Processing math: 100%
Research article Special Issues

Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics


  • Reaction networks are widely used models to describe biochemical processes. Stochastic fluctuations in the counts of biological macromolecules have amplified consequences due to their small population sizes. This makes it necessary to favor stochastic, discrete population, continuous time models. The stationary distributions provide snapshots of the model behavior at the stationary regime, and as such finding their expression in terms of the model parameters is of great interest. The aim of the present paper is to describe when the stationary distributions of the original model, whose state space is potentially infinite, coincide exactly with the stationary distributions of the process truncated to finite subsets of states, up to a normalizing constant. The finite subsets of states we identify are called copies and are inspired by the modular topology of reaction network models. With such a choice we prove a novel graphical characterization of the concept of complex balancing for stochastic models of reaction networks. The results of the paper hold for the commonly used mass-action kinetics but are not restricted to it, and are in fact stated for more general setting.

    Citation: Daniele Cappelletti, Badal Joshi. Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 7649-7668. doi: 10.3934/mbe.2022359

    Related Papers:

    [1] David F. Anderson, Tung D. Nguyen . Results on stochastic reaction networks with non-mass action kinetics. Mathematical Biosciences and Engineering, 2019, 16(4): 2118-2140. doi: 10.3934/mbe.2019103
    [2] David F. Anderson, Jinsu Kim . Mixing times for two classes of stochastically modeled reaction networks. Mathematical Biosciences and Engineering, 2023, 20(3): 4690-4713. doi: 10.3934/mbe.2023217
    [3] Gheorghe Craciun, Stefan Muller, Casian Pantea, Polly Y. Yu . A generalization of Birchs theorem and vertex-balanced steady states for generalized mass-action systems. Mathematical Biosciences and Engineering, 2019, 16(6): 8243-8267. doi: 10.3934/mbe.2019417
    [4] Maya Mincheva, Gheorghe Craciun . Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks. Mathematical Biosciences and Engineering, 2013, 10(4): 1207-1226. doi: 10.3934/mbe.2013.10.1207
    [5] Paula Mercurio, Di Liu . Identifying transition states of chemical kinetic systems using network embedding techniques. Mathematical Biosciences and Engineering, 2021, 18(1): 868-887. doi: 10.3934/mbe.2021046
    [6] Linard Hoessly, Carsten Wiuf . Fast reactions with non-interacting species in stochastic reaction networks. Mathematical Biosciences and Engineering, 2022, 19(3): 2720-2749. doi: 10.3934/mbe.2022124
    [7] Gheorghe Craciun, Matthew D. Johnston, Gábor Szederkényi, Elisa Tonello, János Tóth, Polly Y. Yu . Realizations of kinetic differential equations. Mathematical Biosciences and Engineering, 2020, 17(1): 862-892. doi: 10.3934/mbe.2020046
    [8] Wai-Tong (Louis) Fan, Yifan (Johnny) Yang, Chaojie Yuan . Constrained Langevin approximation for the Togashi-Kaneko model of autocatalytic reactions. Mathematical Biosciences and Engineering, 2023, 20(3): 4322-4352. doi: 10.3934/mbe.2023201
    [9] Giorgos Minas, David A Rand . Parameter sensitivity analysis for biochemical reaction networks. Mathematical Biosciences and Engineering, 2019, 16(5): 3965-3987. doi: 10.3934/mbe.2019196
    [10] Swadesh Pal, Malay Banerjee, Vitaly Volpert . Spatio-temporal Bazykin’s model with space-time nonlocality. Mathematical Biosciences and Engineering, 2020, 17(5): 4801-4824. doi: 10.3934/mbe.2020262
  • Reaction networks are widely used models to describe biochemical processes. Stochastic fluctuations in the counts of biological macromolecules have amplified consequences due to their small population sizes. This makes it necessary to favor stochastic, discrete population, continuous time models. The stationary distributions provide snapshots of the model behavior at the stationary regime, and as such finding their expression in terms of the model parameters is of great interest. The aim of the present paper is to describe when the stationary distributions of the original model, whose state space is potentially infinite, coincide exactly with the stationary distributions of the process truncated to finite subsets of states, up to a normalizing constant. The finite subsets of states we identify are called copies and are inspired by the modular topology of reaction network models. With such a choice we prove a novel graphical characterization of the concept of complex balancing for stochastic models of reaction networks. The results of the paper hold for the commonly used mass-action kinetics but are not restricted to it, and are in fact stated for more general setting.



    Reaction networks are mathematical models for studying the evolution of biochemical systems, widely used in applications. Formally, reaction networks are made of a set of species whose populations evolve in time guided by a set of reactions such as the ones in the example below:

    A+Bκ1κ22C,Aκ3κ4B. (1.1)

    Each species has a discrete number of molecules whose counts change as the molecules react or are produced from reactions. In many circumstances, the counts are sufficiently high so that the granularity may be ignored and it suffices to model species concentrations rather than counts. However, several biological macromolecules such as proteins and enzymes are present only in small numbers (100) and it becomes necessary to both track individual numbers and be mindful of stochastic fluctuations in those numbers. Due to such considerations, Zn is the natural state space for the stochastic model (a continuous-time Markov chain) that we consider here. For readable references on stochastic reaction network models and connections with applications, see [1,2].

    The stationary distribution of a stochastic system, assuming it exists, describes the state of the system over a long period of time. So in a sense it is an attracting steady state of the dynamic model. Having an analytic expression of the stationary distribution in terms of the model parameters is useful for various reasons: first of all, the expected level and the fluctuations of different proteins in stationary biological systems is of great interest. Controlling the stationary regime is also attracting an increasing interest, given the novel technological possibilities of changing the cell DNA and, hence, the rate and the form of the biological reactions occurring inside. Notable examples of how the emerging field of synthetic biology focuses on the control of stationary regimes are given in [3,4,5]. Finally, the approximation of complex biological systems with reaction rates spanning over different orders of magnitudes can be performed if the stationary distribution of the faster subsystem is known [6]. The study of multiscale biological system is of crucial importance and is recognized as one of the key ingredients to unlock the function of genetic and cellular compositions in the Perspectives in Mathematical Biology individuated by the European Society for Mathematical and Theoretical Biology (ESMTB) [7,8].

    The aim of the present paper is twofold: first, typical computational techniques to calculate an approximation of the stationary distribution of interest consists in truncating the (potentially infinite) state space to a smaller finite set, and calculate the stationary distribution of the process restricted to it [9,10]. The approximation error is often hard to calculate. A natural question is whether the same stationary distribution of the full model is also a stationary distribution of the restricted one; if that is the case then the approximation error only concerns the normalizing constant, but not the ratio of the distribution in different states. In the present paper we consider restrictions of the state space to the images of copies (or union thereof). A copy of a reaction network is an embedding of the reaction graph into the state space. Figure 1 illustrates how the state space of

    (1.2)
    Figure 1.  The figures depicts the transition graph of the continuous time Markov chain associated with the reaction network (1.2). Different colors indicate different copies, which are different embedding of the graph (1.2) in the state space Z20.

    can be decomposed into copies.

    We prove that the process restricted to arbitrary unions of copies has the same stationary measure of the original one only in the case of complex balancing. In the case of complex balanced systems, an exact formula for the stationary distributions is already known [11], however the general line of thought can be pursued for other types of restrictions different from copies. In fact, in future work we will use this idea to efficiently calculate exact analytical expressions of stationary distributions that have not appeared in the existing literature.

    Secondly, the paper focuses on the connection between stationary distributions and graphical symmetries. This perspective is based on links between steady states and graphical features of the model which have been fruitfully and extensively studied for deterministic models of reaction networks [12,13,14,15,16,17,18,19,20]. Connections between graphical symmetries of the model and the shape of its stationary distribution have been successfully unlocked in several recent papers [11,21,22,23,24,25]. In this work, we define a new type of graphical symmetry (balancing of copies, as defined below) and we show that it characterizes the notion of complex balancing under general assumptions.

    While our study is valid for the commonly used mass-action kinetics, the scope of our results is not restricted to such kinetics but is much broader. Some general results can be stated for any admissible choice of reaction rates, and others hold for a generalization of mass-action kinetics that we call product-form kinetics, considered for example in [11,26,27,28].

    Let R, R0 and R>0 represent the reals, the non-negative reals and the positive reals, respectively. Let Z, Z0 and Z>0 represent the integers, the non-negative integers and the positive integers, respectively. For vRn, v1=|v1++vn|. For v,wRn, vw (v<w) means that viwi (vi<wi) for all i{1,,n}. For v,wRn, we define

    1{vw}={1,vw0, otherwise.

    If v>0 then v is said to be positive. Similarly, if v0 then v is said to be nonnegative. Finally, suppv denotes the index set of the non-zero components. For example, if v=(0,1,1) then suppv={2,3}.

    If xRn0 and vZn0, we define

    xv=ni=1xvii,andv!=ni=1vi!,

    with the conventions that 0!=1 and 00=1.

    A reaction network is a triple G=(S,C,R), where S is a set of n species, C is a set of m complexes, and RC×C is a set of r reactions, such that (y,y)R for all yC. The complexes are linear combinations of species over N, identified as vectors in Rn. A reaction (y,y)R is denoted by yy, and the vector yy is the corresponding reaction vector. We require that every species has a nonzero coordinate in at least one complex and that every complex appears in at least one reaction. With this convention, there are no "redundant" species or complexes and G is uniquely determined by R. In (1.1), there are n=3 species (A,B,C), m=4 complexes (A+B,2C,A,B), and r=4 reactions.

    A reaction network G=(S,C,R) can be viewed as a graph with node set C and edge set R in a natural manner. We will frequently use the viewpoint of a reaction network as a graph in the rest of the paper.

    A reaction network G is weakly reversible if every reaction yyR is contained in a closed directed path. Moreover, G is reversible if for any reaction yyR, yy is in R. It is clear that each reversible reaction network is also weakly reversible. As an example, the network in (1.1) is reversible, and therefore weakly reversible.

    The stoichiometric subspace of G is the linear subspace of Rn generated by the reaction vectors, namely

    S=span(yy|yyR).

    For vRn, the sets (v+S)Rn0 are called the stoichiometric compatibility classes of G.

    We will consider dynamics of a reaction network with n species both on Rn0 and Zn0. Rn0 is the usual underlying state space for deterministic models, while for classic stochastic models the state space is Zn0. It is worth mentioning that dynamics of a reaction network can also be modeled via stochastic differential equations, an example of stochastic models whose underlying state space is Rn0 [29,30,31,32]. However we do not consider this case in this paper.

    Let G be a reaction network. We want to associate each reaction yyR with a rate function λyy, whose domain is Rn0. Formally, we define as deterministic kinetics the following correspondence between reactions and rate functions:

    Λ:(yy)λyy

    It is required that for each yyR

    λyy(x)>0onlyifsuppysuppx. (2.1)

    The pair (G,Λ) is called continuous reaction system. In the deterministic context, the evolution of the species concentrations z(t)Rn0 is determined by the system of ODEs

    dzdt=yyR(yy)λyy(z). (2.2)

    Condition (2.1) implies that any solution to (2.2) is non-negative at all times for which it is defined. The solution z(t) is also confined to its stoichiometric compatibility class:

    z(t)(z(0)+S)Rn0.

    A state cRn is said to be a steady state of a continuous reaction system (G,Λ) if

    yyR(yy)λyy(c)=0.

    Let G be a reaction network. We now want to associate each reaction yyR with a rate function λyy, whose domain is Zn0. Similarly to before, we define as stochastic kinetics the following correspondence between reactions and rate functions:

    Λ:(yy)λyy.

    We require

    λyy(x)>0onlyifxy. (2.3)

    The pair (G,Λ) is called stochastic reaction system. In this setting, a state x=(x1,,xn)Zn0 represents the counts xi of each species i=1,,n. X(t) represents the state of the system at time t, and is considered to be a continuous-time Markov chain with transition rate from state x to state x given by

    q(x,x)=yyRyy=xxλyy(x).

    Equivalently, each rate function described the rate of occurrence of the associated reaction, and whenever a reaction yy takes place the process X() moves from the current state x to x+yy. Condition (2.3) forces the process X() to the positive orthant.

    We say that xZn0 is accessible from xZn0, if there is a sequence of states (x=u0,u1,,un1,un=x) such that for each consecutive pair of states (ui,ui+1), 0in1, we have q(ui,ui+1)>0. A non-empty set ΓZn0 is an closed irreducible set of (G,Λ) if for all xΓ and all uZn0, u is accessible from x if and only if uΓ [33]. A probability distribution π is a stationary distribution for a continuous-time Markov chain if for all tR0 and xZn0

    P(X(t)=x|X(0)π)=π(x).

    By standard theory on Markov chains, the support of π (that is, the largest set of states where π has a positive value) is a union of closed irreducible sets [33]. Moreover, by standard theory on Markov chains, if the continuous-time Markov chain is non-explosive (in the sense of [33]) then a probability distribution π is stationary if and only if for all states x

    π(x)yyRλyy(x)=yyRπ(x+yy)λyy(x+yy). (2.4)

    We further give the definitions below.

    Definition 2.1. Let (G,Λ) be a discrete reaction system and let μ be a measure on Zn0.

    1) μ is said to be a stationary measure if for all xZn0

    μ(x)yyRλyy(x)=yyRμ(x+yy)λyy(x+yy). (2.5)

    2) μ is said to be a σ-finite measure if μ(x)< for all xZn0. π is said to be a finite measure if xZn0μ(x)<.

    Note that based on the definitions used in this paper, if a stationary measure is a probability distribution is not necessarily a stationary distribution, unless the model is non-explosive. Non-explosiveness of the process can be assessed if it is complex balanced, as stated in Theorem 3.3 below.

    An important choice of kinetics is mass-action kinetics.

    Definition 2.2. Consider a reaction network G=(S,C,R).

    1) A deterministic mass-action system is a continuous reaction system (G,ΛDκ) with

    KDκ(yy)(z)=κyyzy

    for some constants κyyR>0, called rate constants.

    2) A stochastic mass-action system is a discrete reaction system (G,KSκ) with

    KSκ(yy)(x)=κyyx!(xy)!1{xy} (2.6)

    for some constants κyyR>0, called rate constants.

    Many generalizations of mass-action kinetics have been proposed both for stochastic and deterministic models. In the present paper, we will study the following generalization for stochastic models, considered for example in [11,26,27,28]. We define the stochastic product-form kinetics as the discrete kinetics

    KSκ,θ(yy)(x)=κyyni=1yi1j=0θi(xij), (2.7)

    for some constants κyyR>0, called rate constants, and some functions θi:ZR0 satisfying θi(m)=0 if and only if m0. A stochastic reaction system (G,KSκ,θ) will be called stochastic product-form system.

    Note that if θi(m)=m1{m0} for all i, then KSκ,θ=KSκ. Finally, we give the following definition:

    Definition 2.3. A stochastic product-form system is called non-saturating if limmθi(m)= for all 1in.

    The definition of complex balancing dates back to [12,13], where deterministic models were of interest. The definition is the following:

    Definition 3.1. Let (G,Λ) be a continuous reaction system. A state cRn0 is complex balanced if for all complexes yC we have

    yC:yyRλyy(c)=yC:yyRλyy(c).

    In words, c is complex balanced if the sum of rates of reactions "entering" any complex y is equal to the sum of the rates of reactions "exiting" from y, calculated in c. It is known that every complex balanced state is a steady state, while the opposite is not true in general [12]. The following are classical results from [12,13]:

    Theorem 3.1. Let (G,KDκ) be a deterministic mass-action system. If a positive complex balanced state c exists, then G is weakly reversible, all positive steady states are complex balanced, and exactly one positive steady state exists within each stoichiometric compatibility class.

    Theorem 3.1 implies that the positive steady states of a deterministic mass-action system are either all complex balanced, or none of them is. Moreover, in the former case non-positive steady states are necessarily complex balanced as well as shown in [21,Theorem 4]. Mass- action systems are called complex balanced if at least one positive steady state is complex balanced, which by Theorem 3.1 is equivalent to the existence of at least a positive steady state and to all of them being complex balanced.

    In [11] the first connection between complex balancing and stochastic dynamics is performed, and the following result is proven.

    Theorem 3.2. Let G be a reaction network. Assume the deterministic mass-action system (G,KDκ) is complex balanced and cRn>0 is a positive complex balanced state. Then the stochastic reaction system (G,KSκ,θ) with KSκ,θ defined as in (2.7) and the same rate constants as (G,KDκ) has a stationary measure of the form

    μ(x)=cxni=1xij=11θi(j) (3.1)

    If KSκ,θ reduced to mass-action kinetics, then (3.1) becomes proportional to a product-form Poisson distribution, which is a stationary distribution because the process is non-explosive as proven in [34]. In [26,Theorem 4.1] mild conditions implying finiteness of (3.1) and non-explosiveness of the model are derived:

    Theorem 3.3. Let G be a reaction network. Assume the deterministic mass-action system (G,KDκ) is complex balanced and cRn>0 is a positive complex balanced state. Consider a stochastic non-saturating product-form system (G,KSκ,θ) with the same rate constants as (G,KDκ). Then, the associated process X() is non-explosive and the measure (3.1) is finite.

    Theorem 3.1, non-explosiveness of the process and finiteness of (3.1) imply that (3.1) is proportional to a stationary distribution of X(). In [21] a parallel theory on complex balancing is developed for stochastic models. A complex balanced measure is defined as follows.

    Definition 3.2. Let (G,Λ) be a discrete reaction system. A measure μ with support in Zn0 is complex balanced if for all complexes yC and all states xZn0 we have

    yC:yyRλyy(x)μ(x)=yC:yyRλyy(x+yy)μ(x+yy). (3.2)

    In words, a measure is complex balanced if, at any state x, the sum of the ingoing fluxes of reactions "entering" any complex y is equal to the sum of the outgoing fluxes of reactions "exiting" from y. It is not difficult to check that a complex balanced measure is necessary stationary, while the converse does not hold in general. Definition 3.2 was given for probability distributions in [21] and then extended to measures in [23]. In particular, it is proven in [23,Proposition 4.13] that under the hypothesis of mass-action a σ-finite complex balanced measure is finite, hence normalizable to a probability distribution. A strong connection between complex balancing in stochastic and deterministic models exists, as stated below.

    Theorem 3.4. A stochastic non-saturating product-form system (G,KSκ,θ) has a σ-finite, positive, complex balanced measure if and only if the deterministic mass-action system (G,KDκ) with the same choice of rate constants is complex balanced.

    Theorem 3.4 is proven in [21] in the case of stochastic mass-action kinetics. The proof for the more general result above is given in Section 6.

    Here we state and prove our main results. Perhaps it is worth mentioning that our results hold not only for mass-conserving systems with only finitely many accessible states but also for more general systems where the number of accessible states may be infinite.

    Definition 4.1. Let G be a reaction network. A function f:CZn0 is a copy of G if for every yyR, we have f(y)f(y)=yy. We further say that a copy f is active if every yyR is active at f(y).

    Denote by f(C) the image of f. G induces a directed graph on f(C) in a natural manner by associating a directed edge from f(y) to f(y) whenever yyR. With this understanding, the inclusion copy ι:CZn0 corresponds to the geometrically embedded graph of G defined in [35]. We state here a lemma that will be useful to prove our main results.

    Lemma 4.1. Let (G,Λ) be a stochastic reaction system. Let yy be a reaction active at a state x. Then, there exists an injective copy f of G with f(y)=x. Moreover, if Λ is such that λyy(˜x)>0 if and only if ˜xy, then there exists an injective, active copy f of G with f(y)=x.

    Proof. Let h=xy. Since yy is active at x, it follows from (2.3) that hZn0. Consider the function f:CZn0 defined by f(˜y)=h+˜y. We have f(y)=h+y=x. Moreover, for all two complexes ˜y,˜yC we have f(˜y)f(˜y)=˜y˜y, so f is a copy of G and is injective. Moreover, if λyy(˜x)>0 if and only if ˜xy then f is active. Hence, the proof is concluded.

    Definition 4.2. Let (G,Λ) be a stochastic reaction system. Let f be a copy of G and let μ be a measure on Zn0.

    f is active with respect to (Λ,μ) if for all yyR

    μ(f(y))˜y˜yR:f(˜y)=f(y),f(˜y)=f(y)λ˜y˜y(f(y))>0

    f is node balanced with respect to (Λ,μ) if for every xf(C)

    μ(x)yyR:f(y)=xλyy(x)=yyR:f(y)=xμ(f(y))λyy(f(y)) (4.1)

    To understand the meaning of node balancing, let Wf() be the continuous-time Markov chain whose transition graph is the one induced f, and the transition rates are given by

    q(x,x)=yyR:f(y)=x,f(y)=xλyy(x).

    Then, the following holds.

    Proposition 4.2. Let (G,Λ) be a stochastic reaction system. Let f be a copy of G and let μ be a σ-finite measure of Zn0. Then, there exists a stationary distribution of Wf() proportional to the restriction of μ to f(C) if and only if f is node balanced with respect to (Λ,μ).

    Proof. The state space of Wf is f(C), which is finite. Then, there exists a stationary distribution proportional to the restriction of μ to f(C) if and only if the latter is a stationary measure, which is equivalent to (4.1). The proof is then concluded.

    Proposition 4.2 implies that if a copy f is node balanced with respect to (Λ,π), then the restriction of π to f(C) is stationary for Wf. If π is a stationary distribution of X(), then π is stationary for both X() and Wf(). The same holds for union of copies, as expressed in the following corollary.

    Corollary 4.3. Let (G,Λ) be a stochastic reaction system and let μ be a σ-finite measure of Zn0. For some h1, let Φ={f1,,fh} be a set of node balanced copies of G with respect to (Λ,μ).Then, the continuous-time Markov chain on state space Zn0 with transition rates

    q(x,x)=hi=1yyR:fi(y)=x,fi(y)=xλyy(x), (4.2)

    has a stationary distribution π=γμ where γ is a positive real constant.

    Proof. The proof is concluded by simply noting that, due to Proposition 4.2, for every state x and any 1ih we have

    xZn0μ(x)yyR:fi(y)=x,fi(y)=xλyy(x)=xZn0μ(x)yyR:fi(y)=x,fi(y)=xλyy(x).

    Hence, by summing both sides over i we obtain that μ is a stationary measure, hence proportional to a stationary distribution by classic theory of Markov chains.

    Corollary 4.3 implies that the stationary distributions of the full model can be found, up to a normalizing constant, by studying the stationary distributions of the finite continuous time Markov chain with rates 4.2, as long as the copies are node balanced with respect to them. As already mentioned in the introduction, the node balancing of the copies is intimately related to complex balancing, in a way made precise by the results below.

    Theorem 4.4. Let (G,Λ) be a stochastic reaction system and let μ be a measure. Then the following are equivalent:

    (1) every injective copy of G is node balanced with respect to (Λ,μ);

    (2) μ is a complex balanced measure;

    (3) every copy of G is node balanced with respect to (Λ,μ).

    Proof. We will prove that (1) implies (2), which implies (3), which implies (1).

    (1) (2). Fix yC and xZn0. If xy then λyy(x)=0 for all reactions yyR by (2.3). Moreover, for all reactions yyR we have x+yyy which implies λyy(x+yy)=0 and again λyy(x)=0 by (2.3). It follows that (3.2) holds. If xy then by Lemma 4.1 there exists an injective copy f of G with f(x)=y, which is node balanced with respect to (Λ,μ). By injectivity, (4.1) becomes (3.2). In conclusion, (3.2) holds for every yC and every xZn0, and (2) is proven.

    (2) (3). Let f be a copy of G and let xf(C). Then, summing both sides of (3.2) over the complexes y with f(y)=x we obtain (4.1), so (3) holds.

    (3) (1). This is trivially true as injective copies of G are copies of G.

    With more assumptions on the form of the kinetics, a finite number of node balanced copies with respect to (Λ,μ) are sufficient to imply complex balancing of μ, as detailed below.

    Theorem 4.5. Let (G,KSκ,θ) be a stochastic reaction system and let μ be a positive stationary measure. Then there exist positive and finite constants M1,M2 such that the following are equivalent:

    1) all the injective copies of G that intersect the cube [0,M1]n are node balanced with respect to (KSκ,θ,μ);

    2) a closed irreducible set Γ containing a state x with xM2 is such that all copies f of G with f(C)Γ are node balanced with respect to (KSκ,θ,μ);

    3) μ is a complex balanced measure.

    The above theorem states that if enough, finitely many copies are node balanced, then the measure μ is complex balanced. The same holds even if all the copies in a single, large enough, closed irreducible set are node balanced. The proof is given in Section 6. In the following, if f is a copy of G and vZn, we denote by f+v the function defined by (f+v)(y)=f(y)+v for all yC. Note that for vZn0, f+v is a copy of G while this may not always hold for general vZn.

    Theorem 4.6. Let (G,KSκ,θ) be a stochastic product-form system and let μ be of the form (3.1). Then the following are equivalent:

    (1) there is an active, injective, node balanced copy of G with respect to (KSκ,θ,μ);

    (2) μ is a complex balanced measure.

    Proof. In what follows we will use that, by substituting the rate functions with (2.7) and μ with (3.1) and by simplifying, we get that for all yyR and for all ˜xZn0 with ˜xy

    μ(˜x)λyy(˜x)=c˜xκyyni=1xiyij=11θi(j). (4.3)

    (1) (2). Fix yC and let x be such that f(x)=y. By injectivity and node balancing of f we have

    μ(x)yC:yyRλyy(x)=yC:yyRμ(x+yy)λyy(x+yy),

    which by (4.3) becomes

    yC:yyRκyy=yC:yyRcyyκyy, (4.4)

    which is in turn equivalent to (3.2) for mass-action kinetics and hence proves that (G,KDκ) is complex balanced. Moreover, by using (4.3) again we have that for all ˜xZn0 with ˜xy

    yC:yyRμ(˜x+yy)λyy(˜x+yy)=yC:yyRc˜x+yyκyyni=1˜xiyij=11θi(j)=yC:yyRc˜xκyyni=1˜xiyij=11θi(j)=yC:yyRμ(˜x)λyy(˜x),

    where the second equality follows from (4.3). Thus, (2) is proven.

    (2) (1). The existence of an injective, active copy follows from Lemma 4.1, and the fact that is node balanced follows from Theorem 4.4.

    Our last result only concerns mass-action models.

    Theorem 4.7. Let (G,KSκ) be a stochastic mass-action system and let μ be of the form (3.1). Then the following are equivalent:

    (1) there is a copy f of G such that f+v is node balanced with respect to (KSκ,θ,μ) for all vZn0;

    (2) there is an active copy f of G such that f+v is node balanced with respect to (KSκ,θ,μ) for all vΞ where ΞZn0 is such that the only polynomial of degree at most d=max{y1:yyR} that vanishes on Ξ is the zero polynomial;

    (3) μ is a complex balanced measure.

    Proof. We will prove that (1) implies (2), which implies (3), which implies (1). Similarly to what done in the proof of Theorem 4.6, we first note that by substituting the rate functions with (2.6) and μ with (3.1) and by simplifying, we get that for all yyR and for all ˜xZn0 with ˜xy

    μ(˜x)λyy(˜x)=c˜xκyy1(xy)!. (4.5)

    (1) (2). This simply follows from the existence of a set Ξ as described. A finite set Ξ can always be constructed, but to prove (2) it is enough to note that Ξ can be chosen as Zn0, since the only polynomial vanishing on Zn0 is the zero polynomial.

    (2) (3). Let x be in f(C). By definition of node balancing we have

    μ(x+v)yyR:f(y)=xλyy(x+v)=yyR:f(y)=xμ(x+v+yy)λyy(x+v+yy)

    for all vΞ. By multiplying both sides by (x+v)! and by applying (4.5) we get

    yC:f(y)=x(x+v)!(x+vy)!(yC:yyRκyyyC:yyRcyyκyy)=0.

    Note that the above is a polynomial of degree at most d in the variable v and it vanishes for all vΞ. Hence, it is the zero polynomial and as a consequence (G,KDκ) is complex balanced with complex balanced steady state c. Then, by using (4.5) again we have that for all ˜xZn0 with ˜xy

    yC:yyRμ(˜x+yy)λyy(˜x+yy)=yC:yyRc˜x+yyκyy1(˜xy)!=yC:yyRc˜xκyy1(˜xy)!=yC:yyRμ(˜x)λyy(˜x),

    where the second equality follows from (4.5). Thus, (2) is proven.

    (3) (1). This follows from Theorem 4.4.

    We give here an example showing that condition 1 alone does not imply μ is complex balanced, unless μ is assumed to be of the form (3.1).

    Example 4.1. Consider the stochastic mass-action system

    0κ1A,3Aκ22A,

    where the reaction rates have been written above the corresponding reactions. Let π be the unique stationary distribution, whose support is in {mZ:m2}. Since the model is a birth and death chain, π is detailed balanced: for all m2 we have π(m)q(m,m+1)=π(m+1)q(m+1,m) [33]. Specifically, we have

    π(m)λ0A(m)=π(m+1)λ3A2A(m+1) (4.6)

    The deterministically modeled system cannot be complex balanced by Theorem 3.1 because it is not weakly reversible. By [21,Corollary 19], π cannot be complex balanced. However, it follows from (4.6) that the copy defined by f(0)=f(2A)=2 and f(A)=f(3A)=3 is node balanced with respect to (KSκ,π), and the same holds for all the copies f+v with vZ0.

    We end by recapitulating the consequences of our results. The truncation of the state space, as already discussed in the Introduction, is a widely used tool to approximate the stationary distribution of stochastic models. A truncation is specified in two steps: the first is a description of the restricted state space and the second is the definition of the constrained process. In particular, a prescription for the allowed transitions of the process on the boundary of the truncated space needs to be given. As an example, in [9] the outgoing transitions from the truncated state space are redirected to a designated state within the truncated state space. Many other strategies have been studied, as discussed in the review [10]. While truncation methods normally give an approximation to the stationary distribution of the original process, our interest lies in identifying the situations where the approximation is exact, except for a normalizing constant.

    In the context of this paper, we consider as a truncation a finite set of copies {f1,,fh}, as described in (4.2). By Corollary 4.3, if the copies fi:1ih are node balanced with respect to the kinetics and a measure μ, then μ is a stationary distribution of the truncated process, up to normalization. We are interested in understanding whether μ is also equal to a stationary distribution of the original process, up to normalization. A consequence of Theorem 4.5 is that this happens if the truncated state space is large enough and the kinetics have product form, a generalization of mass-action kinetics. Unfortunately, in this case and still by the same theorem, we have that μ is a complex balanced measure. Hence, we knew already that μ is proportional to (3.1) by Theorems 3.2 and 3.4, so the application of truncation methods is not useful. In future works, we will identify exact approximation via different truncation methods related to copies and avail the exact stationary distributions of models that are not complex balanced.

    The second important consequence of our results, mainly from a theoretical point of view, is that we have a further characterization of complex balancing given by Theorems 4.4–4.7.

    The main idea behind the proofs in this section is based upon classical notions of deficiency theory which we will briefly introduce here. We start with giving the definition of deficiency, first given in [13].

    Definition 6.1. The deficiency of a reaction network G=(S,C,R) is the number δ=mdimS, where is the number of connected components of the directed graph (C,R) and S is the stoichiometric subspace of G.

    The following geometric interpretation will be used in our proofs: let {ey}yC be the canonical basis of Rm, where each coordinate is univocally associated with a complex. Further, define dyy=eyey for yyR. Let D=span(dyy:yyR). It is proven in [12,13] that dimD=m. Then, the space D is linearly isomorphic to S if and only if δ=0. Specifically, consider the homomorphism

    φ:RmRneyy. (6.1)

    For yyR, we have φ(dyy)=yy and φ|D:DS is thus a surjective homomorphism. Therefore,

    dimKerφ|D=dimDs=ms=δ, (6.2)

    which implies that φ|D is an isomorphism if and only if δ=0. It further follows that the deficiency is a non-negative number.

    The following classical result is proven in [12,13].

    Theorem 6.1. Let (G,KDκ) be a deterministic mass-action system. If G is weakly reversible and its deficiency is zero, then (G,KDκ) is complex balanced for any choice of rate constants.

    Consider a stochastic reaction system (G,KSκ,θ) with G=(S,C,R) and KSκ,θ as in (2.7). We will follow an idea first proposed in [21], and consider the reaction network ˜G=(˜S,˜C,˜R) with

    ˜S=S{Ay:yC},˜C={y+Ay:yC},and˜R={y+Ayy+Ay:yyR}. (6.3)

    It follows from the construction above that ˜G is weakly reversible if and only if G is weakly reversible. Moreover, the following holds.

    Lemma 6.2. The deficiency of ˜G as defined in (6.3) is zero.

    Proof. Consider the linear homomorphism φ as defined in (6.1), for the reaction network ˜G. Since the vectors associated with the complexes y+Ay are linear independent, φ is an isomorphism and as a consequence the deficiency of ˜G is 0. We associate ˜G with a general mass-action kinetics KSκ,˜θ such that the rate of any reaction y+Ayy+Ay˜R is of the form

    KSκ,˜θ(y+Ayy+Ay)(x,u)=KSκ,θ(yy)(x)uAyforall(x,u)Zn0×Zm0.

    The following is the key result of this paper, on which the proof of the other main theorems are based on.

    Theorem 6.3. Let (G,KSκ,θ) be a stochastic reaction system with KSκ,θ as in (2.7). Let μ be a σ-finite measure on Zn0. Define Υ as the set of pairs (x,y)Zn0×C such that there exists an active, injective copy f of G that is node balanced with respect to (KSκ,θ,μ), satisfies f(y)=x, and fulfils f(C)suppμ. Let ˆR be a subset of reactions whose reaction vectors form a basis of S. Assume that

    1) there exists ˆxZn0 such that (ˆx,y)Υ for all yC;

    2) for all yyRˆR, there exist a sequence of (potentially repeated) reactions {yiyi}hi=1 contained in {yy}ˆR such that

    (a) yy=y1y1;

    (b) hi=1(yiyi)=0;

    (c) for all j=0,1,2,,h1 we have

    (ˆx+ji=1(yiyi),yj+1)Υ.

    Then, the deterministic mass-action system (G,KDκ) is complex balanced.

    Proof. Consider (x,ey)Zn0×Zm0. If xy, then any reaction of the form y+Ayy+Ay can take place and move the process from (x,ey) to (x+yy,ey). From xy it follows that x+yyy hence at (x+yy,ey) any reaction of the form y+Ayy+Ay can take place and move the process to (x+yy,ey), and so on. Since G is weakly reversible it is possible to eventually return to the original state (x,ey), and the closed irreducible set containing (x,ey) is precisely given by {(x+˜yy,e˜y):yCy}, where Cy denotes the connected component of (C,R) containing y. We denote the closed irreducible set containing (x,ey) by Γ(x,ey). Of course the notation is not bijective: for example if xy then Γ(x,ey)=Γ(x+yy,ey). If (x,y)Υ, then be definition there exists an active, injective copy f of G that is node balanced with respect to (KSκ,θ,μ) and that satisfies f(y)=x. Hence, by definition of node balancing, for all ˜yCy we have

    yC:˜yyRμ(x+˜yy)λ˜yy(x+˜yy)=yC:y˜yRμ(x+yy)λy˜y(x+yy),

    which implies

    yC:˜y+A˜yy+Ay˜Rμ(x+˜yy)λ˜y+A˜yy+Ay(x+˜yy,e˜y)=yC:y+Ay˜y+A˜y˜Rμ(x+yy)λy+Ay˜y+A˜y(x+yy,ey).

    Since Γ(x,ey) is finite, it follows from the equation above that μ restricted to Γ(x,ey) is proportional to the unique stationary distribution of (˜G,KSκ,˜θ) with support Γ(x,ey). Since ˜G has deficiency zero by Lemma 6.2 and is weakly reversible because G is weakly reversible, by Theorems 6.1 and 3.1 we have

    μ(x+˜yy)=M(x,ey)˜cAycxxij=11θi(j), (6.4)

    where (c,˜c)Rn>0×Rm>0 is a complex balanced steady state of (˜G,KDκ) and M(x,ey) is a proportionality constant depending on the closed irreducible set Γ(x,ey), hence the notation is not bijective and M(x,ey)=M(x+˜yy,e˜y) for all ˜yCy.

    We now want to show that there exists a vector ˆcRn>0 such that

    ˜cAy˜cAy=ˆcyyfor all yyR. (6.5)

    First, note that (6.5) holds for all yyˆR, because it is equivalent to

    (yy)logˆc=log˜cAylog˜cAyfor all yyˆR.

    The latter has a solution because the reaction vectors of the reactions in ˆR are linearly independent. If yyRˆR then by hypothesis there exists a sequence of reactions {yiyi}hi=1 contained in {yy}ˆR such that y1y1=yy and

    (ˆx+ji=1(yiyi),yj+1)Υ.

    for all j=0,1,2,,h1. Hence, it follows from yy+hi=1(yiyi)=0 and from applying (6.4) recursively that

    M(ˆx,y1)=M(ˆx+yy+hi=1(yiyi),y1)=˜cAyh1˜cAy1M(ˆx+yy+hi=1(yiyi),yh1)=˜cAyh1˜cAy1M(ˆx+yy+h1i=1(yiyi),yh1)=˜cAyh1˜cAyh2˜cAy1˜cAyh1M(ˆx+yy+h1i=1(yiyi),yh2)=˜cAyh1˜cAyh2˜cAy1˜cAyh1M(ˆx+yy+h2i=1(yiyi),yh2)==M(ˆx,y1)hi=1˜cAyi˜cAyi.

    As a consequence,

    hi=1˜cAyi˜cAyi=1.

    If yy appears α times in the sequence {yiyi}hi=1, and all other reactions ˜y˜y appear β˜y˜y times, then

    ˜cAy˜cAy=˜y˜yˆR(˜cA˜y˜cA˜y)β˜y˜yα=ˆc1α˜y˜yˆRβ˜y˜y(yy)=ˆcyy.

    Hence, (6.5) is proven. For all yC we have (ˆx,y)Υ, which implies

    yC:yyRμ(ˆx)λyy(ˆx)=yC:yyRμ(x+yy)λy˜y(x+yy).

    By substituting the expressions for the transition rates and μ (over the image of node balanced copies) and by simplifying we obtain

    yC:yyRcxκyy=yC:yyR˜cAy˜cAycx+yyκyy

    which by (6.5) implies

    yC:yyRκyyni=1(ciˆci)yi=yC:yyRκyyni=1(ciˆci)yi,

    hence a positive complex balanced steady state exists for (G,KDκ), which concludes the proof.

    By Theorem 3.2 we already know that if (G,KDκ) is complex balanced, then there exists a σ-finite, positive, stationary measure for (G,KSκ,θ) given by μ as in (3.1). Checking that μ is complex balanced is not difficult, since by substituting the reaction rates with (2.7) and μ with (3.1), Eq (3.2) simplifies to

    yC:yyRκyy=yC:yyRcyyκyy,

    which holds because (G,KDκ) is complex balanced.

    Conversely, assume a σ-finite, positive, complex balanced measure for (G,KSκ,θ) exists. Then, all copies of G are node balanced with respect to (G,KSκ,θ) by Theorem 4.4. Moreover, since μ is positive then for all copies f of G we have f(C)suppμ. Hence, the proof is concluded by Theorem 6.3 by choosing ˆxZn0 large enough.

    It follows from Theorem 4.4 that (3) implies (1) for any M1. Moreover, the existence of a positive complex balanced measure implies that G is weakly reversible [23,Theorem 4.4] hence Zn0 is union of closed irreducible sets [36]. As a consequence, the existence of a closed irreducible set as in (2) is implied by Theorem 4.4 for any M2.

    Conversely, since μ is positive, it is always possible to choose M1 large enough such that all injective copies intersecting [0,M1]n being node balanced with respect to (KSκ,θ) implies that the assumptions of Theorem 6.3 are satisfied. Hence, for large enough M1 (1) implies that (G,KDκ) is complex balanced. In turn, this implies that μ is proportional to (3.1) on every closed irreducible set because the model is non-explosive by Theorem 3.3 (hence all stationary measures are proportional to each other on every closed irreducible set by standard Markov chain theory [33]). By substituting the reaction rates with (2.7) and μ as proportional to (3.1), Eq (3.2) simplifies to

    yC:yyRκyy=yC:yyRcyyκyy,

    which holds because (G,KDκ) is complex balanced.

    Similarly, since μ is positive, all injective copies in a closed irreducible set with large enough states being node balanced with respect to (KSκ,θ) implies that the assumptions of Theorem 6.3 are satisfied. By following the same reasoning as above, this implies (3) and concludes the proof.

    DC was supported by the MIUR grant ‘Dipartimenti di Eccellenza 2018-2022’ (E11G18000350001). The authors acknowledge support from Politecnico di Torino through the Open Access initiative.

    The authors declare there is no conflict of interest.



    [1] D. F. Anderson, T. G. Kurtz, Continuous time markov chain models for chemical reaction networks, in Design and analysis of biomolecular circuits, Springer, (2011), 3–42. https://doi.org/10.1007/978-1-4419-6766-4_1
    [2] D. Schnoerr, G. Sanguinetti, R. Grima, Approximation and inference methods for stochastic biochemical kineticsa tutorial review, J. Phys. A: Math. Theor., 50 (2017), 093001. https://doi.org/10.1088/1751-8121/aa54d9 doi: 10.1088/1751-8121/aa54d9
    [3] S. Aoki, G. Lillacci, A. Gupta, A. Baumschlager, D. Schweingruber, M. Khammash, A universal biomolecular integral feedback controller for robust perfect adaptation, Nature, 570 (2019), 533–537. https://doi.org/10.1038/s41586-019-1321-1 doi: 10.1038/s41586-019-1321-1
    [4] T. Plesa, G. Stan, T. Ouldridge, W. Bae, Quasi-robust control of biochemical reaction networks via stochastic morphing, J. R. Soc. Interface, 18 (2021), 20200985. https://doi.org/10.1098/rsif.2020.0985 doi: 10.1098/rsif.2020.0985
    [5] J. Kim, G. Enciso, Absolutely robust controllers for chemical reaction networks, J. R. Soc. Interface, 17 (2020), 20200031. https://doi.org/10.1098/rsif.2020.0031 doi: 10.1098/rsif.2020.0031
    [6] H. Kang, T. Kurtz, Separation of time-scales and model reduction for stochastic reaction networks, Ann. Appl. Probab., 23 (2013), 529–583. https://doi.org/10.1214/12-AAP841 doi: 10.1214/12-AAP841
    [7] B. Brook, S. Waters, Mathematical challenges in integrative physiology, J. Math. Biol., 56 (2008), 893–896.
    [8] L. Preziosi, Hybrid and multiscale modelling, J. Math. Biol., 53 (2006), 977–978. https://doi.org/10.1007/s00285-006-0042-x doi: 10.1007/s00285-006-0042-x
    [9] A. Gupta, J. Mikelson, M. Khammash, A finite state projection algorithm for the stationary solution of the chemical master equation, J. Chem. Phys., 147 (2017), 154101. https://doi.org/10.1063/1.5006484 doi: 10.1063/1.5006484
    [10] J. Kuntz, P. Thomas, G. Stan, M. Barahona, Stationary distributions of continuous-time Markov chains: a review of theory and truncation-based approximations, SIAM Rev., 63 (2021), 3–64. https://doi.org/10.1137/19M1289625 doi: 10.1137/19M1289625
    [11] D. Anderson, G. Craciun, T. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, Bull. Math. Biol., 72 (2010), 1947–1970. https://doi.org/10.1007/s11538-010-9517-4 doi: 10.1007/s11538-010-9517-4
    [12] F. Horn, R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal., 47 (1972), 81–116.
    [13] M. Feinberg, Complex balancing in general kinetic systems, Arch. Ration. Mech. Anal., 49 (1972), 187–194. https://doi.org/10.1007/BF00255665 doi: 10.1007/BF00255665
    [14] G. Craciun, A. Dickenstein, A. Shiu, B. Sturmfels, Toric dynamical systems, J. Symb. Comput., 44 (2009), 1551–1565. https://doi.org/10.1016/j.jsc.2008.08.006 doi: 10.1016/j.jsc.2008.08.006
    [15] M. Gopalkrishnan, E. Miller, A. Shiu, A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn., 13 (2014), 758–797. https://doi.org/10.1137/130928170 doi: 10.1137/130928170
    [16] B. Joshi, A. Shiu, Atoms of multistationarity in chemical reaction networks, J. Math. Chem., 51 (2013), 153–178. https://doi.org/10.1007/s10910-012-0072-0 doi: 10.1007/s10910-012-0072-0
    [17] B. Joshi, G. Craciun, Reaction network motifs for static and dynamic absolute concentration robustness, preprint, arXiv: 2201.08428.
    [18] M. Pérez Millán, A. Dickenstein, The structure of MESSI biological systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 1650–1682. https://doi.org/10.1137/17M1113722 doi: 10.1137/17M1113722
    [19] D. Cappelletti, A. Gupta, M. Khammash, A hidden integral structure endows absolute concentration robust systems with resilience to dynamical concentration disturbances, J. R. Soc. Interface, 17 (2020), 20200437. https://doi.org/10.1098/rsif.2020.0437 doi: 10.1098/rsif.2020.0437
    [20] G. Craciun, B. Joshi, C. Pantea, I. Tan, Multistationarity in cyclic sequestration-transmutation networks, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-01021-7 doi: 10.1007/s11538-022-01021-7
    [21] D. Cappelletti, C. Wiuf, Product-form Poisson-like distributions and complex balanced reaction systems, SIAM J. Appl. Math., 76 (2016), 411–432. https://doi.org/10.1137/15M1029916 doi: 10.1137/15M1029916
    [22] B. Joshi, A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1077–1105. https://doi.org/10.3934/dcdsb.2015.20.1077 doi: 10.3934/dcdsb.2015.20.1077
    [23] D. Cappelletti, B. Joshi, Graphically balanced equilibria and stationary measures of reaction networks, SIAM J. Appl. Dyn. Syst., 17 (2018), 2146–2175. https://doi.org/10.1137/17M1153315 doi: 10.1137/17M1153315
    [24] L. Hoessly, C. Mazza, Stationary distributions and condensation in autocatalytic reaction networks, SIAM J. Appl. Math., 79 (2019), 1173–1196. https://doi.org/10.1137/18M1220340 doi: 10.1137/18M1220340
    [25] D. F. Anderson, S. L. Cotter, Product-form stationary distributions for deficiency zero networks with non-mass action kinetics, Bull. Math. Biol., 78 (2016), 2390–2407. https://doi.org/10.1007/s11538-016-0220-y doi: 10.1007/s11538-016-0220-y
    [26] D. Anderson, T. Nguyen, Results on stochastic reaction networks with non-mass action kinetics, Math. Biosci. Eng., 16 (2019), 2118–2140. https://www.aimspress.com/article/doi/10.3934/mbe.2019103
    [27] F. Kelly, Reversibility and Stochastic Networks, Wiley, Chichester, 1979.
    [28] P. Whittle, Systems in Stochastic Equilibrium, John Wiley & Sons, Inc., 1986.
    [29] T. Kurtz, Limit theorems and diffusion approximations for density dependent markov chains, in Stochastic Systems: Modeling, Identification and Optimization, I, Springer, 1976, 67–78.
    [30] T. Kurtz, Strong approximation theorems for density dependent Markov chains, Stoch. Proc. Appl., 6 (1978), 223–240. https://doi.org/10.1016/0304-4149(78)90020-0 doi: 10.1016/0304-4149(78)90020-0
    [31] S. Leite, R. Williams, A constrained Langevin approximation for chemical reaction network, Ann. Appl. Probab., 29 (2019), 1541–1608.
    [32] A. Angius, G. Balbo, M. Beccuti, E. Bibbona, A. Horvath, R. Sirovich, Approximate analysis of biological systems by hybrid switching jump diffusion, Theor. Comput. Sci., 587 (2015), 49–72. https://doi.org/10.1016/j.tcs.2015.03.015 doi: 10.1016/j.tcs.2015.03.015
    [33] J. Norris, Markov Chains, Cambridge university press, 1998.
    [34] D. Anderson, D. Cappelletti, M. Koyama, T. Kurtz, Non-explosivity of stochastically modeled reaction networks that are complex balanced, Bull. Math. Biol., 80 (2018), 2561–2579. https://doi.org/10.1007/s11538-018-0473-8 doi: 10.1007/s11538-018-0473-8
    [35] G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, preprint, arXiv: 1501.02860.
    [36] L. Paulevé, G. Craciun, H. Koeppl, Dynamical properties of discrete reaction networks, J. Math. Biol., 69 (2014), 55–72. https://doi.org/10.1007/s00285-013-0686-2 doi: 10.1007/s00285-013-0686-2
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2245) PDF downloads(75) Cited by(0)

Figures and Tables

Figures(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog