2.1. Notations

Let $\mathbb{Z}^m_{\geq 0}$ represent the m-dimensional lattice of non-negative integers, $\mathbb{R}^m_{\geq 0}$ the subset of $\mathbb{R}^m$ whose coordinates are all non-negative and $\mathbb{R}^m_{\gt 0}$ the subset of $\mathbb{R}^m$ whose coordinates are positive. The set of $n \times m$ matrices with real entries is written $\mathbb{R}^{n} \otimes \mathbb{R}^m$ , where the transpose of a matrix $A$ or a vector $x$ is denoted $A^{\intercal }$ and $x^{\intercal }$ . For two vectors $x, y \in \mathbb{R}^m$ (viewed as $m \times 1$ column matrices), we denote by $x \cdot y^{\intercal }$ the resulting $m \times m$ matrix product. For a set $U \subseteq \mathbb{R}^m$ , let $C_c ^2(U)$ be the set of twice continuously differentiable functions that have compact support on $U$ . Similarly, $C_c ^\infty (U)$ represents the set of infinitely continuously differentiable functions that have compact support on $U$ . Let $X$ and $Y$ be two random variables; we write $X \overset {d}{=} Y$ if they have the same distribution.

2.2. Reaction networks

In this subsection, we provide a brief introduction to the concepts of RNs and derive both deterministic and stochastic formulations of their dynamics. For a comprehensive treatment of this topic, we refer the reader to [Reference Anderson and Kurtz5, Reference Feinberg14].

Definition 2.1. [ Reference Feinberg14 ] A RN is a triple $(\mathcal{S},\mathcal{C},\mathcal{R}) \,=\!:\, \mathcal{N}$ , where $\mathcal{S}$ is the set of species $\mathcal{S}=\{S_1,\ldots ,S_n\}$ , $\mathcal{C} = \{C_1, \ldots , C_m\}$ is the set of complexes and $\mathcal{R}$ is the set of reactions $\mathcal{R}=\{r_1,\ldots ,r_d\}$ . The set of species is considered as an orthogonal basis in the Euclidean space $\mathbb{R}^n$ , meaning that each species is a basis vector. A complex is a non-negative integer linear combination of species, represented as a vector in $\mathbb{Z}_{\geq 0}^n$ . That is, for each $y \in \mathcal{C}$ , we express

\begin{align*} y = (y_1,\ldots , y_n) = \sum _{i=1}^n y_i S_i. \end{align*}

A reaction is an ordered pair $(y, y^{\prime })\in \mathcal{R}$ with $y,y^{\prime }\in \mathcal{C}$ and is typically written as $y\to y^{\prime }$ . In this reaction, the source complex $y$ is consumed and the product complex $y^{\prime }$ is produced.

The molecularity of a reaction refers to the total number of reactant molecules involved in an elementary reaction step. Specifically, for a reaction $y\to y^{\prime } \in \mathcal{R}$ , the molecularity is given by $|y| = \sum \limits _{i=1}^n y_i$ . Correspondingly, we call such reactions unimolecular, bimolecular, three-molecular, or $p$ -molecular reactions provided $|y| = 1, 2, 3$ , or $p$ . The reaction vector for a reaction $r = y \to y^{\prime }$ is defined as $l_r = l_{y \to y'} = y' - y \in \mathbb{Z}^n$ . The stoichiometric matrix $\Gamma$ is an $n \times d$ matrix, where each column represents the reaction vector of a corresponding reaction.

Assuming the law of mass-action, each reaction $r = y\to y^{\prime }$ is assigned a positive rate constant $\kappa _r = \kappa _{y\to y^{\prime }}$ . The rate constant vector is defined as

\begin{align*} \kappa = (\kappa _1,\ldots , \kappa _d) = (\kappa _{r_1},\ldots , \kappa _{r_d}) \in \mathbb{R}_{\gt 0}^d. \end{align*}

Reaction network dynamics can be modelled in various ways. For large molecule counts, ODEs are commonly used. Let $x(t)$ denote the species concentration vector, a time-dependent function in $\mathbb{R}^n$ . Then, it satisfies

(2.1) \begin{equation} \dot{\kern-1pt x} = F(x) = \sum \limits _{r=1}^d l_r \lambda _r(x) , \end{equation}

where $\lambda _r(x)$ is the kinetics – the reaction rate of $y_r \to y_r^{\prime }$ evaluated at $x \in \mathbb{R}^n_{\geq 0}$ . Under the law of mass-action, the kinetics (mass-action kinetics) with rate constant $\kappa _r$ is written as

(2.2) \begin{align} \lambda _r(x) = \kappa _r x^{y_r} =\kappa _r \prod \limits _{i=1}^n x_i^{y_{i,r}}. \end{align}

In contrast, in a system of small molecule counts, random fluctuations are significant, and thus CTMCs are used to model the dynamics of a RN. The species counts, denoted by $X_t$ , is a $\mathbb{Z}_{\geq 0}^n$ -valued CTMC, satisfying the equation

\begin{align*} X(t) = X(0) + \sum _{r = 1}^d Y_{r} \left(\int _0^t \widetilde {\lambda }_{r} (X(s))\right) l_r, \end{align*}

where $\{Y_1,\ldots , Y_d\}$ is a set of i.i.d. Poisson processes with unit rates, and under the law of mass action, the kinetics in the stochastic modelling with rate constant $\kappa _r$ is given by

\begin{equation*}\widetilde {\lambda }_r(x) = \kappa _r \prod \limits _{i=1}^n \frac {x_i!}{(x_i - y_{i,r})!}1_{\{x_i\geq y_{i,r}\}}.\end{equation*}

Throughout this article, we denote by $(\mathcal{N}, \kappa )$ for an RN $\mathcal{N}$ equipped with mass-action kinetics specified by the rate constant vector $\kappa$ .

We conclude this subsection by introducing the definition of subnetworks.

2.3. Langevin approximation

In practice, the CTMCs are typically studied through Monte Carlo simulations. Since the discrete stochastic simulations are generally computationally expensive, diffusion approximations, such as linear noise approximation (LNA) [Reference van Kampen57] and LA [Reference Kurtz35, Reference Kurtz36], are commonly used in numerical computation. Recently, a CLA [Reference Anderson, Higham, Leite and Williams4, Reference Leite and Williams40] was developed to address issues arising when the approximating process in LNA or LA reaches the boundary of the positive orthant. In this article, we focus on the identifiability of LA for stochastically modelled RNs.

As a frequently used diffusion approximation for stochastically modelled RNs, the LA is derived from the chemical Langevin equation (CLE) [Reference Gillespie54] and Itô-type SDE. The LA provides an accurate approximation for stochastic RNs up to the first hitting time of the boundary of the positive orthant and is preferred for capturing nonlinear effects in random fluctuations. Specifically, for an RN $(\mathcal{N}, \kappa )$ , the LA satisfies the following $n$ -dimensional CLE:

(2.3) \begin{align} d X_t = A(X_t) dt + \sigma (X_t) dW_t, \end{align}

for all $t \gt 0$ with an initial value $X_0 \in \mathbb{R}_{\gt 0}^n$ , where $A \colon \mathbb{R}^n \to \mathbb{R}^n$ is the drift vector, $\sigma \colon \mathbb{R}^n \to \mathbb{R}^{n} \otimes \mathbb{R}^{n}$ is the diffusion coefficient and $W$ is a $n$ -dimensional Brownian motion. In the context of LA for stochastic RNs, $A$ is given by

(2.4) \begin{equation} A_i(x) = \sum _{r=1}^d \Gamma _{i,r}\lambda _r(x) = \sum _{r=1}^d (y'_{i,r} - y_{i,r}) \kappa _r x^{y_r}, \quad i = 1,\ldots , n; \end{equation}

and $\sigma$ is given as the unique positive semi-definite square root (existence and uniqueness of square roots for any symmetric positive semi-definite matrix is well established; cf. [Reference Bhatia7].) of the diffusion matrix $B$ , with entries

(2.5) \begin{equation} B_{i,j}(x)= (\sigma (x) \sigma (x)^{\intercal })_{i,j} = \sum _{r=1}^d \Gamma _{ir}\Gamma _{jr} \lambda _r(x) = \sum _{r = 1}^d (y'_{i,r} - y_{i,r}) (y'_{j,r} - y_{j,r}) \kappa _r x^{y_r}, \quad i, j = 1, \ldots , n. \end{equation}

Additionally, we recall that in (2.4) and (2.5), $\Gamma$ denotes the stoichiometric matrix, and $\lambda$ represents the mass-action kinetics defined in (2.2). In particular, when the initial state is important, we use the notation $X_t(x)$ to denote the solution to (2.3) with $X_0 = x$ .

3.1. Linear conjugacy

In the context of RNs, the concept of linear conjugacy was introduced in [Reference Johnston and Siegel29] for deterministically modelled RNs, borrowing ideas from dynamical systems theory [Reference Perko50]. This notion was further developed in subsequent works [Reference Johnston, Siegel and Szederkényi30, Reference Johnston, Siegel and Szederkényi31, Reference Nazareno, Eclarin, Mendoza and Lao46], among others. Earlier work on conjugacy in detailed balanced RNs can be found in [Reference Krambeck34]. Intuitively, two RNs are said to be linearly conjugated if there exists a linear mapping that transforms the trajectories of one system into those of the other. This property is of particular interest because qualitative features of mass-action systems, such as local stability, multistability, or persistence, are preserved under linear conjugacy. Our goal is to adapt the concept of linear conjugacy and extend the results of [Reference Johnston and Siegel29] to the stochastic setting.

Analogous to [Reference Johnston and Siegel29], the following theorem provides a necessary and sufficient condition for the linear conjugacy of two distinct RNs that share the same set of species. The proof is deferred to Subsection 7.4.

Theorem 3.7. Let $\mathcal{N}$ and $\mathcal{N}'$ be two RNs that share a common set of species and $\mathcal{R} \neq \mathcal{R}'$ . Then, there exist rate constant vectors $\kappa$ and $\kappa '$ such that $(\mathcal{N}, \kappa )$ and $(\mathcal{N}', \kappa ')$ are linearly conjugated w.r.t. their SDEs if and only if there exists a matrix $G$ of the form $G = DP$ , where $D$ is a positive diagonal matrix and $P$ is a permutation matrix, such that $\mathcal{N}$ and $\mathcal{N}'$ have the same complexes up to permutation by P, and such that for every source complex $y$ of $\mathcal{N}$ it holds that:

\begin{align*} \mathrm {Cone}_{\mathcal{R}} (y) \cap \mathrm {Cone}^G_{\mathcal{R}'} (y) \neq \emptyset , \end{align*}

where $\mathrm {Cone}_{\mathcal{R}}$ is given by (3.2), and

\begin{align*} \mathrm {Cone}^G_{\mathcal{R}'} (y) \,:\!=\, \left \{\sum _{y\to y'\in \mathcal{R}'}\alpha _{y\to y'} \left (G(y' - y),G (y' - y) \cdot (y' - y)^{\intercal } G^{\intercal }\right )\bigg |\, \alpha _{y\to y'}\gt 0\right \} . \end{align*}

Suppose the matrix $G$ in Theorem3.7 consists of only positive scaling. Then, $G$ is a positive diagonal matrix, with diagonal entries $\{c_1, \ldots , c_n\} \subseteq \mathbb{R}_{\gt 0}$ . The following proposition describes the relationship between the rate constant vectors of linearly conjugate RNs w.r.t. their corresponding SDEs. The proof is presented in Subsection 7.5.

Proposition 3.8. Let $\mathcal{N} = (\mathcal{S}, \mathcal{C}, \mathcal{R})$ and $\mathcal{N}' = (\mathcal{S}, \mathcal{C}', \mathcal{R}')$ be two RNs with the same set of species and source complexes. Suppose that for the rate constant vector $\kappa \in \mathbb{R}_{\gt 0}^d$ , there exists a positive vector $\beta \in \mathbb{R}_{\gt 0}^{d'}$ and a positive diagonal matrix $G$ with diagonal entries $\{c_1, \ldots , c_n\} \subseteq \mathbb{R}_{\gt 0}$ , such that for every source complex $y$ :

(3.5) \begin{align} \sum _{y' \colon y\to y'\in \mathcal{R}}\kappa _{y\to y'}(y' - y) =\sum _{y' \colon y\to y'\in \mathcal{R}'}\beta _{y\to y'} G (y' - y) \end{align}

and

(3.6) \begin{align} \sum _{y' \colon y\to y'\in \mathcal{R}}\kappa _{y\to y'}(y' - y) \cdot (y' - y)^{\intercal } = \sum _{y' \colon y\to y'\in \mathcal{R}'} \beta _{y\to y'} G (y' - y) \cdot (y' - y)^{\intercal } G^{\intercal }. \end{align}

Then, $(\mathcal{N}, \kappa ')$ and $(\mathcal{N}{{'}}, \kappa ')$ are linearly conjugated w.r.t. their SDEs, where for all $y \to y' \in \mathcal{R}'$ ,

(3.7) \begin{equation} \kappa '_{y\to y'} \,:\!=\, \beta _{y\to y'} c^y = \beta _{y\to y'} \prod _{j=1}^n c_j^{y_{j}}. \end{equation}

Example 3.10. Consider the RNs:

\begin{equation*}\mathcal{N}:\quad S_1\xrightarrow {\alpha } 3S_1 \quad \text{and } \quad \mathcal{N'}: \quad S_1\xrightarrow {\alpha '} 2S_1 .\end{equation*}

Following Theorem 3.3 , these RNs are not confoundable w.r.t. their SDEs since we cannot find $\alpha \gt 0$ and $\alpha '\gt 0$ such that:

\begin{equation*} 2\alpha = \alpha ' \quad \text{and } \quad 4\alpha = \alpha ' . \end{equation*}

In order for ( $\mathcal{N},\alpha$ ) and ( $\mathcal{N'},\alpha '$ ) to be linearly conjugated, we need to find $\alpha \gt 0, \alpha '\gt 0$ and $c_1\gt 0$ such that 3.5 and 3.6 are verified, i.e.:

\begin{equation*} 2\alpha = c_1 \alpha ' \quad \text{and } \quad 4\alpha = c_1^2 \alpha ' . \end{equation*}

It is verified for $\alpha = \alpha '$ and $c_1=2$ . It follows that ( $\mathcal{N},\alpha$ ) and ( $\mathcal{N'},\alpha '$ ) are linearly conjugated.

Similarly as for reaction identifiability and confoundability (see Section 5), one can easily obtain the following lemma.

4.1. Special cases of reaction-identifiable RNs

1. (1) Consider RNs, which only consist of the following types of reactions:

   \begin{equation*}S_i\rightarrow S_j, \quad S_i\rightarrow \emptyset , \quad \emptyset \rightarrow S_i ,\end{equation*}
   with chemical species $S_i$ , $i=1,\ldots ,n$ .

   For each complex, the associated reaction vectors are linearly independent, and it follows from [Reference Craciun and Pantea10, Theorem 3.2], that their corresponding RNs always have uniquely identifiable rate constants w.r.t. their ODEs and thus their SDEs. This property extends easily to k-unary chemical reaction networks [Reference Laurence and Robert39], with reactions of the types:

   \begin{equation*}k_i S_i\rightarrow k_j S_j, \quad k_i S_i\rightarrow \emptyset , \quad \emptyset \rightarrow k_i S_i ,\end{equation*}
   for $1 \leq i \neq j \leq N$ , where $k = (k_i)_{1\leq i \leq n} \in \mathbb{Z}^n_{\gt 0}$ .

2. (2) Simple RNs with only one species, have also uniquely identifiable rate constants w.r.t. the SDE, if and only if each source complex is a reactant in at most two reactions. Otherwise, the associated set of vectors in (3.1) will be constituted of three or more vectors in $\mathbb{R}^2$ and will then be linearly dependent.

3. (3) If each reaction in a RN has a distinct source complex, then the corresponding reaction vectors are automatically linearly independent, which in turn guarantees identifiability w.r.t. the ODE.

4.2. Counterexample

Given that RNs with complexes of molecularity of at most one are uniquely identifiable w.r.t. their SDEs, one might expect the same for RNs with reactants of molecularity at most one. The following examples show this is not the case.

5.1. More on SDEs

We start with introducing key concepts of SDEs relevant to this paper and refer the reader to, e.g. [Reference Ikeda and Watanabe28, Reference Øksendal48] for a more systematic study of the topic.

In general, the SDE (2.3) may not admit a unique solution for all $t \gt 0$ . One issue arises from the fact that the diffusion coefficient $\sigma (x) = \sqrt {B (x)}$ may fail to be Lipschitz continuous at the boundary of the positive orthant. In the one-dimensional case, this problem has been addressed in the seminal work [Reference Yamada and Watanabe61], which establishes the strong uniqueness for an SDE with an $\alpha$ -Hölder continuous diffusion coefficient when $\alpha \geq 1/2$ . However, in higher dimensions, where the equation under consideration is likely to reside, uniqueness is no longer guaranteed, as demonstrated in [Reference Swart52].

One possible remedy is to restrict the initial condition to lie strictly within the positive orthant. Even then, a second issue remains, that is the potential explosion of solutions in finite time. Since $\sigma (x)$ exhibits polynomial growth as $|x| \to \infty$ , it may blow up in finite time. This phenomenon is well understood in the one-dimensional setting [Reference Feller16] and has been generalised to higher dimensions in [Reference Khas’minskiĭ32]. We also note that the issue of (non-)explosion of RNs modelled by CTMCs has been investigated in [Reference Anderson, Cappelletti, Koyama and Kurtz2].

Nevertheless, both the drift $A$ and the diffusion coefficient $\sigma$ are Lipschitz continuous on compact subsets of the positive orthant and hence the following result holds, see, e.g. [Reference Ikeda and Watanabe28]. We begin with the standard definition of solutions to SDEs.

This is sufficient for our investigation of identifiability. Moreover, the solution $(X_{t\wedge \tau }(x))_{t \geq 0}$ is a homogeneous Markov process. In the following, we introduce the concept of the infinitesimal generator for homogeneous Markov processes.

Definition 5.3 ([Reference Øksendal48], Definition 7.3.1). Let $(X_t)_{t \geq 0}$ be a homogeneous Markov process with state space $D \subseteq \mathbb{R}^n$ . The infinitesimal generator (or simply the generator) of $X$ is an operator $L$ on $C_c^{\infty } (D)$ , such that for all $f \in C_c^{\infty } (D)$ and $x \in D$ ,

(5.1) \begin{align} L f(x) \,:\!=\, \lim _{t \to 0} \frac {\mathbb{E}[f(X_t(x))] - f(x)}{t}. \end{align}

For a general Markov process, the generator may not always exist—additional properties, such as strong continuity of the process, are required to ensure that the limit in (5.1) exists. In this paper, we focus on the LA for RNs, where the generator exists and takes the form in (5.2).

Then, [Reference Ethier and Kurtz13, Proposition 1.7 in Chapter 4] asserts the following theorem.

Theorem 5.4. Let $(X_t)_{t \geq 0}$ be a Markov process with generator $L$ . Then, it solves the martingale problem for $L$ , namely, for every $f \in C_c^{\infty } (\mathbb{R}^d)$ ,

\begin{align*} M_t \,:\!=\, f(X_t) - \int _0^t L f(X_s) ds \end{align*}

is a martingale adapted to the filtration generated by $(X_t)_{t \geq 0}$ .

If the Markov process $(X_t)_{t \geq 0}$ is the solution to an Itô equation (2.3), with locally Lipschitz continuous coefficients $A$ and $\sigma$ on $\mathbb{R}_{\gt 0}^n$ , then its infinitesimal generator can be expressed as follows:

(5.2) \begin{equation} Lf(x) = \langle A(x), \nabla \rangle f{(x)}+{\frac {1}{2}} (\nabla ^{\intercal } \cdot B (x) \cdot \nabla ) f{(x)} = \sum _{i = 1}^n A_i(x)\frac {\partial }{\partial x_i}f(x)+\frac {1}{2}\sum _{i,j = 1}^n B_{i,j}(x)\frac {\partial ^2}{\partial x_i\partial x_j}f(x), \end{equation}

for all test functions $f\in C^{\infty }_c(\mathbb{R}_{\gt 0}^n)$ , cf. [Reference Øksendal48, Theorem 7.3.3].

The following theorem, concerning the equivalence of martingale problems and SDEs, was first established in [Reference Stroock and Varadhan51] and further studied in [Reference Kurtz38]. The equivalence is given in terms of existence of SDEs through a weak solution, i.e. a filtered probability space and stochastic process that satisfies the SDE in its integral form [Reference Øksendal48, Section 5.5].

Example 5.7. In order to exemplify notation, we again consider Example 1.1 :

\begin{equation*}2S\xleftarrow {1}\emptyset \xrightarrow {4}S\xrightarrow {1} \emptyset \xrightarrow {2}3S , \end{equation*}

\begin{equation*}2S\xleftarrow {4}\emptyset \xrightarrow {1}S \xrightarrow {1} \emptyset \xrightarrow {1}3S .\end{equation*}

The stoichiometric matrix is a row vector: $\Gamma = ( 2,1,-1,3)$ , which simplifies the calculation of both the drift vector and the diffusion matrix, that are $A(x)=12-x$ and $ B(x)=x+26$ for all $x \in \mathbb{R}_{\gt 0}$ . Therefore, the LA for the RN is

\begin{align*} d X_t = (12 - X_t) dt + \sqrt {X_t + 26} d W_t; \end{align*}

and the infinitesimal generator associated to the RN is

\begin{align*} L f(x) = (12 - x) \frac {d}{d x} f(x) + \frac {1}{2} (x + 26) \frac {d^2}{d x^2} f(x). \end{align*}

5.2. Reaction identifiability and confoundability given generator

In analogy to [Reference Craciun and Pantea10], we have the following result, whose proof is deferred to Subsection 7.2.

However, the converse is not necessarily true, as demonstrated in the following example.

Example 5.12. Consider the following birth and death reaction network:

\begin{equation*}\emptyset \xleftarrow {\kappa _1} S\xrightarrow {\kappa _2} 2S,\end{equation*}

which is used to describe the evolution of cancer cells apart from other applications in biology [ Reference Novozhilov, Karev and Koonin47 ].

There is only one source complex $y = S$ , and we have that the set of vectors defined in ( 3.1 ), i.e. $ \{ (1,1)^{\intercal }, (-1,1)^{\intercal }\}$ are linearly independent. Therefore, according to Theorem 5.9 , this RN is reaction-identifiable w.r.t. its generator. However, if we consider $(\kappa _1,\kappa _2) = (1.5,1)$ and $(\kappa _1',\kappa _2') = (2,1.5)$ , then both reaction rates yield the same ODE system:

\begin{align*} \dot{\kern-1pt x} = x-1.5x = 1.5x-2x=-0.5x. \end{align*}

It follows that the RN is not reaction-identifiable w.r.t. its ODE.

If an RN is not reaction-identifiable w.r.t. its generator, then it remains non-reaction-identifiable even with adding additional reactions.

5.3. Confoundability of RNs given generators

In the previous subsection, we consider a given RN and its generator.

In this subsection, we investigate the confoundability of RNs given the generator.

The proof of Theorem5.15 can be found in Subsection 7.3.

Example 5.16. Consider the following two different RNs:

\begin{equation*}S\xleftarrow {5}\emptyset \xrightarrow {1}4S, \quad S\xrightarrow {1}\emptyset , \end{equation*}

\begin{equation*}2S\xleftarrow {3}\emptyset \xrightarrow {1}3S, \quad S\xrightarrow {1}\emptyset .\end{equation*}

Again both RNs give the same ODE

\begin{equation*}\frac {ds}{dt}=9-s.\end{equation*}

Despite the RNs being different, there exists two sets of reaction rates for which the ODEs are the same. Hence, the RN structure cannot be distinguished from the ODE alone. This is called (ODE)-confoundable in [ Reference Craciun9 ]. Furthermore, both RNs also have their SDEs given by the same drift vector and diffusion matrix

\begin{equation*}A(s)=9-s,\quad B(s)=s+21.\end{equation*}

These RNs are thus confoundable w.r.t. their generator. As the corresponding SDE is regular enough, we conclude that these RNs are confoundable w.r.t. the SDE.

Similar to Lemma 5.11, we state the following lemma.

Not surprisingly, the converse of Lemma 5.18 does not hold, as demonstrated in Remark 5.17.

By the definition of the generator, and in a manner similar to Lemma 5.14, the following extension result holds.

7.1. Proof of Theorem3.3

With Theorems5.9 and 5.15 in hand, the validity of Theorem3.3 reduces to establishing the following proposition.

Proof. (1) $\implies$ : Suppose that $\mathcal{N}$ is not reaction-identifiable w.r.t. its generator. Then, there exists distinct rate constant vectors $\kappa$ and $\kappa '$ in $\mathbb{R}^{d}_{\gt 0}$ associated with the same generator on some non-empty bounded open subset $U$ . By Theorem5.6 – the equivalence of the martingale problem and the SDE – we know that the stochastic processes associated to $\kappa$ and $\kappa '$ are weak solutions to the same SDE. Additionally, using Theorem5.2, solutions to this SDE are unique in law up to the stopping time $\tau = \inf \{t \gt 0, X_{t} \notin U\}$ . That is, $\kappa$ and $\kappa '$ provide two RNs with the identical distribution when starting from the same initial state. Hence, $\mathcal{N}$ is not reaction-identifiable w.r.t. its SDE. In other words, $\mathcal{N}$ is reaction-identifiable w.r.t. its generator whenever it is reaction-identifiable w.r.t. its SDE.

$\impliedby$ : Suppose that $\mathcal{N}$ is not reaction-identifiable w.r.t. its SDE, then there exists $\kappa \neq \kappa ' \in \mathbb{R}^{d}_{\gt 0}$ such that the two stochastic processes $(X_t)_{t \geq 0}$ and $(X'_t)_{t \geq 0}$ associated to $(\mathcal{N},\kappa )$ and $(\mathcal{N},\kappa ')$ have the same distribution, when stopped upon reaching the boundary of $U$ . Then, the corresponding generators coincide, as by Definition 5.3,

(7.1) \begin{align} L f(x) = \lim _{t \downarrow 0}\frac {\mathbb{E} [f(X_t (x))] - f(x)}{t} = \lim _{t \downarrow 0}\frac {\mathbb{E} [f(X'_t (x))] - f(x)}{t} = L' f(x), \end{align}

for any $x \in U$ , and test function $f \in C_c^{\infty } (U)$ . This ensures that $\mathcal{N}$ is not reaction-identifiable w.r.t. its generator.

(2) $\implies$ : Suppose that $\mathcal{N}$ and $\mathcal{N}'$ are confoundable w.r.t. their generators. Then there exists $\kappa \in \mathbb{R}^{d}_{\gt 0}$ and $\kappa ' \in \mathbb{R}^{d'}_{\gt 0}$ , such that $(\mathcal{N},\kappa )$ and $(\mathcal{N}',\kappa ')$ have same generator on some non-empty bounded open subset $U$ . By Theorem5.6 – the equivalence of the martingale problem and the SDE – we know that the stochastic processes associated to $(\mathcal{N},\kappa )$ and $(\mathcal{N}',\kappa ')$ are weak solutions to the same SDE, and therefore share the same law up to the stopping time $\tau = \inf \{t \gt 0, X_{t} \notin U\}$ due to Theorem5.2. This proves that $\mathcal{N}$ and $\mathcal{N}'$ are confoundable w.r.t. their SDEs. Therefore, $\mathcal{N}$ and $\mathcal{N}'$ are unconfoundable w.r.t. their generators, whenever, they are unconfoundable w.r.t. their SDEs.

$\impliedby$ : Suppose that $\mathcal{N}$ and $\mathcal{N}'$ are confoundable w.r.t. their SDEs, then there exists $\kappa \in \mathbb{R}^{d}_{\gt 0}$ and $\kappa ' \in \mathbb{R}^{d'}_{\gt 0}$ such that the two stochastic processes $(X_t)_{t \geq 0}$ and $(X'_t)_{t \geq 0}$ associated to $(\mathcal{N},\kappa )$ and $(\mathcal{N}',\kappa ')$ have same distribution, when stopped upon reaching the boundary of $U$ . Thus, equation (7.1) holds for any $x \in U$ , and test function $f \in C_c^{\infty } (U)$ . This ensures that $\mathcal{N}$ and $\mathcal{N}'$ are confoundable w.r.t. their generators. The proof of this proposition is complete.

7.2. Proof of Theorem5.9

We will need the following elementary fact.

Lemma 7.2. Let $Y\subset \mathbb Z_{\ge 0}^n$ be a finite set and let $v_y\in \mathbb R^m$ be vectors for $y\in Y$ . If

\begin{align*} \sum _{y\in Y} v_y\,x^y = 0 \qquad \text{for all } x\in U,\quad U\subseteq \mathbb R^n\text{ open}, \end{align*}

then $v_y=0$ for all $y\in Y$ .

Proof of Theorem 5.9. $(1)\implies (2)$ : Let $\mathcal{N}$ be an RN such that $(2)$ fails. Then, there exist rate constant vectors $\kappa \neq \kappa '$ , whose corresponding generators coincide on a nonempty open set $U$ . Recall that the generators take the form (5.2). The fact that two generators coincide on $C_c^{\infty } (U)$ is equivalent to their coefficients $A_i$ and $B_i$ , which are polynomials, coinciding on $U$ . Since a nonzero polynomial cannot vanish on a nonempty open set, it follows that the coefficients (and hence the generators) coincide on all of $\mathbb{R}^n_{\gt 0}$ (resp. on $C^\infty _c(\mathbb{R}^n_{\gt 0})$ ), see Lemma 7.2. As a consequence, $\mathcal{N}$ is not identifiable w.r.t. its generator; that is $(1)$ fails. This proves the implication $(1) \implies (2)$ .

Notice that in the above argument, we do not actually require the underlying RNs with rate constant vectors $\kappa$ and $\kappa '$ to share the same network structure. Therefore, the proof still justifies Remark 5.10 without any modification.

$(2) \implies (3)$ : Assume that on an open subset $U$ as given the generators are different for all $\kappa \neq \kappa '$ , but that there is a source complex $y \in \mathcal{C}$ such that the set of vectors

\begin{equation*}\left \{\left (y' - y,(y' - y)\cdot (y' - y)^{\intercal }\right ) \big | \, y\to y'\in \mathcal{R}\right \}\end{equation*}

is linearly dependent. Then, there exist some real numbers $\alpha _{y\to y'}$ , not all zero such that

\begin{align*} \left (\sum _{y' \colon y \to y'\in \mathcal{R}} \alpha _{y\to y'}(y' - y),\sum _{y' \colon y \to y'\in \mathcal{R}} \alpha _{y\to y'} (y' - y)\cdot (y' - y)^{\intercal } \right ) = 0. \end{align*}

Hence, we can choose rate constant vectors $\kappa ,\kappa ' \in \mathbb{R}^{d}_{\gt 0}$ such that $\kappa _{y \to y'}- \kappa '_{y \to y'} = \alpha _{y \to y'}$ for all coefficients of reactions $y \to y'\in \mathcal{R}$ and $\kappa _{R} = \kappa '_{R} = 1$ for other reactions $R \in \mathcal{R}$ . Then, recalling (2.4), (2.5), and (5.2), for these rate constant vectors, with $L$ and $L'$ denoting the generator with $k$ and $k'$ , respectively,

\begin{align*} L f(x) - L' f(x) = & \sum _{y\to y'\in \mathcal{R}} \left \langle (y' - y)(\kappa _{y\to y'}-\kappa '_{y\to y'})x^y, \nabla \right \rangle f \\ & + \frac {1}{2} \sum _{y\to y'\in \mathcal{R}} \left (\nabla ^{\intercal } \cdot (y' - y)\cdot (y' - y)^{\intercal }(\kappa _{y\to y'}-\kappa '_{y\to y'}) x^y \cdot \nabla \right ) f = 0. \end{align*}

This contradicts with our assumption, and thus verifies $(2) \implies (3)$ .

$(3) \implies (1)$ : Assume that for all source complexes $y \in \mathcal{C}$ the set of vectors

\begin{equation*}\left \{\left (y' - y,(y' - y)\cdot (y' - y)^{\intercal }\right ) \big | \, y\to y'\in \mathcal{R} \right \}\end{equation*}

is linearly independent. Let $\kappa ,\kappa '\in \mathbb{R}^d_{\gt 0}$ such that $L f(x) - L' f(x) = 0$ , for all test functions $f\in C^{\infty }_c(\mathbb{R}_{\gt 0}^n)$ and $x \in R_{\gt 0}^n$ . In other words,

\begin{equation*}\sum _{y \in \mathcal{C}} \left (\sum _{y' \colon y\to y'\in \mathcal{R}}(y' - y)(\kappa _{y\to y'} - \kappa '_{y\to y'})x^y,\sum _{y' \colon y\to y'\in \mathcal{R}}(y' - y)\cdot (y' - y)^{\intercal }(\kappa _{y\to y'}-k'_{y\to y'})x^y \right )= 0,\end{equation*}

where the right-hand side is the zero-function $\mathbb{R}^n\to \mathbb{R}^{n}\times \mathbb{R}^n \otimes \mathbb{R}^n$ . This yields that

\begin{equation*}\sum _{y\in \mathcal{C}}x^y\sum _{y' \colon y\to y'\in \mathcal{R}} (\kappa _{y\to y'}-\kappa '_{y\to y'}) \left ((y' - y),(y' - y)\cdot (y' - y)^{\intercal }\right )= 0.\end{equation*}

Since these vectors are polynomials in $x$ vanishing on $\mathbb{R}^n_{\gt 0}$ , their polynomial coefficients must be zero. Therefore, for each source complex $y\in \mathcal{C}$ ,

\begin{equation*} \sum _{y' \colon y\to y'\in \mathcal{R}}(\kappa _{y\to y'}-\kappa '_{y\to y'}) \left ((y' - y),(y' - y)\cdot (y' - y)^{\intercal }\right )= 0 . \end{equation*}

By the linear independence, we get that $\kappa _{y\to y'}-\kappa '_{y\to y'}=0$ . This concludes the proof of $(3) \implies (1)$ . The proof of this theorem is complete.

7.3. Proof of Theorem5.15

First, note that the condition $\textrm {Cone}_{\mathcal{R}}(y)\cap \textrm {Cone}_{\mathcal{R}'}(y)\neq \emptyset$ for all $y\in \mathcal{C}\cup \mathcal{C}'$ implies that $\mathcal{N}$ and $\mathcal{N}'$ have the same source complexes: if $y$ is not a source complex of $\mathcal{R}$ , then $\textrm {Cone}_{\mathcal{R}}(y)=\{0\}$ by convention on the empty sum (and similarly for $\mathcal{R}'$ ), but the coefficients of the coordinates in $\textrm {Cone}_{\mathcal{R}'}(y)\subset \mathbb{R}^n \otimes \mathbb{R}^n$ from (3.2) are positive, so the intersection is empty. That is the difference to the ODE case of the proof of [Reference Craciun and Pantea10, Theorem 4.4] that was remarked in [Reference Szederkényi53]. Thus, it suffices to prove the equivalence between generator confoundability and the cone condition.

$\implies$ : Assume $\mathcal{N}$ and $\mathcal{N}'$ are confoundable with respect to their generators. Then there exist rate constant vectors $\kappa \in \mathbb R^{|\mathcal{R}|}_{\gt 0}$ and $\kappa '\in \mathbb R^{|\mathcal{R}'|}_{\gt 0}$ such that, for all $x\in \mathbb R^n_{\gt 0}$ ,

(7.2) \begin{equation} \begin{cases} \displaystyle \sum _{y\in \mathcal{C}\cup \mathcal{C}'} \Bigg ( \sum _{\substack {y':\,y\to y'\in \mathcal{R}}} (y'-y)\,\kappa _{y\to y'} - \sum _{\substack {y':\,y\to y'\in \mathcal{R}'}} (y'-y)\,\kappa '_{y\to y'} \Bigg )x^y = 0,\\[15pt] \displaystyle \sum _{y\in \mathcal{C}\cup \mathcal{C}'} \Bigg ( \sum _{\substack {y':\,y\to y'\in \mathcal{R}}} (y'-y)\cdot (y'-y)^{\intercal }\,\kappa _{y\to y'} - \sum _{\substack {y':\,y\to y'\in \mathcal{R}'}} (y'-y)\cdot (y'-y)^{\intercal }\,\kappa '_{y\to y'} \Bigg )x^y = 0. \end{cases} \end{equation}

Apply Lemma 7.2 to the first (with $m=n$ ) and the second lines simultaneously. We obtain, for every $y\in \mathcal{C}\cup \mathcal{C}'$ ,

(7.3) \begin{equation} \begin{cases} \displaystyle \sum _{\substack {y':\,y\to y'\in \mathcal{R}}} (y'-y)\,\kappa _{y\to y'} = \sum _{\substack {y':\,y\to y'\in \mathcal{R}'}} (y'-y)\,\kappa '_{y\to y'},\\[15pt] \displaystyle \sum _{\substack {y':\,y\to y'\in \mathcal{R}}} (y'-y)\cdot (y'-y)^{\intercal }\,\kappa _{y\to y'} = \sum _{\substack {y':\,y\to y'\in \mathcal{R}'}} (y'-y)\cdot (y'-y)^{\intercal }\,\kappa '_{y\to y'}. \end{cases} \end{equation}

Fix a source complex $y$ . The pair determined by the left-hand sides of (7.3) belongs to $\textrm {Cone}_{\mathcal{R}}(y)$ by definition, while the equal pair determined by the right-hand sides belongs to $\textrm {Cone}_{\mathcal{R}'}(y)$ . Hence,

\begin{align*} \textrm {Cone}_{\mathcal{R}}(y)\cap \textrm {Cone}_{\mathcal{R}'}(y)\neq \emptyset \qquad \text{for every source complex } y. \end{align*}

$\impliedby$ : Assume now that $\mathcal{N}$ and $\mathcal{N}'$ have the same source complexes and that for each source complex $y$ , $\textrm {Cone}_{\mathcal{R}}(y)\cap \textrm {Cone}_{\mathcal{R}'}(y)\neq \emptyset$ . For each such $y$ , choose an element of the intersection and corresponding positive rate subvectors $\kappa ^{(y)}$ (for reactions in $\mathcal{R}$ with source $y$ ) and $\kappa '^{(y)}$ (for reactions in $\mathcal{R}'$ with source $y$ ) such that

(7.4) \begin{equation} \begin{cases} \displaystyle \sum _{\substack {y':\,y\to y'\in \mathcal{R}}} (y'-y)\,\kappa ^{(y)}_{y\to y'} = \sum _{\substack {y':\,y\to y'\in \mathcal{R}'}} (y'-y)\,\kappa '^{(y)}_{y\to y'},\\[15pt] \displaystyle \sum _{\substack {y':\,y\to y'\in \mathcal{R}}} (y'-y)\cdot (y'-y)^{\intercal }\,\kappa ^{(y)}_{y\to y'} = \sum _{\substack {y':\,y\to y'\in \mathcal{R}'}} (y'-y)\cdot (y'-y)^{\intercal }\,\kappa '^{(y)}_{y\to y'}. \end{cases} \end{equation}

Define global rate constant vectors $\kappa \in \mathbb R^{|\mathcal{R}|}_{\gt 0}$ and $\kappa '\in \mathbb R^{|\mathcal{R}'|}_{\gt 0}$ by setting, for each reaction $y\to y'$ ,

\begin{align*} \kappa _{y\to y'} \,:\!=\, \kappa ^{(y)}_{y\to y'}, \qquad \kappa '_{y\to y'} \,:\!=\, \kappa ^{\prime (y)}_{y\to y'}. \end{align*}

This is well defined because each reaction has a unique source complex $y$ . Then, (7.4) implies (7.3) for every source complex $y$ . Multiplying (7.3) by $x^y$ and summing over $y$ yields (7.2) for all $x\in \mathbb R^n_{\gt 0}$ . Therefore, $\mathcal{N}$ and $\mathcal{N}'$ are confoundable with respect to their generators.

7.4. Proof of Theorem3.7

The proof is similar in structure to that of Theorem5.15. First, $(\mathcal{N}, \kappa )$ and $(\mathcal{N}', k')$ are linear conjugates w.r.t. their SDEs, if and only if there exists a linear mapping $h'(x) = G x$ , such that, for any bounded open subset $U \subseteq \mathbb{R}_{\gt 0}^n$ (3.3) holds. Applying the same reasoning as in Theorem5.15, we conclude that $\mathcal{N}$ and $\mathcal{N}'$ must have the same source complexes to be linear conjugate. On the other hand, [Reference Johnston and Siegel29, Lemma 3.1] asserts the matrix $G$ consists of at most positively scaling and reindexing coordinates. Additionally, $h'(X_{t \wedge \tau })$ is a Markov process solving the martingale problem for the generator

\begin{equation*} L^{h'} f(x) = \langle G A(x), \nabla \rangle f + (\nabla ^{\intercal } \cdot G B G^{\intercal } (x) \cdot \nabla ) f . \end{equation*}

Thus, (3.3) is equivalent to equations

\begin{equation*} \begin{cases} & \left (\sum \limits _{y' \colon y\to y'\in \mathcal{R}}(y' - y) \kappa _{y\to y'} - \sum \limits _{y' \colon y\to y'\in \mathcal{R}'}G(y' - y)\kappa '_{y\to y'} \right ) x^y = 0, \\[15pt] & \left (\sum \limits _{y' \colon y\to y'\in \mathcal{R}} (y' - y)\cdot (y' - y)^{\intercal } \kappa _{y\to y'}-\sum \limits _{y' \colon y\to y'\in \mathcal{R}'}G(y' - y)\cdot (y' - y)^{\intercal }G^{\intercal } \kappa '_{y\to y'} \right ) x^y = 0, \end{cases} \end{equation*}

for every source complex $y$ and state $x \in U$ , which holds if and only if $ \textrm {Cone}_{\mathcal{R}} (y) \cap \textrm {Cone}^{G}_{\mathcal{R}'} (y) \neq \emptyset$ . This completes the proof of this theorem.

7.5. Proof of Proposition 3.8

Let $U \subseteq \mathbb{R}_{\gt 0}^n$ be an arbitrary bounded open set. Let $X_{t \wedge \tau } (x)$ and $X_{t\wedge \tau '}' (z)$ be solutions to the LAs for $(\mathcal{N}, \kappa )$ and $(\mathcal{N}', \kappa ')$ , starting at $x \in U$ and $z = h(x) \,:\!=\, G^{-1} x \in h(U)$ , respectively, up to stopping times given by (3.4). Thus, concerning (2.4), (2.5), employing Itô’s formula, and recalling that $G$ is a diagonal matrix, we can write

\begin{align*} f \big (h \big (X_{t\wedge \tau }(x)\big )\big ) & =\, f(z) + \sum _{i = 1}^n \int _0^{t\wedge \tau } \sum _{r = 1}^{d} c_i^{-1} (y'_{i,r} - y_{i,r} ) \kappa _r X_{t\wedge \tau }(x)^{y_r} f^{(i)} \big (h \big (X_{t\wedge \tau }(x)\big )\big ) \\ & \quad + \frac {1}{2}\sum _{i,j = 1}^n \int _0^{t\wedge \tau } \sum _{r = 1}^d c_i^{-1} c_j^{-1} (y'_{i,r} - y_{i,r}) (y'_{j,r} - y_{j,r}) \kappa _{r} X_{t\wedge \tau }(x)^{y_r} f^{(i, j)} \big (X_{t\wedge \tau }(x)\big ) + M_{t \wedge \tau }\\ & = \sum _{i = 1}^n \sum _{y \in \mathcal{C}} h \big (X_{t\wedge \tau }(x)\big )^{y} c^y \sum _{y' \colon y \to y' \in \mathcal{R}} c_i^{-1} \kappa _{y \to y'} (y'_{i} - y_{i} ) \frac {\partial }{\partial z_i} f^{(i)} \big (h \big (X_{t\wedge \tau }(x)\big )\big ) \\ & \quad +\frac {1}{2}\sum _{i,j = 1}^n \sum _{y \in \mathcal{C}} h \big (X_{t\wedge \tau }(x)\big )^y c^y \sum _{y' \colon y \to y' \in \mathcal{R}} c_i^{-1} c_j^{-1} \kappa _{y \to y'} (y_{i}' - y_{i}) (y_{j}' - y_{j}) f^{(i, j)} \big (X_{t\wedge \tau }(x)\big ) + M_{t \wedge \tau }, \end{align*}

where $f^{(i)} (x) \,:\!=\, \frac {\partial }{\partial x_i} f(x)$ , $f^{(i, j)} (x) \,:\!=\, \frac {\partial ^2}{\partial x_i \partial x_j} f(x)$ and $M_{t \wedge \tau }$ is a martingale. Therefore, the Markov process $h (X_{t \wedge \tau } (x)) = G^{-1} X_{t \wedge \tau } (x)$ solves the martingale problem for $L^h$ given by

\begin{align*} L^h f (z) = & \sum _{i = 1}^n \sum _{y \in \mathcal{C}} z^{y} c^y \sum _{y' \colon y \to y' \in \mathcal{R}} c_i^{-1} \kappa _{y \to y'} (y'_{i} - y_{i} ) \frac {\partial }{\partial z_i} f(z) \\ & +\frac {1}{2}\sum _{i,j = 1}^n \sum _{y \in \mathcal{C}} z^y c^y \sum _{y' \colon y \to y' \in \mathcal{R}} c_i^{-1} c_j^{-1} \kappa _{y \to y'} (y_{i}' - y_{i}) (y_{j}' - y_{j}) \frac {\partial ^2}{\partial z_i\partial z_j}f(z), \end{align*}

for all $z = h(x) \in h(U)$ . On the other hand, formulas (3.5) and (3.6) suggest that for all $i,j \in \{1,\ldots , n\}$ and any source complex $y$ ,

\begin{align*} c^y \sum _{y' \colon y\to y'\in \mathcal{R}} c_i^{-1} \kappa _{y\to y'}(y_i' - y_i) = \sum _{y' \colon y\to y'\in \mathcal{R}'} c^y \beta _{y\to y'} (y_i' - y_i) = \sum _{y' \colon y\to y'\in \mathcal{R}'} \kappa '_{y\to y'} (y_i' - y_i) \end{align*}

and

\begin{align*} c^y \sum _{y' \colon y\to y'\in \mathcal{R}} c_i^{-1} c_j^{-1} \kappa _{y \to y'}(y_i' - y_j) \cdot (y'_j - y_j) = \sum _{y' \colon y\to y'\in \mathcal{R}'} \kappa '_{y\to y} (y_i' - y_i) \cdot (y_j' - y_j). \end{align*}

As a consequence, $L^h$ coincides with $L'$ the generator for $X'_{t \wedge \tau '} (h(x)) = X'_{t \wedge \tau '} (z)$

\begin{align*} L' f (z) = & \sum _{i = 1}^n \sum _{y \in \mathcal{C}} z^{y} \sum _{y' \colon y \to y' \in \mathcal{R}'} \!\!\kappa _{y \to y'}' (y'_{i} - y_{i}) \frac {\partial }{\partial z_i} f(z) +\frac {1}{2}\sum _{i,j = 1}^n \sum _{y \in \mathcal{C}} z^{y} \sum _{y' \colon y \to y' \in \mathcal{R}'} \!\!\kappa _{y \to y'}' (y_{i}' - y_{i}) \cdot (y_{j}' - y_{j}) \frac {\partial ^2}{\partial z_i\partial z_j}f(z). \end{align*}

This completes the proof of Proposition 3.8.