Cristian Gratie | Ion Petre
Complete Characterization for the FitPreserving Data Refinement of Mass-Action
Reaction Networks
TUCS Technical Report
No 1128, January 2015
Complete Characterization for the FitPreserving Data Refinement of Mass-Action
Reaction Networks
Cristian Gratie
Computational Biomideling Laboratory, Turku Centre for Computer Science and
Department of Information Technologies, Åbo Akademi University
Joukahaisenkatu 3-5 A, 20520 Turku, Finland
[email protected]
Ion Petre
Computational Biomideling Laboratory, Turku Centre for Computer Science and
Department of Information Technologies, Åbo Akademi University
Joukahaisenkatu 3-5 A, 20520 Turku, Finland
[email protected]
TUCS Technical Report
No 1128, January 2015
Abstract
The systematic development of large biological models often relies on iteratively adding more details to an initial, simplified, abstraction of the modeled
system. In data refinement, species of the original model are substituted with
several variants (subspecies) in the refined one, each with its own individual
behavior. In this context, one can distinguish between structural refinement,
where the aim is to generate meaningful refined reactions, and quantitative
refinement, where one looks for a refined model that better fits available experimental data. The latter is generally a computationally expensive process,
as it requires refitting the model and sometimes additional data.
It is possible to reuse parameter values from the previous iteration by
constructing a fit-preserving refinement, i.e. one that captures the same
species dynamics as the original model, but accounts for the newly introduced
subspecies. We showed in previous work that fit-preserving refinements can
be computed efficiently by providing simple linear constraints on the rate
constants of the refined model, which are sufficient for fit-preservation.
In this paper, we extend our result and provide a complete characterization of fit-preserving refinement, in the form of a necessary and sufficient
condition, applicable for mass-action reaction networks with uniquely identifiable rate constants. Furthermore, we discuss the refinement of a simple
model, the Brusselator, and show that, while constrained to capture the same
species dynamics as the original model, fit-preserving refinements can still be
very different with respect to the behavior of subspecies. Furthermore, our
examples provide an intuition about how experimental data can be used to
guide model fitting by identifying classes of qualitatively distinct behaviors.
Keywords: biomodeling, canonical refinement, fit-preserving refinement,
model fit, parameter estimation, quantitative model refinement
TUCS Laboratory
Computational Biomodeling Laboratory
1
Introduction
The top-down development of large biological models relies on iteratively
adding details to an initial abstraction of the biological phenomena. At
each step, the numerical setup of the mathematical model that describes the
dynamic behavior of the system needs to be fit against existing experimental
data. Model fitting is often a computationally expensive process, thus it
would be beneficial to reuse previously fit values to initialize the parameter
estimation routines for the refined model.
We focus here on reaction-based models that rely on mass-action kinetics
[7]. Data refinement, introduced in [2], refers to the replacement of one (or
more) species with several variants, or subspecies, in the refined model. The
approach presented in [2] for the numerical setup of the refined model aims
to preserve the fit of the original model and consists in 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. We formalized this
idea in [4] as fit-preserving refinement and provided a sufficient condition for
preserving the numerical fit. The condition links the refined parameters to
those of the original model without the need for inspecting the ODEs.
The approach from [4] provides a partial answer to the open problem
formulated in [3] of finding values for the unknown parameters of a partially
specified refined model so that it preserves the numerical fit of the original
model: as long as the known values do not already lead to a violation of the
fit-preservation constraints, the problem has at least one solution, which can
be effectively computed.
Since the proposed condition is only sufficient, for some models it is possible to build fit-preserving refinements for which the parameter values do
not satisfy the condition. We address this issue here and show that all such
models are in fact “pathological” cases that do not have uniquely identifiable
rate constants, a critical requirement for being able to fit the model with
experimental data, see [1].
This paper is an extended version of [4]. In addition to [4], we include here
the full proof of its main result, as well as all mathematical considerations
leading to it. Moreover, we prove that the sufficient condition given in [4] is
in fact a necessary and sufficient condition, i.e. a complete characterization
of fit-preserving refinement, provided that the original model has uniquely
identifiable rate constants. Finally, we illustrate our approach on a new case
study, that of the Brusselator [9], to show that fit-preserving refinement alone
can give rise to very different refined models.
The paper is structured as follows. Section 2 provides an introduction to
chemical reaction networks and the main formal results related to uniquely
identifiable rate constants and solutions of ordinary differential equations. In
Section 3 we formally discuss fit-preserving refinement and prove the main
1
result of this paper. We apply our approach to the study of a simple example,
the Brusselator [9], in Section 4, where we analyze four different refinements
of the model. We discuss the implications of our result in Section 5.
2
Preliminaries
We first fix some notations used throughout the paper. We denote by N the
set of non-negative integers and by R≥0 the set of non-negative 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 can also be seen as 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 will use bold-faced upper-case letters to
denote matrices or functions that have multiple inputs and outputs.
2.1
Solutions of Autonomous Ordinary Differential Equations
Let F : Rn → Rn be a continuously differentiable function. We focus on the
solutions of the following autonomous ordinary differential equation (ODE):
ẋ = F (x)
(1)
with the initial condition x(t0 ) = x0 . Following common practice in the
literature, we will also call this an initial value problem (IVP). We can also
understand equation (1) as a system of differential equations if we consider
the components of x and F (x) separately. A solution of such an equation is
a function x : I → Rn , where I ⊆ R is an interval containing t0 , such that x
is differentiable and, moreover, ẋ(t) = F (x(t)), for all t ∈ I. It is well known
that such IVPs have a unique solution in the neighborhood of t0 , as long as
F satisfies some reasonable assumptions, for example see [6]. Since our work
involves the careful manipulation of these ideas, we provide (without proof)
formal results for both the existence and uniqueness in what follows.
First, let us see that if we are given a solution x of equation (1) with
the given initial condition, then the function x(t − t0 ) satisfies the same
differential equation, with the initial condition specified at 0, i.e. x(0) = x0 .
Thus, it suffices to only consider initial conditions specified for t0 = 0.
Theorem 1. Let F : Rn → Rn be a continuously differentiable function and
x0 ∈ Rn . Then there exists a closed interval [−a, a] and a unique function
x : [−a, a] → Rn such that ẋ(t) = F (x(t)) for all t ∈ [−a, a] and x(0) = x0 .
For a complete proof of Theorem 1, the reader may see [6]. The uniqueness
applies also for extensions of the solution to larger intervals. In particular,
we can consider the maximal time domain for which a solution exists and
2
refer to the corresponding solution as the solution of the equation for the
given initial condition.
Definition 1. Let F : Rn → Rn be a continuously differentiable function.
For every α ∈ Rn , we will use a[α] to refer to the unique, maximal (with
respect to its time domain) solution of the differential equation ȧ = F (a),
with a(0) = α. We will use dom(a[α]) to denote the domain of a[α].
Note that the time domain of a solution a[α] may also depend on the
actual value of α.
2.2
Reaction Networks
We consider in this paper that all reactions are irreversible; any reversible
reaction is replaced by its “left-to-right” and “right-to-left” irreversible reactions. We formalize in the following the notion of a reaction, using both a
rewriting rule style, and a vectorial style.
We first define the notions of species and complexes.
Definition 2. Let S = {S1 , . . . Sm } be a finite set whose elements we call
species. A vector in NS is called a complex over S . If c = [c1 , . . . , cm ]T is
a complex over S , then we say that ci is the multiplicity (or, equivalently,
the stoichiometric coefficient) of species Si in complex c, for all 1 ≤ i ≤ m.
Note that our notion of complex refers to a linear combination of species
that may occur on either side of a reaction. It should not be confused with the
concept of a chemical complex, which would be represented in our notation
through a single species.
We define now reactions and reaction networks.
Definition 3. Let S = {S1 , . . . , Sm } be a set of species. A reaction r over
S is a pair of complexes c, d over S , r = (c, d). If c = [c1 , . . . , cm ]T and
d = [d1 , . . . , dm ]T , reaction r is usually written in the style of a rewriting rule
as
r:
m
X
ci Si →
i=1
m
X
di Si
i=1
For the sake of a compact notation, we will also use, following [1], a vectorial
notation:
r:c→d
To a reaction r we can associate a kinetic rate constant kr ∈ R≥0 ; the kinetic
rate constant is usually indicated on top of the arrow in both conventions for
writing a reaction.
3
We note that even though we indicate the multiplicity of all species, both
on the left-hand side, and on the right-hand side of a reaction, in practice
most of the stoichiometric coefficients are zero. Indeed, chemical kinetics
show (see, e.g., [8]) that reactions where the sum of multiplicities on their
left-hand side is at least three have such a low kinetic rate constant that their
effect is negligible.
Definition 4. A reaction network is a pair N = (S , R), where S is a finite
set of species, S = {S1 , . . . , Sm }, and R is a finite set of reactions over S ,
R = {r1 . . . , rn }, where:
rj :
m
X
cij Si →
i=1
m
X
dij Si ,
1 ≤ j ≤ n.
i=1
We also write each reaction through its vectorial convention:
rj : cj → dj ,
1 ≤ j ≤ n,
where cj = [c1j , . . . , cmj ]T and dj = [d1j , . . . , dmj ]T .
A mass-action reaction network is a tuple M = (S , R, k), where (S , R)
is a reaction network and k ∈ RR
≥0 ; kc→d is called the reaction rate constant
of reaction c → d ∈ R.
The system of ODEs associated to a mass-action reaction network is constructed as follows. We associate to each species Si ∈ S a real function
si : R≥0 → R≥0 , whose intended interpretation is the time-dependent concentration of Si . These functions are defined, in the case of mass-action
kinetics, through the following system of ODEs:
ṡi =
n
X
(dij − cij )kj
m
Y
scqqj ,
1 ≤ i ≤ m.
q=1
j=1
We also write this system of ODEs in a compact, vectorial style as
ṡ =
X
kc→d sc (d − c),
(2)
c→d∈R
ci
where s = [s1 , . . . , sm ]T , ṡ = [ṡ1 , . . . , ṡm ], and sc = m
i=1 si .
Note that (2) is an autonomous ODE; let F (s) be its right hand side.
Since in the case of mass-action kinetics the function F is a multinomial with
respect to the components of s, it is continuously differentiable and, thus,
the existence and uniqueness of the solution s[s0 ] is guaranteed by Theorem
1, for any initial values s0 ∈ RS .
Q
4
2.3
Uniquely Identifiable Rate Constants
Let M = (S , R, k) be a mass-action reaction network. We introduce, in the
spirit of [1], the following notation for the right-hand side of a mass-action
system (2), using the vectorial style notation:
r[R, k](α) =
X
kc→d αc (d − c),
c→d∈R
for all α ∈ RS
≥0 .
Definition 5. [1] A reaction network N = (S , R) is said to have uniquely
identifiable rate constants if, for any two distinct rate constant vectors k1 , k2 ∈
RR
≥0 , it holds that r[R, k1 ] 6= r[R, k2 ].
In words, the fact that a reaction network has uniquely identifiable rate
constants simply means that two distinct rate constant vectors cannot give
rise to exactly the same ordinary differential equation. From a model fitting perspective, such a property is clearly desirable. In its absence, even
very precise measurements would not enable one to find the appropriate rate
constants, since several rate constant vectors can lead to exactly the same
dynamic behavior.
Note that the property from Definition 5 only depends on the set of reactions R. Thus, it is a structural property that ensures a different dynamical
behavior for each possible reaction rate constant vector k.
Definition 6. Let N = (S , R) be a reaction network. A complex c over S
is a source complex in N if it is the left hand side of at least one reaction from
R, i.e. there exists at least one complex d over S such that c → d ∈ R.
Theorem 2. [1] A reaction network N = (S , R) has uniquely identifiable
rate constants if and only if, for each source complex c of N , the reaction
vectors {d − c | c → d ∈ R} are linearly independent.
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 . For each S ∈ S we denote ρ(S) = {S ∈ S 0 | (S, S 0 ) ∈ ρ}.
The intuition of species refinement is formally captured in Definition 7.
Definition 7. [4] Let S and S 0 be two sets of species. A relation ρ ⊆
S ×S 0 is a species refinement relation if and only if it satisfies the following
conditions:
5
1. for each S ∈ S , ρ(S) 6= ∅;
2. for each S 0 ∈ S 0 there exists exactly one S ∈ S such that S 0 ∈ ρ(S).
Intuitively, when ρ(S) = {S10 , . . . , Sr0 }, we mean that species S is refined
and replaced in the refined model by its subspecies S10 , . . . , Sr0 . 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.
A species refinement ρ can also be written as an (S × S 0 )-matrix with
{0, 1} entries as follows:
Mρ = (mS,S 0 )S∈S ,S 0 ∈S 0 ,

1,
if S 0 ∈ ρ(S) ;
mS,S 0 = 
0, otherwise .
(3)
By definition, there is at least a 1-entry on each row of the matrix and
there is exactly one 1-entry on each of its columns. We will refer to Mρ as
the characteristic matrix of the species refinement relation ρ.
The notion of species refinement is extended to complexes, reactions, and
reaction networks as follows.
0
Definition 8. Let S = {S1 , . . . , Sm } and S 0 = {S10 , . . . , Sm
0 } be two sets of
species and ρ ⊆ S × S 0 a species refinement relation.
1. Let c = [c1 , . . . , cm ]T ∈ NS and c0 = [c01 , . . . , c0m0 ] ∈ NS be two complexes over S , respectively S 0 . We say that c0 is a ρ-refinement of c,
denoted c0 ∈ ρ(c), if
0
X
c0j = ci ,
for all 1 ≤ i ≤ m
1≤j≤m0
Sj0 ∈ρ(Si )
or, equivalently, relying on the characteristic matrix, if c = Mρ c0 .
2. Let r : c → d be a reaction over S and r0 : c0 → d0 a reaction over S 0 .
We say that r0 is a ρ-refinement of r, denoted r0 ∈ ρ(r), if c0 ∈ ρ(c)
and d0 ∈ ρ(d).
3. Let N = (S , R) and N 0 = (S 0 , R 0 ) be two reaction networks over S
and S 0 , respectively. We say that N 0 is a ρ-refinement of N , denoted
N 0 ∈ ρ(N ), if
R0 ⊆
[
ρ(r)
and
ρ(r) ∩ R 0 6= ∅, for all r ∈ R.
r∈R
In case R 0 =
S
r∈R
ρ(r), we say that N 0 is the full ρ-refinement of N .
6
4. Let M = (S , R, k) and M 0 = (S 0 , R 0 , k0 ) be two mass-action reaction networks over S and S 0 , respectively. We say that M 0 is a ρrefinement of M , denoted M 0 ∈ ρ(M ), if (S 0 , R 0 ) ∈ ρ(S , R). We say
that M 0 is a full ρ-refinement of M if (S 0 , R 0 ) is the full ρ-refinement
of (S , R).
5. Let σ ∈ RS and σ 0 ∈ RS (thought of as the initial values for the
system of ODEs associated to M and M 0 ). We say that σ 0 is a ρrefinement of σ, denoted σ 0 ∈ ρ(σ), if σ = Mρ σ 0 .
0
The definition given above for the refinement of mass-action reaction
networks is only structural, describing the relationship between their sets
of species and reactions. We introduce now a quantitative counterpart for
the notion of refinement for mass-action reaction networks: fit-preserving
refinement.
Definition 9. Let M = (S , R, k) and M 0 = (S 0 , R 0 , k0 ) be two mass-action
reaction networks and ρ ⊆ S ×S 0 a species refinement relation. For any σ ∈
0
0
0
S0
0
S0
S
RS
≥0 and σ ∈ R≥0 we denote by s[σ] : [0, τ ) → R≥0 (s [σ ] : [0, τ ) → R≥0 )
the vector of the real functions obtained from the ODE system associated to
M (to M 0 , respectively) with initial values σ (σ 0 , respectively).
We say that M 0 is a ρ-fit-preserving refinement of M if M 0 ∈ ρ(M ) and,
0
S0
0
for all σ ∈ RS
≥0 and σ ∈ R≥0 such that σ = Mρ σ , we have that
s[σ](t) = Mρ s0 [σ 0 ](t),
(4)
for all values of t in a suitable right-neighborhood of 0.
We prove in our next result that having the fit-preserving condition (4)
for non-negative initial values implies that the same condition holds even
for arbitrary initial values. Even though in practice the actual values are
usually chosen to be non-negative (they represent concentration levels) this
observation is interesting in itself and potentially useful in formulating more
general results about refinement. Moreover, we show that (4) holds for all
values of t for which both s[σ] and s0 [σ 0 ] are defined.
Theorem 3. Let M = (S , R, k) and M 0 = (S 0 , R 0 , k0 ) be two mass-action
reaction networks and ρ ⊆ S × S 0 a species refinement relation. If M 0 is a
0
ρ-fit-preserving refinement of M , then for all σ ∈ RS , σ 0 ∈ RS such that
σ = Mρ σ 0 we have that s[σ](t) = Mρ s0 [σ 0 ](t), for all values of t for which
both functions are defined, i.e. t ∈ dom(s[σ]) ∩ dom(s0 [σ 0 ]).
Proof. For all σ 0 ∈ RS
≥0 , we know from the definition of fit-preserving re0
finement that s[Mρ σ ](t) = Mρ s0 [σ 0 ](t), for t in a right-neighborhood of
0. Moreover, the equality holds for all values of t for which both functions
0
7
are defined, via Theorem 1, since Mρ s0 [σ 0 ] and s[Mρ σ 0 ] are solutions of the
same IVP.
Since the equality holds in a right-neighborhood of 0, it must be that also
the corresponding rates at 0 are equal, i.e. ṡ[Mρ σ 0 ](0) = Mρ ṡ0 [σ 0 ](0), which
leads to
X
c→d∈R
c
kc→d (Mρ σ 0 ) (d − c) − Mρ
X
c0
kc0 0 →d0 σ 0 (d0 − c0 ) = 0
(5)
c0 →d0 ∈R 0
for all σ 0 ∈ RS
≥0 . But since the left hand side only consists of multinomials
with respect to the components of σ 0 , it follows that (5) holds for any σ 0 ∈
RS . But then we can use the equality to show that Mρ s0 [σ 0 ] satisfies the
0
initial value problem of s[Mρ σ 0 ] for any σ 0 ∈ RS , which leads by Theorem 1
to the desired result.
Note that the result of Theorem 3 relies both on the fact that mass-action
ODEs are multinomials with respect to the species’ concentrations, and on
the behavior of the refinement matrix Mρ . With the stronger requirements
of Theorem 3 we aim to capture the maximal extent to which the refined and
original models can describe the same dynamic behavior.
The problem we focus on is how to effectively construct a fit-preserving
refinement of a given mass-action reaction network M when given the species
refinement relation ρ. The first part is to build the full structural ρ-refinement
of M ; this can be done by constructing ρ(c) × ρ(d) for all reactions c → d of
M . The second part is to set the kinetic rate constants of the refined model
so that it yields a fit-preserving refinement.
A difficulty in checking whether a given numerical setup of the full structural refinement yields a fit-preserving refinement is that the system of ODEs
is non-linear and cannot be solved analytically in general. We recall the following problem of [3], with the formulation given in [4].
Problem 1. [3] Let M be a mass-action reaction network, ρ a species refinement relation, and M 0 a full structural ρ-refinement of M . Assuming
that numerical values of some of the kinetic rate constants of M 0 are fixed,
find a numerical setup for all its other kinetic rate constants so that M 0 is a
fit-preserving refinement of M .
In our previous work [4] we gave a partial answer to Problem 1 in the
form of a sufficient condition on the kinetic rate constants that, if satisfied,
would entail that the considered full structural refinement is fit-preserving.
We now turn this result around and we introduce the class of full structural
refinements satisfying the constraint as canonical refinements. This will allow
us to formulate a stronger result, one that also discusses necessity in addition
to sufficiency.
8
Definition 10. Let M = (S , R, k) and M 0 = (S 0 , R 0 , k0 ) be two massaction reaction networks and ρ ⊆ S × S 0 a species refinement relation. We
say that M 0 is a canonical ρ-refinement of M if M 0 is a full ρ-refinement of
M and, for every c → d ∈ R and every c0 ∈ ρ(c), we have that
!
X
d0 ∈ρ(d)
kc0 0 →d0
c
= 0 kc→d , where
c
!
Q|S |
c
i=1 ci !
.
= Q|S
0|
0
0
c
j=1 cj !
Note that the constraint from Definition 10 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.
Theorem 4. Let M = (S , R, k) and M 0 = (S 0 , R 0 , k0 ) be two reaction
networks such that M 0 is a full ρ-refinement of M .
1. If M 0 is a canonical ρ-refinement of M , then M 0 is a fit-preserving
ρ-refinement of M .
2. If M has uniquely identifiable rate constants, then M 0 is a fit-preserving
ρ-refinement of M if and only if M 0 is a canonical ρ-refinement of M .
Note that part 1 is the one we have already presented, without proof,
in [4]. Here, we are going to give a unified proof of both results, one that
will also highlight the important elements that distinguish the particular
problem of refining mass-action reaction networks from a very broad notion
of refinement, which is captured in Lemma 1.
To see why the scenario presented in Lemma 1 is a very general form of
fit-preserving refinement, assume that we are trying to describe the dynamics
of a complex system, for which we have devised two models. One has n state
parameters and is characterized by ȧ = F (a), while the other one has m
parameters and satisfies ḃ = G(b). In order to compare the two models, we
need a way to relate the solutions of the two ODEs. The key ingredient for
that is a direct correspondence between the states of the two models, which
is given by the function Φ.
In this context, a[Φ(β)] = Φ ◦ b[β] essentially means that we expect
that, given an initial value β in the second model, the solution a for the
first model initialized at the image of β through Φ is the same as the image
of the solution b initialized at β. This is very similar to the condition we
imposed for fit-preserving refinement, with the observation that in our case
the correspondence function Φ is simply the left multiplication with the
species refinement matrix Mρ .
9
In what follows, for a given function Φ : Rm → Rn , we will use Φ0 (x)
to denote the derivative of Φ at x, i.e. the linear operator defined by the
Jacobian matrix:

0
Φ (x) =






∂Φ1
(x)
∂x1
∂Φ2
(x)
∂x1
..
.
∂Φn
(x)
∂x1
∂Φ1
(x)
∂x2
∂Φ2
(x)
∂x2
···
···
..
.
..
.
∂Φn
(x) · · ·
∂x2
∂Φ1
(x)
∂xm
∂Φ2
(x)
∂xm
..
.
∂Φn
(x)
∂xm







.
As before, we still use the dot to denote the derivative with respect to time.
Lemma 1. Let F : Rn → Rn , G : Rm → Rm and Φ : Rm → Rn be continuously differentiable functions. The following two statements are equivalent:
1. For every β ∈ Rm , the (unique maximal) solutions a[Φ(β)] and b[β]
of the initial value problems ȧ = F (a), a(0) = Φ(β), respectively
ḃ = G(b), b(0) = β satisfy a[Φ(β)] = Φ ◦ b[β], i.e. a[Φ(β)](t) =
Φ(b[β](t)) for all t ∈ dom(a[Φ(β)]) ∩ dom(b[β]).
2. F ◦ Φ = Φ0 · G, i.e. for all β ∈ Rm , F (Φ(β)) = Φ0 (β)G(β).
Proof. 1 ⇒ 2: We start from a[Φ(β)] = Φ ◦ b[β]. We can differentiate
the relation to get ȧ[Φ(β)] = Φ0 ◦ b[β] · ḃ[β]. But since a and b are the
solutions of the corresponding differential equations, we can write F ◦ a[β] =
Φ0 ◦ b[β] · G ◦ b[β]. Since the relation holds for all values of t, we can take
t = 0 and use the fact that b[β](0) = β and a[Φ(β)](0) = Φ(β) to get
F (Φ(β)) = Φ0 (β) · G(β), which is the desired result.
2 ⇒ 1: We start with F ◦ Φ = Φ0 · G. Let us see that in this case
Φ ◦ b[β] satisfies a’s ODEs. Indeed, we have (Φ ◦ b[β])0 = Φ0 ◦ b[β] · ḃ[β] =
Φ0 ◦b[β]·G◦b[β] = (Φ0 ·G)◦b[β] = F ◦Φ◦b[β], where we assume that function
composition binds stronger than function products. We have obtained the
desired condition from a’s ODEs. Note also that we have (Φ ◦ b[β])(0) =
Φ(β). Thus, by Theorem 1, it must be that dom(b[β]) ⊆ dom(a[Φ(β)])
and a[Φ(β)](t) = Φ(b[β](t)), for all t ∈ dom(b[β]), which is the desired
result.
The result from Lemma 1 conveys the idea that in order for fit-preserving
refinement to be possible, there should be a strong connection between the
three functions F , G and Φ.
Lemma 2. Let S and S 0 be two finite sets, of sizes m and m0 , respectively,
0
and let ρ ⊆ S × S 0 be a species refinement relation. Then, for any x ∈ RS
≥0
and c ∈ NS it holds that
!
(Mρ x)c =
X
c0 ∈ρ(c)
10
c c0
x .
c0
Proof. Without loss of generality, we can assign names to the elements of S ,
let them be S1 , . . . , Sm . Note that ρ induces a partition of the elements of
S 0 into sets of subspecies, one for each of the species in S . Thus, we can
use Sij0 to refer to the elements in ρ(Si ), with j ranging from 1 to ni = |ρ(Si )|
P
0
and m
i=1 ni = m . Furthermore, we use xij to refer to the element of x that
corresponds to Sij0 . Recall that Mρ computes the sum of subspecies for each
species, i.e.:

Mρ x = 
n1
X
x1j ,
j=1
n2
X
x2j , . . . ,
j=1
nm
X
T
xmj 
.
j=1
Then we can write:


c
(Mρ x) =
m
Y

ni
X

i=1
ci
xij  =
j=1


ni
m 
X
Y
Y
ci !
c0ij 



xij  ,
Qni 0

 0
j=1 cij ! j=1
i=1 [c ,...,c0 ]T ∈Nni

i1
ini
Pni 0
c =ci
j=1 ij
where we have used multinomial expansion. Now, since for different values
of i the corresponding c0ij ’s are independent, we can rewrite the result by
swapping the sum and product as:
c
(Mρ x) =
[c0i1 ,...,c0in ]T ∈Nni
Pni

!
ni
m Y
X
Y
ci !
c c0
c0

xijij  =
x ,
Qni 0
c0
j=1 cij !
i=1
i=1 j=1
0
S0
m
Y
X
!
c ∈N
Mρ c0 =c
i
c0 =ci
j=1 ij
1≤i≤m
which is the desired result.
We are now ready to prove Theorem 4.
of Theorem 4. In what follows we use C = {c ∈ NS | c → d ∈ R} and
0
C 0 = {c0 ∈ NS | c0 → d0 ∈ R 0 } to denote the source complexes of M and
M 0 , respectively. In this case, since M 0 is a full ρ-refinement of M , we can
S
write C 0 = c∈C ρ(c).
We start by noting that the fit-preservation condition from Definition 9 is
an instantiation of statement 1 from Lemma 1 with variables a, b, x replaced
by s, s0 , σ 0 , respectively, and:
X
F (s) =
kc→d sc (d − c),
c→d∈R
0
G(s ) =
0
Φ(σ )
X
0
kc0 →d0 s0c (d0 − c0 ),
c0 →d0 ∈R 0
= Mρ σ 0 ,
Φ0 (σ 0 ) = Mρ .
11
Thus, we can write
∀σ 0 . s[Mρ σ 0 ] = Mρ s0 [σ 0 ] ⇐⇒ ∀σ 0 . F (Mρ σ 0 ) = Mρ G(σ 0 )
Next, in the latter equality we replace the left and right hand sides with
the corresponding mass-action formulas and rely on Lemma 2 to obtain
F (Mρ σ 0 ) =
kc→d (Mρ σ 0 )c (d − c)
X
c→d∈R
!
X
=
kc→d
c
0
σ 0c (d − c)
0
c
X
c0 ∈ρ(c)
c→d∈R
!
X
=
X
kc→d
c∈C d∈NS
c→d∈R
X
=
σ
X
c0 ∈ρ(c)
!
0c0
!
c
kc→d (d − c)
c0
X
d∈NS
c=Mρ c0
c→d∈R
c0 ∈C 0
c
0
σ 0c (d − c)
0
c
and
Mρ G(σ 0 ) = Mρ
0
kc0 →d0 σ 0c (d0 − c0 )
X
c0 →d0 ∈R 0
X
=
X
0
kc0 →d0 σ 0c (Mρ d0 − Mρ c0 )
X
c→d∈R c0 ∈ρ(c) d0 ∈ρ(d)

X
=
c0 ∈C 0
σ
0c0

X
X

d∈NS
kc0 →d0  (d − c) .
d0 ∈ρ(d)
c=Mρ c0
c→d∈R
Moving both terms to the left hand side, we can conclude that M 0 is a
0
fit-preserving refinement of M if and only if, for all σ 0 ∈ NS :

X
c0 ∈C 0
σ
0c0
X
d∈NS
c=Mρ c0
c→d∈R

!
X
c

kc→d −
kc0 →d0  (d − c) = 0.
0
c
d0 ∈ρ(d)
Claim 1 of the theorem, as well as the ⇐ part of 2, follows directly from
the equation above, since the condition for canonical refinement makes the
difference in the parentheses vanish. For the ⇒ part of claim 2 we first use
the fact that the sum on the left hand side is actually a multinomial where
0
σ 0c are monomials, so all the inner sums (as coefficients of the monomials)
must vanish. Next, we deduce the condition for canonical refinement from
the fact that the vectors d−c under the summation are linearly independent,
by Theorem 2, since M has uniquely identifiable rate constants.
12
Note that we can now distinguish several versions of the actual notion of
refinement. In the most general sense, we talk about any continuously differentiable mappings F and G and any continuously differentiable mapping Φ
that defines the desired behavior when going from Rm to Rn . Already in this
very general setting we can impose strong constraints on the three mappings,
as shown in Lemma 1.
Next, if we require that the mapping Φ corresponds to a species refinement, we end up using a very particular kind of linear mapping. The last
specialization step consists in the use of mass-action formulas for F and G.
These intermediary steps open up the opportunity to investigate in future
work the more general versions of fit-preserving refinement.
4
Examples
In this section we give examples of how canonical refinement can be used
for obtaining fit-preserving refinements of mass-action reaction networks. In
addition to the examples provided here, the reader may find our analysis of
the Lotka-Volterra prey-predator model in [5].
First, to see why the result presented in [4] (and here as part 1 of Theorem 4) is only sufficient, consider the following mass-action reaction network:
k
2A −−1→ A ,
k
2A −−2→ 3A .
Such a model can, for example, characterize an ecological model describing the dynamics of some population A. The first reaction could encode the
encounters that result in the death of the weaker individual, whereas the
second reaction can stand for mating interactions.
If we are to write the corresponding ODE for this network, we get:
ȧ = (k2 − k1 )a
Clearly the dynamics of this system is completely characterized by k2 −k1
and it is only this difference that can be fitted against experimental data,
the actual values themselves cannot be determined uniquely. Based on Theorem 2 we expect that the reaction vectors for the unique source complex
of this network are not linearly independent. Indeed, the (unidimensional)
vectors are [−1] and [1] and are linearly dependent. Note that more intricate
examples can also be constructed based on this idea.
Consider now that we have two instances of this mass-action reaction
network, say M and M 0 with rate constants k1 and k2 , respectively k10 and
k20 , such that k2 − k1 = k20 − k10 but k1 6= k10 and k2 6= k20 . A canonical
13
refinement of M , call it Mr , is also a fit-preserving refinement of M , by
Theorem 41. Moreover, since M and M 0 characterize the same dynamical
behavior, it is also the case that Mr is a fit-preserving refinement of M 0 . On
the other hand, Mr is not a canonical refinement for M 0 .
An equivalent (with respect to the associated ODEs) mass-action reaction network can be constructed for this example, by keeping only one of
k1 −k2
the two reactions. We can choose between 2A −−
−→ A (the second reaction
is dropped, but accounted for in the rate constant of the first reaction) or
k2 −k1
2A −−
−→ 3A (the first reaction dropped). Note that the actual choice matters, since it constrains the possible solutions (the kinetic rate constants need
to be nonnegative). Thus, if we know that k1 ≥ k2 , we should use the first
reaction, otherwise we must use the second one. In either case, we obtain
a model that is equivalent to the original one (for the specific relation between k1 and k2 that is assumed) and, moreover, has uniquely identifiable
rate constants.
The second model that we are going to discuss is the Brusselator [9]. The
usual formulation of the model is presented below.
A −−→ X;
2X + Y −−→ 3X;
B + X −−→ Y + D;
X −−→ E.
(6)
All the rate constants are taken to be equal to 1 and, additionally, it is
assumed that the concentrations of A and B are constant, either by being
maintained from outside the system, or as an approximation, considering
that the two concentrations are significantly larger than those of the other
species in the model. The analysis of the Brusselator generally refers to the
time evolution of X and Y , i.e. it is assumed that D and E are taken away
from the system as they are produced.
Note that, in the form presented above, this model is not a mass-action
reaction network in the sense of Definition 4, because of the assumption that
A and B are constant (rather than have their dynamical behavior follow
a mass-action equation). The results from this paper can be generalized
to cover this particular case as well by focusing on the associated ODEs.
However, an important feature of our approach is the fact that canonical refinements (and implicitly fit-preserving refinements) can be computed based
on the model representation alone (reactions and rate constants), without
the need to write down the ODEs. Thus, in what follows we use a different
formulation of the Brusselator, one that preserves the behavior of X and Y
but imposes no constraints on the species, other than the mass-action law.
It is not difficult to see that D and E can safely be removed from the
14
model, since they have no impact on the dynamics of X and Y . Furthermore,
we aim to remove A and B as well, but account for the values of their
(constant) concentrations in the new formulation so that X and Y preserve
their behavior. For the original formulation (6), the associated ODEs are:
ẋ = a + x2 y − bx − x,
ẏ = bx − x2 y,
where we have used lowercase letters to denote the corresponding concentration of each species. We can now rely on the fact that a and b are constant
to obtain the same ODEs via mass-action kinetics for the following model:
a
r1 : ∅ −−→ X
p
r2 : 2X + Y −−→ 3X
b
r3 : X −−→ Y
q
r4 : X −−→ ∅
for p = q = 1.
For this simplified version of the Brusselator, we consider the refinement
of X into X1 and X2 . The full refinement of the reaction network is presented
below:
n
a
a
r1 : ∅ −−1→ X1 , ∅ −−2→ X2 ;

p00

2X1 + Y1 −−
→ 3X1 ,



p02



2X1 + Y1 −−→ X1 + 2X2 ,




10
X + X + Y −p−
→ 3X ,
p
01
2X1 + Y1 −−
→ 2X1 + X2 ,
p03
2X1 + Y1 −−→ 3X2 ,
p11
X1 + X2 + Y1 −−
→ 2X1 + X2 ,
1
2
1
1
r2 :
p12
p13


X1 + X2 + Y1 −−→ X1 + 2X2 , X1 + X2 + Y1 −−→ 3X2 ,



p20
p21


2X2 + Y1 −−
→ 3X1 ,
2X2 + Y1 −−
→ 2X1 + X2 ,




p22
p23

2X2 + Y1 −−→ X1 + 2X2 ,
2X2 + Y1 −−→ 3X2 ;
n
b1
b
r3 : X1 −−
→ Y1 , X2 −−2→ Y1 ;
n
q
q
r4 : X1 −−1→ ∅, X2 −−2→ ∅.
We have grouped the reactions of the refined model by the corresponding reaction in the initial model. Furthermore, we used horizontal lines to
separate the 12 refinements of r2 into groups that share the same left hand
side. In order for the full refinement to be canonical, the following conditions
must be satisfied:
15
Parameter
Sum Case 1
Case 2
Case 3
Case 4
a1 , a2
a 3/4, 1/4
3/4, 1/4
3/4, 1/4
1, 0
p00 , p01 , p02 , p03 p 0, 1, 0, 0
1, 0, 0, 0
1, 0, 0, 0
1, 0, 0, 0
p10 , p11 , p12 , p13 2p 0, 0, 1, 0 0, 1/2, 1/2, 0 0, 1/3, 2/3, 0 0, 0, 1, 0
p20 , p21 , p22 , p23 p 0, 0, 0, 1
0, 0, 0, 1
0, 0, 0, 1
0, 0, 0, 1
b1
b
1
1
1
1
b2
b
1
1
1
1
q1
q
1
1
1
1
q2
q
1
1
1
1
x1 (0), x2 (0)
x(0) 3/4, 1/4
3/4, 1/4
3/4, 1/4
0.99, 0.01
y1 (0)
y(0)
1
1
1
1
Table 1: Parameter setup for each of the four fit-preserving refinements.
a1 + a2 = a,
p00 + p01 + p02 + p03 = p,
p10 + p11 + p12 + p13 = 2p,
p20 + p21 + p22 + p23 = p,
b1 = b2 = b,
q1 = q2 = q.
Note that, for the canonical refinement, the values of b1 , b2 , q1 and q2
are already determined. For the other rate constants we can choose between
different numerical setups, as long as the constraints are satisfied. In what
follows we show that, while any canonical refinement guarantees that X1 +X2
and Y1 behave exactly the same as X and Y , respectively, the behaviors of
X1 and X2 can be very different.
For the original model we fixed the values a = p = q = 1 and b = 2.5.
Furthermore, we considered the initial values x(0) = 2 and y(0) = 2. The
resulting simulation is presented in Figure 1a. We consider four different
fit-preserving refinements, corresponding to the parameter values indicated
in Table 1. Note that the values are presented as the fraction of the overall
sum enforced by canonical refinement.
In addition to the chosen values for each case, we also provide possible
assumptions that can lead to these values. The first three scenarios assume
a 3 : 1 ratio of X1 to X2 in X, both for the initial values and for the X that
is added to the system (the refinement of the first reaction). One possible
interpretation of this setup is that the chemical element X has two isotopes
X1 and X2 , and the naturally available X contains them in the chosen proportions. Furthermore, we read the third reaction of the Brusselator model
16
(a) Original model.
(b) Case 1: Y1 can only be converted to X2 .
(c) Case 2: Y1 can be converted to X1 as well as X2 , no preference.
(d) Case 3: Y1 can be converted to X1 as well as X2 , preference for X2 .
(e) Case 4: Conversion to X2 preferred, only X1 added from the outside.
Figure 1: The Brusselator and its four proposed canonical, fit-preserving
refinements.
as follows: in the presence of two X molecules, a molecule of Y can be converted into X. The presented scenarios will extend this interpretation with
assumptions about the difference between X1 and X2 with respect to this
reaction.
Case 1 In this scenario we assume that, in fact, Y1 can only be converted
to X2 via the third reaction and we encode this assumption in the values
chosen for the rate constants of the refined reactions, i.e., we take p01 = p,
p12 = 2p, p23 = p and pij = 0 for all other refined reactions. Note that the
ODE for X1 becomes:
x˙1 = a − (b + q)x1 .
Thus, the behavior of X1 only depends on the rate constants of the model
and on the initial concentration of X1 . In particular, the steady state is given
by x1 = a/(b + q). This is consistent with the plot from Figure 1b, where
a/(b + q) = 0.25. Moreover, the concentration of X1 rapidly approaches the
equilibrium value and the periodic behavior of X is given by X2 alone.
17
Case 2 We assume now that Y1 can be converted to X1 as well as X2 ,
and the actual outcome depends on the two X molecules that Y interacts
with. In the presence of two molecules of X1 , Y1 will be converted to X1 , i.e.
p00 = p and p01 = p02 = p03 = 0. Similarly, in the presence of two molecules
of X2 , Y1 becomes X2 , i.e. p23 = p and p20 = p21 = p22 = 0. Last, but
not least, when we have X1 and X2 interacting with Y1 , we assume an equal
probability of obtaining X1 or X2 , i.e. p11 = p12 = p and p10 = p13 = 0.
In this scenario we expect that both X1 and X2 exhibit similar behavior
due to the symmetry of the refined reactions. Indeed, we can see in Figure 1c
that the fractions of X1 and X2 in the total concentration of X remains the
same throughout the simulation. To see also formally that this is the case,
let us consider the ODEs for x1 and x2 in this case:
ẋ1 = a1 + px21 y1 + px1 x2 y1 − (b + q)x1 ,
ẋ2 = a2 + px1 x2 y1 + px22 y1 − (b + q)x2 .
Now, we use the fact that x1 + x2 = x (guaranteed by the construction
of the fit-preserving refinement), to write x2 = x − x1 in the first ODE, then
we divide both sides by a1 to get:
x1
x1
ẋ1
= 1 + pxy1 − (b + q) .
a1
a1
a1
Using the same approach for x2 , we obtain that x1 /a1 and x2 /a2 satisfy
the same ODE. If we also assume that the initial values are the same, i.e.
x1 (0)/a1 = x2 (0)/a2 (which is true for our scenario), then we have x1 /a1 =
x2 /a2 . This can be rewritten as x1 /x2 = a1 /a2 and completes the proof of
our claim. If the initial ratio x1 /x2 is not equal to a1 /a2 , then it will converge
to this value.
Case 3 Similarly to the previous scenario, we assume that Y1 is converted to
X1 (respectively X2 ) when it interacts with two molecules of X1 (respectively
X2 ). For the interaction with one X1 molecule and one X2 molecule, we
assume that conversion to X2 is favored, i.e. we take p11 = 2p/3 and p12 =
4p/3. The result from Figure 1d shows that X1 and X2 no longer have the
same proportion in X, and their peak values are no longer synchronized in
time.
Case 4 In this scenario we push the bias of X1 +X2 +Y1 towards producing
only X1 + 2X2 . Furthermore, we change the assumptions for a1 and a2 so
that only X1 is introduced in the system. Moreover, in the initial values,
we take x1 (0) = 0.99x(0) and x2 (0) = 0.01x(0). With these assumptions,
it can be seen in Figure 1e that, even though X2 starts with a very low
18
concentration, it does not disappear from the system, but again a periodic
behavior is reached. Note also that the concentration of X2 periodically
surpasses that of X1 , even though we only add X1 from outside the system.
The four scenarios presented in this section are meant to show that fitpreserving refinement does not generate very restricted behaviors on the refined species, even though their sums are constrained by the dynamics of the
original model. Furthermore, if experimental measurements are available for
the refined species, the observed behavior can be compared with one of the
cases presented here (possibly with other ones as well) so as to guide the
model fitting process.
We conclude this section with a discussion of rate constants identifiability
for the Brusselator. The original model has a single reaction for each source
complex (i.e. there are no two reactions that have the same left hand side)
and, thus, it trivially satisfies the condition of Theorem 2. By Theorem 4 we
can conclude that full refinements of the model are fit-preserving if and only
if they are canonical.
For the full refinement, however, we do have reactions with the same left
hand side, namely refinements of r1 and r2 . We assume that complexes contain the stoichiometric coefficients in the order X1 , X2 , Y1 , and we compute
the rank of the matrices containing the difference vectors as columns. The
matrix for the refinements of r1 is


1 0


R1 =  0 1 
0 0
and has rank 2, which means that the difference vectors are linearly independent. For the refinements of r2 we have three matrices, one for each source
complex:

R21


1
0 −1 −2

1
2
3 
= 0
 ,
−1 −1 −1 −1
R22

R23

2
1
0 −1

0
1
2 
=  −1
 ,
−1 −1 −1 −1

3
2
1
0
−2
−1
0
1 
=

 .
−1 −1 −1 −1
All three matrices have rank 2 and, thus, if we wish to have uniquely
identifiable rate constants, we can keep at most 2 reactions from each group
that shares the same left hand side (i.e. the rate constants for the other reactions should be set to zero). Note that the four scenarios we have presented
do satisfy this requirement.
If we have numerical values for all four rate constants of the refined reactions, we can obtain an equivalent model with uniquely identifiable rate
19
constants, by choosing a suitable generator for the set of difference vectors,
so that the rate constants of the reduced model are still positive. On the
other hand, when trying to fit the model we do not know the rate constants,
so a suitable approach would be to consider (as initialization) one instance
for each set of linearly independent refined reactions and then choose the
best fit.
5
Conclusions
In this paper we improved our previous result from [4] by providing a complete characterization of fit-preserving refinement via a constraint relating
the parameters of the refined model to those of the original model. The
characterization applies as long as the original model has uniquely identifiable rate constants, a requirement that is critical for fitting the model [1].
Whenever applying data refinement to such models, we can deduce that every
numerical setup that is fit-preserving corresponds to a canonical quantitative
refinement. This means that we can initialize parameter estimation routines
with these values and, thus, ensure that the resulting model fits the original
data at least as well as the initial model did.
Furthermore, we have seen in Section 4 that the behavior of refined species
varies greatly with the chosen rate constants, even when the refined model
is constrained to exhibit the same dynamics with respect to the species of
the original model. The full refinement of a model does not have, in general,
uniquely identifiable rate constants. Thus, we need to choose, from each set
of refined reactions that share the same left hand side, a subset of linearly
independent reactions. Considering all possible such refinements and simulating them, we can identify different classes for the behavior of the refined
species and use them for guiding the model fitting process when experimental
data is available.
The very simple constraints used for the canonical refinement rely on the
interplay between multinomial expansion and the formulation of mass-action
dynamics. It would be interesting to investigate fit-preserving refinement for
other kinetic models as well. Furthermore, the proof of our result reveals
several versions of refinement, ranging from very general refinement of ODE
models where the functions constraining the dynamics are not specified, to
the very specific case of mass-action and the additive constraint on the behavior of refined species. These intermediary versions can provide additional
insight for building models via refinement.
An interesting case to consider in future work is the refinement of massaction reaction networks where the chemical structure of species is known,
i.e., there is a distinction between atomic and complex species. In such cases,
one can consider species refinement only for atomic species, then use it to
20
construct the refinement of complex species (see, e.g. [2]). The structural
information can be used in this case to impose additional constraints on the
rate constants of the refined reactions, e.g., by specifying mass conservation
relations.
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[3] Eugen Czeizler, Vladimir Rogojin, and Ion Petre. The phosphorylation
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[5] Cristian Gratie and Ion Petre. Fit-preserving data refinement of massaction reaction networks. Technical report, TUCS, 2014.
[6] Morris William Hirsch, Stephen Smale, and Robert Luke Devaney. Differential equations, dynamical systems and an introduction to chaos, volume 60 of Pure and Applied Mathematics. Academic Press, 2 edition,
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21
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University of Turku
Faculty of Mathematics and Natural Sciences
• Department of Information Technology
• Department of Mathematics
Turku School of Economics
• Institute of Information Systems Sciences
Åbo Akademi University
• Department of Computer Science
• Institute for Advanced Management Systems Research
ISBN 978-952-12-3172-8
ISSN 1239-1891