Fit-Preserving Data Refinement of Mass-Action Reaction Networks ⋆

Fit-Preserving Data Refinement of Mass-Action Reaction Networks ? Cristian Gratie and Ion Petre Computational Biomodeling Laboratory, Turku Centre for Computer Science and Department of Information Technologies, ˚ Abo Akademi University FI-20520 Turku, Finland {cgratie,ipetre}@abo.fi

Abstract. The systematic development of large biological models can benefit from an iterative approach based on a refinement process that gradually adds more details regarding the reactants and/or reactions of the model. We focus here on data refinement, where the species of the initial model are substituted with several subspecies in the refined one, each with its own individual behavior in the model. In this context, we distinguish between structural refinement, where the aim is to generate meaningful refined reactions, and quantitative refinement, where one looks for a data fit at least as good as that of the original model. The latter generally requires refitting the model and additional experimental data, a computationally expensive process. A fit-preserving refinement, i.e. one that captures the same species dynamics as the original model, can serve as a suitable alternative or as initialization for parameter estimation routines. We focus in this paper on the problem of finding all numerical setups that yield fit-preserving refinements of a given model and formulate a sufficient condition for it. Our result suggests a straightforward, computationally efficient automation of the quantitative model refinement process. We illustrate the use of our approach through a discussion of the Lotka-Volterra model for prey-predator dynamics. Keywords: biomodeling, model fit, parameter estimation, quantitative model refinement

1

Introduction

The development of models for large biological systems often starts top-down with an abstraction of the biological phenomena via a relatively small number of chemical reactions that illustrate the main mechanisms of the considered process. A mathematical model is then attached to this abstraction in order to describe the dynamic behavior of the system. The numerical setup for the mathematical ?

Cite as: C. Gratie and I. Petre. Fit-Preserving Data Refinement of Mass-Action Reaction Networks, in A. Beckmann, E. Csuhaj-Varj´ u, and K. Meer (Eds.): Language, Life, Limits, LNCS 8493, pp. 204–213, 2014. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-08019-2_21.

model comes from various computational procedures that fit the model with existing experimental data. The model is then iteratively refined by adding details to it. Refinement can involve the replacement of one (or more) species with subspecies, in which case it is called data refinement, or the replacement of a generic reaction with a set of reactions that gives more details about the same process by providing intermediary steps, in which case it is called process refinement [3]. Instead of refitting the model after every refinement, a computationally expensive process, it would be useful to take advantage of the well-fitted model from the previous step. The iterative step-by-step refinement of a formal specification towards an executable implementation is well established in Computer Science, in connection to software engineering and formal methods, see, e.g., [1]. In Systems Biology, refinement has been considered in the context of rule-based [5] and reaction-based models [9]. The implementation of model refinement using various standard modeling frameworks is discussed in [7]. The method of [9] for the numerical setup of a refined reaction-based model aims to preserve the numerical fit of the original model. The approach applies to the data refinement of models that rely on mass-action kinetics [10] and consists of inspecting the ordinary differential equations (ODEs) of the refined model and assigning parameter values in such a way that the ODEs describing the original model can be recovered as a sum of ODEs from the refined model. To see why this is worthwhile, let us consider refinement from a machine learning perspective. The original model provides us with an approximate characterization of the dynamic behavior of some system. The refined model, by having more independent parameters, should allow us to obtain a better characterization of the same system. Instead, what we are looking for in fit-preserving data refinement is a model that leads to the same dynamics as the original model, but also accounts for the possible subspecies of species from the original model. Such a refinement is for example appropriate if we are already satisfied with the approximation given by the original model (i.e. it falls within measurement error), if existing experimental data is not enough to support the number of parameters of the refined model, or if fitting the refined model is unfeasible resource-wise. But even when we are looking for a better approximation, fitpreserving refinements can serve as initialization for iterative algorithms that estimate the parameters of the refined model. Furthermore, fit-preserving refinement enables us to construct a hierarchy of models to describe the same process using various levels of detail. In this paper we provide a sufficient condition that links the refined parameters to those of the original model and guarantees that the refinement preserves the numerical fit of the original model. The constraints can easily be turned into an automatic procedure for setting the values of the refined parameters without the need for inspecting the ODEs. Our result also addresses an open problem presented in [4]. Given a refined model for which some of the parameters are known or can be measured experimentally, the question is whether there exists a numerical setup for the unknown

parameters such that the refined model preserves the numerical fit of the original model. Our result provides a partial answer to this question: as long as the values that are known do not already violate the fit-preservation constraints that we propose in this paper, there exists at least one solution and, moreover, the corresponding parameter values can be computed automatically. Furthermore, we provide a more precise mathematical framework for data refinement by separating structural refinement (aimed at generating the refined reactions) from quantitative refinement, where the goal is to obtain at least as good a fit as that of the original model. In this framework, fit-preserving refinement becomes a special kind of quantitative refinement, where we look for a computationally efficient numerical setup at the expense of not improving the fit of the model (but not making it worse either). The paper is structured as follows. Section 2 provides an introduction to chemical reaction networks and the formal notation that we use throughout the paper, roughly based on [2]. In Section 3 we formally discuss data refinement and state the main result of this paper. We apply our approach to a simple example in Section 4. We discuss the implications of our result in Section 5.

2

Reaction Networks

We first fix some notations used throughout the paper. We denote by N the set of non-negative integers and by N+ the set of positive integers. We denote by R≥0 the set of non-negative real numbers and by R+ the set of positive real numbers. For two sets X, Y we denote by X Y the set of mappings f : Y → X; for a finite set Y , X Y is also the set of vectors of dimension |Y | with elements from X. Throughout this paper we will always denote vectors with a lower-case bold-faced letter. We consider in this paper only irreversible reactions; any reversible reaction will be replaced by its ‘left-to-right’ and ‘right-to-left’ directions. Reactions are k typically denoted as rewriting rules, such as 2A1 + A2 −−1→ A3 , where k1 denotes the kinetic rate constant. We formalize a reaction in the style of [2] by denoting the species on its left and right hand side, together with their stoichiometric coefficients, using a vectorial notation. Throughout the paper we denote by S = {S1 , S2 , . . . , Sm } a finite alphabet whose elements we refer to as species. A vector in NS is called a complex over S . Note that this notion of complex refers to a linear combination of species that may occur on either side of the reaction. It should not be confused with the concept of a chemical complex, which would be represented in our model through a single species. With this notation, a reaction is defined as a pair of complexes (c, d) ∈ Nm × m N , standing for its left- and right-hand side complexes, resp.; the reaction will also be written as c → d. For our example above, we define S = {A1 , A2 , A3 }, c = [2, 1, 0]T and d = [0, 0, 1]T . For a reaction c → d, we use kc→d to denote its kinetic rate constant. We are now ready to give the formal definition of reaction networks.

Definition 1. A reaction network is a tuple N = (S , C , R, k), where S is a finite set of species, C ⊆ NS is a finite set of complexes, R ⊆ C × C is a finite set of reactions and k : R → R≥0 gives the kinetic rate constant of each reaction from R. Using the notation from [3], let us consider the following reaction network, consisting of n reactions and m species: rj :

m X

kj

cij Ai −−→

i=1

m X

dij Ai , for 1 ≤ j ≤ n .

(1)

i=1

The dynamic behavior of this network, under mass-action kinetics, see, e.g., [10], can be described by the following system of ODEs: a˙ i =

n X

(dij − cij )kj

m Y

acqqj , for 1 ≤ i ≤ m ,

(2)

q=1

j=1

where ai : R≥0 → R≥0 stands for the concentration of species Ai at time t ≥ 0 and a˙ i is used for the derivative of ai with respect to time. Our goal in what follows is to provide a more concise representation of (2) based on the vectorial notation introduced above. Given two vectors x, y ∈ Rn , we will denote xy = Qn yi i=1 xi . With this, (2) can be rewritten as a˙ =

X

kc→d ac (d − c) .

(3)

c→d∈R

Consider now the initial conditions of our system of ODEs. The differential equation x˙ = F (t, x) with initial conditions given by x(t0 ) = x0 , is known to have a unique solution in a neighborhood of t0 as long as the function F is continuously differentiable in a neighborhood of (t0 , x0 ), see [8]. In the case of reaction networks that follow mass-action dynamics, e.g. equation (3) with initial condition a(t0 ) = a0 , the function F is a polynomial, and so it is continuously differentiable on its entire domain. Furthermore, since there is no explicit dependence on time in (3), i.e. the system is autonomous, it follows that solutions are time-invariant in the following sense: given a solution a0 for the initial condition a0 (0) = a0 , the solution for the initial condition a(t0 ) = a0 can be written as a(t) = a0 (t − t0 ). Thus, without any loss of generality, it suffices to only consider the problem with initial conditions specified at t = 0, say a(0) = α. Moreover, since the solution depends (even continuously, see [8]) on the actual initial values α, we will make this explicit by writing the solution a as a[α] whenever its dependence on α is relevant.

3

Fit-Preserving Data Refinement

The data refinement of a reaction network is about adding some details into a network, e.g. through replacing one or more species of the network with a set of

subspecies carrying more detailed and potentially differentiated behavior. In the general setting, we assume to have two sets of species S and S 0 and a relation ρ ⊆ S × S 0 that links each species from S to its corresponding subspecies in S 0 . The intuition of species refinement is formally captured in Definition 2. Definition 2. Let S and S 0 be two sets of species. A relation ρ ⊆ S × S 0 is a species refinement relation iff it satisfies the following conditions: i) for each A ∈ S there exists A0 ∈ S 0 such that (A, A0 ) ∈ ρ; ii) for each A0 ∈ S 0 there exists exactly one A ∈ S such that (A, A0 ) ∈ ρ. Intuitively, when (A, A01 ), . . . , (A, A0r ) are all the elements of ρ with A on their left position, we mean that species A is refined and replaced in the refined model by its subspecies A01 , . . . , A0r . Each species from the original model should be refined to at least one species in the refined model (more than one in the case of non-trivial refinements) and each species of the refined model should correspond to exactly one “parent” species from the original model. 0 The species refinement ρ can also be written as a matrix Mρ ∈ RS ×S : ( 1, if (A, A0 ) ∈ ρ ; Mρ = (mA,A0 )A∈S ,A0 ∈S 0 , with mA,A0 = (4) 0, otherwise . As a convention, we will denote matrices throughout this paper with upper-case, bold-faced letters. Definition 3. Let S and S 0 be two sets of species, and ρ ⊆ S × S 0 a species refinement relation. Let c and c0 be two complexes over S and S 0 , respectively. We say that c0 is a ρ-refinement of c (or simply a refinement if ρ is clearly understood in that context) if, for every species S ∈ S , the stoichiometric coefficients of its subspecies in c0 add up to its stoichiometric coefficient in c. This can be written using matrix notation as Mρ c0 = c. We say that complex c is ρ-refined to the set of complexes C 0 over S 0 if all the elements of C 0 are ρ-refinements of c. The set of all possible ρ-refinements of c is denoted by ∆ρ (c). Finally, we say that reaction (c, d) over S is ρ-refined to the set of reactions R 0 over S 0 if, for every (c0 , d0 ) ∈ R 0 , c0 is a ρ-refinement of c and d0 is a ρrefinement of d. The set of all possible ρ-refinements of (c, d) can be written as ∆ρ (c) × ∆ρ (d). Let us consider again the reaction from the previous section 2A1 + A2 → A3 . Assume we refine A1 to B11 and B12 , A2 to B2 and A3 to B31 , B32 and B33 , i.e. we consider the refinement relation ρ = {(A1 , B11 ), (A1 , B12 ), (A2 , B2 ), (A3 , B31 ), (A3 , B32 ), (A3 , B33 )}. Then the possible refinements of the original reaction are: 2B11 + B2 → B31 , 2B11 + B2 → B32 , 2B11 + B2 → B33 , B11 + B12 + B2 → B31 , B11 +B12 +B2 → B32 , B11 +B12 +B2 → B33 , 2B12 +B2 → B31 , 2B12 +B2 → B32 and 2B12 + B2 → B33 . With Definition 3 we are ready now to introduce the notion of reaction network refinement. Since our notion of a network has two components, the

species-reaction structure and the ODE-based quantitative dynamics, our definition of network refinement is given in two parts: a structural refinement and a fit-preserving refinement. The former one is immediate following Definition 3. Definition 4. Let N = (S , C , R, k) and N 0 = (S 0 , C 0 , R 0 , k 0 ) be two reaction networks and ρ ⊆ S × S 0 a species refinement relation. We say that N 0 is a structural refinement of N if [ [ C0 = ∆ρ (c) and R0 ⊆ ∆ρ (c) × ∆ρ (d) . (5) c∈C

c→d∈R

We say that N 0 is the full structural refinement of N if we have equality in the definition of R 0 in (5). The quantitative part of our notion of network refinement focuses on preserving the experimental data fit of the original model; in other words, the kinetic rate constants of the model obtained through structural refinement should be set so that the dynamics of a species in the original model is identical to the dynamics of the sum of its subspecies in the refined model. We formalize this condition in what follows. Definition 5. Let N = (S , C , R, k) and N 0 = (S 0 , C 0 , R 0 , k 0 ) be two reaction networks and ρ ⊆ S × S 0 a species refinement relation; we denote by Mρ the S0 matrix representation of ρ. Let α ∈ RS ≥0 , β ∈ R≥0 be the initial conditions for 0 N and N , resp. We say that β is a ρ-refinement of α if α = Mρ β. Let the ODEs describing the two reaction networks be: X X 0 a˙ = kc→d ac (d − c) and b˙ = kc0 0 →d0 bc (d0 − c0 ) , (6) c0 →d0 ∈R0

c→d∈R 0

0 S where a : R≥0 → RS ≥0 , b : R≥0 → R≥0 . We say that N is a fit-preserving refinement of N if it is a structural refinement of N and, for any initial conditions S0 α ∈ RS ≥0 and β ∈ R≥0 such that β is a ρ-refinement of α, the solutions a and b of equations (6) satisfy a(t) = Mρ b(t) in a neighborhood of 0.

Note that in Definition 5 we can assume without loss of generality that N 0 is the full structural refinement of N . Indeed, any other structural refinement can be extended to the full one by adding to it the missing refined reactions and then setting their kinetic rate constants to 0. Note also that we required that (6) holds for all values of t in a neighborhood of 0 and not for all t ≥ 0, as in [3]. This formulation is enough for the formal result we are going to prove and, moreover, it is also applicable when the solution of the ODEs is not defined for all non-negative values of t. More precisely, the solution is unique for its full domain of existence. The problem that we focus on is how to effectively construct a fit-preserving refinement of a given reaction network N when given the species refinement relation ρ. The first part is to build the full structural ρ-refinement of N ; this

can be done by constructing ∆ρ (c) × ∆ρ (d) for all reactions (c, d) of N . The second part is to set the kinetic rate constants of the refined model so that it yields a fit-preserving refinement. To check whether a given numerical setup of the full structural refinement yields a fit-preserving refinement seems to require in general to solve the systems of ODEs in Equation (6). A difficulty is that those systems of ODEs are non-linear and cannot be solved analytically in general. We recall the following problem of [4]. Problem 1. [4] Let N be a reaction network, ρ a species refinement relation, and N 0 the full structural ρ-refinement of N . Assuming that numerical values of some of the kinetic rate constants of N 0 are fixed, find a numerical setup for all its other kinetic rate constants so that N 0 is a fit-preserving refinement of N . The following result gives a partial answer, i.e. a sufficient condition, to Problem 1. Theorem 1. Let N = (S , C , R, k) be a reaction network, S 0 a set of subspecies and ρ ⊆ S × S 0 a species refinement relation. Let N 0 be the full structural ρrefinement of N , N 0 = (S 0 , C 0 , R 0 , k 0 ). If, for every c → d ∈ R and for every c0 ∈ ∆ρ (c), we have that X d0 ∈∆ρ (d)

kc0 0 →d0 =

Q|S |     c c i=1 ci ! k , with = c→d Q|S 0 | 0 , c0 c0 c ! j=1

(7)

j

then N 0 is a fit-preserving data refinement of N . Due to space constraints we do no provide the proof here, the reader may find it in our technical report [6]. Note that the constraint (7) is not far from what one would expect. Indeed, the rate constants of all refined reactions that share the same left-hand side c0 depend on the rate constant of the parent reaction, kc→d , and on its left hand side c. The interesting aspect is the linear character of the dependency. We can apply Theorem 1 for assigning rate constants to a given structural network refinement so that we obtain a fit-preserving refinement. Even if we are given a partially specified structural data refinement (i.e. one where several rate constants are fixed to predefined values, such as zero for reactions that are assumed impossible), we can turn it into a fit-preserving refinement as long as the fixed values do not already lead to a violation of Equation (7). This might happen because rate constants are non-negative so there is no way to reduce a sum that already exceeds the value prescribed by (7). In the absence of any information about the rate constants of the refined model, we can choose a symmetric assignment, i.e. all rate constants in any sum described by (7) are set to be equal, see also the discussion in the next section. We can also apply Theorem 1 in order to check that a given structural data refinement is fit-preserving; this only gives a partial answer because the condition of Theorem 1 is only sufficient, but not always necessary. Indeed, it is shown in [2]

that it is possible to have two reaction networks that differ only in the assignment of rate constants, but describe the same dynamics (translate to the same ODEs). Let N1 = (S , C , R, k1 ) and N2 = (S , C , R, k2 ) be two such networks. A fitpreserving data refinement constructed for N1 based on Theorem 1 is also a fit-preserving data refinement of N2 , but it may fail to satisfy (7) for N2 .

4

A Simple Example

In this section we are going to use the constraints from Theorem 1 for the data refinement of a simple reaction network, corresponding to the Lotka-Volterra system for modeling prey-predator dynamics [11]. k

A −−1→ 2A

k

k

A + B −−2→ 2B

B −−3→ ∅

(8)

In the equations, A stands for the prey and B for the predator. The set of species in this case is S = {A, B} and the set of complexes C = {[1, 0]T , [2, 0]T , [1, 1]T , [0, 2]T , [0, 1]T , [0, 0]T }. Now, assume that we can distinguish two different kinds of prey. This corresponds to a species refinement ρ = {(A, A1 ), (A, A2 ), (B, B 0 )}. The set of subspecies is S 0 = {A1 , A2 , B 0 }. The full structural ρ-refinement of the model (with the same notation as in [3]) can be written as r

A1 −−2→ A1 + A2 ,

r

A2 −−5→ A1 + A2 ,

A1 −−1→ 2A1 , A2 −−4→ 2A1 , 0

r7

0

A1 + B −−→ 2B ,

r

A1 −−3→ 2A2 ,

r

A2 −−6→ 2A2 ,

0

r8

0

A2 + B −−→ 2B ,

r

r

0

(9)

r9

B −−→ ∅ .

If we identify each reaction by the name of its rate constant, we can write the corresponding refinement of each reaction from the original model: k1 → {r1 , r2 , r3 , r4 , r5 , r6 }, k2 → {r7 , r8 }, k3 → {r9 }. The sums given by Equation (7) in this case are the following:    [1, 0]T [1, 0]T r1 + r2 + r3 = k1 = k1 , r4 + r5 + r6 = k1 = k1 , [1, 0, 0]T [0, 1, 0]T     [1, 1]T [1, 1]T (10) r7 = k = k , r = k2 = k2 , 2 2 8 [1, 0, 1]T [0, 1, 1]T   [0, 1]T r9 = k3 = k3 . [0, 0, 1]T 

Equation (10) can now be used in a versatile way to construct very different refined models. For example, assume that all reactions involving both subspecies A1 and A2 should be eliminated from the refined model based on biological arguments. We can easily remove them from the model and still obtain a fitpreserving refinement by taking r2 = r3 = r4 = r5 = 0 and then consider the problem of finding values for the remaining rate constants so that (10) is satisfied.

In this particular case, the solution obtained is r1 = r6 = k1 , r7 = r8 = k2 , and r9 = k3 . As another example, assume that we may allow reactions involving both A1 and A2 , but with no quantitative distinction between them; in this case the constants in each of the sums of Equation (10) could be set equal, as follows, and as set also in [3]: r1 = r2 = r3 = k1 /3, r4 = r5 = r6 = k1 /3, r7 = k2 , r8 = k2 , r9 = k3 . A similar approach can be taken when some of the rate constants of the refined model are known or can be measured/estimated experimentally. In such cases, for each sum in (7), all known rate constants are subtracted, and the remainder, if non-negative, can be split (perhaps evenly) between the unknown rate constants. If, e.g., we know from the literature that r1 = r4 = k1 /2, then we can obtain a fit-preserving refinement by choosing any value for the remaining constants so that r2 + r3 = k1 /2 and r5 + r6 = k1 /2.

5

Conclusions

In this paper we introduced a new mathematical framework for the notion of reaction network model and model refinement. We focused on the problem of finding fit-preserving refinements and we proposed a sufficient condition for the rate constants of a structural refinement of a reaction network so that the resulting model describes the same dynamics as the original model (with respect to the species of the original model). Our result is versatile and can be combined with partial information about some of the rate constants of the refined model. In particular, this leads to an algorithmic assignment of rate constant values that can be performed automatically and be used as initialization for parameter estimation software. Since the constraint (7) of Theorem 1 only partitions the rate constants of the refined model into sets that should add up to particular values that depend on the original model, this means that in general there is still a lot of freedom left in the choice of the actual values. On one hand, this is useful for the initialization of parameter estimation algorithms, since it provides room for randomization in order to avoid local optima. On the other hand, this also raises an interesting question for further research, namely whether among all possible assignments there are some that offer other desirable properties in addition to fit-preservation. Such properties may, for example, be based on the internal structure of species and some desired conservation laws and, in such cases, the symmetric assignment of the remaining values might no longer be the most appropriate thing to do. An implicit assumption in our considerations is that the kinetic constants are fixed throughout the model dynamics. This may not be the case if some of them depend on temperature and the model includes, e.g., some exothermic reactions. It seems an interesting question whether our approach can be extended to such models. Our result also provides a partial answer to the open question of [4] on the existence of rate constant values that can turn a structural refinement (possibly

with some of the values fixed in advance based on existing literature) into a fit-preserving one with respect to the original model. The answer that follows from Theorem 1 is that such an assignment is guaranteed to exist, as long as the already fixed values do not lead to a violation of the fit-preservation constraints (this can happen for example if the sum of the fixed values already exceeds the required value, since the rate constants can only take non-negative values). One possible use of fit-preserving refinement that we have mentioned in the introduction of this paper is the construction of hierarchical models to capture different levels of detail for the same biological process. Refinement in this case allows us to go from one model to a more detailed one. The fit-preservation constraints, on the other hand, enable us to go backward, i.e. obtain a more general model from a detailed one. In this context, it is also interesting to investigate whether it is also possible to retrieve the more general model after the detailed one has been refitted (and the fit-preservation constraints may not hold anymore). The proof of our result relies on the interplay between multinomial expansion and the formulation of mass-action dynamics to yield very simple constraints. On the other hand, it would be interesting to investigate fit-preserving data refinement for other kinetic models as well. Another problem that remains open is to find necessary conditions for the numerical setup of refined models, thus aiming for a full solution of Problem 1.

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