
(1.1) is false, then we can already deduce that the system does not satisfy
the detailed balance condition. In Section 8 we show that it is not possible
to formulate a necessary condition for the detailed balance in terms of two
measurements that are not mutually reversed, as they are in (1.1). In Section
8 we also show that, to be able prove that (1.1) implies the detailed balance
condition without any stability assumption on the rates, one needs to verify
(1.1) for a number of reciprocal measurements that is of the order of $L/2$
when $L$ is large. Here, $L$ is the size of the system. Hence, when $L$ is
large, this approach would not be feasible, at least from the practical point
of view.

We wish to point out that the theory of biochemical systems is a well
developed area and a huge amount of work has been done (also for non-linear
systems) in order to relate the behaviour of the solutions to chemical
networks in terms of topological properties of the graph, see [6, 7, 8, 9].

For the reader’s convenience we state here the main result of this paper in an
informal way. A precise statement is given in Theorem 5.5.

######  Theorem 1.1.

Consider a biochemical network without cut vertices for which property
$\eqref{eq:intro p(DB)}$ holds and is stable under small, admissible
perturbations of the chemical rates. Then, the detailed balance condition
holds.

Here, admissible perturbations are changes that keep zero reaction rates which
were originally zero and keep the detailed balance property on a set of
reactions for which the detailed balance property is expected to hold, see
Section 4.

###  1.1 Plan of the paper

In Section 2 we introduce the definition of linear biochemical system, we
include the definition of detailed balance in this context and we introduce
some basic results, useful to understand if a system satisfies the detailed
balance condition or not in particular situations.

In Section 3 we introduce the definition of pathwise detailed balance, we
explain its relations with the detailed balance condition and with equality
(1.1). We also provide a graphical interpretation of the pathwise detailed
balance condition, that will require us to introduce also the concept of
pathwise symmetry.

In Section 4 we introduce the concept of stability of a property in a suitably
defined class of networks. In this section we also explain in detail the
biological motivation for the definition of class that we propose and the
reasons why the stability property is relevant for biological applications.
The stability allows to give a precise mathematical definition of the concept
of robustness and fine-tuning of the parameters. Finally, we prove that the
pathwise detailed balance condition is stable for a set of classes.

In Section 5 we prove that the existence of a cut vertex is a necessary
property for the stability of the pathwise detailed balance condition. This
implies the instability of the pathwise detailed balance property for all the
classes of networks that do not have a cut vertex.

In Section 6 we give an interpretation of (1.1) in terms of stochastic
processes. In Section 7 we extend the concept of detailed balance to
nonconservative systems, i.e. systems without mass conservation. In Section 8
we explain why we concentrate on reverse measurements of the form (1.1).
Finally, we provide some concluding remarks in Section 9.

###  1.2 Notations

We define $\mathbb{R}_{+}:=(0,\infty)$, $\mathbb{R}_{*}=[0,\infty)$ and write

$\textbf{e}_{n}=(1,\dots,1)^{T}\in\mathbb{R}^{n}.$ (1.2)

Vectors will always be column vectors, i.e. $v\in\mathbb{R}^{n}$ is such that
$v=(v_{1}\dots,v_{n})^{T}$. We denote the scalar product between two vectors
$v_{1},v_{2}$ as $\langle v_{1},v_{2}\rangle$. We use the notation
$v_{1}\otimes v_{2}$ to indicate the tensor product between the two vectors
$v_{1},v_{2}.$ We use the following notation

$D=\operatorname{diag}(\\{v_{i}\\}_{i=1}^{n})$

to denote the diagonal matrix $D\in\mathbb{R}^{n\times n}$ with $D_{ii}=v_{i}$
and $D_{ij}=0$ for $i\neq j$. Furthermore, we write $\left\|A\right\|$ for the
norm of a matrix $A\in\mathbb{R}^{n\times m}$ with respect to any matrix norm.
Finally, for a matrix $A\in\mathbb{R}^{n\times m}$ we denote by
$A^{T}\in\mathbb{R}^{m\times n}$ its transpose.

##  2 Linear biochemical systems and detailed balance

In this section we give a precise definition of the biochemical systems that
we consider in this paper. Then we introduce the notion of detailed balance as
well as some necessary or sufficient conditions that allow, in some specific
contexts, to check if the detailed balance condition holds (see Section 2.3).
We also explain why it is relevant to analyse biochemical systems that do not
satisfy the detailed balance condition (see Section 2.4). Before doing that we
recall some notions of graph theory that are used in this paper (see Section
2.1).

###  2.1 Notions of graph theory

We recall here some notions of graph theory that will be used in this paper,
see e.g. [4]. Let $\mathcal{G}=(V,E)$ be a directed graph with vertices $V$
and edges $E\subset V\times V$. Given an edge $e=(e_{1},e_{2})$ we say that
its ends are the vertices $e_{1}$ and $e_{2}$. We use the following notation
for the reverse edge, $e^{*}:=(e_{2},e_{1})$. We say that $\mathcal{G}$ is a
symmetric directed graph if $e=(e_{1},e_{2})\in E$ implies that
$e^{*}=(e_{2},e_{1})\in E$. Notice that we can associate to $\mathcal{G}$ a
(undirected) graph, by replacing every directed edge $(e_{1},e_{2})\in E$ with
an undirected edge and removing the reverse edge $(e_{2},e_{1})$. Finally, a
graph is the complete graph if $E=V^{2}$.

We recall now the definition of walk, path and cycle. A walk $w$ in
$\mathcal{G}$ is a finite non-null sequence $v_{0}e_{1}v_{1}e_{2}v_{2}\dots
e_{k}v_{k}$ whose terms are alternatively vertices and edges such that, for
$1\leq i\leq k$, the ends of $e_{1}$ are $v_{i-1}$ and $v_{i}$, i.e.
$e_{1}=(v_{i-1},v_{i})$. If the first vertex of $w$ is $v_{0}$ and the last is
$v_{k}$, we say that $w$ is a walk from $v_{0}$ to $v_{k}$, or a
$(v_{0},v_{k})$-walk. Moreover, $\ell(w):=k$ is the length of the walk. We
denote the set of walks in $\mathcal{G}$ by $W$. Furthermore, we write
$W_{vu}^{(n)}$ for the set of $(v,u)$-walks of length $n$. If the edges and
the vertices of a walk $w$ are distinct, then $w$ is called a path. A walk is
closed if it has positive length and its origin and terminus are the same. A
closed walk, without repeated edges, whose origin and internal vertices are
distinct is a cycle. Let $w$ be a walk and let $v\in V$. We say that $v\in w$
if the sequence $w$ contains the vertex $v$. Similarly we say that an edge $e$
is such that $e\in w$ if $e$ belongs to the sequence $w$. Moreover, if $e\in
w$, we denote with $w\setminus\\{e\\}$ the set of the edges in $w$ different
from $e.$ Finally we denote with $w_{1}\cap w_{2}$ the set of the vertices
that belong both to $w_{1}$ and to $w_{2}$. A $(v_{i},v_{j})$-section of a
walk $v_{0}e_{1}v_{1}e_{2}v_{2}\dots e_{k}v_{k}$ is a walk that is a
subsequence $v_{i}e_{j+1}v_{i+1},\dots e_{j}v_{j}$ of consecutive terms of
$w$. We will use the notation $w_{(v_{i},v_{j})}$ to indicate the
$(v_{i},v_{j})$-section of $w$. Let $\gamma$ be a walk. Assume that
$v_{1},v_{2}\in\gamma$ then we use the following notation

$\operatorname{dist}_{\gamma}(v_{1},v_{2})=\ell(\gamma_{(v_{1},v_{2})})$ (2.1)

Let $w_{1}$ be a $(v_{0},v_{k})$-walk and let $w_{2}$ be a
$(v_{k},v_{n})$-walk. Then $w:=w_{1}\oplus w_{2}$ is the walk obtained by
concatenating $w_{1}$ and $w_{2}$, i.e. $w=v_{0}\ldots v_{k}\ldots v_{n}$.

Two vertices $u$ and $v$ of $\mathcal{G}$ are said to be connected if there is
a $(u,v)$-path in $\mathcal{G}$. If every pair of vertices in a graph are
connected then we say that the graph $\mathcal{G}$ is a connected graph. A
tree is a connected undirected graph with no cycles. A tree is a spanning tree
of a graph $\mathcal{G}$ if it includes every vertex of $\mathcal{G}$ and is a
subgraph of $\mathcal{G}$. A cycle graph is a graph that consists only of a
cycle.

###  2.2 Linear biochemical systems

We consider a finite chemical system and denote its state space by
$\Omega:=\\{1,\dots,L\\}$, where $L$ is the number of states in the system.
Furthermore, we assume the reactions are linear. Hence we deal only with
reactions of the form

$(i)\rightarrow(j)$

that take place at rate $A^{i}_{j}\geq 0$.

The dynamics of the concentrations
$n(t)=(n_{1}(t),\ldots,n_{L}(t))^{T}\in\mathbb{R}_{*}^{L}$ is given by the
following system of ODEs

$\frac{dn}{dt}=An,\quad n(0)=n_{0},$ (2.2)

with $n_{0}\in\mathbb{R}_{*}^{L}$. In this paper we assume that the matrix
$A\in\mathbb{R}^{L\times L}$ is Markovian, i.e.

$A_{ii}=-\sum_{k\in\Omega\backslash\\{i\\}}A_{ki}=-\sum_{k\in\Omega\backslash\\{i\\}}A^{i}_{k}\
\ \text{for all }i\in\Omega\ \ ,\ \ A_{ki}=A^{i}_{k}\geq 0\ \text{for }k\neq
i.$ (2.3)

The Markovianity property of $A\in\mathbb{R}^{L\times L}$ implies that
$\textbf{e}_{L}^{T}A=0$, where $\textbf{e}_{L}$ is given by (1.2), hence the
total number of elements in the system is conserved. Another way of
interpreting the solution $n(t)$ to the system of ODEs is the probability
distribution of a pure-jump Markov process with rates $A_{ij}$ and initial
distribution $n_{0}$.

The system of ODEs (2.2) induces a graph structure on the state space
$\Omega$. More precisely, we define the set of edges $E\subset\Omega^{2}$ by

$\displaystyle E=\left\\{(i,j)\in\Omega^{2}\ :\ A^{i}_{j}\neq 0\right\\}.$

The graph $\mathcal{G}(A)=(\Omega,E)$ is directed. Notice that this graph
includes also self-loops. If we associate to each directed edge $(i,j)$ the
weight $A^{i}_{j}$, then $\mathcal{G}(A)$ is a weighted and directed graph. We
will refer to the weighted graph (i.e. including the values of the chemical
rates) as biochemical network.

In this paper we assume that the graph $\mathcal{G}(A)$ is connected, i.e. the
undirected underlying graph obtained by replacing all directed edges of
$\mathcal{G}(A)$ with undirected edges is a connected graph.

Moreover, we will assume that the matrix $A$ is ergodic, i.e. we assume that
the underlying Markov process is irreducible, in other words, every two states
can be connected by a walk in the graph associated to $A$, and hence the
system of ODEs (2.2) has a unique steady state
$N=(N_{1},\dots,N_{L})^{T}\in\mathbb{R}_{+}^{L}$ with
$\|N\|_{1}=\sum_{i=1}^{L}N_{i}=1$ such that $AN=0$. Furthermore,
$n(t)\rightarrow N$ as $t\rightarrow\infty$ due to the Perron-Frobenius
theorem, see e.g. [10, Chapter VIII]. For notational reasons it is therefore
useful to define the following set

$\mathcal{A}(L):=\\{A\in\mathbb{R}^{L\times L}:A\text{ is Markovian and
ergodic}\\}.$

To simplify the notation, we sometimes abbreviate $\mathcal{A}(L)$ with
$\mathcal{A}$. We now define the response functions in terms of the matrix
$A$. Given $i,j\in\Omega$ we define the response function $R_{ij}$ as

$R_{ij}(t):=\langle e_{j},e^{tA}e_{i}\rangle\text{ for }i,j\in\Omega.$

In other words $R_{ij}$ is the $j$-th component of the solution $n(t)$ of the
ODE (2.2) with initial condition $n_{0}=e_{i}$. Here, $e_{i}$ is the $i$-th
vector of the canonical basis in $\mathbb{R}^{L}$. Therefore, as indicated in
the introduction, the response function $R_{ij}(t)$ measures the response of
the chemical system to a signal, which is just the injection of the substance
$i$ at time zero, by measuring the changes in time of the concentration of the
substance $j$. We notice that this formula for the response function is in
agreement with the expression of response functions corresponding to systems
of ODEs derived in [12, Lemma 2.9] by considering the chemical system as a
compartment with influxes in $i$ and outfluxes in $j$.

The response function $R_{ij}$ has an interpretation in the theory of hidden
Markov processes. Indeed, these are stochastic processes for which there is an
underlying Markov process, but only a small number of states are observable.
For instance, suppose that in the Markov process with probability densities
solving (2.2) only the states $1$ and $2$ are observable. Then, the response
functions $R_{12},\,R_{21}$ will be observable, but not the other response
functions $R_{ij}$ with $(i,j)\neq(1,2),\,(2,1)$. It is then natural to derive
a condition involving only $R_{12},\,R_{21}$ in order to determine if the
detailed balance condition holds. Hidden Markov processes have been
extensively applied in mathematical biology (see for instance [21]).

In the paper it will be, in some cases, useful to divide a biochemical systems
in compartments and to analyse the concentrations of elements in each of the
compartments. To this end it is useful to introduce the following notation.
Let $A\in\mathbb{R}^{L\times L}$. Let $X=\\{\alpha_{1},\dots,\alpha_{n}\\}$ be
a partition of $\\{1,\dots,L\\}$, i.e. $\Omega=\cup_{i=1}^{n}\alpha_{i}$ with
$\alpha_{i}\cap\alpha_{j}=\emptyset$ for every $i\neq j$. We define the
following blocks of the matrix $A.$ Assume that $\beta,\gamma\in X$. Then we
define the matrix $A_{\beta\gamma}\in\mathbb{R}^{|\gamma|\times|\beta|}$ as

$(A_{\beta\gamma})_{ji}:=A_{ji}\quad\text{ for }\quad i\in\gamma,\ j\in\beta.$
(2.4)

The matrix $A_{\beta\gamma}$ encodes the interactions between the compartment
$\beta$ and $\gamma$. Moreover, we define the matrix $E_{\beta}$ as

$(E_{\beta})_{ji}:=A_{ji}\quad\text{ for }i,j\in\beta,\text{ with }i\neq
j\text{ and }(E_{\beta})_{ii}:=-\sum_{j\in\beta}A_{ji}.$ (2.5)

Therefore the matrix $E_{\beta}$ describes the evolution within the
compartment $\beta$ ignoring the interaction with states outside $\beta$.
Finally, given a compartment $\beta$ we define the matrix

$C_{\beta}=\operatorname{diag}\left(\left\\{\sum_{\alpha\in
X\setminus\\{\beta\\}}\sum_{j\in\alpha}A_{ji}\right\\}_{i\in\beta}\right).$
(2.6)

The matrix $C_{\beta}$ describes the loss of elements from the compartment
$\beta$ to any other compartment.

###  2.3 Detailed balance

In this section we introduce the definition of detailed balance for the matrix
$A$ and explain that, the fact that the matrix $A$ satisfies or not the
detailed balance condition, is related to the topology of the graph
$\mathcal{G}(A)$ induced by the matrix $A$.

######  Definition 2.1 (Detailed balance).

We say that a matrix $A\in\mathcal{A}(L)$ satisfies the detailed balance
condition if $A_{ij}N_{j}=A_{ji}N_{i}$ for all $i,j\in\Omega$, where
$N\in\mathbb{R}_{+}^{L\times L}$ is the normalized steady state of $A$.

We now give an equivalent condition of detailed balance in terms of the matrix
$B=S^{-1}AS$, where

$\displaystyle
S:=\operatorname{diag}\left(\\{\sqrt{N_{j}}\\}_{j\in\Omega}\right).$ (2.7)

Here, $N$ is the unique (with $\|N\|_{1}=1$) steady state of the ergodic
matrix $A$. Let us mention that the matrix $B=S^{-1}AS$ has the same left and
a right eigenvector $(\sqrt{N}_{i})_{i=1}^{L}$ with respect to the eigenvalue
zero.

######  Proposition 2.2.

Assume $A\in\mathcal{A}(L)$ satisfies the detailed balance condition. Then the
matrix $B=S^{-1}AS$ is symmetric and such that
$\text{ker\,}(B)=\text{ker\,}(B^{T})=\operatorname{span}(v)$ where
$v_{i}=\sqrt{N_{i}}$ for $i\in\Omega$.

Vice versa, assume that $B\in\mathbb{R}^{L\times L}$ is symmetric and such
that $\text{ker\,}(B)=\text{ker\,}(B^{T})=\operatorname{span}(v)$. Then the
matrix $A:=S_{v}BS_{v}^{-1}$, with

$S_{v}:=\operatorname{diag}(\\{v_{i}\\}_{i\in\Omega}),$ (2.8)

has the unique steady state $N\in\mathbb{R}_{+}^{L}$ with $N_{i}=(v_{i})^{2}$
for $i\in\Omega$. Moreover $A$ satisfies the detailed balance condition.

###### Proof.

The first statement follows by the definition of $B$ and by the fact that $A$
satisfies the detailed balance condition.

Assume instead $B=B^{T}$ and
$\text{ker\,}(B)=\text{ker\,}(B^{T})=\operatorname{span}(v)$. Then by
definition the matrix $A$ is such that $AN=0$ for $N\in\mathbb{R}^{L}$ with
$N_{i}=(v_{i})^{2}$. Moreover assume that there exists a vector $w$ such that
$Aw=0$. Then $S^{-1}w$ would be in $\text{ker\,}(B)=\operatorname{span}(v)$.
Therefore, up to a constant, $w=N$. Hence $S=S_{v}$ and $A$ satisfies the
detailed balance condition. ∎

Since we are considering linear biochemical systems, as explained in Section
2, we can associate to every system a graph $\mathcal{G}$ on the set of the
states $\Omega$. This is very useful since from the geometry of the graphs we
can, in some cases, infer that the system satisfies the detailed balance
property. In the case of non-linear system the correspondence between ODE
systems and graphs is more involved and the analysis of the detailed balance
property in the non-linear systems is therefore more challenging (see [6]).

######  Remark 2.3.

Let $L=2$. Consider a matrix $A\in\mathcal{A}(2)$. If $A^{1}_{2}>0$ and
$A^{2}_{1}>0$, then $A$ satisfies the detailed balance condition.

We provide now a sufficient condition that guarantees that (2.2) satisfies the
detailed balance condition. This result is a particular case of Proposition
14.3.3 in [9] and we refer there for the proof.

######  Proposition 2.4 (Sufficient geometrical condition for detailed
balance).

Assume that $A\in\mathcal{A}$. Assume that the graph $\mathcal{G}(A)$ is a
connected directed symmetric graph. Let $\mathcal{G}$ be the corresponding
(undirected) graph. If $\mathcal{G}$ is a tree, then $A$ satisfies the
detailed balance condition.

Consider a matrix $A\in\mathcal{A}$ that defines a connected and reversible
graph $\mathcal{G}$. Consider any spanning tree $\mathcal{T}$ of
$\mathcal{G}$. Then, by Proposition 2.4, we know that the matrix
$A_{\mathcal{T}}\in\mathbb{R}^{L\times L}$, corresponding to the tree,
satisfies the detailed balance condition. Therefore there exists a vector
$\mu\in\mathbb{R}_{+}^{L}$ with $\sum_{i=1}^{L}\mu_{i}=1$ such that
$B=S^{-1}A_{\mathcal{T}}S$ is symmetric, where $S$ is given by (2.7) with
respect to $\mu$. Notice that the vector $\mu$ is not necessary the steady
state of $A$, unless, as we will see in the next corollary, $A$ satisfies the
detailed balance condition. In the following text we refer to the vector $\mu$
as the energy vector associated with the spanning tree $\mathcal{T}$. This
terminology refers to the fact that $\mu$ can be written as
$(e^{-E_{1}},\ldots,e^{-E_{L}})/Z$, where $E_{i}$ is the free energy of the
state $i$ and $Z$ a normalization constant (see [28]).

We now present a useful corollary of Proposition 2.4.

######  Corollary 2.5.

Assume that $A\in\mathcal{A}$ and assume that the corresponding graph
$\mathcal{G}=\mathcal{G}(A)$ is a connected directed symmetric graph. Let
$N\in\mathbb{R}_{+}^{L}$ be the steady state of $A$. Then the three following
conditions are equivalent:

  1. (a)

$A$ satisfies the detailed balance condition.

  2. (b)

Consider any spanning tree $\mathcal{T}$ of $\mathcal{G}$. Let $\mu$ be the
energy vector associated with $\mathcal{T}$ and consider the matrix
$S:=\operatorname{diag}\left(\\{\sqrt{\mu_{j}}\\}_{j\in\Omega}\right)$. Then
$B=S^{-1}AS$ is symmetric.

  3. (c)

The energy vector associated with any spanning tree of $\mathcal{G}$ is equal
to the steady state of $A$.

###### Proof.

We prove that $(a)$ implies $(c)$. By definition of detailed balance $(a)$
implies that $A_{ij}N_{j}=A_{ji}N_{i}$ for every $i,j\in\Omega$. On the other
hand, Proposition 2.4 implies that for any spanning tree $\mathcal{T}$ of
$\mathcal{G}$ we have that
${(A_{\mathcal{T}})}_{ij}\mu_{j}={(A_{\mathcal{T}})}_{ji}\mu_{i}$ for every
$i,j\in\Omega$. Notice that for every $(i,j)\in\mathcal{T}$, we have that
$A_{ij}={(A_{\mathcal{T}})}_{ij}$ and $A_{ji}={(A_{\mathcal{T}})}_{ji}$. Hence
for every $i,j\in\Omega$ we have that $N_{j}/N_{i}=\mu_{j}/\mu_{i}$, hence due
to the normalization of $\mu$ and $N$ it follows that $N=\mu$. This implies
$(c)$.

We now prove that $(c)$ implies $(b)$. Therefore assume that $(c)$ holds. Then
consider a spanning tree $\mathcal{T}$. For every $(i,j)\in\mathcal{T}$ we
have that
$A_{ij}N_{j}={(A_{\mathcal{T}})}_{ij}N_{j}={(A_{\mathcal{T}})}_{ji}N_{i}=A_{ij}N_{j}$.
Hence the detailed balance condition holds on the edges of $\mathcal{G}$ that
belong also to $\mathcal{T}$. Repeating this argument for all the spanning
trees of $\mathcal{G}$ we deduce that $A$ satisfies the detailed balance
condition. By Proposition 2.2 statement $(b)$ follows.

Finally, we prove that $(b)$ implies $(a)$. From $(b)$ we have that for every
spanning tree $\mathcal{T}$ it holds that
$A_{ji}{\mu_{\mathcal{T}}}_{i}=A_{ij}{\mu_{\mathcal{T}}}_{j}$ for every
$i,j\in\Omega$, where $\mu_{\mathcal{T}}$ is the energy vector associated with
$\mathcal{T}$. This implies that $\mu_{\mathcal{T}}=N$ and $A$ satisfies
detailed balance. ∎

###  2.4 Lack of detailed balance in biochemical systems

In this section we explain why it is relevant to study biochemical systems for
which the detailed balance condition does not hold. We now recall that the
reversibility in time of the quantum mechanical equations imply that in all
the physical processes the detailed balance condition holds (see [5]).
Therefore, in principle, one might think that the detailed balance condition
holds for all biochemical systems. However, most of the biochemical systems
are open systems, where the constant influxes/outfluxes (for instance of
ATP/ADP) keep the system out of equilibrium.

If these outfluxes/influxes of molecules are the dominant factors determining
the concentrations of some of the molecules in the system, it is possible to
approximate the evolution of the concentration of the chemical substances in
the systems by a reduced biochemical network. In this reduced network the
evolution of the substances whose concentrations are determined by external
fluxes can be ignored and the concentrations of the remaining substances are
determined via a biochemical system that does not have detailed balance.

We can illustrate this issue with an example. Suppose that one of the
reactions in the biochemical system is

$A+B\leftrightarrows C+D.$ (2.9)

We assume that the reaction takes place at constant temperature and constant
pressure. We will denote the chemical reaction rates for the direct and
inverse reaction respectively as $K_{+}$ and $K_{-}$. Then the detailed
balance condition for the reaction (2.9) yields

$K_{-}N_{A}N_{B}=K_{+}N_{C}N_{D}$

where $N_{A},N_{B},N_{C},N_{D}$ are the equilibrium concentrations at given
pressure and temperature.

On the other hand, suppose that the system is open, in particular assume that
there is a an influx and outflux respectively of substances $A$ and $C$ in the
system. This results in having an approximately constant concentration of
substances $n_{A}$, $n_{C}$ in the system. Where in general we can have
$N_{A}\neq n_{A}$, $N_{C}\neq n_{C}$. In particular, also the stationary
values would differ. Namely, the stationary values of $B,C$ in the open system
are $\overline{N}_{C}\neq N_{C}$, $\overline{N}_{D}\neq N_{D}$. Moreover, in
general, we would have that

$K_{-}n_{A}\overline{N}_{B}\neq K_{+}n_{C}\overline{N}_{D}.$ (2.10)

Due to the fact that the concentrations $n_{A}$ and $n_{C}$ are constant we
can replace the reaction (2.9) by the effective reaction

$B\leftrightarrows D.$ (2.11)

The effective reaction rates are $\overline{K}_{-}=K_{-}n_{A}$ and
$\overline{K}_{+}=K_{+}n_{C}$. Then (2.10) implies that, in general, we have

$\overline{K}_{-}\overline{N}_{B}\neq\overline{K}_{+}\overline{N}_{D}.$

Hence the effective reaction (2.11) does not satisfy the detailed balance
condition.

This argument explains why it is reasonable, in certain situations, to
consider systems that do not satisfy the detailed balance condition. The lack
of detailed balance is linked to the presence of fluxes injecting and removing
substances to the system, hence keeping the system out of equilibrium.

##  3 Pathwise detailed balance

In this section we introduce the definition of pathwise detailed balance and
of pathwise symmetry. In Section 3.1 we study the relation between these two
properties. Moreover we show that a matrix $A$ satisfies the pathwise detailed
balance property if and only if (1.1) holds. In Section 3.2 we also prove
that, if a matrix $A$ satisfies the detailed balance condition, then it
satisfies the pathwise detailed balance condition. Moreover, we show via an
example, Example 3.12, that the pathwise detailed balance condition does not
imply the detailed balance condition.

###  3.1 Pathwise detailed balance and pathwise symmetry

We start this section by giving the definition of pathwise detailed balance.

######  Definition 3.1 (Pathwise detailed balance).

We say that a matrix $A\in\mathcal{A}(L)$ satisfies the pathwise detailed
balance condition with respect to two states $i,\,j\in\Omega$, $i\neq j$ such
that

$\langle e_{i},A^{n}e_{j}\rangle=\frac{N_{i}}{N_{j}}\langle
e_{j},A^{n}e_{i}\rangle\quad\text{ for every }n\in\mathbb{N},$ (3.1)

where $N$ is the steady state of the matrix $A$.

If a matrix $A$ satisfies the pathwise detailed balance condition with respect
to the states $(i,j)$, we can assume without loss of generality that
$(i,j)=(1,2)$. Furthermore, when we need to specify that a matrix $A$
satisfies the pathwise detailed balance condition with respect to two specific
states, say $1,2$, we will say that it satisfies the $(12)$-pathwise detailed
balance. As we did for the detailed balance condition, we want to relate the
pathwise detailed balance condition of $A$ with the symmetry properties of the
rescaled matrix $S^{-1}AS$. To this end we introduce the following definition.

######  Definition 3.2 (Pathwise symmetry).

We say that a matrix $B\in\mathbb{R}^{L\times L}$ is pathwise symmetric with
respect to two vertices $i,\,j\in\Omega$, $i\neq j$ if

$\langle e_{i},B^{n}e_{j}\rangle=\langle e_{j},B^{n}e_{i}\rangle\quad\text{
for every }n\in\mathbb{N}.$ (3.2)

As before, we can assume without loss of generality that $(i,j)=(1,2)$.
Furthermore, when we need to specify that a matrix $B$ satisfies pathwise
symmetry with respect to two specific states $1,2\in\Omega$ we say that it
satisfies the $(12)$-pathwise symmetry.

In the following we study the equivalence of pathwise detailed balance and
pathwise symmetry. In particular, we prove that, if a matrix $A$ satisfies the
pathwise detailed balance condition, then the matrix $B=S^{-1}AS$, with $S$
defined by (2.7) is pathwise symmetric. Furthermore, we provide additional
conditions on a pathwise symmetric matrix $B$ that guarantee that there exists
a matrix $A$ satisfying the pathwise detailed balance condition and
$B=S^{-1}AS$.

######  Proposition 3.3.

Assume that $A\in\mathcal{A}(L)$ satisfies the $(12)$-pathwise detailed
balance condition. Then the matrix $B:=S^{-1}AS$, with $S$ defined as in (2.7)
with respect to $N$, the steady state of $A$, is such that
$\text{ker\,}(B)=\text{ker\,}(B^{T})=\operatorname{span}(v)$ with
$v_{i}=\sqrt{N_{i}}$ for every $i\in\\{1,\dots,L\\}$. Moreover $B$ is
$(12)$-pathwise symmetric.

Vice-versa assume that $B\in\mathbb{R}^{L\times L}$ is ergodic and is such
that $\text{ker\,}(B)=\text{ker\,}(B^{T})=\operatorname{span}(v)$ and assume
$B$ is $(12)$-pathwise symmetric. Then $A:=S_{v}BS_{v}^{-1}$, where $S_{v}$ is
given by (2.8), is such that $A\in\mathcal{A}$ and $A$ satisfies the
$(12)$-pathwise detailed balance condition.

###### Proof.

We firstly assume that $A\in\mathcal{A}$ satisfies the pathwise detailed
balance condition. Then by definition of $B$ we have that $AN=0$ implies
$BS^{-1}N=0$ as well as $(S^{-1}N)^{T}B=0$. Hence
$S^{-1}N=v\in\text{ker\,}(B)\cap\text{ker\,}(B^{T})$. Now assume that $w\neq
v$ is such that $Bw=0$. We obtain that $ASw=0$ which implies that $Sw=cN$ for
some $c>0$. Hence $w\in\operatorname{span}(v)$. Moreover, we can argue
similarly when $B^{T}w=0$. Thus, we deduce that
$\text{ker\,}(B)=\text{ker\,}(B^{T})=\operatorname{span}(v)$. Finally $B$ is
pathwise symmetric, indeed for every $n\in\mathbb{N}$

$\displaystyle\langle e_{2},B^{n}e_{1}\rangle=\langle
e_{1},(S^{-1}AS)^{n}e_{2}\rangle=\frac{\sqrt{N_{2}}}{\sqrt{N_{1}}}\langle
e_{1},A^{n}e_{2}\rangle=\frac{\sqrt{N_{1}}}{\sqrt{N_{2}}}\langle
e_{2},A^{n}e_{1}\rangle=\langle e_{2},(S^{-1}AS)^{n}e_{1}\rangle=\langle
e_{2},B^{n}e_{1}\rangle.$

Assume now that $B$ is such that
$\text{ker\,}(B)=\text{ker\,}(B^{T})=\operatorname{span}(v)$ and is pathwise
symmetric. Then, by the definition of $A$ and the properties of $B$ we deduce
that $\text{ker\,}(A)=\operatorname{span}(S_{v}v)$. Moreover, since $v^{T}B=0$
we have that $e^{T}A=e^{T}S_{v}BS_{v}^{-1}=v^{T}BS_{v}^{-1}=0$. Hence
$A\in\mathcal{A}$ and the steady state of $A$ is the vector $N$ such that
$N_{i}=v_{i}^{2}$ for every $i=1\dots L$. Then the path symmetry of $B$
implies that for every $n\in\mathbb{N}$ we have

$\displaystyle\frac{v_{2}}{v_{1}}\langle e_{1},A^{n}e_{2}\rangle$
$\displaystyle=\langle e_{1},(S_{v}^{-1}AS_{v})^{n}e_{2}\rangle=\langle
e_{1},B^{n}e_{2}\rangle=\langle e_{2},B^{n}e_{1}\rangle=\langle
e_{2},(S_{v}^{-1}AS_{v})^{n}e_{1}\rangle$
$\displaystyle=\frac{v_{1}}{v_{2}}\langle e_{2},A^{n}e_{1}\rangle.$

Hence $A$ satisfies the pathwise detailed balance condition. ∎

We now give an interpretation of the detailed balance condition in terms of
graphs. The following lemma will allow us to interpret the pathwise detailed
balance property in terms of the rates of the walks of the graph
$\mathcal{G}=\mathcal{G}(A)$ induced by $A$. In particular, we can associate
to any walk $w$ a number via the following map
$\textbf{a}_{A}:W\rightarrow\mathbb{R}$ such that

$\textbf{a}_{A}(w)=\prod_{e=(e_{1},e_{2})\in w}A^{e_{1}}_{e_{2}}.$ (3.3)

Let us mention that also self-loops are included in the above product.

######  Lemma 3.4.

Consider $A\in\mathcal{A}(L)$ and $\mathcal{G}$ be induced by $A$. Let
$1,2\in\Omega$. Then for every $n\geq 1$ we have that

$\langle e_{1},A^{n}e_{2}\rangle=\sum_{w\in W_{12}^{(n)}}\textbf{a}_{A}(w)$

where $\textbf{a}_{A}$ is given by (3.3).

###### Proof.

The statement follows from the definition of $A^{n}$. ∎

######  Remark 3.5.

When the matrix $A$ is Markovian end ergodic, Lemma 3.4 implies that $A$
satisfies the $(12)$-pathwise detailed balance condition if and only if, for
every $n\geq 1$ we have that

$\Delta_{n}(A):=N_{1}\langle e_{2},A^{n}e_{1}\rangle-N_{2}\langle
e_{1},A^{n}e_{2}\rangle=N_{1}\sum_{w\in
W_{12}^{(n)}}\textbf{a}_{A}(w)-N_{2}\sum_{w\in
W^{(n)}_{21}}\textbf{a}_{A}(w)=0.$ (3.4)

where $\textbf{a}_{A}$ is given by (3.3).

######  Remark 3.6.

As a consequence of Remark 3.5 and of Proposition 3.3 we deduce that $A$
satisfies the pathwise detailed balance property if the graph corresponding to
$B=S^{-1}AS$ is such that the sum of the weights of the $(1,2)$-walks of
length $n$ is equal to the sum of the weights of the $(2,1)$-walks of length
$n$.

Indeed Proposition 3.3 guarantees that if $A$ satisfies the $(12)$-pathwise
detailed balance condition, then $B$ is $(12)$-pathwise symmetric. Consider
the weighted graph induced by the coefficients of $B$, i.e. $\mathcal{G}(B)$.
Then (3.2) holds. By Lemma 3.4 we have that the condition (3.2) is equivalent
to

$\sum_{w\in W_{12}^{(n)}}\textbf{a}_{B}(w)=\sum_{w\in
W_{21}^{(n)}}\textbf{a}_{B}(w),\quad\forall n\geq 1,$ (3.5)

where $\textbf{a}_{B}(w)$ is the function defined in (3.3) with respect to the
matrix $B$.

As suggested by the graphical interpretation of the pathwise detailed balance
property, the following lemma states that, to prove that the pathwise detailed
balance condition holds for a matrix $A$, it is enough to check the property
(3.1) for $n\leq L-1$.

######  Lemma 3.7.

Let $A\in\mathcal{A}(L)$. If (3.1) holds for every $n\leq L-1$, then (3.1)
holds for every $n\in\mathbb{N}$.

###### Proof.

This follows by Cayley-Hamilton Theorem (see [13]), which guarantees that for
every $n\geq L$ we have $A^{n}\in\operatorname{span}(A^{i})_{i=1}^{L-1}$. ∎

In the following proposition we state that the pathwise detailed balance
condition is equivalent to (1.1).

######  Proposition 3.8.

Assume that $A\in\mathcal{A}(L)$ with steady state $N\in\mathbb{R}_{+}^{L}$.
If the matrix $A$ satisfies the $(12)$-pathwise detailed balance condition,
then

$R_{12}(t)=\frac{N_{2}}{N_{1}}R_{21}(t)\text{ for all }t\geq 0.$ (3.6)

Vice-versa, if there exists a $c>0$ such that

$R_{12}(t)=cR_{21}(t)\text{ for all }t\geq 0,$ (3.7)

then $c=N_{2}/N_{1}$ and $A$ satisfies the $(12)$-pathwise detailed balance
condition.

###### Proof.

Assume that the matrix $A$ satisfies the pathwise detailed balance condition.
Then (3.1) holds for any $n\in\mathbb{N}$, hence for every $t>0$ we have that
$\langle e_{2},e^{tA}e_{1}\rangle=\frac{N_{2}}{N_{1}}\langle
e_{1},e^{tA}e_{2}\rangle$ and (3.7) follows.

Vice-versa, assume that (3.6) holds. Then $\langle
e_{2},e^{tA}e_{1}\rangle=c\langle e_{1},e^{tA}e_{2}\rangle$. Since the matrix
$A$ is ergodic and the steady state is $N$ we deduce that
$e^{tA}e_{1}\rightarrow N$ and $e^{tA}e_{2}\rightarrow N$ as
$t\rightarrow\infty.$ Hence $\langle e_{2},e^{tA}e_{1}\rangle\rightarrow
N_{2}$ and $\langle e_{1},e^{tA}e_{2}\rangle\rightarrow N_{1}$ as
$t\rightarrow\infty.$ Hence $c=N_{2}/N_{1}$. Using the Taylor expansion of
$e^{tA}$ and identifying powers of $t$ we deduce that for every $n\geq 0$

$\langle e_{2},A^{n}e_{1}\rangle=\frac{N_{2}}{N_{1}}\langle
e_{1},A^{n}e_{2}\rangle.$

∎

###  3.2 Pathwise detailed balance and detailed balance

In this section we study the relation between the detailed balance condition
and the pathwise detailed balance condition. First of all we prove that the
detailed balance condition implies the pathwise detailed balance condition,
see Proposition 3.9. Then we prove that, if the graph $\mathcal{G}(A)$ is a
cycle graph, then the matrix $A$ satisfies the detailed balance condition if
and only if $A$ satisfies the pathwise detailed balance condition. In
particular this implies, see Lemma 3.11, that, when $L=3$ the pathwise
detailed balance condition is equivalent to the detailed balance condition.

######  Proposition 3.9.

Assume that the matrix $A\in\mathcal{A}$ satisfies the detailed balance
condition. Then $A$ satisfies the pathwise detailed balance condition with
respect to any states $i,j\in\Omega$, $i\neq j$.

###### Proof.

The matrix $A$ satisfies the detailed balance condition, hence the matrix
$B=S^{-1}AS$ is symmetric by Proposition 2.2. Here, the matrix $S$ is given in
(2.7) via the steady state $N$. Therefore for every $i,j\in\Omega$ we have
that

$\displaystyle\frac{\sqrt{N_{j}}}{\sqrt{N_{i}}}\langle
e_{i},A^{n}e_{j}\rangle$ $\displaystyle=\langle
e_{i},(S^{-1}AS)^{n}e_{j}\rangle=\langle e_{i},B^{n}e_{j}\rangle=\langle
e_{j},B^{n}e_{i}\rangle=\langle e_{j},(S^{-1}AS)^{n}e_{i}\rangle$
$\displaystyle=\frac{\sqrt{N_{i}}}{\sqrt{N_{j}}}\langle
e_{j},A^{n}e_{i}\rangle,$

for every $n\in\mathbb{N}$. ∎

As a second step we prove that if $L\geq 3$, then the detailed balance
property is equivalent to the pathwise detailed balance property, if the graph
induced by $A$ is a cycle graph.

######  Lemma 3.10.

Let $L\geq 3$. Assume that the graph $\mathcal{G}=\mathcal{G}(A)$ induced by
the matrix $A$ is a directed symmetric cycle graph. Then the matrix $A$
satisfies the pathwise detailed balance condition if and only if it satisfies
the detailed balance condition.

###### Proof.

Consider the spanning tree defined by
$\mathcal{T}=(V,E\setminus\\{(1,2),(2,1)\\}$ and consider the associated
energy vector $\overline{N}$. Define the matrix
$S:=\operatorname{diag}\left(\\{\sqrt{\overline{N}_{j}}\\}_{j\in V}\right)$
and the matrix $B:=S^{-1}AS$. Due to the fact that the detailed balance
condition holds on $\mathcal{T}$ we have that $B_{ij}=B_{ji}$ for every
$(i,j)\neq(1,2).$ The pathwise detailed balance condition implies that

$\displaystyle\langle e_{1},B^{L-1}e_{2}\rangle$ $\displaystyle=\langle
S^{-1}e_{1},A^{L-1}Se_{2}\rangle=\frac{\sqrt{\overline{N}_{2}}}{\sqrt{\overline{N}_{1}}}\langle
e_{1},A^{L-1}e_{2}\rangle=\frac{\sqrt{\overline{N}_{2}}}{\sqrt{\overline{N}_{1}}}\frac{N_{1}}{N_{2}}\langle
e_{2},A^{L-1}e_{1}\rangle$
$\displaystyle=\frac{\overline{N}_{2}}{\overline{N}_{1}}\frac{N_{1}}{N_{2}}\langle
e_{2},B^{L-1}e_{1}\rangle$

as well as $\langle
e_{1},Be_{2}\rangle=\frac{\overline{N}_{2}}{\overline{N}_{1}}\frac{N_{1}}{N_{2}}\langle
e_{2},Be_{1}\rangle.$

Now notice that

$\langle e_{2},B^{L-1}e_{1}\rangle=B_{j_{1}1}B_{j_{2}j_{1}}\dots
B_{2,j_{L-1}}=\langle
e_{1},B^{L-1}e_{2}\rangle=\frac{\overline{N}_{2}}{\overline{N}_{1}}\frac{N_{1}}{N_{2}}\langle
e_{2},B^{L-1}e_{1}\rangle.$

This implies that
$1=\frac{\overline{N}_{2}}{\overline{N}_{1}}\frac{N_{1}}{N_{2}}$, hence
$B_{12}=B_{21}$. This implies the detailed balance condition for $A$. ∎

Finally, if $L=3$, then the detailed balance property is equivalent to the
pathwise detailed balance property.

######  Corollary 3.11.

Assume that $A\in\mathcal{A}(3)$. Then, the matrix $A$ satisfies the detailed
balance condition if and only if it satisfies the pathwise detailed balance
condition.

###### Proof.

For $L=3$ a connected graph is either a cycle or a tree. In the first case the
claim follows from Lemma 3.10. In the second case detailed balance is always
satisfied due to Proposition 2.4. ∎

If $L>3$ then the fact that the matrix $A$ satisfies the pathwise detailed
balance condition does not imply that $A$ satisfies the detailed balance
condition, as we show in the following example.

######  Example 3.12.

Consider the matrix

$A=\left(\begin{matrix}&-3&1&1&1\\\ &1&-3&1&1\\\ &1&1&-3&2\\\ &1&1&1&-4\\\
\end{matrix}\right).$

See Figure 1 to visualize the graph induced by the matrix $A$. This matrix has
steady state $N=\frac{1}{5}(5/4,5/4,3/2,1)$. However, its spanning tree with
edges $\\{(1,3),(1,2),(2,4)\\}$ corresponding to the matrix

$A_{\mathcal{T}}=\left(\begin{matrix}&-2&1&1&0\\\ &1&-2&0&1\\\ &1&0&-1&0\\\
&0&1&0&-1\\\ \end{matrix}\right).$

and has energy vector $\frac{1}{4}(1,1,1,1)$. By Corollary 2.5 $A$ does not
have the detailed balance property. However, $\langle
e_{2},A^{n}e_{1}\rangle=\frac{N_{1}}{N_{2}}\langle
e_{1},A^{n}e_{2}\rangle=\langle e_{1},A^{n}e_{2}\rangle$ for $n=1,2,3$.

Figure 1: Graph corresponding to the matrix $A$. In red and marked with the
symbol $//$ we have the only reaction without detailed balance of the network.

This example shows that for $L>3$ the pathwise detailed balance condition does
not imply the detailed balance condition. Nevertheless, as we stated in the
introduction pathwise detailed balance will be a sufficient condition for
detailed balance if we assume a topological property of the graph as well as a
suitable defined stability concept on a class of graphs. This is the reason to
introduce the definition of stability classes of graphs in the next section.

##  4 Stability classes of chemical networks

In this section we introduce the definition of classes of networks. This
definition will allow us to speak about the stability of certain properties
(in particular the pathwise detailed balance condition) of a specific
representative of the class.

###  4.1 Stability classes

In this section we give the definition of classes of networks. This definition
has a biological rationale. Specifically, the network classes are defined in
such a way that a significant change in the architecture of the network is
needed to transform one class into another. The class of a network could
change in time due to evolution, but this would require a significantly large
number of mutations or a relevant mutation.

We now precise what we mean exactly with significant changes in the network,
that would require important mutations to take place. In the case of linear
networks, that we consider in this paper, we assume that significant changes
are:

  1. (a)

the addition of a new substance, that interacts in a non-trivial manner with
some of the substances previously available in the network;

  2. (b)

the removal of one of the substances of the network, including all the edges
connecting this substance with any of the substances of the network;

  3. (c)

the creation of a new edge connecting two of the previously existing
substances in the network;

  4. (d)

the removal of one edge connecting two of the substances present in the
network;

  5. (e)

the transformation of some of the edge connections for which the detailed
balance condition holds to a new connection where the detailed balance
condition does not hold.

Therefore we say that the architecture of a network is characterized by the
set of substances, the set of edges connecting them, and the set of
connections for which detailed balance holds. Notice that, any of the
modifications of the network indicated in the points (a)-(e), implies a
modification of the architecture of the network. We assume that important
modifications as the ones in (a)-(e) require rare significant mutations or a
very large number of small mutations.

We stress that the definition of architecture of the network defines a
topological space on the set of the chemical rates of the network. There are
other possible topological spaces on networks that allow more general
modifications as the one we introduce in (4.5) below. For instance, it might
allow the formation of new weak reactions between substances that were not
interacting before. Some of these alternative topologies can be justified by
the way in which evolution is expected to produce chemical pathways [24].

We define the classes of networks as a set of chemical networks sharing the
same architecture. More precisely we have the following definition of class of
matrices.

######  Definition 4.1 (Class of matrices).

Assume $L\geq 3$. Let

$\mathcal{S}:=\\{1,\dots L\\}^{2}=\Omega^{2}.$ (4.1)

Assume that $E_{B}\subset\mathcal{S}$, $E_{n-B}\subset\mathcal{S}$ are such
that $E_{B}\cap E_{n-B}=\emptyset.$ Let $E_{N}:=\mathcal{S}\setminus(E_{B}\cup
E_{n-B})$ We define the class of matrices $\mathcal{A}_{E_{N},E_{B}}$ as

$\mathcal{A}_{E_{N},E_{B}}:=\left\\{\ \parbox[c]{303.53267pt}{
$A\in\mathcal{A}(L)$ with normalized steady state $N\in\mathbb{R}_{+}^{L}:\
A_{\alpha_{2}\alpha_{1}}=A_{\alpha_{1},\alpha_{2}}=0,\\\ \forall\alpha\in
E_{N}$ and
$A_{\alpha_{2}\alpha_{1}}N_{\alpha_{1}}=A_{\alpha_{1}\alpha_{2}}N_{\alpha_{2}},\forall\alpha\in
E_{B}$ }\ \right\\}.$ (4.2)

Since to every matrix $A\in\mathcal{A}_{E_{N},E_{B}}$ we can associate a graph
$\mathcal{G}(A)$, a class of matrices corresponds to a class of networks.
According to Definition 4.1, a class of networks is characterized by the set
of nodes $\Omega=\\{1,\dots,L\\}$, a set of edges for which the detailed
balance condition holds, that is $E_{B}$, and a subset of edges for which the
detailed balance condition is not imposed, that is $E_{n-B}$. The network is
then uniquely characterized by means of the chemical coefficients
$K_{\alpha}>0$ for the reaction $\alpha\in E_{B}\cup E_{n-B}$. We have that
$K_{\alpha}=0$ if $\alpha\in E_{N}$.

######  Example 4.2.

Let $A_{1},A_{2}\in\mathcal{A}$ satisfy the detailed balance condition and
assume that $\mathcal{G}(A_{1})$ and $\mathcal{G}(A_{2})$ are complete graphs.
Then $A_{1},A_{2}\in\mathcal{A}_{\emptyset,\mathcal{S}}$.

Notice that in each network class we can define a distance between two
networks characterized by chemical coefficients $\\{K_{\alpha}\\}_{\alpha\in
E_{B}\cup E_{n-B}}$ and $\\{\overline{K}_{\alpha}\\}_{\alpha\in E_{B}\cup
E_{n-B}}$, using for instance the euclidean distance between these vectors. It
is natural to assume that most of the mutations are small, i.e. they preserve
the architecture of the network and just modify the coefficients
$\\{K_{\alpha}\\}$ by a small amount. On the other hand, some rare mutations
or a combinations of several small mutations can lead to a change in the
architecture of the network.

The significant modifications in (a)-(e) have been chosen with the idea that
such changes would require important mutations (or long sequences of small
mutations). We assume that addition or elimination of substances or edges in
the network requires substantial changes. The same applies also to
modifications of reactions with detailed balance that result in reactions
without detailed balance and vice versa.

This might be expected because the reactions in which the detailed balance
condition holds are usually associated to interactions with some additional
molecules, typically ATP, see Section 2.4. The introduction of this additional
molecule in one reaction can be expected to be a significant change. This is
the reason why we include the set of reactions for which detailed balance
holds or fails as part of our definition of classes of graphs.

###  4.2 Stability/instability of a property in a stability class

In this section we introduce the concept of stability of a certain property,
say $(P)$, in a class of graphs. As explained in Section 4.1, our definition
of network class allows to define a concept of distance between two networks
in the same class. It turns out that it allows also to define suitable
concepts of stability and instability that will be relevant for us later.

The two reciprocal measurements $R_{12}$ and $R_{21}$ are not sufficient in
general to determine if a system satisfies the detailed balance condition or
not. In Section 8 we show that measurements that are not reciprocal are less
efficient to determine the detailed balance condition compared to the
reciprocal measurements $R_{12}$ and $R_{21}$. Moreover, we show that in order
to be able to deduce detailed balance from pathwise detailed balance we need a
number of measurements of the order of $L$, recalling that $L$ is the size of
the state space $\Omega$. This means that, from the practical point of view it
is impossible to prove detailed balance from pathwise detailed balance for
large networks. This is the reason why, instead, we focus on analyzing the
stability of pathwise detailed balance.

Consider a class $\mathcal{A}_{E_{N},E_{B}}$ for some choice of
$E_{N},E_{B}\subset\mathcal{S}$ where $\mathcal{S}$ is as in (4.1). Given any
$K\in\mathcal{A}_{E_{N},E_{B}}$ we can define a ball around $K$ in the class
of matrices $\mathcal{A}_{E_{N},E_{B}}$ by

$B_{\delta}(K):=\\{A\in\mathcal{A}_{E_{N},E_{B}}:\|K-A\|<\delta\\}$ (4.3)

for some $0<\delta<\infty$. This induces a topology that we denote by
$\mathcal{T}(E_{N},E_{B})$.

Alternatively, we can also define the following topology, abbreviated by
$\mathcal{T}_{w}(E_{B})$, induced by the following open balls around matrices
$K\in\mathcal{A}_{E_{N},E_{B}}$

$B_{\delta}(K):=\\{A\in\mathcal{A}_{E_{B}^{*}}:\|K-A\|<\delta,\quad
E_{B}\subset E_{B}^{*}\\},$ (4.4)

where we define

$\displaystyle\mathcal{A}_{E_{B}}:=\\{A\in\mathcal{A}(L)\text{ with normalized
steady state
}N\in\mathbb{R}_{+}^{L}:A_{\alpha_{2}\alpha_{1}}N_{\alpha_{1}}=A_{\alpha_{1}\alpha_{2}}N_{\alpha_{2}},\forall\alpha\in
E_{B}\\}.$ (4.5)

The balls $B_{\delta}(K)$ in the topology $\mathcal{T}_{w}(E_{B})$ are larger
than the ones in the topology $\mathcal{T}(E_{N},E_{B})$. Indeed, in the
definition of $B_{\delta}(K)$ for the topology $\mathcal{T}_{w}(E_{B})$ we
take into account also of the possible formation of a reaction between
substances that, according to the rates in $K$, are not interacting. Therefore
even if $\mathcal{G}(K)$ does not have a certain edge, a small perturbation of
$K$ in the topology $\mathcal{T}_{w}(E_{B})$ might create this edge with a
small reaction rate. Nevertheless, all edges in which detailed balance holds
are required to satisfy detailed balance. This weaker topology is suitable for
biochemical networks in which evolution can produce the formation of weak
reactions between substances of the network that were not interacting before,
as for instance in [24].

######  Definition 4.3.

Assume $L>1$ and let $E_{N},E_{B}\subset\mathcal{S}$ be as in Definition 4.1.
Assume that $A\in\mathcal{A}_{E_{N},E_{B}}\subset\mathbb{R}^{L\times L}$
satisfies the property $(P)$. Then the property $(P)$ of $A$ is called stable
in the class $\mathcal{A}_{E_{N},E_{B}}$ if there exists an open set
$\mathcal{O}$ of the topology $\mathcal{T}$ containing $A$ and such that
$\forall\bar{A}\in\mathcal{O}$ property $(P)$ holds. Otherwise, the property
$(P)$ is unstable.

In this paper the topology mentioned in Definition 4.3 is either
$\mathcal{T}(E_{N},E_{B})$ or $\mathcal{T}(E_{B})$. But it would be possible
to consider also other topologies.

Notice that the concepts of stability and instability introduced above is
rather natural from the point of view of biological applications. Indeed, as
explained in Section 4.1, small mutations can be expected to modify the
chemical rates within one given network class, without modifying the class,
i.e. the network architecture.

Since the specific values of the coefficients characterizing a network are
determined by evolution, we cannot expect to have an unstable property $(P)$
for a network, because small mutations would modify the coefficients and would
modify the properties of the network. In other words, if a network has a
certain unstable property, it would require fine-tuning of the parameters. As
a consequence, in the following, we will assume that the only networks that
appear in biological applications are the ones which are stable, in the sense
of Definition 4.3 with respect to a topology $\mathcal{T}$. The specific
topology would depend on the way in which evolution generates different
networks.

Notice that, in some particular biochemical systems, some specific properties
of the proteins involved in the reaction networks or the DNA template yielding
these proteins could provide additional restrictions to the values of the
chemical coefficients $\\{K_{\alpha}\\}$. In such cases, additional
constraints should be added to the class of chemical networks, but in
principle an approach similar to the one used in this paper should apply to
that case.

We conclude this section by proving that the set of matrices that do not
satisfy the detailed balance condition is an open subset of $\mathcal{A}$. In
other words we prove that the lack of detailed balance is stable under small
changes of the coefficients $K_{\alpha}$ that characterize the graph, induced
by mutations.

######  Proposition 4.4.

Let $L>1$. The set

$\mathcal{A}_{n-B}:=\\{A\in\mathcal{A}:A\text{ does not satisfy the detailed
balance condition}\\}$

is an open subset of $\mathcal{A}$.

###### Proof.

We prove that the set of Markovian ergodic matrices satisfying the detailed
balance condition is closed. To this end, consider a converging sequence of
matrices $A_{n}\in\mathcal{A}$, such that $A_{n}\to A$, with
$A\in\mathcal{A}$, as $n\rightarrow\infty$ and such that $A_{n}$ satisfy the
detailed balance condition for each $n\in\mathbb{N}$. We want to prove that
also the limit $A$ satisfies the detailed balance condition. Since the
eigenvalue $\lambda=0$ corresponding to the steady state $N$ of $A$ is simple
we have $\mu_{n}\to N$ as $n\rightarrow\infty$, where $\mu_{n}$ is the steady
state of $A_{n}$. Passing to the limit in the equality

$\displaystyle{(A_{n})}_{ij}{(\mu_{n})}_{j}={(A_{n})}_{ji}{(\mu_{n})}_{i}$

for all $i,\,j\in\Omega$ yields the detailed balance condition for $A$. Hence
the set of matrices satisfying the detailed balance condition is closed. In
particular, $\mathcal{A}_{n-B}$ is open. ∎

###  4.3 Classes of graphs for which the pathwise detailed balance condition
is stable

In this section we introduce a special class of matrices that do not
necessarily satisfy the detailed balance condition, but that satisfy the
pathwise detailed balance condition in a stable manner. As we will see in this
section, this class of graphs is characterized by a topological feature of the
corresponding network, namely the existence of a cut vertex.

We start this section by recalling the definition of cut vertex.

######  Definition 4.5 (Cut vertex).

Let $\mathcal{G}=(V,E)$ be a connected graph. A vertex $X\in V$ is a cut
vertex if $E$ can be partitioned into two non empty subsets $E_{1}$ and
$E_{2}$ such that $\mathcal{G}[E_{1}]$ and $\mathcal{G}[E_{2}]$ have just the
vertex $X$ in common.

We now introduce a class of matrices that correspond to a graph that has a cut
vertex that partitions $E$ in such a way that the edges that do not satisfy
the detailed balance condition belong all to the same part of the graph. More
precisely, the definition is the following.

######  Definition 4.6 (A family of stability classes with cut vertices).

Let $L>3$. Let $E_{N},E_{B}\subset\mathcal{S}$ be as in Definition 4.1. We say
that $\mathcal{A}_{E_{N},E_{B}}$ is a stability class with cut vertices if
$E_{N},\,E_{B}$ are such that every $\mathcal{G}(A)$, with
$A\in\mathcal{A}_{E_{N},E_{B}}$, has a cut vertex $X$ such that either
$E_{1}\subset E_{B}$ or $E_{2}\subset E_{B}$ where $E_{1},E_{2}$ are the two
sets in Definition 4.5. We denote by $\mathcal{C}_{E_{N},E_{B}}$ this class of
matrices.

######  Example 4.7.

Figure 2: Graph corresponding to the matrix $A$. In red and marked with the
symbol $//$ we have the only reaction without detailed balance of the network.

Let $L=5$. Assume that $E_{N}=\\{(1,4),(2,5)\\}$ and assume that
$E_{n-B}=\\{(4,5)\\}$. See Figure 2 for a visualization of the network. Then
$\mathcal{A}_{E_{N},E_{B}}$ is a stability class with the cut vertex $X=3$.
Hence $\mathcal{A}_{E_{N},E_{B}}=\mathcal{C}_{E_{N},E_{B}}$.

######  Theorem 4.8.

Let $L>3$. Let $E_{N},E_{B}\subset\mathcal{S}$ be as in Definition 4.1 and
such that the class $\mathcal{A}_{E_{N},E_{B}}=\mathcal{C}_{E_{N},E_{B}}$ is a
stability class with cut vertices. Every matrix
$A\in\mathcal{C}_{E_{N},E_{B}}$ satisfies the $(12)$-pathwise detailed balance
condition with $1,2\in\mathcal{G}[E_{1}]$.

###### Proof.

Let $A\in\mathcal{A}_{E_{N},E_{B}}$ and let $N$ be the associated (normalized)
steady state. Define the matrix as $B:=S^{-1}AS$, where

$S:=\operatorname{diag}\left(\\{\sqrt{N_{j}}\\}_{j=1}^{L}\right).$

Notice that by definition $B\in\mathcal{A}_{E_{N},E_{B}}$. Consider the graph
$\mathcal{G}(B)$ associated with $B$ and
$\text{ker\,}(B)=\text{ker\,}(B^{T})=\operatorname{span}(v)$ with
$v_{i}=\sqrt{N_{i}}$ for every $i=1,\dots,L$. We will prove that $B$ is
pathwise symmetric. By Proposition 3.3 this will imply that $A$ satisfies the
pathwise detailed balance condition. The graph $\mathcal{G}(B)$ has a cut
vertex $X$ by assumption. Moreover, $E_{1}\subset E_{B}$, where $E_{1}$ is as
in Definition 4.5 and $1,2\in\mathcal{G}[E_{1}]$.

As a consequence, if the walk $w\in W^{(n)}_{12}$ for some $n\geq 1$ is such
that $X\notin w$ this implies that $\forall e\in w$, we have $e\in E_{B}$. The
same holds for its reverse walk $w^{*}\in W^{(n)}_{21}$. Therefore, if
$X\notin w$, then

$\textbf{a}_{B}(w)=\textbf{a}_{B}(w^{*}).$

Therefore, for every $n\geq 1$

$\sum_{w\in W_{12}^{(n)},\ X\notin
w}\textbf{a}_{B}(w)=\sum_{w^{*}:w\in\Pi_{12}^{(n)},\ X\notin
w}\textbf{a}_{B}(w^{*})=\sum_{w\in\Pi_{21}^{(n)},\ X\notin
w}\textbf{a}_{B}(w).$

To conclude, we prove that the same holds for walks that contain the cut
vertex $X$ , hence we need to prove that

$\sum_{w\in W_{12}^{(n)},\ X\in w}\textbf{a}_{B}(w)=\sum_{w\in W_{21}^{(n)},\
X\in w}\textbf{a}_{B}(w).$ (4.6)

To this end we consider a walk $w\in W_{12}^{(n)}$ with $X\in w$. By the
definition of cut vertex, since $1,2\in E_{1}$ we have that $w$ must pass
through $X$ at least two times. Hence there exists a $(X,X)$-section of $w$
that has length larger or equal than one. In other words this $(X,X)$-section
of $w$ is a cycle. Hence the walk $w$ contains at least a cycle $c_{X}$ such
that $X\in c_{X}$.

We assume, for the moment, that all the walks $w$ such that $X\in w$ of length
less than $L-1$ contain only one cycle, we denote it with $c_{X}$, i.e.
$w=w_{(1,X)}\oplus c_{X}\oplus w_{(X,2)}$. Let us define the walk
$\overline{w}$ as $\overline{w}=w_{(1,X)}\oplus c_{X}^{*}\oplus w_{(X,2)}$ and
the walk $\overline{w}^{*}=w^{*}_{(1,X)}\oplus c_{X}\oplus w^{*}_{(X,2)}$,
where we recall that we denote with $w^{*}$ the reverse of the walk $w$.
Notice that $\overline{w}\in W_{12}^{(n)}$ and that $w^{*},\overline{w}^{*}\in
W_{21}^{(n)}$.

We recall that by the definition of stability class with cut vertices
(Definition 4.6) we have that every $e\in w_{(1,X)}$ is such that $e\in E_{B}$
and every $e\in w_{(X,2)}$ is such that $e\in E_{B}$. As a consequence
$\textbf{a}_{B}(w_{(X,1)})=\textbf{a}_{B}(w_{(1,X)})$ and
$\textbf{a}_{B}(w_{(2,X})=\textbf{a}_{B}(w_{(X,2)})$. Hence, we have that

$\displaystyle\textbf{a}_{B}(w)+\textbf{a}_{B}(\overline{w})$
$\displaystyle=\left(\textbf{a}_{B}(c^{*}_{X})+\textbf{a}_{B}(c_{X})\right)\textbf{a}_{B}(w_{(1,X)})\textbf{a}_{B}(w_{(X,2)})$
$\displaystyle=\left(\textbf{a}_{B}(c^{*}_{X})+\textbf{a}_{B}(c_{X})\right)\textbf{a}_{B}(w_{(X,1)})\textbf{a}_{B}(w_{(2,X)})=\textbf{a}_{B}(w^{*})+\textbf{a}_{B}(\overline{w}^{*}).$

Hence,

$\displaystyle\sum_{w\in W_{12}^{(n)}:c_{X}\in w}\textbf{a}_{B}(w)+\sum_{w\in
W_{12}^{(n)}:c^{*}_{X}\in w}\textbf{a}_{B}(w)=\sum_{w\in
W_{21}^{(n)}:c^{*}_{X}\in w}\textbf{a}_{B}(w)+\sum_{w\in W_{21}^{(n)}:c_{X}\in
w}\textbf{a}_{B}(w).$

This argument can be repeated for walks crossing $X$ more than twice, namely
$2m$ times for $m\geq 2$. Implying $\langle e_{1},B^{n}e_{2}\rangle=\langle
e_{2},B^{n}e_{1}\rangle$ for every $n\in\mathbb{N}$. Therefore, by the
definition of the matrix $B$ this implies that the matrix $A$ satisfies the
pathwise detailed balance condition. ∎

We provide now an alternative proof of Theorem 4.8. While in the proof above
we used the topological structure of the graph with a cut vertex, in the
following proof we will use the structure of the matrix $A$ that corresponds
to the graph with a cut vertex. Since the Theorem has been already proven
above in this proof we consider only the case in which the matrix $A$ has
steady state $N=(1,\dots,1)/L$ and we write only the main ideas behind the
proof without technical details.

###### Alternative proof of Theorem 4.8.

Let $X\in\Omega$ be a cut vertex. Let $\alpha$ and $\beta$ be the vertices of
the graphs $\mathcal{G}[E_{1}]$, $\mathcal{G}[E_{2}]$ in which $X$ divides the
graph $\mathcal{G}$. Moreover we know that $E_{1}\subset E_{B}$. Therefore,
using the notation (2.5), the matrix $E_{\alpha\cup\\{X\\}}$ satisfies the
detailed balance condition. This implies that $E_{\alpha}^{T}=E_{\alpha}$ and
that $A_{\alpha X}=A_{X\alpha}$. As a consequence we can rewrite the system of
ODEs (2.2) as

$\displaystyle\frac{dn_{\alpha}}{dt}=$ $\displaystyle
E_{\alpha}n_{\alpha}-C_{\alpha}n_{\alpha}+A_{\alpha{X}}n_{X}$
$\displaystyle\frac{dn_{X}}{dt}=$ $\displaystyle A^{T}_{\alpha
X}n_{\alpha}+A^{T}_{X\beta}n_{\beta}-\sum_{i\in\alpha}A_{iX}n_{X}-\sum_{i\in\beta}A_{iX}n_{X}$
$\displaystyle\frac{dn_{\beta}}{dt}=$ $\displaystyle
E_{\beta}n_{\beta}-C_{\beta}n_{\beta}+A_{\beta{X}}n_{X}$

where (up-on reordering) $n=(n_{1},n_{X},n_{2})$ and where
$n(0)=(n_{0},0,\textbf{0})$.

We solve the system of ODEs by performing the Laplace transform to all the
terms and we obtain that for $z>0$

$\displaystyle\left(z-(E_{\alpha}-C_{\alpha})\right)\hat{n}_{\alpha}(z)$
$\displaystyle=n_{0}+A_{\alpha X}\hat{n}_{X}(z)$ $\displaystyle
z\hat{n}_{X}(z)$ $\displaystyle=A^{T}_{\alpha
X}\hat{n}_{\alpha}(z)+A_{X\beta}\hat{n}_{\beta}(z)-a\hat{n}_{X}$
$\displaystyle\left(z-(E_{\beta}-C_{\beta})\right)\hat{n}_{\beta}(z)$
$\displaystyle=A_{\beta X}\hat{n}_{X}(z)$

where $a=\sum_{i\in\alpha}A_{iX}+\sum_{i\in\beta}A_{iX}$.

Solving these equations we deduce that for $z>0$

$\left(z-(E_{\alpha}-C_{\alpha})-\frac{(1+A_{X\beta}\lambda(z))A_{\alpha
X}\otimes A_{\alpha X}}{z+a}\right)\hat{n}_{\alpha}(z)=n_{0}$ (4.7)

where

$\lambda(z):=\frac{1}{z+a}\left(z-E_{\beta}+C_{\beta}-\frac{A_{\beta X}\otimes
A_{X\beta}}{z+a}\right)^{-1}A_{\beta X}$

Now notice that the matrix

$z-(E_{\alpha}-C_{\alpha})-\frac{(1+A_{X\beta}\lambda(z))A_{\alpha X}\otimes
A_{\alpha X}}{z+a}$

is symmetric. Hence the pathwise detailed balance condition holds for every
couple of vertices in $\alpha$. ∎

The interest of this alternative proof is that it shows that the existence of
a cut vertex is connected with a specific form of the response function,
namely the Laplace transform of the response function satisfies (4.7). It
would be interesting to understand if all the response function of this form
correspond to networks with a cut vertex. This would allow to detect the
existence of a cut vertex using only the response functions.

Theorem 4.8 immediately implies that the matrices in a stability class with
cut vertices satisfy the pathwise detailed balance condition in a stable way.

######  Corollary 4.9.

Let $\mathcal{A}_{E_{N},E_{B}}=\mathcal{C}_{E_{N},E_{B}}$ be a stability class
with cut vertices. Consider any matrix $A\in\mathcal{C}_{E_{N},E_{B}}$ that
satisfies the pathwise detailed balance. Then the pathwise detailed balance
condition is stable in the class $\mathcal{C}_{E_{N},E_{B}}$ with respect to
the topology $\mathcal{T}(E_{N},E_{B})$.

In particular Corollary 4.9 shows that there exists a stability class of
matrices satisfying the pathwise detailed balance condition and that do not
satisfy the detailed balance condition. So if the architecture of a graph is
known and belongs to the stability class with cut vertices, then it is not
possible to understand if the system satisfies the detailed balance condition
having information only about the two measurements $R_{12}$ and $R_{21}$.

######  Remark 4.10.

We stress that the assumption that $E_{1}\subset E_{B}$ in Theorem 4.8 is
necessary. Indeed consider the matrix

$A:=\left(\begin{matrix}&-3&1&1&1&0\\\ &1&-3&1&1&0\\\ &1&1&-3&2&0\\\
&1&1&1&-5&1\\\ &0&0&0&1&-1\end{matrix}\right),$

see Figure 3 for the corresponding graph. The normalized steady state of $A$
is $N=\frac{1}{24}(5,5,6,4,4)$. Instead the energy vector of the spanning tree
with edges

$\\{(1,3),(1,2),(2,4),(4,5)\\}$

is $\mu=\frac{1}{5}(1,1,1,1,1)$. Hence by Corollary 2.5 $A$ does not satisfy
the detailed balance condition. However $A$ satisfies the $(12)$-pathwise
detailed balance. Nevertheless, even if the graph $\mathcal{G}(A)$ has a cut
vertex, the pathwise detailed balance property is unstable for $A$.

Figure 3: Graph corresponding to the matrix $A$. In red and marked with the
$//$ symbol we have the only reaction without detailed balance of the network.

##  5 Stable pathwise detailed balance without detailed balance implies the
existence of a cut vertex

In this section we prove that the graph $\mathcal{G}(A)$ of a matrix
$A\in\mathcal{A}$ has necessarily a cut vertex, if $A$ satisfies the pathwise
detailed balance condition in a stable manner, but does not have detailed
balance.

To prove this result we assume that $A$ does not satisfy the detailed balance
condition, but it satisfies the pathwise detailed balance property. Moreover,
we will assume that $\mathcal{G}(A)$ does not have a cut vertex. We will then
construct a perturbation $D$ of the matrix $A$ such that $A+D$ does not
satisfy the pathwise detailed balance condition. This implies that the
pathwise detailed balance property is unstable and concludes the proof.

To construct the aforementioned perturbation we need to prove that if
$\mathcal{G}(V,E)$ is an undirected graph without cut vertices, there exists
always a path that connects two vertices $I,F\in V$ and that contains a given
vertex $\alpha=(P,Q)$. This will be the object of Section 5.1.

###  5.1 Construction of a suitable path in a graph without cut vertices

Since in this section we are interested in the topological properties of a
graph we will always consider undirected graphs. Given $I,F\in V$ and
$\alpha\in E$, let us denote with $\Pi_{\alpha}(I,F)$ the set of paths with
origin $I$ and terminus $F$ that contain the edge $\alpha=(P,Q).$ The aim of
this section is to prove that $\Pi_{\alpha}(I,F)\neq\emptyset$. Notice that
since the graph is undirected $\alpha=\alpha^{*}=(Q,P)$, hence
$\Pi_{\alpha}(I,F)=\Pi_{\alpha^{*}}(F,I)$.

######  Theorem 5.1.

Assume $\mathcal{G}=(V,E)$ is an undirected connected graph that does not
contain cut vertices. Let $\alpha=(P,Q)\in E$. For every $I,F\in V$ with
$I\neq F$, we have that

$\Pi_{\alpha}(I,F)\neq\emptyset.$

Given the vertices $X,Y$ we use the notation $\mathcal{P}_{\overline{V}}(X,Y)$
to denote the set of the paths connecting $X$ and $Y$ that do not contain the
vertices in $\overline{V}\subset V$.

######  Lemma 5.2.

Assume $\mathcal{G}=(V,E)$ is an undirected connected graph that does not
contain cut vertices. Let $\alpha=(P,Q)\in E$. For every $I,F\in V$ with
$I\neq F$ we have that at least one of the sets

$\mathcal{P}_{\\{F,Q\\}}(I,P),\,\mathcal{P}_{\\{F,P\\}}(I,Q),\,\mathcal{P}_{\\{I,Q\\}}(F,P),\,\mathcal{P}_{\\{I,P\\}}(F,Q)$
(5.1)

is non empty.

###### Proof.

Since $\mathcal{G}$ does not contain cut vertices the vertex $F$ is not a cut
vertex. Hence the graph $\mathcal{G}_{F}:=(V_{F},E_{F})$ with
$V_{F}=V\setminus\\{F\\}$ and $E_{F}=\\{e=(e_{1},e_{2})\in E:e_{1},e_{2}\in
V_{F}\\}$ is connected. By connectivity the set of the paths going from $I$ to
$P$ and not containing $F$ and the set of the paths going from $I$ to $Q$ that
do not contain $F$ are nonempty. If there exists a path $\pi$ connecting $I$
with $P$ that does not contain $F$, but contains $\alpha$, i.e. $\pi=I\dots
Q\alpha P$, then the $(I,Q)$ section of $\pi$ in
$\mathcal{P}_{\\{F,P\\}}(I,Q)$. Otherwise, $\mathcal{P}_{\\{F,Q\\}}(I,P)$ is
not empty. Hence, one of the two sets $\mathcal{P}_{\\{F,Q\\}}(I,P)$ or
$\mathcal{P}_{\\{F,P\\}}(I,Q)$ is not empty. ∎

######  Lemma 5.3.

Assume $\mathcal{G}=(V,E)$ is an undirected connected graph that does not
contain cut vertices. Let $\alpha=(P,Q)\in E$. Assume that for some $I,F\in
V$, with $I\neq F$ it holds that $\mathcal{P}_{\\{F,Q\\}}(I,P)\neq\emptyset$
and $\mathcal{P}_{\\{F,P\\}}(I,Q)=\emptyset$. Then
$\mathcal{P}_{\\{P,I\\}}(Q,F)\neq\emptyset$.

###### Proof.

Assume by contradiction that $\mathcal{P}_{\\{I,P\\}}(Q,F)=\emptyset$. This
implies that all the paths connecting $F$ with $Q$ must contain $P$. Indeed
they cannot contain $I$ due to the assumption
$\mathcal{P}_{\\{F,P\\}}(I,Q)=\emptyset$. Hence $P$ is a cut vertex. This is a
contradiction, therefore $\mathcal{P}_{\\{I,P\\}}(Q,F)\neq\emptyset$. ∎

######  Lemma 5.4.

Assume $\mathcal{G}=(V,E)$ is an undirected connected graph that does not
contain cut vertices. Let $\alpha=(P,Q)\in E$ and let $I,F\in V$ be such that
$I\neq F$. Assume that

$\mathcal{P}_{\\{F,Q\\}}(I,P)\neq\emptyset,\
\mathcal{P}_{\\{F,P\\}}(I,Q)=\emptyset,\
\mathcal{P}_{\\{I,P\\}}(Q,F)\neq\emptyset.$ (5.2)

Then $\Pi_{\alpha}(I,F)\neq\emptyset$.

###### Proof.

Assume that (5.2) holds. If there exists two paths
$\gamma_{1}\in\mathcal{P}_{\\{F,Q\\}}(I,P)$ and
$\gamma_{2}\in\mathcal{P}_{\\{I,P\\}}(Q,F)$ such that
$\gamma_{1}\cap\gamma_{2}=\emptyset$. Then the path $\pi=\gamma_{1}\oplus
P\alpha Q\oplus\gamma_{2}$ belongs to the set $\Pi_{\alpha}(I,F)$. Hence the
desired result follows in this case.

Assume, instead, that for every couple of paths $(\gamma_{1},\gamma_{2})$ with
$\gamma_{1}\in\mathcal{P}_{\\{F,Q\\}}(I,P)$ and
$\gamma_{2}\in\mathcal{P}_{\\{I,P\\}}(Q,F)$ we have that
$\gamma_{1}\cap\gamma_{2}\neq\emptyset$. Consider a vertex
$X\in\gamma_{1}\cap\gamma_{2}$ and define $\overline{\gamma}_{1}$ as the
$(I,X)$-section of $\gamma_{1}$ and $\overline{\gamma}_{2}$ as the
$(X,Q)$-section of $\gamma_{2}$. Then
$w=\overline{\gamma}_{1}\oplus\overline{\gamma}_{2}$ is a walk connecting $I$
with $Q$ and does not contain $P,F$. Since we can extract a path $\pi$ from
the walk $w$ this implies that $P_{\\{F,P\\}}(I,Q)\neq\emptyset$, leading to a
contradiction. Hence there exists two paths
$\gamma_{1}\in\mathcal{P}_{\\{F,Q\\}}(I,P)$ and
$\gamma_{2}\in\mathcal{P}_{\\{I,P\\}}(Q,F)$ such that
$\gamma_{1}\cap\gamma_{2}=\emptyset$. As a consequence, as shown before, it
follows that $\Pi_{\alpha}(I,F)\neq\emptyset$. ∎

###### Proof of Theorem 5.1.

Lemma 5.2 implies that at least one of the four sets in (5.1) is not empty.
Therefore, without loss of generality we can assume that
$\mathcal{P}_{\\{F,Q\\}}(I,P)\neq\emptyset$. Now we have two possibilities,
either $\mathcal{P}_{\\{F,P\\}}(I,Q)\neq\emptyset$ or
$\mathcal{P}_{\\{F,P\\}}(I,Q)=\emptyset$. If
$\mathcal{P}_{\\{F,P\\}}(I,Q)=\emptyset$ holds, then Lemma 5.3 implies that
$\mathcal{P}_{\\{I,P\\}}(Q,F)\neq\emptyset$. Hence Lemma 5.4 implies that
$\Pi_{\alpha}(I,F)\neq\emptyset$. If instead we have that
$\mathcal{P}_{\\{F,P\\}}(I,Q)\neq\emptyset$, then we have three options:

  1. (a)

either
$\mathcal{P}_{\\{I,P\\}}(F,Q),\,\mathcal{P}_{\\{I,Q\\}}(F,P)\neq\emptyset$

  2. (b)

or
$\mathcal{P}_{\\{I,P\\}}(F,Q)=\emptyset,\,\mathcal{P}_{\\{I,Q\\}}(F,P)\neq\emptyset$

  3. (c)

or
$\mathcal{P}_{\\{I,P\\}}(F,Q)\neq\emptyset,\,\mathcal{P}_{\\{I,Q\\}}(F,P)=\emptyset$.

If one of the two last cases occurs, we can repeat the argument above
relabelling $(I,P)$ with $(F,Q)$ in case (c) and $(I,P)$ with $(F,P)$ in case
(b). In both cases we deduce that $\Pi_{\alpha}(I,F)\neq\emptyset$. It remains
to study the first case, in which all the sets (5.1) are non empty.

Therefore we assume from now on that all the sets (5.1) are non empty. We use
the following notation

$\mathcal{P}(I,F;P,Q):=\mathcal{P}_{\\{F,Q\\}}(I,P)\cup\mathcal{P}_{\\{F,P\\}}(I,Q)\cup\mathcal{P}_{\\{I,P\\}}(Q,F)\cup\mathcal{P}_{\\{I,Q\\}}(P,F).$

Let

$D(I,F):=\min_{p\in\mathcal{P}(I,F;P,Q)}\ell(p)$

We aim at proving that $\Pi_{\alpha}(I,F)\neq\emptyset$ by induction over
$D(I,F)$.

Assume that $D(I,F)=1$. Then there exists a path $\pi_{1}$ of length $1$ in
the set $\mathcal{P}(I,F;P,Q)$. Without loss of generality we assume that this
path $\pi_{1}$ belongs to the set $\mathcal{P}_{\\{F,Q\\}}(I,P)$. Hence, since
$\ell(\pi_{1})=1$ we have that $\pi_{1}=I(I,P)P.$ By assumption we have that
$\mathcal{P}_{\\{I,P\\}}(Q,F)\neq\emptyset$. Hence we can consider a
$\pi_{2}\in\mathcal{P}_{\\{I,P\\}}(Q,F)$. The path $\pi_{2}$ does not contain
the edge $\alpha$ and does not contain the vertices $I,P$. Hence
$\pi_{1}\oplus P\alpha Q\oplus\pi_{2}$ is a path from $I$ to $F$ that contains
$\alpha$. Hence $\Pi_{\alpha}(I,F)\neq\emptyset$.

We now assume that $D(I,F;P,Q)=d+1$ with $d\geq 1$. Hence there exists a path
$\pi\in\mathcal{P}(I,F)$ such that $\ell(\pi)=d+1$. Without loss of
generality, we assume that $\pi\in\mathcal{P}_{\\{F,Q\\}}(I,P)$. As a
consequence there exists a $X\in V$, with $X\neq P$, such that
$\pi=I(I,X)X\dots P$. Notice that, since $\pi\in\mathcal{P}_{\\{F,Q\\}}(I,P)$
we have that $X\neq F$, $X\neq Q$ and $X\neq I$. Notice, that by construction
$\mathcal{P}_{\\{F,Q\\}}(X,P)\neq\emptyset$, and therefore
$\mathcal{P}(X,F;P,Q)\neq\emptyset$. As a consequence we can compute $D(X,F)$
and we have that $D(X,F)=d$ because the $(X,P)$-section of $\pi$ has length
$d$ by construction and belongs to $\mathcal{P}(X,F;P,Q)$. The induction
hypothesis implies that the set $\Pi_{\alpha}(X,F)\neq\emptyset$. We have two
possibilities

  1. (a)

$\forall\gamma\in\Pi_{\alpha}(X,F)$ we have that $I\in\gamma$;

  2. (b)

$\exists\gamma\in\Pi_{\alpha}(X,F)$ such that $I\notin\gamma$.

If $(b)$ holds then $I(I,X)X\oplus\gamma$ is a path that belongs to the set
$\Pi_{\alpha}(I,F)$ and the desired result follows.

Assume now that $(a)$ holds. Then we have the following four possibilities

  1. (1)

there exists a $\gamma$ of the form $\gamma=\gamma_{1}\oplus\gamma_{2}\oplus
P\alpha Q\oplus\gamma_{3}$ with $\gamma_{1}\in\mathcal{P}_{\\{P,Q,F\\}}(X,I)$,
$\gamma_{2}\in\mathcal{P}_{\\{X,Q,F\\}}(I,P)$ and
$\gamma_{3}\in\mathcal{P}_{\\{P,I,X\\}}(Q,F)$;

  2. (2)

there exists a $\gamma$ of the form $\gamma=\gamma_{1}\oplus\gamma_{2}\oplus
Q\alpha P\oplus\gamma_{3}$ with $\gamma_{1}\in\mathcal{P}_{\\{P,Q,F\\}}(X,I)$,
$\gamma_{2}\in\mathcal{P}_{\\{X,P,F\\}}(I,Q)$ and
$\gamma_{3}\in\mathcal{P}_{\\{Q,I,X\\}}(P,F)$;

  3. (3)

there exists a $\gamma$ of the form $\gamma=\gamma_{1}\oplus P\alpha
Q\oplus\gamma_{2}\oplus\gamma_{3}$ with
$\gamma_{1}\in\mathcal{P}_{\\{I,F,Q\\}}(X,P)$,
$\gamma_{2}\in\mathcal{P}_{\\{F,P,X\\}}(Q,I)$ and
$\gamma_{3}\in\mathcal{P}_{\\{P,Q,X\\}}(I,F)$;

  4. (4)

there exists a $\gamma$ of the form $\gamma=\gamma_{1}\oplus Q\alpha
P\oplus\gamma_{2}\oplus\gamma_{3}$ with
$\gamma_{1}\in\mathcal{P}_{\\{P,I,F\\}}P(X,Q)$,
$\gamma_{2}\in\mathcal{P}_{\\{Q,X,F\\}}(P,I)$ and
$\gamma_{3}\in\mathcal{P}_{\\{X,P,Q\\}}P(I,F)$.

Analysis of the case (1) and (2). Assume that $1$ holds. Then
$\gamma_{2}\oplus P\alpha Q\oplus\gamma_{3}\in\Pi_{\alpha}(I,F)$ and the
desired conclusion follows. Similarly, if $2$ holds then $\gamma_{2}\oplus
Q\alpha P\oplus\gamma_{3}\in\Pi_{\alpha}(I,F)$.

Analysis of the case (3). Assume now that $3$ holds. Then we have two
possibilities

  1. (a)

$\exists\beta\in\mathcal{P}_{\\{X,P\\}}(Q,F)$ such that $I\notin\beta$;

  2. (b)

$\forall\beta\in\mathcal{P}_{\\{X,P\\}}(Q,F)$ we have that $I\in\beta$.

Assume that $(a)$ holds. Then we analyse all the possible intersections of
$\gamma$ with $\beta$.

  * •

If $\beta\cap\gamma_{1}=\emptyset$, then $I(I,X)X\oplus\gamma_{1}\oplus
P\alpha Q\oplus\beta\in\Pi_{\alpha}(I,F)$.

  * •

If $\beta\cap\gamma_{1}\neq\emptyset$ and $\beta\cap\gamma_{2}=\emptyset$,
then there exists a $Y\in\gamma_{1}\cap\beta$ such that

$\displaystyle\operatorname{dist}_{\beta}(F,Y)=\min\\{\operatorname{dist}_{\beta}(F,Z)\,:\,Z\in\gamma_{1}\cap\beta\\}.$
(5.3)

We stress that here we are using the notation (2.1) for
$\operatorname{dist}_{\beta}(F,Z)$. We obtain that $\gamma_{2}\oplus Q\alpha
P\oplus{\gamma_{1}}_{(P,Y)}\oplus\beta_{(Y,F)}\in\Pi_{\alpha}(I,F)$. Note that
$\beta_{(Y,F)}\cap{\gamma_{1}}_{(P,Y)}=\\{Y\\}$ by (5.3).

  * •

If $\beta\cap\gamma_{1}\neq\emptyset$ and $\beta\cap\gamma_{2}\neq\emptyset$,
then there exists a $Y_{1}\in\gamma_{1}\cap\beta$ and a
$Y_{2}\in\gamma_{2}\cap\beta$ such that

$\displaystyle\operatorname{dist}_{\beta}(F,Y_{1})$
$\displaystyle=\min\\{\operatorname{dist}_{\beta}(F,Z)\,:\,Z\in\gamma_{1}\cap\beta\\},$
$\displaystyle\operatorname{dist}_{\beta}(F,Y_{2})$
$\displaystyle=\min\\{\operatorname{dist}_{\beta}(F,Z)\,:\,Z\in\gamma_{2}\cap\beta\\}.$

Notice that since $\gamma$ is a path $\gamma_{1}\cap\gamma_{2}=\emptyset$.
Then $Y_{1}\neq Y_{2}$. If
$\operatorname{dist}_{\beta}(F,Y_{2})>\operatorname{dist}_{\beta}(F,Y_{1})$,
then $\gamma_{2}\oplus Q\alpha
P\oplus{\gamma_{1}}_{(P,Y_{1})}\oplus{\beta}_{(Y_{1},F)}\in\Pi_{\alpha}(I,F)$.
If instead
$\operatorname{dist}_{\beta}(F,Y_{2})<\operatorname{dist}_{\beta}(F,Y_{1})$,
then $I(I,X)X\oplus\gamma_{1}\oplus P\alpha
Q\oplus{\gamma_{2}}_{(Q,Y_{2})}\oplus\beta_{(Y_{2},F)}\in\Pi_{\alpha}(I,F)$.

In all the possible cases we have $\Pi_{\alpha}(I,F)\neq\emptyset$ , hence if
$(a)$ holds the result follows.

Let us assume now that $(b)$ holds. We prove now that $(b)$ implies that for
every $\eta\in\mathcal{P}_{\\{I,P\\}}(Q,F)$ we have that $X\in\eta$. Hence
$\gamma_{1}\cap\eta\neq\emptyset$. This can be proven by contradiction. Assume
that there exists $\eta\in\mathcal{P}_{\\{I,P\\}}(Q,F)$ such that
$X\notin\eta$. This implies that
$\eta\in\mathcal{P}_{\\{I,P,X\\}}(Q,F)\subset\mathcal{P}_{\\{P,X\\}}(Q,F)$.
Hence assumption $(b)$ implies $I\in\eta$, which is a contradiction. Therefore
we have that every $\eta\in\mathcal{P}_{\\{I,P\\}}(Q,F)$ is such that
$X\in\eta$. In particular this implies that $\gamma_{1}\cap\eta\neq\emptyset$.
Also in this cases we have two possibilities.

  * •

If there exists a $\eta\in\mathcal{P}_{\\{I,P\\}}(Q,F)$ such that
$\eta\cap\gamma_{1}\neq\emptyset$ and $\eta\cap\gamma_{2}=\emptyset$. Let
$Y\in\gamma_{1}\cap\eta$ be such that

$\displaystyle\operatorname{dist}_{\eta}(F,Y)=\min\\{\operatorname{dist}_{\eta}(F,Z)\,:\,Z\in\gamma_{1}\cap\eta\\}.$
(5.4)

then $\gamma_{2}\oplus Q\alpha
P\oplus{\gamma_{1}}_{(P,Y)}\oplus\eta_{(Y,F)}\in\Pi_{\alpha}(I,F)$.

  * •

Assume every $\eta\in\mathcal{P}_{\\{I,P\\}}(Q,F)$ is such that
$\eta\cap\gamma_{1}\neq\emptyset$ and $\eta\cap\gamma_{2}\neq\emptyset$. In
this case consider one of the paths $\eta\in\mathcal{P}_{\\{I,P\\}}(Q,F)$. Let
$Y_{1}\in\eta\cap\gamma_{1}$ and $Y_{2}\in\eta\cap\gamma_{2}$

$\displaystyle\operatorname{dist}_{\eta}(F,Y_{1})$
$\displaystyle=\min\\{\operatorname{dist}_{\eta}(F,Z)\,:\,Z\in\gamma_{1}\cap\eta\\},$
$\displaystyle\operatorname{dist}_{\eta}(F,Y_{2})$
$\displaystyle=\min\\{\operatorname{dist}_{\eta}(F,Z)\,:\,Z\in\gamma_{2}\cap\eta\\}.$

Notice that since $\gamma$ is a path $\gamma_{1}\cap\gamma_{2}=\emptyset$.
Then $Y_{1}\neq Y_{2}$. If
$\operatorname{dist}_{\eta}(F,Y_{2})<\operatorname{dist}_{\eta}(F,Y_{1})$,
then $I(I,X)X\oplus\gamma_{1}\oplus P\alpha
Q\oplus{\gamma_{2}}_{(Q,Y_{2})}\oplus\eta_{(Y_{2},F)}\in\Pi_{\alpha}(I,F)$. If
instead
$\operatorname{dist}_{\eta}(F,Y_{2})>\operatorname{dist}_{\eta}(F,Y_{1})$,
then $\gamma_{2}\oplus Q\alpha
P\oplus{\gamma_{1}}_{(P,Y_{1})}\oplus\eta_{(Y_{1},F)}\in\Pi_{\alpha}(I,F)$.

Hence $(b)$ implies that $\Pi_{\alpha}(I,F)\neq\emptyset.$ Hence the desired
conclusion follows if assumption (3) holds.

Analysis of the case (4). This case can be analysed as the case $(3)$
relabelling $P$ with $Q$. ∎

###  5.2 Stable pathwise detailed balance implies the existence of a cut
vertex

In this section we prove that stable pathwise detailed balance implies the
existence of a cut vertex.

######  Theorem 5.5.

Let $A\in\mathcal{A}_{E_{N},E_{B}}$ and let $\mathcal{G}=\mathcal{G}(A)$ be
the graph associated to $A$, where $E_{N},\,E_{B}\subset\mathcal{S}$ are as in
Definition 4.1. Assume that $A$ satisfies the pathwise detailed balance
condition and assume that this property is stable in
$\mathcal{A}_{E_{N},E_{B}}$ with respect to the topology
$\mathcal{T}(E_{N},E_{B})$. Then we have two possibilities

  1. (a)

either the matrix $A$ satisfies the detailed balance condition;

  2. (b)

or the graph $\mathcal{G}$ has a cut vertex.

######  Remark 5.6.

Notice that in Theorem 5.5 we allow perturbations merely in the stronger
topology $\mathcal{T}(E_{N},E_{B})$. As a consequence when we can also
consider perturbations of the matrix $A$ in the weaker topology
$\mathcal{T}_{w}(E_{B})$.

Before writing the proof of this theorem we give an informal explanation of
the proof. We assume that there exists a matrix $A$ that satisfies the
pathwise detailed balance condition in a stable manner and is such that the
graph induced by $A$, $\mathcal{G}(A)$ does not have a cut vertex. We prove
that this implies that $A$ satisfies the detailed balance condition.

We argue by contradiction. Assume that there exists an edge $\alpha$ on which
the detailed balance condition does not hold. Since $A$ satisfies the pathwise
detailed balance condition $\Delta_{n}(A)=0$ in a stable manner, we have
$\Delta_{n}(\overline{A})=0$ for $\overline{A}=A+D$, where $D$ is any
admissible, sufficiently small perturbation. However, we prove that
$\Delta_{n}(\overline{A})\neq 0$ for a certain choice of $D$, leading to a
contradiction. We now explain how we construct this suitable perturbation $D$.

Notice that the sums in $\Delta_{n}(\overline{A})$ involve positive and
negative terms. It is therefore not easy to find a perturbation avoiding
cancellations. To this end, we perturb all the rates of the matrix $A$ that
correspond to edges, different from $\alpha$, of a specific path
$\pi\in\Pi_{\alpha}(1,2)$. Theorem 5.1 guarantees the existence of such a
path. Furthermore, we choose the perturbation of these edges in such a way
that the matrix $D$ has steady state $N$ and satisfies the detailed balance
condition with respect to $N$.

Differentiating $\Delta_{n}(\overline{A})$ with respect to the non-zero
entries of $D$ allows to reduce the sums in $\Delta_{n}(\overline{A})$ to a
single term. This term turns out to be non-zero due to the fact that in
$\alpha$ the detailed balance condition fails. This implies that
$\Delta_{n}(\overline{A})\neq 0$ leading to a contradiction.

###### Proof of Theorem 5.5.

Consider a matrix $A\in\mathcal{A}_{E_{N},E_{B}}$ and assume that the pathwise
detailed balance condition holds in a stable manner in
$\mathcal{A}_{E_{N},E_{B}}$. Furthermore, assume that the graph $\mathcal{G}$
does not have a cut vertex. We want to prove that this implies that $A$
satisfies the detailed balance condition. To this end we proceed by
contradiction and assume that there exists an edge $\alpha=(P,Q)$ of the graph
$\mathcal{G}$ that does not satisfy the detailed balance condition. In other
words we have that $A_{PQ}N_{Q}\neq A_{QP}N_{P}$, where $N$ is the normalized
steady state of $A$.

Since the graph $\mathcal{G}$ does not have cut vertices, Theorem 5.1 implies
that for every $I,F\in\Omega$ we have that $\Pi_{\alpha}(I,F)\neq\emptyset$.
Let us consider a path $\pi\in\Pi_{\alpha}(1,2)$ for some $1,2\in\Omega$ with
$1\neq 2$. We aim at perturbing all the rates of the edges that belong to the
path $\pi$.

Given an edge $e=(e_{1},e_{2})\in\pi$ such that $e\neq\alpha$, we define the
matrix $D(e)$ as

$\displaystyle\begin{split}D_{ij}(e)&=0\quad\forall(i,j)\neq
e,e^{*},(e_{1},e_{1}),(e_{2},e_{2}),\\\
D_{e_{1}e_{2}}(e)&=\frac{N_{e_{1}}}{N_{e_{2}}},\ D_{e_{2},e_{1}}(e)=1,\
D_{e_{1}e_{1}}=-\frac{N_{e_{1}}}{N_{e_{2}}},\
D_{e_{2}e_{2}}(e)=-1.\end{split}$ (5.5)

Notice that $\textbf{e}^{T}_{L}D(e)=0$ and that $D(e)N=0$. Therefore, also in
this case $N$ is the unique steady state of
$\overline{A}(e)=A+\varepsilon(e)D(e)$, where $\varepsilon(e)>0$. Moreover
$\overline{A}(e)\in\mathcal{A}_{E_{N},E_{B}}$.

Let

$D(\pi):=\sum_{e\in\pi:e\neq\alpha}\varepsilon(e)D(e).$

We define the perturbation $\overline{A}(\pi)$ of $A$ on
$\pi\setminus\\{\alpha\\}$ as

$\overline{A}(\pi)=A+D(\pi).$ (5.6)

By construction $N$ is the unique steady state of $\overline{A}(\pi)$ and
$\overline{A}(\pi)\in\mathcal{A}_{E_{N},E_{B}}$.

Let $n:=\ell(\pi)$. Since $A$ satisfies the pathwise detailed balance
condition we know that $\Delta_{n}(A)=0$, where $\Delta_{n}(A)$ is defined as
in (3.4). The fact that the pathwise detailed balance condition is stable in
$\mathcal{A}_{E_{N},E_{B}}$ implies that also the matrix
$\overline{A}(e)\in\mathcal{A}_{E_{N},E_{B}}$ satisfies the pathwise detailed
balance condition, hence $\Delta_{n}(\overline{A}(\pi))=0$, if all
$\varepsilon(e)$ in the definition of $D(\pi)$ are sufficiently small.

On the other hand,

$\displaystyle\Delta_{n}(\overline{A}(\pi))=N_{1}\sum_{w\in
W_{12}^{(n)}}\textbf{a}_{\overline{A}(\pi)}(w)-N_{2}\sum_{w\in
W^{(n)}_{21}}\textbf{a}_{\overline{A}(\pi)}(w)$ (5.7)

where $W_{12}^{(n)}$ is the set of the walks from $1$ to $2$ of length $n$ and
$\textbf{a}_{\overline{A}(\pi)}(w)$ is given by (3.3).

Now notice that

$\displaystyle\textbf{a}_{\overline{A}(\pi)}(w)$
$\displaystyle=\prod_{e=(e_{1},e_{2})\in
w}\overline{A}^{e_{1}}_{e_{2}}(\pi)=\prod_{e=(e_{1},e_{2})\in
w}\left(A^{e_{1}}_{e_{2}}+D^{e_{1}}_{e_{2}}(\pi)\right)$
$\displaystyle=\prod_{e=(e_{1},e_{2})\in
w}A^{e_{1}}_{e_{2}}+\prod_{e=(e_{1},e_{2})\in
w}D^{e_{1}}_{e_{2}}(\pi)+\sum_{E(w)\subset w}\ \prod_{e\in
E(w)}D^{e_{1}}_{e_{2}}(\pi)\prod_{e\in w\setminus E(w)}A^{e_{1}}_{e_{2}}.$
(5.8)

Here we are using the notation $E(w)\subset w$ to say that $E(w)$ is a subset
of the set of the edges of $w$.

We now compute $\textbf{a}_{\overline{A}(\pi)}(\pi)$ applying (5.2) to $w=\pi$

$\displaystyle\textbf{a}_{\overline{A}(\pi)}(\pi)$
$\displaystyle=\prod_{e=(e_{1},e_{2})\in\pi}A^{e_{1}}_{e_{2}}+\prod_{e=(e_{1},e_{2})\in\pi}D^{e_{1}}_{e_{2}}(\pi)+\sum_{E(\pi)\subset\pi}\
\prod_{e\in E(\pi)}D^{e_{1}}_{e_{2}}(\pi)\prod_{e\in\pi\setminus
E(\pi)}A^{e_{1}}_{e_{2}}$
$\displaystyle=\prod_{e=(e_{1},e_{2})\in\pi}A^{e_{1}}_{e_{2}}+\sum_{E(\pi)\subset\pi}\
\prod_{e\in E(\pi)}D^{e_{1}}_{e_{2}}(\pi)\prod_{e\in\pi\setminus
E(\pi)}A^{e_{1}}_{e_{2}}.$

The last equality is due to the fact that
$\prod_{e=(e_{1},e_{2})\in\pi}D^{e_{1}}_{e_{2}}(\pi)=0$ because
$D^{\alpha_{1}}_{\alpha_{2}}(\pi)=D^{\alpha_{2}}_{\alpha_{1}}(\pi)=0$.
Moreover we can rewrite $\textbf{a}_{\overline{A}(\pi)}(\pi)$ as

$\displaystyle\textbf{a}_{\overline{A}(\pi)}(\pi)=\textbf{a}_{A}(\pi)+A^{\alpha_{1}}_{\alpha_{2}}\prod_{e\in\pi:e\neq\alpha}D^{e_{1}}_{e_{2}}(\pi)+\sum_{\\{E(\pi)\subset\pi:E(\pi)\neq\pi\setminus\\{\alpha\\}\\}}\quad\prod_{e\in
E(\pi)}D^{e_{1}}_{e_{2}}(\pi)\prod_{e\in\pi\setminus
E(\pi)}A^{e_{1}}_{e_{2}}.$ (5.9)

Similarly,

$\displaystyle\textbf{a}_{\overline{A}(\pi)}(\pi^{*})=\textbf{a}_{A}(\pi^{*})+A^{\alpha_{2}}_{\alpha_{1}}\prod_{e\in\pi^{*}:e\neq\alpha^{*}}D^{e_{1}}_{e_{2}}(\pi)+\sum_{\\{E(\pi^{*})\subset\pi:E(\pi^{*})\neq\pi\setminus\\{\alpha^{*}\\}\\}}\prod_{e\in
E(\pi^{*})}D^{e_{1}}_{e_{2}}(\pi)\prod_{e\in\pi\setminus
E(\pi^{*})}A^{e_{1}}_{e_{2}}.$ (5.10)

Now notice that, since $D(\pi)$ satisfies the detailed balance condition with
respect to the steady state $N$ we have that

$\displaystyle
N_{1}A^{\alpha_{1}}_{\alpha_{2}}\prod_{e\in\pi:e\neq\alpha}D^{e_{1}}_{e_{2}}(\pi)=A^{\alpha_{1}}_{\alpha_{2}}\frac{N_{\alpha_{1}}N_{2}}{N_{\alpha_{2}}}\prod_{e\in\pi^{*}:e\neq\alpha^{*}}D^{e_{1}}_{e_{2}}(\pi).$

Therefore

$\displaystyle
N_{1}A^{\alpha_{1}}_{\alpha_{2}}\prod_{e\in\pi:e\neq\alpha}D^{e_{1}}_{e_{2}}(\pi)-N_{2}A^{\alpha_{1}}_{\alpha_{2}}\prod_{e\in\pi^{*}:e\neq\alpha}D^{e_{1}}_{e_{2}}(\pi)=N_{2}\prod_{e\in\pi^{*}:e\neq\alpha^{*}}D^{e_{1}}_{e_{2}}(\pi)\left(\frac{N_{\alpha_{1}}}{N_{\alpha_{2}}}A^{\alpha_{1}}_{\alpha_{2}}-A^{\alpha_{2}}_{\alpha_{1}}\right)\neq
0.$

As a consequence we notice that

$\left(\prod_{e\in\pi:\alpha\neq
e}\frac{d}{d\varepsilon(e)}\right)\left(N_{1}\textbf{a}_{\overline{A}(\pi)}(\pi)-N_{2}\textbf{a}_{\overline{A}(\pi)}(\pi^{*})\right)=N_{2}\left(\frac{N_{\alpha_{1}}}{N_{\alpha_{2}}}A^{\alpha_{1}}_{\alpha_{2}}-A^{\alpha_{2}}_{\alpha_{1}}\right)\neq
0.$ (5.11)

Indeed we have that

$\left(\prod_{e\in\pi:\alpha\neq
e}\frac{d}{d\varepsilon(e)}\right)\textbf{a}_{A}(\pi^{*})=\left(\prod_{e\in\pi:\alpha\neq
e}\frac{d}{d\varepsilon(e)}\right)\textbf{a}_{A}(\pi)=0$

as well as

$\left(\prod_{e\in\pi:\alpha\neq
e}\frac{d}{d\varepsilon(e)}\right)\sum_{\\{E(\pi)\subset\pi:E(\pi)\neq\pi\setminus\\{\alpha\\}\\}}\
\prod_{e\in E(\pi)}D^{e_{1}}_{e_{2}}(\pi)\prod_{e\in\pi\setminus
E(\pi)}A^{e_{1}}_{e_{2}}=0$

and

$\left(\prod_{e\in\pi:\alpha\neq
e}\frac{d}{d\varepsilon(e)}\right)\sum_{\\{E(\pi^{*})\subset\pi^{*}:E(\pi^{*})\neq\pi^{*}\setminus\\{\alpha^{*}\\}\\}}\
\prod_{e\in E(\pi^{*})}D^{e_{1}}_{e_{2}}(\pi^{*})\prod_{e\in\pi^{*}\setminus
E(\pi^{*})}A^{e_{1}}_{e_{2}}=0.$

Notice that here we are using the important assumption that $\pi$ is a path,
hence $\alpha^{*}\notin\pi$, since $\alpha\in\pi$. This is the reason why we
need Theorem 5.1.

Now notice that for every $w\in W_{12}^{(n)}$, $w\neq\pi$ there exists
$e\in\pi\setminus\\{\alpha\\}$ such that $e\notin w$. Therefore we have first
of all that, if $w\neq\pi$, then

$\prod_{e\in w}D^{e_{1}}_{e_{2}}(\pi)=0,$

as well as

$\left(\prod_{e\in\pi:\alpha\neq
e}\frac{d}{d\varepsilon(e)}\right)\sum_{E(w)\subset w}\ \prod_{e\in
E(w)}D^{e_{1}}_{e_{2}}(\pi)\prod_{e\in w\setminus E(w)}A^{e_{1}}_{e_{2}}=0.$

Finally since $\textbf{a}_{A}(w)$ does not depend on $D(\pi)$ we also have
that

$\left(\prod_{e\in\pi:\alpha\neq
e}\frac{d}{d\varepsilon(e)}\right)\textbf{a}_{A}(w)=0.$

The above equalities, together with (5.7), (5.9), (5.10), (5.11) and (5.2),
that

$\displaystyle\left(\prod_{e\in\pi:\alpha\neq
e}\frac{d}{d\varepsilon(e)}\right)\Delta_{n}(\overline{A})$
$\displaystyle=\left(\prod_{e\in\pi:\alpha\neq
e}\frac{d}{d\varepsilon(e)}\right)\left(N_{1}\textbf{a}_{\overline{A}(\pi)}(\pi)-N_{2}\textbf{a}_{\overline{A}(\pi)}(\pi^{*})\right)$
$\displaystyle=N_{2}\left(\frac{N_{\alpha_{1}}}{N_{\alpha_{2}}}A^{\alpha_{1}}_{\alpha_{2}}-A^{\alpha_{2}}_{\alpha_{1}}\right)\neq
0.$

This implies that $\Delta_{n}(\overline{A})\neq 0$ for sufficiently small
$\varepsilon(e)=0$ for every $e\in\pi\setminus\\{\alpha\\}$. This contradicts
$\Delta_{n}(\overline{A})=0$ and concludes the proof. ∎

Theorem 5.5 implies that, if a matrix does not have a cut vertex and satisfy
pathwise detailed balance in a stable manner, then it satisfies the detailed
balance condition. Hence we have the following corollary of Theorem 5.5.

######  Corollary 5.7.

Let $L>3$. Assume that the matrix $A\in\mathbb{R}^{L\times L}$ belongs to the
class $\mathcal{A}_{\\{\emptyset,\\{(1,2),(2,1)\\}\\}}$. Assume that $A$
satisfies the pathwise detailed balance condition in a stable manner with
respect to the topology $\mathcal{T}(\emptyset,\\{(1,2),(2,1)\\})$. Then, $A$
satisfies the detailed balance condition.

###### Proof.

The graph $\mathcal{G}(A)$ is complete because $E_{N}=\emptyset$, hence it
does not contain a cut vertex. Theorem 5.5 implies the statement. ∎

##  6 Approximating the response functions using the time series of the Markov
process associated to the biochemical network

A possible way to measure the response function $R_{12}(t)$ is to inject
substance $1$ in the system and measure substance $2$. If we have an
homogeneous system of cells this would give us $R_{12}$. On the other hand,
one could imagine other types of experiments, where the observed quantity is a
time series of events. In the case of chemical systems modelled by equations
with the form (2.2), a natural time series is the sequence of the times in
which some of the states of the system take palace, say $1$ and $2$. In this
section we will indicate how to detect the property (1.1) in such time series.
Specifically, we explain how to identify (1.1) in a single realization of the
(hidden) Markov process associated to (2.2) with probability one.

First of all, we recall that a chemical network given by ODEs of the form
(2.2), with a Markovian matrix $A$, is associated with a pure-jump Markov
process $\\{X_{t}\\}_{t\geq 0}$ with transition probability rate $A_{ji}dt$
from state $i$ to $j$ in the time interval $[t,t+dt]$.

The response function $R_{21}$ is related to the process $(X_{t})_{t\geq 0}$,
initialized at time $t=0$ as $X_{0}$ with probability density
$n_{0}=(0,1,0,\ldots,0)$, by the formula

$\displaystyle\mathbb{P}(X_{t}=1\mid X_{0}=2)=n_{1}(t)=R_{21}(t).$

Recall that the Markov process is translation invariant with respect to time,
in particular we have

$\displaystyle\mathbb{P}(X_{t_{2}}=1\mid
X_{t_{1}}=2)=R_{21}(t_{2}-t_{1}),\quad t_{2}>t_{1}\geq 0.$

We recall that the stability of the property (1.1) implies detailed balance
condition for systems without cut vertices, see Theorem 5.5.

In principle, to compute the response functions $R_{12}$, $R_{21}$, would
require a detailed knowledge of the underlying Markov process $X_{t}$, or in
other words of the rates of all the reactions taking place in the biochemical
network. On the other hand, the two functions $R_{12}(t)$ and $R_{21}(t)$ can
be computed in several different ways by means of realizations of stochastic
processes. We first consider the case in which, to reconstruct the response
functions, we use multiple independent realizations of the process and apply
the Law of Large Numbers, (see e.g. [10]).

######  Theorem 6.1.

Consider infinitely many independent measurements $X_{t}^{k}$ and
$\bar{X}_{t}^{k}$, $k\in\mathbb{N}$, of the process $(X_{t})_{t\geq 0}$ with
$X_{0}=1$ and of the process $(\bar{X}_{t})_{t\geq 0}$ with $\bar{X}_{0}=2$,
respectively. Then we have

$\displaystyle\lim_{N\to\infty}\dfrac{1}{N}\sum_{k=1}^{N}\mathbbm{1}_{\\{X_{t}^{k}=2\\}}=R_{12}(t),\quad\lim_{N\to\infty}\dfrac{1}{N}\sum_{k=1}^{N}\mathbbm{1}_{\\{\bar{X}_{t}^{k}=2\\}}=R_{21}(t).$

The above theorem allows us to check if the condition
$R_{12}(t)/R_{21}(t)=const.$, cf. (1.1), holds.

Let us now explain how we can, alternatively, verify condition (1.1) using a
single realization of the process using the ergodicity assumption. Let us fix
two times $t_{2}>t_{1}>0$. Consider the process $(X_{t})_{t\geq 0}$ starting
at $X_{0}=1$. We then make the following experiments $N$ times leading to an
approximation of $R_{12}(t)$.

  1. (1)

Wait for time $t_{1}$ and measure if $X_{t_{1}}=2$ or $X_{t_{1}}\neq 2$.

  2. (2)

Wait until time $t_{2}>t_{1}$ and measure if $X_{t_{2}}=2$ or $X_{t_{2}}\neq
2$.

  3. (3)

Wait now until the process reaches the state $1$. Due to the ergodicity of the
process this will happen in finite time, say $\tau$.

  4. (4)

Restart the process at time $t_{2}+\tau$ and go back to Step (1).

Let us mention that this yields $N$ independent measurements
$(X^{k}_{t_{1}})_{k=1}^{N}$, $(X^{k}_{t_{2}})_{k=1}^{N}$ of the process
$(X_{t})_{t\geq 0}$ starting in $X_{0}=1$ due to the strong Markov property.
Accordingly, in the same realization we can wait until the process reaches the
state $2$. Again, this happens in finite time due to the ergodicity. Then, we
can repeat the above procedure exchanging $1$ and $2$. This yields $N$
independent measurements $(\bar{X}^{k}_{t_{1}})_{k=1}^{N}$,
$(\bar{X}^{k}_{t_{2}})_{k=1}^{N}$ of the process $(\bar{X}_{t})_{t\geq 0}$
starting in $\bar{X}_{0}=2$. Using Theorem 6.1 this yields approximations of
the response functions $R_{12}(t),\,R_{21}(t)$ for $t=t_{1},\,t_{2}$. If the
equality $R_{12}(t_{1})/R_{21}(t_{1})=R_{12}(t_{2})/R_{21}(t_{2})$ fails, we
deduce that detailed balance fails. Otherwise, we repeat the experiments for
different times. Notice that with a sufficiently large number of times
$t_{1},t_{2}$ we can approximate the response functions with arbitrary
accuracy.

The use of time series suggested above requires very long measurements. Most
likely there is a way of optimizing the information in a time series, in order
to obtain $R_{12}(t)$ and $R_{21}(t)$. However we are not going to study this
in this paper.

##  7 Extended detailed balance condition

All the results in the previous sections are for Markovian matrices $A$. From
the biological point of view this means that we restrict our attention to
chemical systems that are isolated, because we assume that the total
concentration of substances in the system does not change in time.

In this section we illustrate how it is possible to extend the previous
results to the case of chemical systems where the number of substance in the
population might vary, due to the presence of sources and sinks. In
particular, we extend the definition of detailed balance to chemical networks
with sources and sinks and we prove that, if a matrix $A$ satisfies the
extended detailed balance condition, then it satisfies also the pathwise
detailed balance condition.

To this end we adopt the following notation in (2.4)-(2.6).

######  Definition 7.1 (Matrix with sources and sinks).

Let $A\in\mathbb{R}^{L\times L}$ be a Markovian matrix. Assume that the matrix
$A$ is such that there exist a partition
$\\{\alpha,\alpha_{in},\alpha_{out}\\}$ of $\Omega$, such that

$A_{\alpha,\alpha_{in}}\neq\textbf{0},\quad
A_{\alpha_{in},\alpha}=\textbf{0},\quad
A_{\alpha_{out},\alpha}\neq\textbf{0},\quad
A_{\alpha,\alpha_{out}}=\textbf{0}\quad
A_{\alpha_{out},\alpha_{in}}=\textbf{0},\quad
A_{\alpha_{in},\alpha_{out}}=\textbf{0}.$ (7.1)

Finally, we assume that the dynamics in each of the compartments is Markovian.
Hence we assume that $e_{|\alpha|}^{T}E_{\alpha}=0$,
$e_{|\alpha_{in}|}^{T}E_{\alpha_{in}}=0$ and
$e_{|\alpha_{out}|}^{T}E_{\alpha_{out}}=0$. Then, we call $\alpha_{in}$ the
source and $\alpha_{out}$ the sink of the chemical network.

Notice that, if $A$ is a matrix with sources and sinks, then the assumptions
(7.1) on $A$ imply that

$C_{\alpha}=\operatorname{diag}\left(\left\\{\sum_{j\in\alpha_{out}}A^{i}_{j}\right\\}_{i\in\alpha}\right),\quad
C_{\alpha_{in}}=\operatorname{diag}\left(\left\\{\sum_{j\in\alpha}A^{i}_{j}\right\\}_{i\in\alpha_{in}}\right)\text{
and }C_{\alpha_{out}}=\textbf{0}.$

Using the notations introduced above and the assumptions on $A$ and on the set
of compartments $X=\\{\alpha,\alpha_{in},\alpha_{out}\\}$ we can rewrite (2.2)
as the following equation for
$n(t)=(n_{\alpha},n_{\alpha_{in}},n_{\alpha_{out}})$,

$\displaystyle\frac{dn_{\alpha}(t)}{dt}$
$\displaystyle=E_{\alpha}n_{\alpha}(t)-C_{\alpha}n_{\alpha}(t)+A_{\alpha,\alpha_{in}}n_{\alpha_{in}}(t)$
$\displaystyle\frac{dn_{\alpha_{in}}(t)}{dt}$
$\displaystyle=E_{\alpha_{in}}n_{\alpha_{in}}(t)-C_{\alpha_{in}}n_{\alpha_{in}}(t)$
$\displaystyle\frac{dn_{\alpha_{out}}(t)}{dt}$
$\displaystyle=E_{\alpha_{out}}n_{\alpha_{out}}(t)+A_{\alpha_{out},\alpha}n_{\alpha}(t).$

We refer to [12] for more details on the reduction of systems of ODEs of the
form (2.2) in system of equations for the density of elements in a
compartment.

######  Definition 7.2 (Extended detailed balance).

Assume that $A\in\mathbb{R}^{L\times L}$ is a Markovian matrix with sources
and sinks. We say that the matrix $A$ satisfies the extended detailed balance
condition if the matrix $E_{\alpha}$ satisfies the detailed balance condition.
Namely, there exists an $N\in\mathbb{R}_{+}^{|\alpha|}$ such that
$E_{\alpha}N=0$ and such that the matrix
$B_{\alpha}:=S_{\alpha}^{-1}E_{\alpha}S_{\alpha}$, where

$S_{\alpha}:=\operatorname{diag}\left(\\{\sqrt{N_{j}}\\}_{j\in\alpha}\right)$

is symmetric.

In particular notice that if the ODE (2.2) satisfies the detailed balance
condition then $E_{\alpha}=A$ and $n_{\alpha}=n$.

When a matrix $A$ has sources and sinks then we define for $i,j\in\alpha$

$R_{ij}(t):=\langle
e_{j},e^{t\left(E_{\alpha}-C_{\alpha}\right)}e_{i}\rangle\text{ for
}i,j\in\alpha$ (7.2)

as the response function of $j$ to a signal $i$. We refer to [12] for more
details on response functions.

######  Theorem 7.3.

Assume that the matrix $A$ has sources and sinks and it satisfies the extended
detailed balance condition for $\alpha\subset\Omega$. Assume $1,2\in\alpha$.
Then

$R_{21}(t)=\frac{N_{1}}{N_{2}}R_{12}(t)\quad\text{ for all }t\geq 0$ (7.3)

where the response functions $R_{12},R_{21}$ are given by (7.2) and
$N\in\mathbb{R}_{+}^{|\alpha|}$ is such that $E_{\alpha}N=0$.

###### Proof.

The extended detailed balance condition implies that

$\displaystyle R_{21}(t)$ $\displaystyle=\langle
e_{1},e^{t\left(E_{\alpha}-C_{\alpha}\right)}e_{2}\rangle=\langle
e_{1},e^{tS_{\alpha}\left(B_{\alpha}-S_{\alpha}^{-1}C_{\alpha}S_{\alpha}\right)S_{\alpha}^{-1}}e_{2}\rangle=\langle
e_{1},e^{tS_{\alpha}\left(B_{\alpha}-S_{\alpha}^{-1}C_{\alpha}S_{\alpha}\right)S_{\alpha}^{-1}}e_{2}\rangle$
$\displaystyle=\langle
S_{\alpha}e_{1},e^{t\left(B_{\alpha}-S_{\alpha}^{-1}C_{\alpha}S_{\alpha}\right)}S_{\alpha}^{-1}e_{2}\rangle=\frac{\sqrt{N_{1}}}{\sqrt{N_{2}}}\langle
e_{1},e^{t\left(B_{\alpha}-S_{\alpha}^{-1}C_{\alpha}S_{\alpha}\right)}e_{2}\rangle$
$\displaystyle=\frac{\sqrt{N_{1}}}{\sqrt{N_{2}}}\langle
e^{t\left(B_{\alpha}-S_{\alpha}^{-1}C_{\alpha}S_{\alpha}\right)^{T}}e_{1},e_{2}\rangle=\frac{N_{1}}{N_{2}}\frac{\sqrt{N_{2}}}{\sqrt{N_{1}}}\langle
e_{2},e^{t\left(B_{\alpha}-S_{\alpha}^{-1}C_{\alpha}S_{\alpha}\right)}e_{1}\rangle$
$\displaystyle=\frac{N_{1}}{N_{2}}\langle
e_{2},e^{tS_{\alpha}\left(B_{\alpha}-S_{\alpha}^{-1}C_{\alpha}S_{\alpha}\right)S_{\alpha}^{-1}}e_{1}\rangle=\frac{N_{1}}{N_{2}}\langle
e_{2},e^{t\left(E_{\alpha}-C_{\alpha}\right)}e_{1}\rangle=\frac{N_{1}}{N_{2}}R_{12}(t).$

∎

##  8 Alternative choices of measurements

In the previous section we study if the detailed balance condition of a matrix
$A$ can be detected by reciprocal measurements $R_{12}$, $R_{21}$ or non-
reciprocal measurements. In this section we explain why we decided to focus
our attention only on these reverse measurements. More precisely, we consider
the set of response functions $R_{12},R_{13}$ corresponding to matrices
$A\in\mathbb{R}^{4\times 4}$ that corresponds to a complete graph
$\mathcal{G}(A)$. We will show that, the fact that $A$ satisfies the detailed
balance condition does not lead to any restriction on the response functions
$R_{12},R_{13}$. Hence this type of measurements are not useful in order to
detect if the matrix $A$ satisfies or not detailed balance.

Before doing that we write a heuristic computation that give some insights on
the dimensionality of different sets of matrices. For instance, we consider
the set of matrices satisfying the detailed balance condition respectively the
pathwise detailed balance condition. These heuristic computations are in
agreement with the results in the previous sections and allow us to see the
notion of stability from a "geometrical perspective". For the purposes of this
section we assume that the manifolds of matrices satisfying the detailed
balance property respectively the pathwise detailed balance condition are well
defined.

Consider a matrix $A\in\mathcal{A}(L)$ corresponding to a complete graph
$\mathcal{G}(A)$. We recall that $\mathcal{A}$ is the set of Markovian and
ergodic matrices.

We use the notation

$\mathcal{B}:=\\{A\in\mathcal{A}:A\text{ satisfies the detailed balance
condition}\\}$

Finally, we denote with

$\mathcal{C}:=\\{A\in\mathcal{A}:A\text{ satisfies the pathwise detailed
balance condition}\\}$

Notice that since the matrices in $\mathcal{A}$ are Markovian we have that

$\dim(\mathcal{A})=L^{2}-L=L(L-1).$

We now compute the dimension of the manifold $\mathcal{B}$. Notice that for
every edge $e$ we can chose freely a reaction rate, while the reaction rate of
the reverse edge $e^{*}$ is determined by the steady state and by the reaction
rate of $e$. Therefore, the dimension of $\mathcal{B}$ is given by the number
of edges in the graph, that is $\frac{L(L-1)}{2}$, plus the possible choices
for the steady state, that is $L-1$. In total we have

$\dim(\mathcal{B})=\frac{L(L-1)}{2}+L-1=\frac{1}{2}(L-1)(L+2).$

Finally $\dim(\mathcal{C})$ is $\dim(\mathcal{A})$ minus the $L-1$ constraints
(assuming that they are all independent) of the pathwise detailed balance
condition $+1$, due to the choice of the constant $c$ in the pathwise detailed
balance condition

$\dim(\mathcal{C})=L(L-1)-(L-1)+1=L(L-2)+2.$

Notice that for $L=3$ we have $\dim(\mathcal{B})=\dim(\mathcal{C})=5$ as
expected from Corollary 3.11, where the equivalence between pathwise detailed
balance and detailed balance is proven. Instead, for $L=4$ we have
$9=\dim(\mathcal{B})<\dim(\mathcal{C})=10$, as expected from Example 3.12,
where it is proven that pathwise detailed balance is not equivalent to
detailed balance.

Notice that for generic $L>3$ we have
$\dim(\mathcal{A})>\dim(\mathcal{C})>\dim(\mathcal{B})$. As a consequence of
the fact that $\dim(\mathcal{A})>\dim(\mathcal{C})$ we can expect that, given
a matrix $A\in\mathcal{C}$, every open set $U\subset\mathcal{A}$ with $A\in U$
satisfies $U\cap(\mathcal{A}\setminus\mathcal{C})\neq\emptyset$. In other
words, this geometrical argument suggests that the pathwise detailed balance
condition is unstable in the class of complete graphs with at least one edge
without detailed balance. This is exactly what is proven in Corollary 5.7.

Moreover, let $\mathcal{C}_{d}$ be the submanifold of $\mathcal{A}$ for which
pathwise detailed balance holds for $d$ couple of points. Then we compute
$\dim(\mathcal{C}_{d})$ as we did before with $\mathcal{C}$, but taking into
account that now we have $d(L-1)$ constraints due to the pathwise detailed
balance condition and we have $d$ different constants. Here we assume that all
$d$ measurements give $d$ independent constraints. Hence,

$\dim(\mathcal{C}_{d})=L(L-1)-d(L-1)+d.$

Assume now that $L$ tends to infinity. We want to understand how many couples
of reverse measurements (hence of the form $R_{ij},R_{ji}$) we should do in
order to have $\dim(\mathcal{C}_{d})=\dim(\mathcal{B})$. It turns out that

$d=\left(L^{2}/2-3/2L+1\right)/(L-2).$

Hence as $L\rightarrow\infty$ we have $d\approx L/2$. This implies that, in
order to be able to prove the detailed balance condition, performing reverse
measurements we need to make a number of measurements of the order of $L$.
Therefore, from the practical point of view this approach is not feasible.
This is why in this paper we focus instead on studying the stability of the
pathwise detailed balance condition for some classes of matrices.

Finally, consider the Markovian matrices $A\in\mathbb{R}^{L\times L}$ such
that $\mathcal{G}(A)$ is a cycle graph. We denote this set of matrices as
$\mathcal{A}_{c}$. Notice that, these matrices (as well as their corresponding
steady state) are determined by the weights of the edges along the cycle,
hence

$\dim(\mathcal{A}_{c})=2L.$

Notice that, under these assumptions, the pathwise detailed balance condition
provides only two constraint, indeed (3.1) is non trivial only for $n=1$ and
$n=L-1$. Hence, including the degree of freedom given by the steady state, we
get

$\dim\left(\\{A\in\mathcal{A}_{c}:A\text{ satisfies the pathwise detailed
balance condition}\\}\right)=2L-2+1=2L-1.$

Finally the dimension of the set of the matrices in $\mathcal{A}_{c}$ that
satisfy the detailed balance condition can be computed as above, namely it is
the sum of the number of edges plus the number of degree of freedom provided
by the steady state, namely

$\dim\left(\\{A\in\mathcal{A}_{c}:A\text{ satisfies the detailed balance
condition}\\}\right)=L+L-1=2L-1.$

Therefore, the dimension of the set of matrices in $\mathcal{A}_{c}$ that
satisfy the detailed balance condition is equal to the dimension of the set of
matrices in $\mathcal{A}_{c}$ that satisfy the pathwise detailed balance
condition. This is in agreement with the equivalence of the two properties
(detailed balance and pathwise detailed balance), proven in Lemma 3.10.

We now consider different sets of possible response functions. We assume that
$L=4$ and that the set of measurements is
$\mathcal{M}:=\\{(1,2),(1,3)\\}\subset\\{1,\dots,4\\}^{2}$, i.e. we consider
the response functions

$R_{12}(t)=\langle e_{2},e^{tA}e_{1}\rangle\text{ and }R_{13}(t)=\langle
e_{3},e^{tA}e_{1}\rangle.$

Since we are considering $L=4$ the response function $R_{12}$ is characterized
by the constants $\langle e_{2},Ae_{1}\rangle$, $\langle e_{2}$ ,
$A^{2}e_{1}\rangle,\langle e_{2},A^{3}e_{1}\rangle$ while the response
function $R_{13}$ is characterized by the constants $\langle
e_{3},Ae_{1}\rangle$, $\langle e_{3}$ , $A^{2}e_{1}\rangle,\langle
e_{3},A^{3}e_{1}\rangle$.

Let $F:\mathcal{A}\rightarrow\mathbb{R}^{6}$ be given by

$F:A\mapsto(\langle e_{2},Ae_{1}\rangle,\langle
e_{2},A^{2}e_{1}\rangle,\langle e_{2},A^{3}e_{1}\rangle\langle
e_{3},Ae_{1}\rangle,\langle e_{3},A^{2}e_{1}\rangle,\langle
e_{3},A^{3}e_{1}\rangle)$

Let

$A_{0}=\left(\begin{matrix}&-3&1&1&1\\\ &1&-3&1&1\\\ &1&1&-3&1\\\
&1&1&1&-3\end{matrix}\right).$

We want to show that $F(B_{\delta}(A_{0}))$, with
$B_{\delta}(A_{0})\subset\mathcal{A}_{\emptyset,\\{1,\dots,4\\}^{2}}$ is an
open set in $\mathbb{R}^{6}$. To this end we show that the linearization of
$F$ around $A_{0}$ is of full rank. Notice that

$DF(A_{0})(\xi)=(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})$

where $\xi=(\xi_{12},\xi_{13},\xi_{14},\xi_{23},\xi_{24},\xi_{34})$ is the
vector of the values that characterize a $4\times 4$ symmetric matrix in
$\mathcal{A}$, and where

$\displaystyle x_{1}$ $\displaystyle=\xi_{12},$ $\displaystyle x_{2}$
$\displaystyle=\xi_{13}+\xi_{32}+\xi_{14}+\xi_{42},$ $\displaystyle x_{3}$
$\displaystyle=4\xi_{13}+7\xi_{12}+4\xi_{14}+2\xi_{34}+2\xi_{42}+2\xi_{32},$
$\displaystyle x_{4}$ $\displaystyle=\xi_{13},$ $\displaystyle x_{5}$
$\displaystyle=\xi_{12}+\xi_{23}+\xi_{14}+\xi_{43}$ $\displaystyle x_{6}$
$\displaystyle=7\xi_{13}+3\xi_{12}+3\xi_{14}+3\xi_{34}+2\xi_{42}+3\xi_{32}.$

Then

$DF(A_{0})(\xi)=M\xi$

where

$M:=\left(\begin{matrix}&1&0&0&0&0&0\\\ &0&1&1&1&1&0\\\ &7&4&4&2&2&2\\\
&0&1&0&0&0&0\\\ &1&0&1&1&0&1\\\ &3&7&3&3&2&3\end{matrix}\right)$

Since $\det M\neq 0$ the desired conclusion follows: we can cover an open set
of coefficients of the response functions $R_{12}$ and $R_{13}$ with matrices
that satisfy the detailed balance condition. Hence the fact that a matrix $A$
is symmetric (or satisfies the detailed balance condition) does not impose any
condition on the response functions $R_{12},R_{13}$. It is therefore
convenient to consider the relation between the response functions $R_{12}$
and $R_{21}$, as we do in this paper. Indeed in that case we have that the
detailed balance condition of $A$ imposes strong conditions on the relation
between $R_{12}$ and $R_{21}$. See Proposition 3.9 and Proposition 3.8.

##  9 Concluding remarks

We conclude summarizing the main results that we obtained in this paper and
indicate possible continuations of this work. The aim of this paper is to
understand whether a biochemical network satisfies the detailed balance
condition by performing a couple of reciprocal measurements ($R_{ij}$ and
$R_{ji}$). The advantage of this approach is that it does not require a
detailed knowledge of the rates of the reactions of the network. We proved
that the pathwise detailed balance condition is a necessary condition for
detailed balance to hold. We studied the stability property of the pathwise
detailed balance condition in suitably defined classes of biochemical
networks. Interestingly, it turns out that the stability of the pathwise
detailed balance property is strictly related with the topological properties
of the network, in particular to the existence of cut vertices. In particular,
we proved that if a network does not contain a cut vertex and satisfies the
pathwise detailed balance in a stable way, then it satisfies the detailed
balance condition.

The definition of stability introduced in this paper provides a possible
rigorous definition of robustness for biochemical networks. In this paper we
refer essentially always to the stability of the pathwise detailed balance
condition, but similar definitions of stability/robustness could also be
extended to other contexts. This definition could be modified in order to
adapt to the knowledge available about the evolution of biochemical networks
(see [24]).

Finally, it would be interesting to generalize the results of this paper to
non-linear reaction networks. The challenge in this case is that one cannot
rely on a straightforward connection with graph theory, as we did in this
paper. It would also be interesting to analyse the case in which the set of
states $\Omega$ is not discrete. For instance, it is relevant to determine if
the molecular concentrations in a cell are due to passive transport processes,
like diffusion, or active processes, for instance the action of molecular
motors.

Acknowledgements The authors gratefully acknowledge the support by the
Deutsche Forschungsgemeinschaft (DFG) through the collaborative research
centre "The mathematics of emerging effects" (CRC 1060, Project-ID 211504053).
B. Kepka is funded by the Bonn International Graduate School of Mathematics at
the Hausdorff Center for Mathematics (EXC 2047/1, Project-ID 390685813). E.
Franco and J. J. L. Velázquez are funded by the DFG under Germany’s Excellence
Strategy-EXC2047/1-390685813. The funders had no role in study design,
analysis, decision to publish, or preparation of the manuscript.

## References

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