# On the quasi-steady-state assumption in enzyme kinetics: rigorous analysis

Vasiliki Bitsouni<EMAIL_ADDRESS>Department of Mathematics,
National and Kapodistrian University of Athens, Panepistimioupolis, GR-15784
Athens, Greece  Nikolaos Gialelis<EMAIL_ADDRESS>Department of
Mathematics,
National and Kapodistrian University of Athens, Panepistimioupolis, GR-15784
Athens, Greece  Ioannis G. Stratis<EMAIL_ADDRESS>Department of
Mathematics,
National and Kapodistrian University of Athens, Panepistimioupolis, GR-15784
Athens, Greece

###### Abstract

We study, from a purely quantitative point of view, the quasi-steady-state
assumption for the fundamental mathematical model of the general enzymatic
reaction: we re-establish, on a rigorous basis, certain already known results
and we propose a novel approach to the subject, that makes waivable an
ambiguous, widely used, practice. In particular, we define the two parts of
the assumption in a quantitative fashion, we employ a simple algorithm for the
proper scaling of the corresponding problem which naturally provides us with
the necessary and sufficient information, and we comment, among other issues,
on a dispensable third part of the assumption.

Keywords: enzyme kinetics, quasi-steady-state assumption, standard, reverse,
scaling algorithm

MSC: 92C45, 92E20, 34D15, 34D20, 34E10, 34E15, 37N25.

##  1 Introduction

The study of the fundamental mathematical model for the kinetics of the
general enzymatic reaction with chemical equation

$S+E\rightarrow E+P,$ (1.1)

where $S$ is the substrate, $P$ is the product and $E$ is the enzyme that
catalyses it, has a long history, which we briefly present below.

Already since 1894, Fischer [8] derived the lock and key model for the
interpretation of biocatalysis. Already prior to 1901, Brown suggested an
intermediate step in the enzymatic reaction that is described by (1.1), where
the substrate forms a complex with the enzyme before the beginning of the
catalysis, an idea that was eventually published in 1902 [13]. Thus, it had
already been realised by that time, that enzymatic biochemical reactions
should take place in at least two stages, and in fact these stages should have
different time scales. Based on this idea, combined with conversations he had
with Bodenstein, Henri published in 1902 [12] and then in 1903 [23] an initial
version of a reliable differential equation for the description of the
kinetics of the enzyme reaction with chemical equation given by (1.1), an idea
he had conceived as early as 1901. A decade later, in 1913, Michaelis and
Menten [16] (translated in English in [1]), extracted this equation by using a
more detailed and analytical form that makes use of the rapid equilibrium
assumption; they interpreted it convincingly and studied it thoroughly. In
particular, using as an example the invertase-catalysed hydrolysis of sucrose
into glucose and fructose, they studied (1.1) through the chemical mechanism

$S+E\,\underset{k_{-1}}{\overset{k_{1}}{\rightleftharpoons}}\,C\xrightarrow{k_{2}}E+P,$
(1.2)

where $k_{1},k_{-1},k_{2}\,>0$ and $C$ represents the substrate-enzyme
complex, and indirectly concluded that, when

$\left[S\right]={\left[S\right]}_{0}\gg{\left[E\right]}_{0}=\left[E\right]\text{
and }\left[C\right]={\left[C\right]}_{0}=0,\text{ for }t=0,$ (1.3)

a condition acceptable in enzymatic reactions, then for the rate $\upsilon$ of
the enzymatic reaction with chemical equation (1.1) it holds that

$\upsilon\approx\frac{\upsilon_{\sup}\left[S\right]}{K_{{\rm
dis}}+\left[S\right]}\,,$ (1.4)

where

$\upsilon_{\sup}\coloneqq k_{2}{\left[E\right]}_{0}\,,$

and

$K_{{\rm dis}}\coloneqq\frac{k_{-1}}{k_{1}\,}.$ (1.5)

$K_{{\rm dis}}$ is the constant that is nowadays called the dissociation
constant (of the complex).

On the other hand, Van Slyke and Cullen, working in parallel with Michaelis
and Menten, but studying urease-catalysed hydrolysis of urea to ammonia and
carbon dioxide, used - instead of (1.2) - the chemical mechanism

$S+E\xrightarrow{k_{1}}C\xrightarrow{k_{2}}E+P$

and concluded in 1914 [6] to

$\upsilon\approx\frac{\upsilon_{\sup}\left[S\right]}{K_{\\!V\\!SC}+\left[S\right]},$
(1.6)

instead of (1.4), where

$K_{\\!V\\!SC}\coloneqq\frac{k_{2}}{k_{1}}\,.$ (1.7)

$K_{\\!V\\!SC}$ is a constant that is now known as the Van Slyke-Cullen
constant.

In 1925, Briggs and Haldane [7] published a short note where they composed the
ideas of Michaelis & Menten and Van Slyke & Cullen through a raw first version
of a new assumption, known presently as the standard quasi-steady-state
assumption. In particular, they improved (1.4) and (1.6), demonstrating that

$\upsilon\approx\frac{\upsilon_{\sup}\left[S\right]}{K_{\\!M}+\left[S\right]}\,,$
(1.8)

where

$K_{\\!M}\coloneqq K_{{\rm dis}}+K_{\\!V\\!SC}=\frac{k_{-1}+k_{2}}{k_{1}}\,.$
(1.9)

It is a standard expression nowadays, that “(1.8) characterises the Michaelis-
Menten kinetics”, and the constant (1.9) is called the Michaelis-Menten
constant.

Lineweaver and Burk in 1934 [11] established (1.8) in the form

$\frac{1}{\upsilon}\approx\frac{1}{\upsilon_{\sup}}+\frac{K_{\\!M}}{\upsilon_{\sup}}\frac{1}{\left[S\right]}\,,$

as a tool for experimental calculation of the values $\upsilon_{\sup}$ and
$K_{\\!M}$.

Already since the beginning of the second half of the 20th century and
throughout it, many researchers have dealt with the validity of the quasi-
steady-state assumption and the determination of the two time scales of the
model through the application of perturbation methods. However, it was much
later, in 1988 and in 1989, when Segel [2], and Segel and Slemrod [3],
respectively, showed that if

$\left(K_{\\!M}+{\left[S\right]}_{0}\right)\gg{\left[E\right]}_{0}\text{ and
}{\left[C\right]}_{0}=0,$ (1.10)

then there are indeed two time scales, which recorded as follows

$t_{C}=\frac{1}{k_{1}\left(K_{\\!M}+{\left[S\right]}_{0}\right)}\ll\frac{K_{\\!M}+{\left[S\right]}_{0}}{k_{2}{\left[E\right]}_{0}}=t_{S},$

and it holds that

$\upsilon\approx\begin{cases}0,&\text{ for times comparable to }t_{C},\\\
\text{as in (\ref{MMeq})},&\text{ for times comparable to }t_{S}.\end{cases}$
(1.11)

In fact, (1.10) is more general than (1.3) since it allows

$\frac{{\left[E\right]}_{0}}{{\left[S\right]}_{0}}=O{\left(1\right)},\text{ as
}\frac{{\left[E\right]}_{0}}{K_{\\!M}+{\left[S\right]}_{0}}\rightarrow 0^{+},$

or even

${\left[E\right]}_{0}\gg{\left[S\right]}_{0}.$

In 1997, Schnell and Mendoza [20] captured the solution of the Michaelis-
Menten kinetics equation in closed form, by using the Lambert $W$ function,
and in particular a special case of it, which is defined by its inverse as
follows

$W^{-1}{\left(x\right)}=x\,\exp{\left\\{x\right\\}},\text{ }\forall x\geq 0.$

In addition, via the aforementioned work of Segel and Slemrod, an initial form
of another hypothesis was introduced for the first time, the reverse quasi-
steady-state assumption, and it was shown that, when

${\left[E\right]}_{0}\gg K_{\\!M}\text{ and }{\left[C\right]}_{0}=0,$ (1.12)

then there are again two time scales,

$\tilde{t}_{S}=\frac{1}{k_{1}{\left[E\right]}_{0}}\ll\frac{1}{k_{2}}=\tilde{t}_{C},$

and it holds that

$\upsilon\approx\begin{cases}k_{1}{\left[E\right]}_{0}\left[S\right],&\text{
for times comparable to }\tilde{t}_{S},\\\ 0,&\text{ for times comparable to
}\tilde{t}_{C}.\end{cases}$ (1.13)

About a decade later, in 2000, Schnell and Maini [21] found that (1.12) is not
sufficient for (1.13) to hold; on the contrary, the new case should have the
form

${\left[E\right]}_{0}\gg K_{\\!M},\text{ }{\left[E\right]}_{0}\gg
K_{\\!M}\text{ and }{\left[C\right]}_{0}=0.$ (1.14)

Finally, let us mention that in 1996, with the work of Borghans, Boer and
Segel [18], the total substrate concentration, $\left[T\right]$ is introduced,
i.e., the sum of the concentration of the unbound/free substrate plus the
concentration of the bound substrate in the form of complex with the enzyme,
that is

$\left[T\right]\coloneqq\left[S\right]+\left[C\right],$ (1.15)

to describe an alleged third hypothesis that shares common ground with both
the previous ones, the so called total quasi-steady-state assumption, and
since then several researchers have adopted, and dealt with this hypothesis.

In this work, our novel results are

  1. 1.

We propose a general and simple algorithm for the proper scaling of every
problem with non negative solutions in a bounded domain, and we essentially
employ it in our case. Until now, only a “rough” rule for the non
dimensionalisation process is utilised in applications, which states that the
scales considered for the variables of a problem are chosen so that they
should be roughly of the same order of magnitude of the respective variables
themselves [14]. The proposed procedure is as follows:

    1. I.

Identification of the bounded feasible region - i.e., the range of the
dependent variables - of the problem.

    2. II.

Scaling of the dependent variables of the problem by their respective supremum
feasible values - which do exist, since the non negative solutions exist in a
bounded domain.

    3. III.

Natural scaling of the independent variables by gathering the remaining terms
of the previous step.

By this algorithm,

     * $\circ$

the dependent variables are comparable with each other, since they all range
onto $\left[0,1\right]$,

     * $\circ$

any scale of the independent variables follows naturally by the process, hence
there is no need of the - somehow unjustified - approach of “an estimate of
the minimum value for which the variable undergoes a significant change in
magnitude” (see, e.g., [4], [2] and [3]), which is widely adopted thenceforth,
for the choice of the largest of the two time scales (in our case),

     * $\circ$

the quantity $\varepsilon$, that characterises both the standard and the
reverse quasi-steady-state assumptions, arises effortless by the problem
itself.

  2. 2.

We clarify the purely quantitative nature of the standard and the reverse
quasi-steady-state assumptions, as opposed to the aforementioned qualitative
one. In particular, ($sQSSA$) and ($rQSSA$) do not serve for the validation of
the standard and the reverse, respectively, quasi-steady-state assumptions:
they define them.

  3. 3.

We relinquish the, so called, total quasi-steady-state assumption, by showing
that there is no substantive third hypothesis, but only a different approach
to the first two. We note that such a duality, characterised by a positive
parameter $\varepsilon$, that either tends to $0$ or to $\infty$, is common in
applications, for instance in the study of Hamiltonian systems possessing
either a relatively small or a relatively large Hamiltonian.

As far as the mathematical tools employed in the present work are concerned,
we note the following:

  * •

For the sake of brevity, we neither state nor discuss the necessary concepts
and fundamental results regarding a Cauchy (initial value) problem for vector
first order autonomous, or not, nonlinear ordinary differential equations.
These issues would certainly comprise the existence, the uniqueness, the
extendibility, the regularity, the continuous and smooth dependence of the
solutions on the data (initial condition, vector field determining the
differential equation, possible parameters appearing in the Cauchy problem).
Additional principal issues would include notions of stability, local and
global techniques for studying it, and essential results of the qualitative
theory of ODEs, in general. There is a huge literature on these topics;
indicatively, we refer to [15], [5], [22], and [10].

  * •

A powerful technique for problems with relatively small (or large) parameters
is the Method of Matched Asymptotics, where approximate solutions, accurate in
one region of the problem domain, are matched to different approximate
solutions, accurate in another region. This subject is discussed in many
books, see, e.g., [17], [9], [4], and [14].

##  2 Principal analysis of the problem

In this section, we introduce the main problem and proceed to its basic
analysis, that comprises the identification of the feasible regions, the well
posedness of the problem, the determination of the invariant sets, as well as
the simplification and the stability analysis of the problem.

###  2.1 Cauchy problem

Employing the chemical mechanism (1.2) along with the Law of Mass Action [19],
we arrive at the equations

$\displaystyle\dfrac{\mathrm{d}\left[S\right]}{\mathrm{d}t}$
$\displaystyle=-k_{1}\left[S\right]\left[E\right]+k_{-1}\left[C\right],$
(2.1a) $\displaystyle\dfrac{\mathrm{d}\left[E\right]}{\mathrm{d}t}$
$\displaystyle=-k_{1}\left[S\right]\left[E\right]+\left(k_{-1}+k_{2}\right)\left[C\right],$
(2.1b) $\displaystyle\dfrac{\mathrm{d}\left[C\right]}{\mathrm{d}t}$
$\displaystyle=k_{1}\left[S\right]\left[E\right]-\left(k_{-1}+k_{2}\right)\left[C\right],$
(2.1c) $\displaystyle\dfrac{\mathrm{d}\left[P\right]}{\mathrm{d}t}$
$\displaystyle=k_{2}\left[C\right],$ (2.1d)

and the corresponding Cauchy problem reads:

Given
${\left[S\right]}_{0},{\left[E\right]}_{0},{\left[C\right]}_{0},{\left[P\right]}_{0}\,\geq
0$, we seek an interval $\mathcal{I}\subseteq\mathbb{R}$ with
$0\in\mathcal{I}$, and a function
$\left(\left[S\right],\left[E\right],\left[C\right],\left[P\right]\right)\colon\mathcal{I}\rightarrow{\left[0,\infty\right)}^{4}$,
such that
$\left(\left[S\right],\left[E\right],\left[C\right],\left[P\right]\right)$
satisfies both (2.1) in
${\left(\mathcal{I}\setminus\left\\{0\right\\}\right)}^{\circ}$ and
$\left(\left[S\right],\left[E\right],\left[C\right],\left[P\right]\right)=\left({\left[S\right]}_{0},{\left[E\right]}_{0},{\left[C\right]}_{0},{\left[P\right]}_{0}\right)$,
for $t=0$.

For a solution of (2.1) it holds that

$\dfrac{\mathrm{d}\left[S\right]}{\mathrm{d}t}+\dfrac{\mathrm{d}\left[C\right]}{\mathrm{d}t}+\dfrac{\mathrm{d}\left[P\right]}{\mathrm{d}t}=0,$

or, equivalently,

$\left[S\right]+\left[C\right]+\left[P\right]={\left[S\right]}_{0}+{\left[C\right]}_{0}+{\left[P\right]}_{0}\eqqcolon
A_{1},$ (2.2)

due to the initial condition of (2.1), as well as that

$\dfrac{\mathrm{d}\left[E\right]}{\mathrm{d}t}+\dfrac{\mathrm{d}\left[C\right]}{\mathrm{d}t}=0,$

or, equivalently,

$\left[E\right]+\left[C\right]={\left[E\right]}_{0}+{\left[C\right]}_{0}\eqqcolon
A_{2}.$ (2.3)

From (2.2) and (2.3) combined with the non negativity of the components of the
solutions of (2.1), we conclude that

$\left[S\right]\leq A_{1},\text{ }\left[E\right]\leq A_{2},\text{
}\left[C\right]\leq\min{\left\\{A_{1},A_{2}\right\\}}\text{ and
}\left[P\right]\leq A_{1}.$ (2.4)

In addition, from (2.1c) together with the bounds for $\left[S\right]$ and
$\left[E\right]$ in (2.4) we have that

$\left[C\right]\leq\frac{A_{1}A_{2}}{K_{\\!M}},$ (2.5)

where $K_{\\!M}$ is defined as in (1.9), whereas the rest of the equations of
(2.1) do not include further related information. Thus, from (2.4) and (2.5)
we finally get that

$\left[S\right]\leq A_{1},\text{ }\left[E\right]\leq A_{2},\text{
}\left[C\right]\leq\min{\left\\{A_{1},A_{2},\frac{A_{1}A_{2}}{K_{\\!M}}\right\\}}\eqqcolon
A_{3}\text{ and }\left[P\right]\leq A_{1}.$ (2.6)

In the light of (2.6), we set

$\Omega_{0}\coloneqq\left\\{\left(s,e,c,p\right)\in\left[0,A_{1}\right]\times\left[0,A_{2}\right]\times\left[0,A_{3}\right]\times\left[0,A_{1}\right]\,\big{|}\,s+c+p=A_{1},\text{
}e+c=A_{2}\right\\}$

and we can therefore consider an equivalent to (2.1) problem as follows:

Given
${\left[S\right]}_{0},{\left[E\right]}_{0},{\left[C\right]}_{0},{\left[P\right]}_{0}\,\geq
0$, we are looking for an interval $\mathcal{I}\subseteq\mathbb{R}$ with
$0\in\mathcal{I}$ and a function
$\left(\left[S\right],\left[E\right],\left[C\right],\left[P\right]\right)\colon\mathcal{I}\rightarrow\Omega_{0}$,
such that
$\left(\left[S\right],\left[E\right],\left[C\right],\left[P\right]\right)$ to
satisfy both (2.1) in
${\left(\mathcal{I}\setminus\left\\{0\right\\}\right)}^{\circ}$ and
$\left(\left[S\right],\left[E\right],\left[C\right],\left[P\right]\right)=\left({\left[S\right]}_{0},{\left[E\right]}_{0},{\left[C\right]}_{0},{\left[P\right]}_{0}\right)$
for $t=0$.

Employing standard arguments of the theory of Ordinary Differential Equations,
we can conclude that (2.1) is globally well posed, with an infinitely smooth
solution in an interval $\widetilde{\mathcal{I}}\subseteq\mathbb{R}$, where

$\widetilde{\mathcal{I}}=\mathbb{R},\text{ or
}\widetilde{\mathcal{I}}=\left[-a,\infty\right)\,\text{for some
}a\in\left[0,\infty\right).$

In addition, when $A_{2}=0$, the unique solution is the constant

$\left(S,E,C,P\right)=\left({\left[S\right]}_{0},0,0,{\left[P\right]}_{0}\right),\text{
}\forall t\in\mathbb{R}.$

Thus, when $A_{2}=0$, $\Omega_{0}$ reduces to

$\Omega_{0}=\left\\{\left(s,0,0,p\right)\in\left[0,A_{1}\right]\times{\left\\{0\right\\}}^{2}\times\left[0,A_{1}\right]\,\big{|}\,s+p=A_{1}\right\\}$

which is invariant (in particular, every singleton
$\left\\{\left(s,0,0,A_{1}-s\right)\right\\}$ for $s\in\left[0,A_{1}\right]$
is invariant), whereas $\Omega_{0}$ is positively invariant, when $A_{2}>0$.

###  2.2 A simpler equivalent problem

Given that (2.2) and (2.3) hold, we conclude that system (2.1) can be
equivalently reduced to

$\displaystyle\dfrac{\mathrm{d}\left[S\right]}{\mathrm{d}t}$
$\displaystyle=-k_{1}A_{2}\left[S\right]+k_{1}\left[S\right]\left[C\right]+k_{-1}\left[C\right],$
(2.7a) $\displaystyle\dfrac{\mathrm{d}\left[C\right]}{\mathrm{d}t}$
$\displaystyle=k_{1}A_{2}\left[S\right]-k_{1}\left[S\right]\left[C\right]-\left(k_{-1}+k_{2}\right)\left[C\right].$
(2.7b)

Let us now study the above subsystem. Using (2.7b) combined with the bound of
$\left[S\right]$ in (2.6) we have that

$\left[C\right]\leq\frac{A_{1}A_{2}}{K_{\\!M}+A_{1}}\,.$ (2.8)

Therefore, from the bound of $\left[C\right]$ in (2.6) and from (2.8) we
eventually get that

$\left[C\right]\leq\min{\left\\{A_{1},A_{2},\frac{A_{1}A_{2}}{K_{\\!M}},\frac{A_{1}A_{2}}{K_{\\!M}+A_{1}}\right\\}}=\min{\left\\{A_{1},\frac{A_{1}A_{2}}{K_{\\!M}+A_{1}}\right\\}}\eqqcolon
A_{4}.$ (2.9)

In fact, key to what follows are the immediately verifiable inferences

$\framebox{$A_{2}\leq K_{\\!M}+A_{1}\Rightarrow
A_{4}=\dfrac{A_{1}A_{2}}{K_{\\!M}+A_{1}}$},$ (2.10)

and on the other hand,

$\framebox{$A_{2}\geq K_{\\!M}+A_{1}\Rightarrow A_{4}=A_{1}$}.$ (2.11)

Now, in the light of the bound for $\left[S\right]$ in (2.6) and of (2.9), we
set

$\Omega_{1}\coloneqq\left\\{\left(s,c\right)\in\left[0,A_{1}\right]\times\left[0,A_{4}\right]\,\big{|}\,s+c\leq
A_{1}\right\\}$

and so we can consider the equivalent, to (2.1), problem as follows:

Given
${\left[S\right]}_{0},{\left[E\right]}_{0},{\left[C\right]}_{0},{\left[P\right]}_{0}\,\geq
0$, we seek an interval $\mathcal{I}\subseteq\mathbb{R}$ with
$0\in\mathcal{I}$, and a function
$\left(\left[S\right],\left[C\right]\right)\colon\mathcal{I}\rightarrow\Omega_{1}$,
such that $\left(\left[S\right],\left[C\right]\right)$ satisfies both (2.7) in
${\left(\mathcal{I}\setminus\left\\{0\right\\}\right)}^{\circ}$, and
$\left(\left[S\right],\left[C\right]\right)=\left({\left[S\right]}_{0},{\left[C\right]}_{0}\right)$
for $t=0$.

###  2.3 Stability analysis

First, we can easily deduce that

$\begin{cases}\left(s,0,0,A_{1}-s\right)\text{ where
}s\in\left[0,A_{1}\right],&\text{ when }A_{2}=0,\\\
\left(0,e,0,A_{1}\right)\text{ where }e\in\left[0,A_{2}\right],&\text{ when
}A_{2}>0,\end{cases}$

are the steady states of (2.1). However, we immediately conclude that it makes
sense to study their stability only for the non-trivial case, where

$A_{1}>0\text{ and }A_{2}>0.$

It is sufficient though, as usually, to study the stability of
$\left(0,0\right)$, as a steady state of (2.2), when $A_{1}>0$ and $A_{2}>0$.

As for the local stability of $\left(0,0\right)$, we calculate the Jacobi
matrix:

$J{\left(s,c\right)}=\begin{pmatrix}k_{1}\left(c-A_{2}\right)&k_{1}s+k_{-1}\\\
k_{1}\left(A_{2}-c\right)&-k_{1}s-\left(k_{-1}+k_{2}\right)\end{pmatrix}.$

Its eigenvalues at $\left(0,0\right)$, are

$\lambda_{\pm}=\frac{1}{2}\left(-k_{1}A_{2}-\left(k_{-1}+k_{2}\right)\pm{\left({\left(k_{1}A_{2}+\left(k_{-1}+k_{2}\right)\right)}^{2}-4k_{1}k_{2}A_{2}\right)}^{\frac{1}{2}}\right).$

Since

${\left(k_{1}A_{2}+\left(k_{-1}+k_{2}\right)\right)}^{2}-4k_{1}k_{2}A_{2}={\left(k_{1}A_{2}+\left(k_{-1}-k_{2}\right)\right)}^{2}+4k_{-1}k_{2}\geq
0,$

the origin $\left(0,0\right)$ is locally asymptotically stable for (2.2),
since

$\lambda_{\pm}<0.$

In fact, we can also find, as usually, a local approach to the solution close
to $\left(0,0\right)$; we omit it for the sake of brevity.

As for the global stability of $\left(0,0\right)$, we can apply the Bendixson-
Dulac Negative Criterion with

$\displaystyle g\,\colon{\left(\mathbb{R}_{+}\right)}^{2}$
$\displaystyle\rightarrow\left(0,\infty\right)$
$\displaystyle\left(s,c\right)$ $\displaystyle\mapsto
g{\left(s,c\right)}=\dfrac{1}{sc},$

thereby obtaining the desired result, since in
${\left(\Omega_{1}\right)}^{\circ}$ it holds that

$\mathrm{div}{\left[\left(\frac{-k_{1}A_{2}}{\left[C\right]}+k_{1}+\frac{k_{-1}}{\left[S\right]},\frac{k_{1}A_{2}}{\left[C\right]}-k_{1}-\frac{\left(k_{-1}+k_{2}\right)}{\left[S\right]}\right)\right]}=-\frac{k_{-1}}{{\left[S\right]}^{2}}-\frac{k_{1}A_{2}}{{\left[C\right]}^{2}}<0.$

##  3 The standard quasi-steady-state assumption

The standard quasi-steady-state assumption is

$\framebox{$A_{1}>0$ and $0<A_{2}\ll K_{\\!M}+A_{1}$},$ ($sQSSA$)

or, equivalently,

$A_{1}>0\text{ and }0<A_{2}\ll K_{\\!M},\text{ or }0<A_{2}\ll A_{1}\text{ (or
these two together)},$

and provided that it holds, we study (2.2).

We consider two approaches for examining the assumption, the free substrate
approach, where the concentration dynamics of the unbound substrate,
$\left[S\right]$, is studied, and the total substrate approach, where the
concentration dynamics of the total substrate, $\left[T\right]$, is studied,
as defined in (1.15).

###  3.1 Free substrate approach

Using only ($sQSSA$) we will show that

  1. 1.

Problem (2.2), and therefore problem (2.1) as well, has inherently two time
scales which we will determine. In fact, ($sQSSA$) owes its name to the
existence of the above time scales. In particular, except for a short initial
time interval, where the enzymatic reaction with chemical equation (1.1) is
not evolving, i.e., $\upsilon\approx 0$, during the rest of the time the
enzymatic reaction is at a “steady state”, in which (1.8) holds.

  2. 2.

There is a good uniform approximation in closed form to the solution of (2.2),
and therefore to (2.2) as well, which we will determine.

To highlight the above time scales, the first and basic step is scaling (2.2).
Thus, as usually, due to the bound of $\left[S\right]$ in (2.6) and the
relation (2.9), we choose the dimensionless dependent variables as

$S_{\alpha}{\left(t_{\alpha}\right)}\coloneqq\frac{1}{A_{1}}\left[S\right]{\left(\frac{t}{t_{*}}\right)}\text{
and
}C_{\alpha}{\left(t_{\alpha}\right)}\coloneqq\frac{1}{A_{4}}\left[C\right]{\left(\frac{t}{t_{*}}\right)},$

where we have chosen an arbitrary, for the time being, time scale $t_{*}>0$
for the scaling, i.e.,

$t_{\alpha}\coloneqq\frac{t}{t_{*}},$

the determination of which will arise in a natural manner during the process.
We note, however, that - given ($sQSSA$) - it follows from (2.10) that

$\framebox{$A_{4}=\varepsilon A_{1}\ll A_{1}$}$

and

$C_{\alpha}{\left(t_{\alpha}\right)}=\frac{1}{\varepsilon
A_{1}}\left[C\right]{\left(\frac{t}{t_{*}}\right)},$

where

$0\,\overset{\text{(\ref{sQSSA})}}{<}\,\varepsilon\coloneqq\frac{A_{2}}{K_{\\!M}+A_{1}}\,\overset{\text{(\ref{sQSSA})}}{\ll}\,1,$
(3.1)

i.e., equivalently,

$0<\varepsilon\approx 0.$ (3.2)

A first conclusion is that the possible change of $\left[S\right]$ is much
larger than the corresponding one of $\left[C\right]$. Now, (2.7) will take
the following form

$\displaystyle\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=k_{1}A_{2}t_{*}\left(-S_{\alpha}+\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}+\frac{\rho}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha}\right),$
(3.3a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=k_{1}\left(K_{\\!M}+A_{1}\right)t_{*}\left(S_{\alpha}-\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{1}{1+\sigma}C_{\alpha}\right),$
(3.3b)

where

$\sigma\coloneqq\frac{A_{1}}{K_{\\!M}}\text{ and }\rho\coloneqq\frac{K_{\rm
dis}}{K_{\\!V\\!SC}}=\frac{k_{-1}}{k_{2}},$ (3.4)

where $K_{\rm dis}$ and $K_{\\!V\\!SC}$ are as in (1.5) and (1.7),
respectively.

Observing (3.3), we define

$\frac{1}{k_{1}\left(K_{\\!M}+A_{1}\right)}\eqqcolon
t_{1}\,\overset{\text{(\ref{epsln})}}{\ll}\,\frac{t_{1}}{\varepsilon}=\frac{1}{k_{1}A_{2}}\eqqcolon
t_{2},$ (3.5)

to conclude that

$t_{*}=t_{1}\Rightarrow\begin{cases}k_{1}A_{2}t_{*}=\varepsilon,\\\
k_{1}\left(K_{\\!M}+A_{1}\right)t_{*}=1,\end{cases}\text{ and
}\,\,\,t_{*}=t_{2}\Rightarrow\begin{cases}k_{1}A_{2}t_{*}=1,\\\
k_{1}\left(K_{\\!M}+A_{1}\right)t_{*}=\frac{1}{\varepsilon},\end{cases}$

and so (3.3) takes the following form:

  * •

If $t_{\alpha}=\dfrac{t}{t_{1}}$, then

$\displaystyle\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\varepsilon\left(-S_{\alpha}+\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}+\frac{\rho}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha}\right),$
(3.6a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=S_{\alpha}-\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{1}{1+\sigma}C_{\alpha}.$
(3.6b)

  * •

If $t_{\alpha}=\dfrac{t}{t_{2}}$, then

$\displaystyle\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=-S_{\alpha}+\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}+\frac{\rho}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha},$
(3.7a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\frac{1}{\varepsilon}\left(S_{\alpha}-\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{1}{1+\sigma}C_{\alpha}\right).$
(3.7b)

Setting

$\Omega_{\varepsilon}\coloneqq\left\\{\left(s,c\right)\in{\left[0,1\right]}^{2}\,\big{|}\,s+\varepsilon
c\leq 1\right\\},$

the scaled version of (2.2) is:

Given ${\left[S\right]}_{0},{\left[C\right]}_{0},{\left[P\right]}_{0}\,\geq 0$
and $\varepsilon>0$, we seek an interval $\mathcal{I}\subseteq\mathbb{R}$ with
$0\in\mathcal{I}$, and a function
$\left(S_{\alpha},C_{\alpha}\right)\colon\mathcal{I}\rightarrow\Omega_{\varepsilon}$,
such that $\left(S_{\alpha},C_{\alpha}\right)$ satisfying both (3.6) if
$t_{\alpha}=\dfrac{t}{t_{1}}$ or (3.7) if $t_{\alpha}=\dfrac{t}{t_{2}}$ in
${\left(\mathcal{I}\setminus\left\\{0\right\\}\right)}^{\circ}$ and
$\left(S_{\alpha},C_{\alpha}\right)=\left(\dfrac{{\left[S\right]}_{0}}{A_{1}},\dfrac{{\left[C\right]}_{0}}{\varepsilon
A_{1}}\right)$, for $t_{\alpha}=0$.

Figure 1: The invariant set $\Omega_{\varepsilon}$ of problem (3.1). We notice
that $\Omega_{\varepsilon}\rightarrow{\left[0,1\right]}^{2}$ as
$\varepsilon\rightarrow 0^{+}$.

We study separately each of the two versions of (3.1) in order to find an
inner and outer, respectively, approximation to the solution
$\left(\left[S\right],\left[C\right]\right)$ of (2.2), i.e., one approximation
for times comparable to $t_{1}$ and another one for times comparable to
$t_{2}$, respectively. In more detail:

  * •

Looking at (2.2) as a perturbed problem, with perturbation $\varepsilon>0$
close to $0$, we have the following information on (3.6a)

$\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}\,=\,\underbrace{\varepsilon\left(\underbrace{-S_{\alpha}+\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}+\frac{\rho}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha}}_{\begin{subarray}{c}=\,O{\left(1\right)}\text{
uniformly (with respect to $t_{\alpha}$) as }\varepsilon\rightarrow 0^{+},\\\
\text{since }\left(S_{\alpha},C_{\alpha},\right)\,=\,O{\left(1\right)}\text{
uniformly as }\varepsilon\rightarrow 0^{+},\\\ \text{since
}\left(S_{\alpha},C_{\alpha}\right)\,\in\,\Omega_{\varepsilon}\text{
}\forall\varepsilon>0\end{subarray}}\right)}_{=\,O{\left(\varepsilon\right)}\text{
uniformly as }\varepsilon\rightarrow 0^{+}},$

as well as

$\underbrace{\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}}_{\begin{subarray}{c}=\,O{\left(1\right)}\text{
uniformly as }\varepsilon\rightarrow 0^{+},\\\ \text{when
}t_{\alpha}\,=\,O{\left(1\right)}\text{ uniformly as }\varepsilon\rightarrow
0^{+},\\\ \text{since }S_{\alpha}\,=\,O{\left(1\right)}\text{ uniformly as
}\varepsilon\rightarrow 0^{+}\\\ \text{combined with the definition of the
derivative}\end{subarray}}\,=\,\underbrace{\dots}_{=\,O{\left(\varepsilon\right)}\text{
uniformly as }\varepsilon\rightarrow 0^{+}},$

thus

$\underbrace{\frac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}}_{\begin{subarray}{c}=\,O{\left(\varepsilon\right)}\text{
uniformly as }\varepsilon\rightarrow 0^{+},\\\ \text{when
}t_{\alpha}\,=\,O{\left(1\right)}\text{ uniformly as }\varepsilon\rightarrow
0^{+}\end{subarray}}.$

Thus, due to (3.2) it follows that

$\frac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}\approx 0,\text{ when
}\exists B>0\text{ independent of }\varepsilon,\text{ such that:
}\left|t_{\alpha}\right|\leq B,$ (3.8)

and due to the initial condition of (3.1) we eventually have that

$S_{\alpha}\approx\frac{{\left[S\right]}_{0}}{A_{1}},\text{ when }\exists
B>0:\text{ }\left|t_{\alpha}\right|\leq B.$

If we insert the above approximate equality in (3.6b), then the later becomes
an approximate linear differential equation, the solution of which is

$C_{\alpha}\approx\frac{\left(1+\sigma\right)S_{\alpha}}{1+\sigma
S_{\alpha}}+\left(\dfrac{{\left[C\right]}_{0}}{\varepsilon
A_{1}}-\frac{\left(1+\sigma\right)S_{\alpha}}{1+\sigma
S_{\alpha}}\right)\exp{\left\\{-\frac{1+\sigma
S_{\alpha}}{1+\sigma}\,t_{\alpha}\right\\}},\text{ when }\exists B>0:\text{
}\left|t_{\alpha}\right|\leq B,$

given the initial condition of (3.1).
Therefore, the inner approximation, $\left({\left[S\right]}_{{\rm
in}},{\left[C\right]}_{{\rm in}}\right)$, of the solution
$\left(\left[S\right],\left[C\right]\right)$ of (2.2), i.e., the approximation
for those $t$ for which it holds that

$\text{\framebox{$\exists B>0$ independent of $\varepsilon$, such that:
$\left|t\right|\leq Bt_{1}$}},$

is

$\left({\left[S\right]}_{{\rm in}},{\left[C\right]}_{{\rm
in}}\right)=\left({\left[S\right]}_{0},\frac{A_{2}{\left[S\right]}_{0}}{K_{\\!M}+{\left[S\right]}_{0}}+\left({\left[C\right]}_{0}-\frac{A_{2}{\left[S\right]}_{0}}{K_{\\!M}+{\left[S\right]}_{0}}\right)\exp{\left\\{-k_{1}\left(K_{\\!M}+{\left[S\right]}_{0}\right)t\right\\}}\right).$

  * •

For (3.7b) we have that

$\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}=\underbrace{\frac{1}{\varepsilon}\left(\underbrace{S_{\alpha}-\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{1}{1+\sigma}C_{\alpha}}_{=\,O{\left(1\right)}\text{
uniformly as }\varepsilon\rightarrow
0^{+}}\right)}_{=\,O{\left(\dfrac{1}{\varepsilon}\right)}\text{ uniformly as
}\varepsilon\rightarrow 0^{+}}$

as well as

$\underbrace{\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}}_{\begin{subarray}{c}=\,O{\left(1\right)}\text{
uniformly as }\varepsilon\rightarrow 0^{+},\\\ \text{when
}t_{\alpha}\,=\,O{\left(1\right)}\text{ uniformly as }\varepsilon\rightarrow
0^{+}\end{subarray}}=\underbrace{\dots}_{=\,O{\left(\dfrac{1}{\varepsilon}\right)}\text{
uniformly as }\varepsilon\rightarrow 0^{+}},$

therefore

$\underbrace{S_{\alpha}-\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{1}{1+\sigma}C_{\alpha}}_{\begin{subarray}{c}=\,O{\left(\varepsilon\right)}\text{
uniformly as }\varepsilon\rightarrow 0^{+},\\\ \text{when
}t_{\alpha}\,=\,O{\left(1\right)}\text{ uniformly as }\varepsilon\rightarrow
0^{+}\end{subarray}}.$

Thus, it follows that

$\begin{split}\varepsilon\frac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}=S_{\alpha}-\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{1}{1+\sigma}C_{\alpha}\approx
0,&\\\ &\text{when }\exists B>0\text{ independent of }\varepsilon,\text{ such
that: }\left|t_{\alpha}\right|\leq B,\end{split}$ (3.9)

i.e.,

$C_{\alpha}\approx\frac{\left(1+\sigma\right)S_{\alpha}}{1+\sigma
S_{\alpha}},\text{ when }\exists B>0\text{ independent of }\varepsilon,\text{
such that: }\left|t_{\alpha}\right|\leq B.$

If we insert the above approximate equality in (3.7a), then the later becomes
an approximate separable non linear differential equation, i.e.,

$\frac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}\approx-\frac{S_{\alpha}}{\left(1+\rho\right)\left(1+\sigma
S_{\alpha}\right)},\text{when }\exists B>0\text{ independent of
}\varepsilon,\text{ such that: }\left|t_{\alpha}\right|\leq B,$ (3.10)

the solution of which is

$S_{\alpha}\approx\frac{1}{\sigma}W{\left(\sigma\ell\exp{\left\\{\sigma\ell-\frac{1}{1+\rho}\,t_{\alpha}\right\\}}\right)},\text{
when }\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B,$

where $W$ is the aforementioned Lambert function, and $\ell\geq 0$ is a
constant that remains to be determined.
Therefore, the outer approximation, $\left({\left[S\right]}_{{\rm
out}},{\left[C\right]}_{{\rm out}}\right)$, of the solution
$\left(\left[S\right],\left[C\right]\right)$ of (2.2), i.e., the approximation
for those $t$ for which it holds that

$\text{\framebox{$\exists B>0$ independent of $\varepsilon$, such that:
$\left|t\right|\leq Bt_{2}$}},$

is

$\begin{split}{\left[S\right]}_{{\rm out}}&=K_{\\!M}W{\left(\frac{\ell
A_{1}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left(\ell
A_{1}-k_{2}A_{2}t\right)\right\\}}\right)}\\\ {\left[C\right]}_{{\rm
out}}&=\frac{A_{2}W{\left(\frac{\ell
A_{1}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left(\ell
A_{1}-k_{2}A_{2}t\right)\right\\}}\right)}}{1+W{\left(\frac{\ell
A_{1}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left(\ell
A_{1}-k_{2}A_{2}t\right)\right\\}}\right)}}.\end{split}$

We can now utilise the matching technique in order to find a uniform
approximation of the solution $\left(\left[S\right],\left[C\right]\right)$ of
(2.2) from the individual approximations $\left({\left[S\right]}_{{\rm
in}},{\left[C\right]}_{{\rm in}}\right)$ and $\left({\left[S\right]}_{{\rm
out}},{\left[C\right]}_{{\rm out}}\right)$. First, choosing a time scale
between $t_{1}$ and $t_{2}$, e.g.

$\frac{t_{1}}{\varepsilon^{\frac{1}{2}}}\in\left(t_{1},t_{2}\right),$

we find easily that the common limit resulting from the matching condition of
the two individual solutions should be

$L\coloneqq\left({\left[S\right]}_{0},\frac{A_{2}{\left[S\right]}_{0}}{K_{\\!M}+{\left[S\right]}_{0}}\right).$

Therefore

$\ell=\frac{{\left[S\right]}_{0}}{A_{1}}$

and thus a uniform approximation $\left({\left[S\right]}_{{\rm
un}},{\left[C\right]}_{{\rm un}}\right)$, of
$\left(\left[S\right],\left[C\right]\right)$ is

$\left({\left[S\right]}_{{\rm un}},{\left[C\right]}_{{\rm
un}}\right)=\left({\left[S\right]}_{{\rm in}},{\left[C\right]}_{{\rm
in}}\right)+\left({\left[S\right]}_{{\rm out}},{\left[C\right]}_{{\rm
out}}\right)-L,$

i.e., in more detail,

$\begin{split}{\left[S\right]}_{{\rm
un}}&=K_{\\!M}W{\left(\frac{{\left[S\right]}_{0}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left({\left[S\right]}_{0}-k_{2}A_{2}t\right)\right\\}}\right)},\\\
{\left[C\right]}_{{\rm
un}}&=\frac{A_{2}W{\left(\frac{{\left[S\right]}_{0}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left({\left[S\right]}_{0}-k_{2}A_{2}t\right)\right\\}}\right)}}{1+W{\left(\frac{{\left[S\right]}_{0}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left({\left[S\right]}_{0}-k_{2}A_{2}t\right)\right\\}}\right)}}+\left({\left[C\right]}_{0}-\frac{A_{2}{\left[S\right]}_{0}}{K_{\\!M}+{\left[S\right]}_{0}}\right)\exp{\left\\{-k_{1}\left(K_{\\!M}+{\left[S\right]}_{0}\right)t\right\\}}.\end{split}$
(3.11)

###  3.2 Total substrate approach

Since $\left(\left[S\right],\left[C\right]\right)\in\Omega_{1}$, then
$\left[T\right]\leq A_{1}$, where $\left[T\right]$ is as in (1.15). Thus, we
introduce, as usually, the dimensionless dependent variable

$T_{\alpha}{\left(t_{\alpha}\right)}\coloneqq\frac{1}{A_{1}}{\left(\left[S\right]+\left[C\right]\right)}{\left(t_{\alpha}\right)}={\left(S_{\alpha}+\varepsilon
C_{\alpha}\right)}{\left(t_{\alpha}\right)}.$

It is easily verified that (3.6) and (3.7) will take the following forms:

  * •

If $t_{\alpha}=\dfrac{t}{t_{1}}$, then

$\displaystyle\dfrac{\mathrm{d}T_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=-\frac{\varepsilon}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha},$
(3.12a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\frac{\varepsilon\sigma}{1+\sigma}{C_{\alpha}}^{2}-\left(\frac{1}{1+\sigma}+\varepsilon\right)C_{\alpha}-\frac{\sigma}{1+\sigma}C_{\alpha}T_{\alpha}+T_{\alpha}.$
(3.12b)

  * •

If $t_{\alpha}=\dfrac{t}{t_{2}}$, then

$\displaystyle\dfrac{\mathrm{d}T_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=-\frac{1}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha},$
(3.13a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\frac{\sigma}{1+\sigma}{C_{\alpha}}^{2}-\left(\frac{1}{\varepsilon\left(1+\sigma\right)}+1\right)C_{\alpha}-\frac{\sigma}{\varepsilon\left(1+\sigma\right)}C_{\alpha}T_{\alpha}+\frac{1}{\varepsilon}T_{\alpha}.$
(3.13b)

So we have the following scaled problem:

Given ${\left[S\right]}_{0},{\left[C\right]}_{0},{\left[P\right]}_{0}\,\geq 0$
and $\varepsilon>0$, we seek an interval $\mathcal{I}\subseteq\mathbb{R}$ with
$0\in\mathcal{I}$, and a function
$\left(T_{\alpha},C_{\alpha}\right)\colon\mathcal{I}\rightarrow{\left[0,1\right]}^{2}$,
such that $\left(T_{\alpha},C_{\alpha}\right)$ satisfies both (3.12) if
$t_{\alpha}=\dfrac{t}{t_{1}}$, or (3.13) if $t_{\alpha}=\dfrac{t}{t_{2}}$ in
${\left(\mathcal{I}\setminus\left\\{0\right\\}\right)}^{\circ}$, and
$\left(T_{\alpha},C_{\alpha}\right)=\left(\dfrac{{\left[S\right]}_{0}}{A_{1}}+\dfrac{{\left[C\right]}_{0}}{\varepsilon
A_{1}},\dfrac{{\left[C\right]}_{0}}{\varepsilon A_{1}}\right)$, for
$t_{\alpha}=0$.

Working as with problem (3.1), we conclude for problem (3.2) now, the
following:

  * •

(3.12a) gives that

$\frac{\mathrm{d}T_{\alpha}}{\mathrm{d}t_{\alpha}}\approx 0,\text{ when
}\exists B>0\text{ independent of }\varepsilon,\text{ such that:
}\left|t_{\alpha}\right|\leq B,$

and due to the initial condition of (3.2) we have that

$T_{\alpha}\approx\frac{{\left[S\right]}_{0}}{A_{1}}+\frac{{\left[C\right]}_{0}}{\varepsilon
A_{1}},\text{ when }\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B.$

Inserting the above approximate equality into (3.12b), which in turn takes the
following approximate form

$\begin{split}\frac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}\approx-\left(\frac{\sigma}{1+\sigma}T_{\alpha}+\frac{1}{1+\sigma}\right)C_{\alpha}+T_{\alpha},&\\\
&\text{when }\exists B>0\text{ independent of }\varepsilon,\text{ such that:
}\left|t_{\alpha}\right|\leq B,\end{split}$

then the later becomes an approximate linear differential equation, the
solution of which is

$C_{\alpha}\approx\frac{\left(1+\sigma\right)T_{\alpha}}{1+\sigma
T_{\alpha}}+\left(\dfrac{{\left[C\right]}_{0}}{\varepsilon
A_{1}}-\frac{\left(1+\sigma\right)T_{\alpha}}{1+\sigma
T_{\alpha}}\right)\exp{\left\\{-\frac{1+\sigma
T_{\alpha}}{1+\sigma}\,t_{\alpha}\right\\}},\text{ when }\exists B>0:\text{
}\left|t_{\alpha}\right|\leq B,$

given the initial condition of (3.2).
Therefore, the inner approximation, $\left({\left[T\right]}_{{\rm
in}},{\left[C\right]}_{{\rm in}}\right)$, of
$\left(\left[T\right],\left[C\right]\right)$ is

$\left({\left[T\right]}_{{\rm in}},{\left[C\right]}_{{\rm
in}}\right)=\left({\left[T\right]}_{0},\frac{A_{2}{\left[T\right]}_{0}}{K_{\\!M}+{\left[T\right]}_{0}}+\left({\left[C\right]}_{0}-\frac{A_{2}{\left[T\right]}_{0}}{K_{\\!M}+{\left[T\right]}_{0}}\right)\exp{\left\\{-k_{1}\left(K_{\\!M}+{\left[T\right]}_{0}\right)t\right\\}}\right),$

where

${\left[T\right]}_{0}\coloneqq{\left[S\right]}_{0}+{\left[C\right]}_{0}.$

  * •

From (3.13b) we get that

$\begin{split}\varepsilon\left(\frac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}-\frac{\sigma}{1+\sigma}{C_{\alpha}}^{2}+C_{\alpha}\right)=-\frac{1}{1+\sigma}C_{\alpha}-\frac{\sigma}{1+\sigma}C_{\alpha}T_{\alpha}+T_{\alpha}\approx
0,&\\\ &\text{when }\exists B>0\text{ independent of }\varepsilon,\text{ such
that: }\left|t_{\alpha}\right|\leq B,\end{split}$

i.e.,

$C_{\alpha}\approx\frac{\left(1+\sigma\right)T_{\alpha}}{1+\sigma
T_{\alpha}},\text{ when }\exists B>0\text{ independent of }\varepsilon,\text{
such that: }\left|t_{\alpha}\right|\leq B.$

If we insert the above approximate equality into (3.13a), then the later
becomes an approximate separable non linear differential equation, namely

$\frac{\mathrm{d}T_{\alpha}}{\mathrm{d}t_{\alpha}}\approx-\frac{T_{\alpha}}{\left(1+\rho\right)\left(1+\sigma
T_{\alpha}\right)},\text{ when }\exists B>0\text{ independent of
}\varepsilon,\text{ such that: }\left|t_{\alpha}\right|\leq B,$

the solution of which is

$T_{\alpha}\approx\frac{1}{\sigma}W\left(\sigma\ell\exp{\left\\{\sigma\ell-\frac{1}{1+\rho}\,t_{\alpha}\right\\}}\right),\text{
when }\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B,$

where $\ell\geq 0$ is a constant that remains to be determined. Therefore, the
outer approximation $\left({\left[T\right]}_{{\rm out}},{\left[C\right]}_{{\rm
out}}\right)$ of $\left(\left[T\right],\left[C\right]\right)$, is

$\begin{split}{\left[T\right]}_{{\rm out}}&=K_{\\!M}W{\left(\frac{\ell
A_{1}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left(\ell
A_{1}-k_{2}A_{2}t\right)\right\\}}\right)}\\\ {\left[C\right]}_{{\rm
out}}&=\frac{A_{2}W{\left(\frac{\ell
A_{1}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left(\ell
A_{1}-k_{2}A_{2}t\right)\right\\}}\right)}}{1+W{\left(\frac{\ell
A_{1}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left(\ell
A_{1}-k_{2}A_{2}t\right)\right\\}}\right)}}.\end{split}$

Finally, with a similar reasoning as for the uniform approximation of the
solution of (2.2), we have that

$\ell=\frac{{\left[T\right]}_{0}}{A_{1}}$

and also that the uniform approximation, $\left({\left[T\right]}_{{\rm
un}},{\left[C\right]}_{{\rm un}}\right)$, of
$\left(\left[T\right],\left[C\right]\right)$ is

$\begin{split}{\left[T\right]}_{{\rm
un}}&=K_{\\!M}W{\left(\frac{{\left[T\right]}_{0}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left({\left[T\right]}_{0}-k_{2}A_{2}t\right)\right\\}}\right)}\\\
{\left[C\right]}_{{\rm
un}}&=\frac{A_{2}W{\left(\frac{{\left[T\right]}_{0}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left({\left[T\right]}_{0}-k_{2}A_{2}t\right)\right\\}}\right)}}{1+W{\left(\frac{{\left[T\right]}_{0}}{K_{\\!M}}\exp{\left\\{\frac{1}{K_{\\!M}}\left({\left[T\right]}_{0}-k_{2}A_{2}t\right)\right\\}}\right)}}+\left({\left[T\right]}_{0}-\frac{A_{2}{\left[T\right]}_{0}}{K_{\\!M}+{\left[T\right]}_{0}}\right)\exp{\left\\{-k_{1}\left(K_{\\!M}+{\left[T\right]}_{0}\right)t\right\\}}.\end{split}$
(3.14)

###  3.3 Conclusions

We showed that given ($sQSSA$) there is a $t_{1}>0$ such that

$\frac{\mathrm{d}\left[S\right]}{\mathrm{d}t}\approx 0,\text{ when }\exists
B>0\text{ independent of }\varepsilon,\text{ such that: }\left|t\right|\leq
Bt_{1},$

which arises directly from (3.8), as well as that there is a $t_{2}\gg t_{1}$
such that

$\frac{\mathrm{d}\left[C\right]}{\mathrm{d}t}\approx 0,\text{ when }\exists
B>0\text{ independent of }\varepsilon,\text{ such that: }\left|t\right|\leq
Bt_{2},$

which in turn results from (3.9).
In fact, due to (3.10) it holds that

$\frac{\mathrm{d}\left[S\right]}{\mathrm{d}t}\approx-\frac{k_{2}A_{2}\left[S\right]}{K_{\\!M}+\left[S\right]},\text{
when }\exists B>0:\text{ }\left|t\right|\leq Bt_{2}.$

Hence, we can conclude that

$\upsilon\approx\begin{cases}0,&\text{ when }\exists B>0:\text{
}\left|t\right|\leq Bt_{1},\\\
\dfrac{k_{2}A_{2}\left[S\right]}{K_{\\!M}+\left[S\right]},&\text{ when
}\exists B>0:\text{ }\left|t\right|\leq Bt_{2},\end{cases}$

where $\upsilon$ is the rate of the chemical reaction with chemical equation
(1.1). The above approximation for $t$ comparable to $t_{2}$ is none other
than the Michaelis-Menten approximation for the kinetics of the aforementioned
chemical reaction, as already commented in (1.8).

Figure 2: An approximation for the kinetics of the chemical reaction (1.1)
given that ($sQSSA$) holds, for times comparable to $t_{2}$.

Furthermore, comparing the approximate solution of the free substrate
$\left({\left[S\right]}_{{\rm un}},{\left[C\right]}_{{\rm un}}\right)$ of
(3.11) with the approximate solution of the total substrate
$\left({\left[T\right]}_{{\rm un}},{\left[C\right]}_{{\rm un}}\right)$ of
(3.14), these two should be in agreement. Indeed, it is sufficient to observe
that

$\left[T\right]\approx\left[S\right],$

as

$T_{\alpha}=S_{\alpha}+\varepsilon
C_{\alpha}\overset{\text{(\ref{epsln2})}}{\approx}S_{\alpha}\Rightarrow
T\approx S.$

###  3.4 Numerical solution

We proceed to the numerical solution of the problem, as shown in Figure 3, in
Figure 4 and in Figure 5, to verify our conclusions. For the numerical values
of the constants and the initial conditions we follow the work of Segel in
1988 [2]. The values are given in the table below.

Parameter | Value | Unit
---|---|---
$k_{-1}$ | $25$ | $s^{-1}$
$k_{1}$ | $4\cdot 10^{6}$ | $M^{-1}s^{-1}$
$k_{2}$ | $15$ | $s^{-1}$
${\left[S\right]}_{0}$ | $10^{-5}$ | $M$
${\left[E\right]}_{0}$ | $10^{-8}$ | $M$
${\left[C\right]}_{0}$ | $0$ | $M$
${\left[P\right]}_{0}$ | $0$ | $M$

We calculate

$K_{\\!M}={\left[S\right]}_{0}=A_{1}\text{ and
}A_{2}={\left[E\right]}_{0}=10^{-3}{\left[S\right]}_{0}=10^{-3}A_{1},$

i.e.,

$\varepsilon=5\cdot 10^{-4}$

and

$t_{1}=1,25\cdot 10^{-2}s\text{ and }t_{2}=25s.$

(a)

(b)

(c)

(d)

Figure 3: Plot of $\left[S\right]$, $\left[E\right]$, $\left[C\right]$ and
$\left[P\right]$ of problem (2.1) for non negative times, given that ($sQSSA$)
holds. We see that $\left[S\right]$ and $\left[C\right]$ are of different
order of magnitude, as well as that there are two distinct phases of the
evolution of the phenomenon.

(a)

(b)

(c)

(d)

Figure 4: Plots of $S_{\alpha}$ and $C_{\alpha}$ of problem (3.1) for non
negative times, given that ($sQSSA$) holds. In (a) and (b) time is measured
based on $t_{1}$, whereas in (c) and (d) based on $t_{2}$.

(a)

(b)

(c)

(d)

Figure 5: Plots of the inner and outer approximations of ${\left[S\right]}$
and $\left[C\right]$ of problem (2.1), for non negative times, given that
($sQSSA$) holds.

##  4 The reverse quasi-steady-state assumption

The reverse quasi-steady-state assumption is the following:

$\framebox{$A_{1}>0$ and $A_{2}\gg K_{\\!M}+A_{1}$},$ ($rQSSA$)

or, equivalently,

$A_{1}>0,\text{ }A_{2}\gg K_{\\!M}\text{ and }A_{2}\gg A_{1},$

and given that this holds we study problem (2.2).
We notice that the inequality $A_{2}\gg K_{\\!M}+A_{1}$ of ($rQSSA$) is the
reverse of the one corresponding to ($sQSSA$), hence the name of the first. As
for the analysis of ($sQSSA$), here, as well, we consider two approaches for
the study of case ($rQSSA$), the free substrate approach and the total
substrate approach.

###  4.1 Free substrate approach

Using only ($rQSSA$) we will show that:

  1. 1.

Problem (2.2), and therefore problem (2.1) as well, has inherently two time
scales which we will determine. In particular, except for a short initial time
interval where the enzymatic reaction with chemical equation (1.1) is evolving
with rate $\upsilon$, showing approximately linear behaviour with respect to
$\left[S\right]$, and $\upsilon\approx k_{1}A_{2}\left[S\right]$ as in (1.13),
during the rest of the time the enzymatic reaction does not evolve, i.e.
$\upsilon\approx 0$.

  2. 2.

There is a good uniform approximation in closed form to the solution of (2.2),
and therefore to (2.1) as well, which we will determine.

As usually, we use the dimensionless dependent variables

$S_{\alpha}{\left(t_{\alpha}\right)}\coloneqq\frac{1}{A_{1}}\left[S\right]{\left(\frac{t}{t_{*}}\right)}\text{
and
}C_{\alpha}{\left(t_{\alpha}\right)}\coloneqq\frac{1}{A_{4}}\left[C\right]{\left(\frac{t}{t_{*}}\right)},$

where we have chosen an arbitrary, for the time being, time scale $t_{*}>0$
for the scaling.
We notice, however, that given ($rQSSA$) it follows from (2.11) that

$\framebox{$A_{4}=A_{1}$}$

and

$C_{\alpha}{\left(t_{\alpha}\right)}=\frac{1}{A_{1}}\left[C\right]{\left(\frac{t}{t_{*}}\right)}.$

A first conclusion is that the possible change of $\left[S\right]$ is
comparable to the corresponding of $\left[C\right]$.
We set

$0\,\overset{\text{(\ref{rQSSA})}}{<}\,\eta\coloneqq\frac{A_{1}}{A_{2}}<\frac{K_{\\!M}+A_{1}}{A_{2}}=\frac{1}{\varepsilon}\,\overset{\text{(\ref{rQSSA})}}{\ll}\,1,$
(4.1)

where $\varepsilon$ is as in (3.1), i.e., equivalently

$0<\eta<\frac{1}{\varepsilon}\approx 0,$ (4.2)

and so (2.7) will take the following form

$\displaystyle\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\frac{k_{1}\left(K_{\\!M}+A_{1}\right)A_{2}}{A_{1}}t_{*}\left(-\frac{\sigma}{1+\sigma}S_{\alpha}+\frac{\eta\sigma}{1+\sigma}S_{\alpha}C_{\alpha}+\frac{\eta\rho}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha}\right),$
(4.3a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\frac{k_{1}\left(K_{\\!M}+A_{1}\right)A_{2}}{A_{1}}t_{*}\left(\frac{\sigma}{1+\sigma}S_{\alpha}-\frac{\eta\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{\eta}{1+\sigma}C_{\alpha}\right),$
(4.3b)

where $\sigma$ and $\rho$ are as in (3.4).

Observing (4.3), we define

$\frac{A_{1}}{k_{1}\left(K_{\\!M}+A_{1}\right)A_{2}}\eqqcolon
t_{1}\,\overset{\text{(\ref{etavareps})}}{\ll}\,\frac{t_{1}}{\eta}=\frac{1}{k_{1}\left(K_{\\!M}+A_{1}\right)}\eqqcolon
t_{2},$ (4.4)

to conclude that

$t_{*}=t_{1}\Rightarrow\frac{k_{1}\left(K_{\\!M}+A_{1}\right)A_{2}}{A_{1}}t_{*}=1\text{
and
}t_{*}=t_{2}\Rightarrow\frac{k_{1}\left(K_{\\!M}+A_{1}\right)A_{2}}{A_{1}}t_{*}=\frac{1}{\eta},$

and so (4.3) to get the following forms:

  * •

If $t_{\alpha}=\dfrac{t}{t_{1}}$, then

$\displaystyle\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=-\frac{\sigma}{1+\sigma}S_{\alpha}+\frac{\eta\sigma}{1+\sigma}S_{\alpha}C_{\alpha}+\frac{\eta\rho}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha},$
(4.5a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\frac{\sigma}{1+\sigma}S_{\alpha}-\frac{\eta\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{\eta}{1+\sigma}C_{\alpha}.$
(4.5b)

  * •

If $t_{\alpha}=\dfrac{t}{t_{2}}$, then

$\displaystyle\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=-\frac{\sigma}{\eta\left(1+\sigma\right)}S_{\alpha}+\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}+\frac{\rho}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha},$
(4.6a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\frac{\sigma}{\eta\left(1+\sigma\right)}S_{\alpha}-\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{1}{1+\sigma}C_{\alpha}.$
(4.6b)

Setting

$\Omega\coloneqq\left\\{\left(s,c\right)\in{\left[0,1\right]}^{2}\,\big{|}\,s+c\leq
1\right\\},$

the scaled version of (2.2) will be as follows:

Given ${\left[S\right]}_{0},{\left[C\right]}_{0},{\left[P\right]}_{0}\,\geq 0$
and $\eta>0$, we seek an interval $\mathcal{I}\subseteq\mathbb{R}$ with
$0\in\mathcal{I}$, and a function
$\left(S_{\alpha},C_{\alpha}\right)\colon\mathcal{I}\rightarrow\Omega$, such
that $\left(S_{\alpha},C_{\alpha}\right)$ satisfies both (4.5) if
$t_{\alpha}=\dfrac{t}{t_{1}}$, or (4.6) if $t_{\alpha}=\dfrac{t}{t_{2}}$ in
${\left(\mathcal{I}\setminus\left\\{0\right\\}\right)}^{\circ}$, and
$\left(S_{\alpha},C_{\alpha}\right)=\left(\dfrac{{\left[S\right]}_{0}}{A_{1}},\dfrac{{\left[C\right]}_{0}}{A_{1}}\right)$,
for $t_{\alpha}=0$.

Figure 6: The invariant set $\Omega$ of problem (4.1).

We study each of two versions of (4.1) separately:

  * •

From (4.5a), which due to (4.2) takes the approximate linear form

$\frac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}\approx-\frac{\sigma}{1+\sigma}S_{\alpha},\text{
when }\exists B>0\text{ independent of }\eta,\text{ such that:
}\left|t_{\alpha}\right|\leq B,$ (4.7)

we get, due to the initial condition of (4.1), that

$S_{\alpha}\approx\frac{{\left[S\right]}_{0}}{A_{1}}\exp{\left\\{-\frac{\sigma}{1+\sigma}\,t_{\alpha}\right\\}},\text{
when }\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B.$

If we insert the above approximate equality in (4.5b), which will now have the
approximate form

$\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}\approx\frac{\sigma}{1+\sigma}S_{\alpha},\text{
when }\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B,$

then we get that

$C_{\alpha}\approx\frac{{\left[C\right]}_{0}}{A_{1}}+\frac{{\left[S\right]}_{0}}{A_{1}}\left(1-\exp{\left\\{-\frac{\sigma}{1+\sigma}\,t_{\alpha}\right\\}}\right),\text{
when }\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B,$

given the initial condition of (4.1). Therefore, the inner approximation,
$\left({\left[S\right]}_{{\rm in}},{\left[C\right]}_{{\rm in}}\right)$, of
$\left(\left[S\right],\left[C\right]\right)$ is

$\left({\left[S\right]}_{{\rm in}},{\left[C\right]}_{{\rm
in}}\right)=\left({\left[S\right]}_{0}\exp{\left\\{-k_{1}A_{2}t\right\\}},{\left[C\right]}_{0}+{\left[S\right]}_{0}\left(1-\exp{\left\\{-k_{1}A_{2}t\right\\}}\right)\right).$

  * •

From (4.6a) we have that

$\begin{split}\eta\left(\frac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}-\frac{\sigma}{1+\sigma}S_{\alpha}C_{\alpha}-\frac{\rho}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha}\right)=-\frac{\sigma}{1+\sigma}S_{\alpha}\approx
0,&\\\ &\text{when }\exists B>0\text{ independent of }\eta,\text{ such that:
}\left|t_{\alpha}\right|\leq B,\end{split}$ (4.8)

i.e.,

$S_{\alpha}\approx 0,\text{ when }\exists B>0:\text{
}\left|t_{\alpha}\right|\leq B,$

thus

$\frac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}\approx 0,\text{ when
}\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B.$

If we insert the above approximate equality in the sum of (4.6a) and (4.6b),
then the following approximate linear differential equation arises

$\frac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}\approx-\frac{1}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha},\text{
when }\exists B>0\text{ independent of }\eta,\text{ such that:
}\left|t_{\alpha}\right|\leq B,$

the solution of which is

$C_{\alpha}\approx\ell\exp{\left\\{-\frac{1}{(1+\rho)(1+\sigma)}\,t_{\alpha}\right\\}},\text{
when }\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B,$

where $\ell\geq 0$ a constant that remains to be determined. Therefore, the
external approximation, $\left({\left[S\right]}_{{\rm
out}},{\left[C\right]}_{{\rm out}}\right)$, of
$\left(\left[S\right],\left[C\right]\right)$ is

$\left({\left[S\right]}_{{\rm out}},{\left[C\right]}_{{\rm
out}}\right)=\left(0,\ell A_{1}\exp{\left\\{-k_{2}t\right\\}}\right).$

Finally, as usually, we find that

$\ell=\frac{{\left[S\right]}_{0}}{A_{1}}+\frac{{\left[C\right]}_{0}}{A_{1}},$

as well as that the uniform approximation, $\left({\left[S\right]}_{{\rm
un}},{\left[C\right]}_{{\rm un}}\right)$, of
$\left(\left[S\right],\left[C\right]\right)$ is

$\left({\left[S\right]}_{{\rm un}},{\left[C\right]}_{{\rm
un}}\right)=\left({\left[S\right]}_{0}\exp{\left\\{-k_{1}A_{2}t\right\\}},{\left[C\right]}_{0}\exp{\left\\{-k_{2}t\right\\}}+{\left[S\right]}_{0}\left(\exp{\left\\{-k_{2}t\right\\}}-\exp{\left\\{-k_{1}A_{2}t\right\\}}\right)\right).$
(4.9)

###  4.2 Total substrate approach

Since $\left(\left[S\right],\left[C\right]\right)\in\Omega_{1}$, we have that
$\left[T\right]\leq A_{1}$. Thus, we introduce, as usually, the dimensionless
dependent variable

$T_{\alpha}{\left(t_{\alpha}\right)}\coloneqq\frac{1}{A_{1}}{\left(\left[S\right]+\left[C\right]\right)}{\left(t_{\alpha}\right)}={\left(S_{\alpha}+C_{\alpha}\right)}{\left(t_{\alpha}\right)}.$

It is easily verified that (4.5) and (4.6) will get the following forms:

  * •

If $t_{\alpha}=\dfrac{t}{t_{1}}$, then

$\displaystyle\dfrac{\mathrm{d}T_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=-\frac{\eta}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha},$
(4.10a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\frac{\eta\sigma}{1+\sigma}{C_{\alpha}}^{2}-\frac{\eta+\sigma}{1+\sigma}C_{\alpha}-\frac{\eta\sigma}{1+\sigma}C_{\alpha}T_{\alpha}+\frac{\sigma}{1+\sigma}T_{\alpha}.$
(4.10b)

  * •

If $t_{\alpha}=\dfrac{t}{t_{2}}$, then

$\displaystyle\dfrac{\mathrm{d}T_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=-\frac{1}{\left(1+\rho\right)\left(1+\sigma\right)}C_{\alpha},$
(4.11a) $\displaystyle\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=\frac{\sigma}{1+\sigma}{C_{\alpha}}^{2}-\frac{\eta+\sigma}{\eta\left(1+\sigma\right)}C_{\alpha}-\frac{\sigma}{1+\sigma}C_{\alpha}T_{\alpha}+\frac{\sigma}{\eta\left(1+\sigma\right)}T_{\alpha}.$
(4.11b)

So we have the following scaled problem:

Given ${\left[S\right]}_{0},{\left[C\right]}_{0},{\left[P\right]}_{0}\,\geq 0$
and $\eta>0$, we seek an interval $\mathcal{I}\subseteq\mathbb{R}$ with
$0\in\mathcal{I}$, and a function
$\left(T_{\alpha},C_{\alpha}\right)\colon\mathcal{I}\rightarrow{\left[0,1\right]}^{2}$,
such that $\left(T_{\alpha},C_{\alpha}\right)$ satisfies both (4.10) if
$t_{\alpha}=\dfrac{t}{t_{1}}$, or (4.11) if $t_{\alpha}=\dfrac{t}{t_{2}}$ in
${\left(\mathcal{I}\setminus\left\\{0\right\\}\right)}^{\circ}$, and
$\left(T_{\alpha},C_{\alpha}\right)=\left(\dfrac{{\left[T\right]}_{0}}{A_{1}},\dfrac{{\left[C\right]}_{0}}{A_{1}}\right)$,
for $t_{\alpha}=0$.

Working as with problem (4.1), we conclude, now for problem (4.2), the
following:

  * •

(4.10a) gives that

$\frac{\mathrm{d}T_{\alpha}}{\mathrm{d}t_{\alpha}}\approx 0,\text{ when
}\exists B>0\text{ independent of }\eta,\text{ such that:
}\left|t_{\alpha}\right|\leq B,$

and due to the initial condition of (4.2) we have that

$T_{\alpha}\approx\frac{{\left[T\right]}_{0}}{A_{1}},\text{ when }\exists
B>0:\text{ }\left|t_{\alpha}\right|\leq B.$

If we insert the above approximate equality into (4.10b), which will now have
the approximate linear form

$\dfrac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}\approx-\frac{\sigma}{1+\sigma}C_{\alpha}+\frac{\sigma}{1+\sigma}T_{\alpha},\text{
when }\exists B>0\text{ independent of }\eta,\text{ such that:
}\left|t_{\alpha}\right|\leq B,$

then we will get

$C_{\alpha}\approx
T_{\alpha}+\left(\frac{{\left[C\right]}_{0}}{A_{1}}-T_{\alpha}\right)\exp{\left\\{-\frac{\sigma}{1+\sigma}\,t_{\alpha}\right\\}},\text{
when }\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B,$

given the initial condition (4.2).
Therefore, the initial condition, $\left({\left[T\right]}_{{\rm
in}},{\left[C\right]}_{{\rm in}}\right)$, of
$\left(\left[T\right],\left[C\right]\right)$ is

$\left({\left[T\right]}_{{\rm in}},{\left[C\right]}_{{\rm
in}}\right)=\left({\left[T\right]}_{0},{\left[T\right]}_{0}+\left({\left[C\right]}_{0}-{\left[T\right]}_{0}\right)\exp{\left\\{-k_{1}A_{2}t\right\\}}\right).$

  * •

From (4.11b) we obtain that

$\begin{split}\eta\left(\frac{\mathrm{d}C_{\alpha}}{\mathrm{d}t_{\alpha}}-\frac{\sigma}{1+\sigma}{C_{\alpha}}^{2}+\frac{1}{1+\sigma}C_{\alpha}-\frac{\sigma}{1+\sigma}C_{\alpha}T_{\alpha}\right)=\frac{\sigma}{1+\sigma}\left(-C_{\alpha}+T_{\alpha}\right)\approx
0,&\\\ &\text{when }\exists B>0\text{ independent of }\eta,\text{ such that:
}\left|t_{\alpha}\right|\leq B,\end{split}$

i.e.,

$T_{\alpha}\approx C_{\alpha},\text{ when }\exists B>0\text{ independent of
}\eta,\text{ such that: }\left|t_{\alpha}\right|\leq B.$

If we insert the above approximate equality in (4.11a), then the later becomes
an approximate linear differential equation, which is none other than

$\frac{\mathrm{d}T_{\alpha}}{\mathrm{d}t_{\alpha}}\approx-\frac{1}{\left(1+\rho\right)\left(1+\sigma\right)}T_{\alpha},\text{
when }\exists B>0\text{ independent of }\eta,\text{ such that:
}\left|t_{\alpha}\right|\leq B,$

the solution of which is

$T_{\alpha}\approx\ell\exp{\left\\{-\frac{1}{(1+\rho)(1+\sigma)}\,t_{\alpha}\right\\}},\text{
when }\exists B>0:\text{ }\left|t_{\alpha}\right|\leq B,$

where $\ell\geq 0$ a constant that remains to be determined. Therefore, the
outer approximation, $\left({\left[T\right]}_{{\rm
out}},{\left[C\right]}_{{\rm out}}\right)$, of
$\left(\left[T\right],\left[C\right]\right)$ is

$\begin{split}\left({\left[T\right]}_{{\rm out}},{\left[C\right]}_{{\rm
out}}\right)=\left(\ell A_{1}\exp{\left\\{-k_{2}t\right\\}},\ell
A_{1}\exp{\left\\{-k_{2}t\right\\}}\right).\end{split}$

Finally, as usually, we find that

$\ell=\frac{{\left[T\right]}_{0}}{A_{1}},$

as well as that the uniform approximation, $\left({\left[T\right]}_{{\rm
un}},{\left[C\right]}_{{\rm un}}\right)$, of
$\left(\left[T\right],\left[C\right]\right)$ is

$\left({\left[T\right]}_{{\rm un}},{\left[C\right]}_{{\rm
un}}\right)=\left({\left[T\right]}_{0}\exp{\left\\{-k_{2}t\right\\}},{\left[T\right]}_{0}\exp{\left\\{-k_{2}t\right\\}}+\left({\left[C\right]}_{0}-{\left[T\right]}_{0}\right)\exp{\left\\{-k_{1}A_{2}t\right\\}}\right).$
(4.12)

###  4.3 Conclusions

Although for the previous analysis it was used that $0<\eta\approx 0,$ i.e.,

$A_{1}\ll A_{2},$

nevertheless we emphasise that also the relation

$K_{\\!M}\ll A_{2},$

although somehow “obscure”, plays an essential role in distinguishing
($rQSSA$) from ($sQSSA$). Indeed, let

$K_{\\!M}\nll A_{2},$

i.e.,

$K_{\\!M}\gg A_{2}\text{ or }K_{\\!M}\approx A_{2}.$

Then, given $A_{1}\ll A_{2}$, we could finally have that

$A_{2}\ll K_{\\!M}+A_{1},$

i.e., ($sQSSA$), if in addition it holds that $K_{\\!M}\gg A_{2}\gg A_{1}$, or
otherwise that

$A_{2}\approx K_{\\!M}+A_{1},$

which does not fall under any case, if additionally $K_{\\!M}\approx A_{2}\gg
A_{1}$.

In addition, we showed that given ($rQSSA$) there is $t_{1}>0$ such that

$\frac{\mathrm{d}\left[S\right]}{\mathrm{d}t}\approx-
k_{1}A_{2}\left[S\right],\text{ when }\exists B>0\text{ independent of
}\eta,\text{ such that: }\left|t\right|\leq Bt_{1},$

which arises directly from (4.7), as well as that there is $t_{2}\gg t_{1}$
such that

$\frac{\mathrm{d}\left[S\right]}{\mathrm{d}t}\approx\left[S\right]\approx
0,\text{ when }\exists B>0\text{ independent of }\eta,\text{ such that:
}\left|t\right|\leq Bt_{2},$

which in turn results from (4.8). i.e., we can conclude that

$\upsilon\approx\begin{cases}k_{1}A_{2}\left[S\right],&\text{ when }\exists
B>0:\text{ }\left|t\right|\leq Bt_{1},\\\ 0,&\text{ when }\exists B>0:\text{
}\left|t\right|\leq Bt_{2},\end{cases}$

where $\upsilon$ stands for the rate of the chemical reaction with chemical
equation (1.1), as we have already mentioned.

Figure 7: An approximation for the kinetics of the chemical reaction (1.1)
given that ($rQSSA$) holds, for times comparable to $t_{1}$.

Finally, comparing the approximate solution of the free substrate
$\left({\left[S\right]}_{{\rm un}},{\left[C\right]}_{{\rm un}}\right)$ of
(4.9) with the approximate solution of the total substrate
$\left({\left[T\right]}_{{\rm un}},{\left[C\right]}_{{\rm un}}\right)$ of
(4.12), we easily observe by definition in (1.15) that they are in agreement.

###  4.4 Numerical solution

We proceed to the numerical solution of the problem, as shown in Figure 8, in
Figure 9 and in Figure 10, to verify our conclusions. The numerical values of
the constants and the initial conditions are given in the table below.

Parameter | Value | Unit
---|---|---
$k_{-1}$ | $25$ | $s^{-1}$
$k_{1}$ | $4\cdot 10^{6}$ | $M^{-1}s^{-1}$
$k_{2}$ | $15$ | $s^{-1}$
${\left[S\right]}_{0}$ | $10^{-5}$ | $M$
${\left[E\right]}_{0}$ | $10^{-2}$ | $M$
${\left[C\right]}_{0}$ | $0$ | $M$
${\left[P\right]}_{0}$ | $0$ | $M$

We calculate

$K_{\\!M}={\left[S\right]}_{0}=A_{1}\text{ and
}A_{2}={\left[E\right]}_{0}=10^{3}{\left[S\right]}_{0}=10^{3}A_{1},$

i.e.,

$\varepsilon=500,\text{ }\frac{1}{\varepsilon}=2\cdot 10^{-3}\text{ and
}\eta=10^{-3}$

and

$t_{1}=1,25\cdot 10^{-5}s\text{ and }t_{2}=1,25\cdot 10^{-2}s.$

(a)

(b)

(c)

(d)

Figure 8: Plots of $\left[S\right]$, $\left[E\right]$, $\left[C\right]$ and
$\left[P\right]$ of problem (2.1) for non negative times, given that ($rQSSA$)
holds. We see that $\left[S\right]$ and $\left[C\right]$ are of the same order
of magnitude, as well as that there are two distinct phases of the evolution
of the phenomenon.

(a)

(b)

(c)

(d)

Figure 9: Plots of $S_{\alpha}$ and $C_{\alpha}$ of problem (4.1) for non
negative times, given that ($rQSSA$) holds. In (a) and (b) time is measured
based on $t_{1}$, whereas (c) and (d) based on $t_{2}$.

(a)

(b)

(c)

(d)

Figure 10: Plots of the inner and outer approximations of ${\left[S\right]}$,
$\left[E\right]$ and $\left[C\right]$ of problem (2.1), for non negative
times, given that ($sQSSA$) holds. The inner and outer approximation of
${\left[E\right]}$ are given by the relations ${\left[E\right]}_{{\rm
in}}=A_{2}-{\left[C\right]}_{{\rm in}}$ and ${\left[E\right]}_{{\rm
out}}=A_{2}-{\left[C\right]}_{{\rm out}}$, respectively, due to (2.3).

##  5 Discussion

We employed a simple scaling algorithm for the rigorous treatment of the
quasi-steady-state assumption.

We note that such an algorithm can be utilised in every problem with non
negative solutions in a bounded domain, e.g., the simple classical $SIR$
problem of Epidemiology

$\displaystyle\dfrac{\mathrm{d}S}{\mathrm{d}t}$ $\displaystyle=-\beta SI,$
$\displaystyle\dfrac{\mathrm{d}I}{\mathrm{d}t}$ $\displaystyle=-\gamma I+\beta
SI,$ $\displaystyle\dfrac{\mathrm{d}R}{\mathrm{d}t}$ $\displaystyle=\gamma I,$

for $\beta,\gamma\,>0$, where the feasible region is the set

$\left\\{\left(s,i,r\right)\in{\left[0,N_{0}\right]}^{3}\,\big{|}\,s+i+r=N_{0}\right\\},$

hence every dependent variable, $S$, $I$ and $R$, is scaled by $N_{0}$ and the
above system then becomes

$\displaystyle\frac{1}{\gamma}\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t}$
$\displaystyle=-\dfrac{\beta N_{0}}{\gamma}S_{\alpha}I_{\alpha},$
$\displaystyle\frac{1}{\gamma}\dfrac{\mathrm{d}I_{\alpha}}{\mathrm{d}t}$
$\displaystyle=-I_{\alpha}+\dfrac{\beta N_{0}}{\gamma}S_{\alpha}I_{\alpha},$
$\displaystyle\frac{1}{\gamma}\dfrac{\mathrm{d}R_{\alpha}}{\mathrm{d}t}$
$\displaystyle=I_{\alpha}.$

By such an approach we naturally obtain the time scale to be
$\dfrac{1}{\gamma}$ and, using the well known non dimensionalised quantity
$\mathcal{R}_{0}=\dfrac{\beta N_{0}}{\gamma}$, the fully scaled equations
finally get the form

$\displaystyle\dfrac{\mathrm{d}S_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=-\mathcal{R}_{0}S_{\alpha}I_{\alpha},$
$\displaystyle\dfrac{\mathrm{d}I_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=-I_{\alpha}+\mathcal{R}_{0}S_{\alpha}I_{\alpha},$
$\displaystyle\dfrac{\mathrm{d}R_{\alpha}}{\mathrm{d}t_{\alpha}}$
$\displaystyle=I_{\alpha}.$

Returning to our problem, from the basic mathematical analysis of (2.1) we
were able to sleekly generate the quantity $\varepsilon$ that characterises
both ($sQSSA$) and ($rQSSA$). Moreover, we naturally determined two pairs of
distinctive time scales, each pair of which is characteristic for one of the
aforementioned two assumptions.

(a)

(b)

(c)

Figure 11: For a good approximation of the solution of (2.1) for the case
where $\varepsilon\sim 1$, a sophisticated extrapolation technique is required
(work by the present authors in progress), than just a linear combination of
the approximations of the solution for ($sQSSA$) and ($rQSSA$).

We further obtained a good approximation of the solution in closed form, for
both the cases where $\varepsilon\to 0^{+}$ and $\varepsilon\to\infty$, which
we can communally write as

$\frac{\varepsilon}{1+\varepsilon}\times\left(\text{approximation for
}\varepsilon\to\infty\right)\,+\,\left(1-\frac{\varepsilon}{1+\varepsilon}\right)\times\left(\text{approximation
for }\varepsilon\to 0^{+}\right).$

We emphasise that the above linear combination is far from being a good
approximation of the solution for the case where $\varepsilon\sim 1$, as it is
illustrated in Figure 11. Such an approximation requires a much more
sophisticated extrapolation technique, the study of which lies beyond the
scope of the present work.

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