Indistinguishability and identifiability of kinetic models for the MurC reaction in peptidoglycan biosynthesis.

Abstract

An important question in Systems Biology is the design of experiments that enable discrimination between two (or more) competing chemical pathway models or biological mechanisms. In this paper analysis is performed between two different models describing the kinetic mechanism of a three-substrate three-product reaction, namely the MurC reaction in the cytoplasmic phase of peptidoglycan biosynthesis. One model involves ordered substrate binding and ordered release of the three products; the competing model also assumes ordered substrate binding, but with fast release of the three products. The two versions are shown to be distinguishable; however, if standard quasi-steady-state assumptions are made distinguishability cannot be determined. Once model structure uniqueness is ensured the experimenter must determine if it is possible to successfully recover rate constant values given the experiment observations, a process known as structural identifiability. Structural identifiability analysis is carried out for both models to determine which of the unknown reaction parameters can be determined uniquely, or otherwise, from the ideal system outputs. This structural analysis forms an integrated step towards the modelling of the full pathway of the cytoplasmic phase of peptidoglycan biosynthesis.

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Author(s): J.G. Hattersley, J. Pérez-Velázquez, M.J. Chappell, D.
Bearup, D. Ropeb, C. Dowson, T. Bugg, N.D. Evans
Article Title: Indistinguishability and identifiability of kinetic models for
the MurC reaction in peptidoglycan biosynthesis
Year of publication: 2011
Link to published article:
http://dx.doi.org/10.1016/j.cmpb.2010.07.009
Publisher statement: “NOTICE: this is the author’s version of a work
that was accepted for publication in Computer Methods and Programs
in Biomedicine. Changes resulting from the publishing process, such
as peer review, editing, corrections, structural formatting, and other
quality control mechanisms may not be reflected in this document.
Changes may have been made to this work since it was submitted for
publication. A definitive version was subsequently published in
Computer Methods and Programs in Biomedicine, VOL:104, ISSUE:2,
November 2011, DOI: 10.1016/j.cmpb.2010.07.009”

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Indistinguishability and identifiability of kinetic models for the MurC
reaction in peptidoglycan biosynthesis
J.G. Hattersleya, J. Pérez-Velázqueza, M.J. Chappella, D. Bearupb, D. Roperb, C. Dowsonb,
T. Buggc, N.D. Evans∗,a
aSchool of Engineering, University of Warwick, Coventry, CV4 7AL, UK (e-mail: Neil.Evans@warwick.ac.uk ).
bSchool of Biological Sciences, University of Warwick, Coventry, UK
cSchool of Chemistry, University of Warwick, Coventry, UK
Abstract
An important question in Systems Biology is the design of experiments that enable discrimination
between two (or more) competing chemical pathway models or biological mechanisms. In this
paper analysis is performed between two different models describing the kinetic mechanism of
a three-substrate three-product reaction, namely the MurC reaction in the cytoplasmic phase of
peptidoglycan biosynthesis. One model involves ordered substrate binding and ordered release of the
three products; the competing model also assumes ordered substrate binding, but with fast release
of the three products. The two versions are shown to be distinguishable; however, if standard quasi
steady-state assumptions are made distinguishability can not be determined. Once model structure
uniqueness is ensured the experimenter must determine if it is possible to successfully recover rate
constant values given the experiment observations, a process known as structural identifiability.
Structural identifiability analysis is carried out for both models to determine which of the unknown
reaction parameters can be determined uniquely, or otherwise, from the ideal system outputs.
This structural analysis forms an integrated step towards the modelling of the full pathway of the
cytoplasmic phase of peptidoglycan biosynthesis.
Key words:
Indistinguishability, Identifiability, Experiment design, MurC, Biomedical systems,
Parameter identification
∗Corresponding author
Preprint submitted to Computer Methods and Programs in Biomedicine9th June 2010

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2
1. Introduction
In the biological sciences it is becoming increasingly common to collect data in high-throughput
experiments on several scales: genomic, proteomic, or metabolic (Snoep and Westerhoff [30], Sauer
et al. [27]). These data hold the promise of identifying the mechanisms of interactions that com-
prise large-scale regulatory biochemical networks. An important step in this work is the design of
experiments to allow discrimination between two (or more) competing pathway models or biologi-
cal mechanisms. Structural indistinguishability provides a formal approach to distinguish between
competing model mechanisms (Kholodenko et al. [19]).
In Systems Biology mathematical models that are generated invariably include large numbers
of state variables with numerous model parameters, many of which are unknown, or cannot be
directly measured. With such highly complex systems there are often few direct measurements
that can be made and limited access for input perturbation to elucidate system dynamics. These
limitations cause problems when investigating the existence of hidden pathways or attempting to
estimate unknown parameters. Identifiability analysis provides a formal approach to determine
what additional inputs and/or measurements are necessary in order to reduce, or remove, these
limitations and permit the derivation of models that can be used for practical purposes with greater
confidence (Snoep and Westerhoff [30], Sauer et al. [27], Kholodenko et al. [19]).
Structural indistinguishability for system models is concerned with determining the uniqueness
between possible candidates for the model (or mechanism) structure (Evans et al. [13]).The
analysis is concerned with whether the underlying possibilities for the parameterised mathematical
model can be distinguished using the inputs (perturbations or interventions) and observations (or
measurements) available for the biological system under investigation.
In chemical kinetics it is key to characterise reaction mechanisms, however there are often
several different process models that are consistent with the available data. These mechanisms may
be described by the same mathematical representation (see Érdi and Tóth [9], Espenson [10]) but
without formal analysis of the mathematical model a reaction mechanism’s validity is only disproved
by showing inconsistency with available data. Whilst this problem has been recognised [5, 6],
structural indistinguishability is not routinely applied to chemical kinetics experiments and model

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3
development. For linear systems structural indistinguishability analysis is generally exhaustive
with all competing mechanisms generated from a given one (see Godfrey and Chapman [16]). For
nonlinear systems, approaches are generally only for pairs of candidate models, though in some cases
a parameterised family of such candidates can be generated. There has been limited application
of structural indistinguishability analysis to the chemical kinetic models. In Schnell et al. [28] the
issues of distinguishability with respect to biochemical kinetics was considered via application of
structural indistinguishability to classical models for a single-enzyme, single-substrate reaction. In
addition, simple kinetic models are incorporated into studies of structural analysis methods, often
with Michaelis-Menten type reparameterisation (see Saccomani et al. [25], Bellu et al. [3]; however,
to the authors’ knowledge this is the first time indistinguishability and identifiability have been
applied to a full (three-substrate/three-product) enzyme kinetic model.
Structural identifiability arises in the inverse problem of inferring from the known, or assumed,
properties of a biological system, estimates for the corresponding rate constants and other parame-
ters; as such it can be considered as a special case of the structural indistinguishability problem.
Structural identifiability analysis considers the uniqueness of the unknown model parameters from
the input-output structure corresponding to proposed experiments to collect data for parameter
estimation. This is an important, but often overlooked, prerequisite to experiment design, system
identification and parameter estimation, since estimates for structurally unidentifiable parameters
are effectively meaningless. If parameter estimates are to be used to inform intervention or inhibition
strategies, or other critical decisions, then it is essential that the parameters be uniquely identi-
fiable. Numerous techniques for performing a structural identifiability analysis on linear parametric
models exist (see Godfrey and DiStefano III [17], Walter [34]). In comparison, there are relatively
few techniques available for nonlinear systems such as the Taylor series approach (Pohjanpalo [23]),
similarity transformation based approaches (Tunali and Tarn [31], Vajda et al. [32], Evans et al.
[12]) and differential algebra techniques (Ljung and Glad [20], Saccomani, Audoly, and D’Angio
[26]). Unfortunately for systems with a complex structure significant computational problems can
arise even for relatively low dimensional models. At present there has been relatively little work on
techniques for large-scale, highly complex systems, which are typical in Systems Biology. As shall be

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4
shown, the analytic identifiability approaches can generate computationally intractable solutions;
therefore, an alternative numerical approach that uses the sensitivity of the observation functions
to changes in the parameters is implemented to suggest the local identifiability of the parameters
and associated model (Fisher [15], Jacquez [18]).
The purpose of this paper is to explore the possible effectiveness of using structural indistingui-
shability and identifiability techniques in model discrimination within Systems Biology networks,
using MurC as a case study; to this end, a structural indistinguishability analysis is performed bet-
ween two different models describing the kinetic mechanism of the MurC (Ter-ter) reaction in the
cytoplasmic phase of bacterial peptidoglycan biosynthesis (BPB). One model assumes the kinetic
mechanism proposed by Emanuele et al. [8], which involves step-by-step release of the three pro-
ducts; the competing model assumes that the release of the products is simultaneous. A structural
identifiability analysis is also carried out for both models to ensure that the model output uniquely
determines the unknown parameters.
2. The Models
Prior to describing the mathematical models of the Ter-ter enzyme reaction, it is necessary to
review the basic biology and context of the MurC reaction. The cell wall of many bacteria is com-
posed of peptidoglycan, which is made up of a combination of peptide bonds and carbohydrates.
Peptidoglycan serves a structural role in the bacterial cell wall, giving rigidity, as well as counte-
racting the osmotic pressure of the cytoplasm (El Zoeiby et al. [7]). The BPB pathway (shown in
Fig. 1) is a significant target in the development of antibacterial agents (Walsh [33]). A detailed
understanding of the biosynthesis pathway is essential for the development of new strategies for
antibacterial action to compensate for the emergence of clinical resistance to penicillin antibiotics
in S. aureus, S. pneuminiae and Gram-negative pathogens such as P. aeruginosa (Marmor et al.
[21], Anderson et al. [1]), and the emergence of vancomycin resistance in Enterococci, together with
the lack of new classes of antibacterial agent. Although this pathway is quite well known, charac-
terisation, especially of the later lipid linked steps, has been hampered by difficulties in making the
natural substrates and there remain some uncertainties within the wider reaction network. As such

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2.1Model 1: Ordered release of products
5
this pathway is ideal for a feasibility study of the effectiveness of structural indistinguishability and
identifiability in mechanism discrimination, model formulation and experiment design (Snoep and
Westerhoff [30], Sauer et al. [27]).
Within the biosynthetic pathway for bacterial peptidoglycan there is already a reasonable (but
not complete) understanding of the cytoplasmic pathways.Whilst a basic pathway scheme is
recognised for this region, there are still a number of steps within the pathway where competing
reaction schemes may exist (e.g., multiple isoforms of an enzyme within the same cell) and also
where feedback inhibition may play an important (but not yet fully understood) role. The study
of the cytoplasmic phase of the biosynthetic process comprises a comprehensive understanding of
the MurA to MurF reactions of the full pathway (see Fig. 1). In this paper the focus is on MurC.
The MurC reaction to be modelled is a three-substrates, three-products enzyme catalysed reac-
tion. It is assumed that the kinetic mechanism is as proposed by Emanuele et al. [8], see Fig. 2,
which involves step-by-step release of the three products, with a competing model that assumes
that the release of the products is simultaneous. Note however that the mathematical model does
not apply to MurC alone, but to any general three-substrates three-products ordered mechanism
(potentially MurD, E, and F, if they are ordered).
2.1. Model 1: Ordered release of products
From Emanuele et al. [8] it is known that the third substrate (L-Ala) binds before the reaction
starts. Thus the reaction can be written as the following set of chemical steps:
E + S1
k1
− ?
? −
r1
C1
C1+ S2
k2
− ?
? −
r2
C2
C2+ S3
k3
− ?
? −
r3
C3
k4
− ?
? −
r4
C4
k5
− ?
? −
r5
C5+ P
C5
k6
− ?
? −
r6
C6+ Q
C6
k7
− ?
? −
r7
E + R
where:

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2.2Model 2: Fast release of products
6
1. Enzyme (E), E
2. Substrate 1 (ATP), S1
3. Substrate 2 (UNAM), S2
4. Substrate 3 (L-Ala), S3
5. Complex E-ATP, C1
6. Binary Complex E-ATP-UNAM, C2
7. Ternary Complex E-ATP-UNAM-L-Ala, C3
8. Complex E-ADP-UNAMA-Pi, C4
9. There are three products: ADP (R), UNAMA and Pi
Note that the order of the release of the products is not determined in this scheme, it is only known
that ADP releases at the end, the release order of the other two products (UNAMA and Pi) is not
known and therefore they have been generically called P and Q. The complexes ERP and ER are
C5and C6, respectively.
2.2. Model 2: Fast release of products
The second model assumes that the release of the products is fast (instantaneous and simulta-
neous). The corresponding set of chemical steps is as follows:
E + S1
k1
− ?
? −
r1
C1
C1+ S2
k2
− ?
? −
r2
C2
C2+ S3
k3
− ?
? −
r3
C3
k4
− ?
? −
r4
C4
k5
− ?
? −
r5
E + P + Q + R
2.2.1. Equations
The time evolution of the reaction is obtained by applying the Law of Mass Action to yield a
set of 13 coupled nonlinear ordinary differential equations with 14 parameters for Model 1 and 11
coupled nonlinear differential equations with 10 parameters for Model 2. The system equations for
both models are defined below.

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2.2 Model 2: Fast release of products
7
These are the core equations for both models
S?
1(t) = −k1E(t)S1(t) + r1C1(t),
C?
1(t) = k1E(t)S1(t) − r1C1(t) − k2C1(t)S2(t) + r2C2(t),
S?
2(t) = −k2C1(t)S2(t) + r2C2(t),
C?
2(t) = k2C1(t)S2(t) − r2C2(t) − k3C2(t)S3(t) + r3C3(t),
S?
3(t) = −k3C2(t)S3(t) + r3C3(t)
C?
3(t) = k3C2(t)S3(t) − r3C3(t) − k4C3(t) + r4C4(t)
Model 1 core equations plus:
E?(t) = −k1E(t)S1(t) + r1C1(t) + k7C6(t) − r7E(t)R(t),
C?
4(t) = k4C3(t) − r4C4(t) − k5C4(t) + r5C5(t)P(t),
C?
5(t) = k5C4(t) − r5C5(t)P(t) − k6C5(t) + r6C6(t)Q(t),
C?
6(t) = k6C5(t) − r6C6(t)Q(t) − k7C6(t) + r7E(t)R(t),
P?(t) = k5C4(t) − r5C5(t)P(t),
Q?(t) = k6C5(t) − r6C6(t)Q(t),
R?(t) = k7C6(t) − r7E(t)R(t).
Model 2 core equations plus:
E?(t) = −k1ES1(t) + r1C1(t) + k5C4(t) − r5EPQR(t),
C4(t)?= k4C3(t) − r4C4(t) − k5C4(t) + r5EPQR(t),
P?(t) = k5C4(t) − r5EPQR(t),
Q?(t) = k5C4(t) − r5EPQR(t),
R?(t) = k5C4(t) − r5EPQR(t).

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8
2.2.2. Experimental Observations
In biochemical progress curve experiments, concentrations are measured by the absorbance of
light at one or more wavelengths (see Fersht [14], Nölting [22]). If the molar absorptivities of the
measured components at these wavelengths are known, then biochemists use Beer’s law to determine
the concentrations.
Thus, in matrix notation,
y y y = ? ? ?x x x
(1)
where y y y is the vector of observed absorbances at different wavelengths, x x x is the vector of concentra-
tions of species absorbing at these wavelengths, and ? ? ? is a matrix of molar absorptivities. In these
experiments, the kinetic parameters are determined from expressions for the species concentrations
as a function of time. The concentration of the substrate or product is recorded in time after the
initial fast transient and for a sufficiently long period to allow the reaction to approach equilibrium.
While they are less common now, progress curve experiments were widely used in the early period of
enzyme kinetics when Victor Henri was active in the field (early 20th Century). It is assumed that
progress curves are available and therefore can be related to the models presented in the previous
section.
3. Methods
3.1. Structural Indistinguishability
Since an indistinguishability analysis can be seen as a generalisation of the identifiability pro-
blem, it can be studied by modifying existing approaches for identifiability. Here a modification of
the Taylor series approach for identifiability is used.
Consider two uncontrolled systems of the form:

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3.1Structural Indistinguishability
9
Σ(p p p)









˙ x ˙ x ˙ x(t,p p p) = f f f(x x x(t,p p p),p p p), x x x(0,p p p) = x x x0(p p p)
y y y(t,p p p) = h h h(x x x(t,p p p),p p p)
(2)
˜Σ(˜ p p p)

˙˜ x x x(t,˜ p p p) =˜f f f(˜ x x x(t,˜ p p p),˜ p p p),
˜ x x x(0,˜ p p p) = ˜ x x x0(˜ p p p)
˜ y y y(t,˜ p p p) =˜h h h(˜ x x x(t,˜ p p p),˜ p p p)
(3)
where p ∈ Ω ⊆ Rqand ˜ p p p ∈˜Ω ⊆ R˜ q, both open subsets consisting of the admissible parameter
vectors for the two systems respectively; f f f(·,p p p) and h h h(·,p p p) are analytic on M(p p p), an open and
connected subset of Rnsuch that x x x0(p p p) ∈ M(p p p);˜f˜f˜f(·,p p p) and˜h˜h˜h(·,p p p) are analytic on M(˜ p ˜ p ˜ p), an open
and connected subset of R˜ nsuch that ˜ x ˜ x ˜ x0(˜ p) ∈ M(˜ p ˜ p ˜ p); p p p and ˜ p ˜ p ˜ p are constant parameter vectors; Ω and
˜Ω are the sets of admissible parameter vectors for the two models (2) and (3), respectively; x x x(t,p p p)
and ˜ x ˜ x ˜ x(t, ˜ p ˜ p ˜ p) are the state variables for each model, which are the different species concentrations
whose values are governed by the system of differential equations comprising the model, (2) and
(3), respectively. These kinetics, and hence the solutions x x x(t,p p p) and ˜ x ˜ x ˜ x(t, ˜ p ˜ p ˜ p), are dependent on the
particular parameter vectors p p p ∈ Ω and ˜ p ˜ p ˜ p ∈˜Ω used in the models.
The indistinguishability problem arises because, in general, it is not possible to measure all
reactants in a given chemical reaction. An experiment that is used to collect measurements of the
process gives rise to an output structure for the model, the resulting output, or measurement vectors
are y y y(t,p p p) = (y1(t,p p p),...,yr(t,p p p))Tand ˜ y ˜ y ˜ y(t, ˜ p ˜ p ˜ p) = (y1(t, ˜ p ˜ p ˜ p),...,yr(t, ˜ p ˜ p ˜ p))T, respectively, and it is
these vectors that are compared with the collected experimental data during subsequent parameter
estimation. Suppose that there exists a p p p ∈ Ω and a ˜ p ˜ p ˜ p ∈˜Ω such that y y y(t,p p p) = ˜ y ˜ y ˜ y(t, ˜ p ˜ p ˜ p) for all t ≥ 0.
Then it is not possible to distinguish between the model given by (2) with parameter vector p p p
(i.e., Σ(p p p)) and the model given by (3) with parameter vector ˜ p ˜ p ˜ p (i.e.,˜Σ(˜ p ˜ p ˜ p)) from their outputs.
Therefore, even with perfect data (continuous measurements that are noise-free and error-free) it
is not possible to distinguish between the reaction schemes modelled by Σ(p p p) and˜Σ(˜ p ˜ p ˜ p) from the
proposed experiment. In this case the models Σ(p p p) and˜Σ(˜ p ˜ p ˜ p) are said to be indistinguishable, written
as Σ(p p p) ∼˜Σ(˜ p ˜ p ˜ p). The models are therefore structurally indistinguishable (Σ ∼˜Σ) if for almost all
p p p,there exists a ˜ p ˜ p ˜ p such that (Σ ∼˜Σ) and for almost all ˜ p ˜ p ˜ p such that˜Σ(˜ p ˜ p ˜ p) ∼ Σ(p p p).

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3.2Structural Identifiability
10
3.2. Structural Identifiability
Prior to discussing the methods of determining identifiability, it is necessary to formally define
identifiability of a single parameter and the relationship between parameter identifiability and
model structure identifiability. A review of identifiability techniques can be found in Godfrey and
DiStefano III [17]. For a generic parameter vector (p p p) which is a member of the set of all admissible
parameter vectors Ω, the individual parameter, pi, is referred to as locally identifiable if there exists
a neighbourhood of vectors around p p p, N(p p p), such that, if ˜ p ˜ p ˜ p ∈ N(p p p) ⊆ Ω and
y y y(t,p p p) = y y y(t, ˜ p ˜ p ˜ p) ∀ t ≥ 0,
(4)
then pi= ˜ pi. If the neighbourhood of the locally defined set equates to the set of all admissible
vectors (N(p p p) = Ω) then the parameter piis said to be globally identifiable. Finally, if the parameter
pi is not locally identifiable, then it is unidentifiable and the experiment that this input-output
relation corresponds to cannot be used to distinguish between piand ? pi.
The identifiability of an individual parameter determines the identifiability of the model struc-
ture; the following definitions succinctly describe this relationship [11]:
Definition 1. A model is said to be structurally globally identifiable (SGI) if, for generic p ∈ Ω, all
of the parameters pi(in p p p) are globally identifiable.
Definition 2. (SLI). A model is said to be structurally locally identifiable (SLI) if, for generic p ∈ Ω,
all of the parameters pi(in p p p) are locally identifiable and at least one is not globally identifiable.
Definition 3. A model is said to be structurally unidentifiable (SUI) if, for generic p ∈ Ω, any of
the parameters pi(in p p p) is unidentifiable.
For a more complete definition, please refer to the papers by Chappell et al. [4] and Evans et al.
[12]. There are several techniques available to determine the identifiability of nonlinear systems
(e.g. differential algebra methods [25] or the Similarity Transformation approach [12]); however,
the Taylor series approach of Pohjanpalo [23] can be used for experiments which can produce
time series data. The basis of the Taylor series approach is that the components of the output or
observation function yi(t,p p p) and its successive time derivatives are evaluated at some known time
point (usually an initial condition), i.e.
y(t,p p p) = y(0,p p p) + y(1)(0,p p p)t + ... + y(n)(0,p p p)tn
n!+ ...

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3.3Quasi-Steady State Assumptions
11
where
y(n)(0,p p p) =
dn
dtny(0,p p p).
These derivatives are thus expressed solely in terms of the system parameters p p p (and ˜ p ˜ p ˜ p for the
indistinguishability case). Since the coefficients in the Taylor series expansion are unique and, in
principle, measurable, the identifiability problem reduces to determining the number of solutions
for the system parameters in a set of algebraic equations that are, in general, non-linear in the
parameters. Failure to obtain a result using this approach does not necessarily imply that the
reaction schemes are distinguishable, and further tests may be required.
3.3. Quasi-Steady State Assumptions
In Model 1 there are 13 state variables corresponding to the concentrations of the enzyme (E),
three substrates (S1,S2,S3), three products (P,Q,R) and complexes (C1, C2, C3, C4, C5, C6). In
Model 2 there are 11 state variables corresponding to the concentrations of the enzyme (E), the
three substrates (S1,S2,S3), three products (P,Q,R) and complexes (C1, C2, C3and C4). After
simplifying the models, by assuming the rate the complexes are formed are significantly faster than
other reactions, the right-hand side of both sets of model equations is given by a set of 6 differential
equations (all of them are defined by the same function) for the substrates and products. The
quasi-steady-state equations for the two models are:
QSS Model 1: Let
f(S1,S2,S3,P,Q,R) =A
B
(5)

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3.3Quasi-Steady State Assumptions
12
where
A = E0(r1r2r3r4r5r6r7PQR − k1k2k3k4k5k6k7S1S2S3)
B = k4k5k6k7r1r2+ k5k6k7r1r2r3+ k6k7r1r2r3r4+ k1k4k5k6k7r2S1+ k1k5k6k7r2r3S1
+ k1k6k7r2r3r4S1+ k1k2k4k5k6k7S1S2+ k1k2k5k6k7r3S1S2+ k1k2k6k7r3r4S1S2
+ k3k4k5k6k7r1S3+ k1k3k4k5k6k7S1S3+ k2k3k4k5k6k7S2S3+ k1k2k3k4k5k6S1S2S3
+ k1k2k3k4k5k7S1S2S3+ k1k2k3k4k6k7S1S2S3+ k1k2k3k5k6k7S1S2S3
+ k1k2k3k6k7r4S1S2S3+ k1k2k3k4k5r6QS1S2S3+ r7(k6+ r6Q)R(r1r2(k4k5+ r3(k5+ r4))
+ k3k4k5(r1+ k2S2)S3) + r5P(k7r1r2r3r4+ r1r2r3r4r7R + k1k7r2r3r4S1+ k1k2k7r3r4S1S2
+ k1k2k3k4k7S1S2S3+ k1k2k3k7r4S1S2S3+ r6Q(r1r2r3r4+ k1S1(r2r3r4+ k2S2(r3r4+ k3(k4+ r4)S3))
+ r7R(k4r1r2+ r1r2r3+ r1r2r4+ r1r3r4+ r2r3r4+ k3r1(k4+ r4)S3+ k2S2(r3r4+ k3(k4+ r4)S3))))
the reduced system consists of six equations, each with right-hand side given by f:
S?
i= f(S1,S2,S3,P,Q,R)
and
P?= Q?= R?= −f(S1,S2,S3,P,Q,R).
QSS Model 2: Let
g(S1,S2,S3,P,Q,R) =C
D
(6)
where
C = E0(r1r2r3r4r5PQR − k1k2k3k4k5S1S2S3)
D = k4k5r1r2+ k5r1r2r3+ r1r2r3r4+ k3k4k5r1S3+ k2k3k4k5S2S3+ r5PQR(k4r1r2+ r1r2r3
+ r1r2r4+ r1r3r4+ r2r3r4+ k3r1(k4+ r4)S3+ k2S2(r3r4+ k3(k4+ r4)S3)) + k1S1(k4k5r2+ k5r2r3
+ r2r3r4+ k3k4k5S3+ k2S2(k4k5+ r3(k5+ r4) + k3(k4+ k5+ r4)S3))
with the QSS system described by six equations, given by g (S1,S2and S3) or −g (P,Q and R).

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3.4Numerical Identifiability
13
The quasi-steady assumption is a particular case of a simplification which relies on conservation
laws; finding all conservation laws applicable for Model 1 and 2 it was possible to find reduced
versions of both models. Using this method it is possible to choose to keep certain variables, for
example only substrates, or only complexes.
3.4. Numerical Identifiability
For complex systems the symbolic analysis, as described in Section 3.2 is often difficult to
perform manually, and as is the case for the two models presented (Section 2), intractable to
symbolic computational.In such cases an alternative method is presented by Pohjanpalo [24]
which employs the derivatives of the observations functions to determine the local identifiability of
the unknown parameter vector by generating a potentially infinite Jacobian matrix, evaluated at a
known time point (t0) i.e.
J = (Jij(t0)) =



∂y1(t0,p p p)
∂p1
...
···
...
∂y1(t0,p p p)
∂pq
...
∂ym(t0,p p p)
∂p1
∂y(1)
1
∂p1
...
···
∂ym(t0,p p p)
∂pq
∂y(1)
1
∂pq
...
(t0,p p p)
···
...
(t0,p p p)
∂y(n)
1
(t0,p p p)
∂p1
...
···
...
∂y(n)
1
(t0,p p p)
∂pq
...



(7)
where yiis the ithobservation, pj the jthelement of the parameter vector p, and the superscript
(n) donates the nthderivative. For the nonlinear models presented above (Sections 2.1 and 2.2)
due to the number of parameters to be estimated and the complexity of the model, the Jacobian
matrix, as defined by (7) could not be calculated to sufficient dimensions to obtain an appropriate
rank measure.1
A less general approach that is applicable to large complex systems is to use numerical methods
1To determine the rank a minimum of the 10th Taylor series co-efficient is required for the fast model and 14th for
the ordered release model; however, evaluating at time t = 0 results in zero initial conditions for all but the enzyme
and substrates, therefore higher order co-efficients are required.

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14
to estimate the sensitivity matrix, where the sensitivity matrix is defined as the matrix of first
derivatives, with respect to the parameter vector p p p, of the observation at each measurement time-
point
S = (Sij) =
?∂yi(t t t,p p p)
∂pj
?
=



∂y1(t1,p p p)
∂p1
...
···
...
∂y1(t1,p p p)
∂pn
...
∂ym(tr,p p p)
∂p1
···
∂ym(tr,p p p)
∂pn



(8)
where tr is the rthobservation sample.The sensitivity matrix can be estimated using finite-
differencing methods; however, a more robust and accurate result can be obtained with Automatic
Differentiation. The local identifiability of the system can be determined from the rank of the
numerical sensitivity matrix by the following rules:
• if rank(S) = number of parameters, consistently, then the model is structurally identifiable,
else
• if rank(S) < number of parameters, the model is unidentifiable.
If the model is unidentifiable the unidentifiable parameters are indicated by the dependent columns
of the sensitivity matrix.
4. Results
4.1. Structural Indistinguishability Analysis
Using the Taylor series approach it can be shown that in order for the outputs to be equal at
least one of the parameters would genuinely need to be zero and therefore it can be safely concluded
that the two models are distinguishable. This is the case if S1and P1are measured simultaneously,
or other combinations of the products P,Q and R. Due to the complexity of the resulting equations
it is not possible to provide a detailed list of these results; however, a worked example can be found
in Appendix A.
For the reduced quasi steady-state model, as described in Section 3.3, it was not possible to
perform the required symbolic manipulation to offer any information with respect to the model’s
indistinguishability due to the increased equation complexity.