Theory of the reactant-stationary kinetics for zymogen activation coupled to an enzyme catalyzed reaction

A theoretical analysis is performed on the nonlinear ordinary diﬀerential equations that govern the dynamics of a coupled enzyme catalyzed reaction. The reaction consists of a primary non-observable zymogen activation reaction that it is coupled to an indicator (observable) reaction, where the product of the ﬁrst reaction is the enzyme of the indicator reaction. Both reactions are governed by the Michaelis–Menten reaction mechanism. Using singular perturbation methods, we derive asymptotic solutions that are valid under the quasi-steady-state and reactant-stationary assumptions. In particular, we obtain closed form solutions that are analogous to the Schnell–Mendoza equation for Michaelis–Menten type reactions. These closed-form solutions approximate the evolution of the observable reaction and provide the mathematical link necessary to measure the enzyme activity of the non-observable reaction. Conditions for the validity of the asymptotic solutions are also derived and demonstrate that these asymptotic expressions are applicable under the reactant-stationary kinetics. in coupled enzyme assays.


Introduction
Many enzyme catalyzed reactions that occur in physiological processes require an activation step, in which a precursor of a zymogen (inactive enzyme precursor or pro-enzyme) is converted to an active enzyme. This process, known generally as zymogen activation [1], is typically the first step in a cascade of coupled enzyme catalyzed reactions since the newly activated enzyme is then free to either (1) bind with a substrate to yield a product through an enzyme catalyzed reaction or (2) activate an additional enzyme [2]. The activation step of the zymogen is itself an enzyme catalyzed reaction, since the inactive enzyme precursor is activated by an active enzyme. The active enzyme can be generated by enzyme-catalyzed proteolosis or enzyme activation by phosphorylation [3]. For example, the digestive enzyme trypsin, which is the activate form of trypsinogen, is activated by the enzyme enterokinase; trypsin can then bind with trypsinogen to convert remaining trypsinogen into trypsin [2]. Likewise, plasminogen is activated by streptokinase to form plasmin (enzyme), which then binds with and degrades fibrin (subsrate) to break down clots in blood coagulation [4]. Regardless of the reactants, the preliminary zymogen activation step coupled with another secondary enzyme-catalyzed reaction can be expressed with the following reaction mechanism (1)-(2) in which the primary enzyme, E 1 , reacts with the zymogen E i 2 to form an intermediate complex C 1 following the Michaelis-Menten (MM) [5] reaction mechanism. The product of the primary reaction is thus the activated form of the secondary enzyme E 2 . In the secondary reaction the substrate (S) binds with the enzyme (E 2 ) which forms a complex (C 2 ) and synthesizes the product (P ) in a catalytic reaction step In the above chemical pathways, k 1 , k −1 , k 3 , k −3 are microscopic rate constants, and k 2 , k 4 are catalytic constants.
The mass action equations governing (1)-(2) are inherently nonlinear. However, much of the literature in which reactions of this form are analyzed mathe- 5 matically invokes the assumption of pseudo-first-order (PFO) kinetics [6,1,7,4,2]. Of course, the PFO kinetic models have the mathematical advantage of being linear; thus, closed-form solutions are easily obtained through the use of Laplace transforms. To date however, a nonlinear analysis of zymogen activation in a coupled enzyme assay has not been performed. 10 Another interesting aspect of coupled enzyme catalyzed reactions with a zymogen activation step (1)- (2) is the quantification of the catalytic conversion of zymogen in vitro. If the activation step (1) is not detectable experimentally (i.e., non-observable), the secondary reaction step (2) is selected to be an easily observable reaction in the enzyme kinetic assay. This is done with the goal of 15 measuring the enzyme activity of the non-observable reaction by analyzing the progress curves of the secondary observable reaction. In this case, the secondary reaction step (2) is known as the indicator reaction. Traditionally, coupled enzyme assays are designed so that the product of the non-observable reaction is a substrate for the secondary enzyme in the indicator reaction (see [8] for 20 specific applications). This type of assay is well-studied [9,10,11,12] and is known as a coupled (or auxiliary) enzyme assay or coupled sequential enzyme assay.
Zymogen activation coupled to an enzyme catalyzed reaction (1)-(2) occurs naturally in coagulation cascades [13]. As a distinct example, the activation of protein C (P C) by thrombin (T ) follows a reaction consistent with (1): where "AP C" denotes the activated form of P C. In the experimental assay, the activated enzyme APC then catalyzes a substrate (S). Assuming S is specific to AP C and does not bind with T , the secondary observable reaction follows the form of (2) Experimentally, the kinetics of the non-observable reaction is measured by decoupling the analysis of progress curves by adding excessive concentrations of 25 the primary enzyme, thus making the first reaction PFO [13]. However, it has been demonstrated that excessive concentrations of the initial enzyme E 1 is not sufficient to guarantee the validity of PFO kinetic models. What is necessary is that initial concentration of zymogem for (1) be much less than the Michaelis constant of the primary reaction [14]. Thus, from an experimental point of 30 view, it is difficult to ensure the validity of the PFO model when the Michaelis constant is unknown and consequently to be determined.
It is well-established that under appropriate experimental conditions the MM reaction mechanism will obey quasi-steady-state (QSS) kinetics, and the rate of substrate depletion for the reaction is described by the MM equatioṅ where s is the concentration of S, K M = (k − +k cat )/k is the Michaelis constant, 35 and V = k cat e 0 is limiting rate of the reaction (5), which is dependent on both the catalytic constant k cat and the initial concentration of E (the initial concentration of E is denoted as e 0 ). Note that the zymogen activation step in (1)-(2) is a single enzyme, single substrate reaction, where the substrate is effectively the zymogen.

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Of great interest to both theoreticians [15,16]  The inverse problem presents a unique challenge for both experimentalists 55 and theorists in coupled enzyme assays like (1)- (2). First, the parameters that govern the enzyme activity of the non-observable reaction must somehow be determined from the indicator reaction, since progress curves from a typical in vitro laboratory experiment can only be generated for the indicator reaction.
Second, a reduced model for the zymogen activation coupled to an enzyme 60 catalyzed reaction (1)-(2) must be developed. The reduced model should: (1) decrease the number of variables, and (2) lessen the number of parameters needed to describe the time course of the complete zymogen activation coupled to an enzyme catalyzed reaction reaction.

Goals of this paper
For the single enzyme, single substrate MM reaction mechanism (5) where ε is very small (i.e., ε 1), and is proportional to the ratio of the fast timescale to the slow timescale. Differential equations in the form of (7) are called singularly perturbed differential equations, and they are ubiquitous in mathematical chemistry [17,18,19] and biology [20]. Thus, central to deriving a reduced model for the zymogen activation coupled to an enzyme catalyzed 80 reaction (using slow manifold projection) is the estimation of the slow and fast timescales for the non-observable reaction. This is challenging for coupled reactions, since the time to completion of the indicator reaction can occur before, after, or at approximately the same time as the non-observable reaction. Furthermore, it is unlikely that the relative speeds and completion time of the 85 non-observable reaction will be known. Thus, there is a need derive a reduced model that is general enough so that its validity is certain regardless of which reaction is fastest. Finally, the most desirable reduced model will be one in which a closed form solution is obtainable so that the reduced model may be expressed as an explicit function of time. This will eliminate the need to gener-90 ate explicit progress curves for substrate depletion of the primary reaction since the time course of substrate is unknown in coupled enzyme assays.

Structure of this paper
As mentioned previously, the theoretical reduction analysis of zymogen activation reactions has been limited to PFO kinetics models. Such models have 95 limited validity in time course experiments [14], and the aim of this work is first and foremost to take a necessary "first step" in the nonlinear analysis of such reactions. First, we will introduce proper scaling techniques that can be employed in a general methodology to more complicated reaction. In Section 3, we will show how to estimate timescales based on these scaling methods, as well as how 100 to formulate reduced model from the analysis of these timescales (Section 4).
The reduced model is analogous to (7), and admits closed-form solutions in the form of a Schnell-Mendoza equation [21]. Conditions for the validity of the reduced model will be established, and timescale estimates will be derived. In addition, we will exploit the geometry of the mathematical structure [22,23] in 105 extreme situations when the speeds of the reactions are significantly disparate.
This will allow us to "simplify" the reduced model and obtain asymptotic solutions that are in some ways easier in form than both the general reduced model  Let us consider the zymogen activation coupled to an enzyme catalyzed reaction (1)-(2). In reaction (1), the zymogen E i 2 is effectively a substrate. To distinguish between substrates and enzymes in (1)-(2), we will change notation by replacing E i 2 with S 1 in (1), and S with S 2 in (2). Under this notation, applying the law of mass action to zymogen activation coupled to an enzyme catalyzed reaction yields seven rate equationṡ where lowercase letters represent concentrations of the corresponding uppercase species. Typically, laboratory enzyme assays present the following initial 115 conditions (e 1 , s 1 , c 1 , e 2 , s 2 , c 2 , p) | t=0 = e 0 1 , s 0 1 , 0, 0, s 0 2 , 0, 0 .
By examining the system of rate equations (8), the zymogen activation coupled to an enzyme catalyzed reaction obeys three conservation laws: Mathematically speaking, the solution trajectory to (8) must lie on the intersection of the hyperplanes defined in (10), which means the original sevendimensional problem can be reduced to a four-dimensional problem. Using (10a) and (10b) to decouple the enzyme concentrations, the redundancies in the system (8) are eliminated to yielḋ where e 1 (t), e 2 (t) and p(t) are readily calculated once s 1 (t), c 1 (t), s 2 (t) and c 2 (t) are known.

Rate expressions for the non-observable enzyme catalyzed reaction
The rate equations (11a)-(11b) are uncoupled from (11c)-(11d). These rate 120 equations have the same structure to those of the single substrate, single enzyme reaction following the MM mechanism. Therefore, it is possible to derive rate equations to model the zymogen activation coupled to an enzyme catalyzed reaction, and estimate its kinetic parameters using the general theory of the reactant-stationary assumption (RSA, [24]). The rate equations for the non- is assumed to be in a QSSċ The timescale t c1 is the time associated with the initial transient buildup of c 1 In the above equation, K M1 = (k −1 + k 2 )/k 1 is the Michaelis constant for the zymogen activation step (1). The quasi-steady-state assumption (QSSA, 12), in combination with (11a)-(11b), leads to the derivation of the well-known rate expressions In (14b), V 1 = k 2 e 0 1 is the limiting rate of the zymogen reaction. Note that 135 the mass action equations (11a)-(11b) are reduced to a differential-algebraic equation systems with one single differential equation for s 1 in (14a)-(14b).
Since equations (14a) and (14b) are only valid after the initial transient, t c1 , it is necessary to define a boundary condition for s 1 at t = t c1 . This is equivalent to the initial experimental condition for the initial rate or time 140 course experiments. To find this condition, it can be assumed that there is a negligible decrease in s 1 during the initial transient. This is known as the RSA, and is expressed as The RSA provides an initial condition for (11a) under the variable transfor- From the perspective of asymptotic theory, Schnell and Mendoza [21] have provided a piecewise solution for the MM reaction in terms of a fast transient solution for s 1 , valid for t ≤ t c1 , as well as a QSS solution for s 1 , valid for t > t c1 : From the earlier work of Segel [25], we have a fast solution, valid when t ≤ t c1 , for the complex c 1 , as well as a QSS solution which is valid for t > t c1 Collectively, equations (17a) and (18b) constitute an asymptotic solution that serves as an accurate approximation to the full time course of (11), provided the appropriate qualifiers (i.e, the RSA and the QSSA) are obeyed.
In addition to the timescale t c1 , which quantifies the length of the initial fast transient (build-up of c 1 ), the time it takes for the majority of the substrate s 1 155 to deplete is given by t s1 . Although there are several methods for estimating the significant timescales of chemical reactions [26], we employ the heuristic method proposed by Segel [25], and approximate the depletion time to be effectively the total depletion of s 1 (the total depletion is s 0 1 ) divided by the maximum rate of substrate of depletion after t c1 : Generally speaking, t s1 is a reasonable measure of how long it takes for the non-observable reaction to complete. is assumed to be in a QSS an the difference between the rate of C 2 depletion is approximately equal to the rate C 2 formation. It was originally proposed that In other words, it was assumed that the c 1 -nullcline should be considered a 185 good approximation to the slow manifold M ε if the timescale accounting for the build-up of c 1 was small compared to the timescale accounting for the duration of the reaction.
As for the validity of the RSA, Segel [27] proposed that one could assume little change in s 1 (an almost straight phase space trajectory towards the slow 190 manifold) if the depletion of s 1 over the timescale t c1 is minimal: Since |ṡ 1 | ≤ s 0 1 e 0 1 , the strict inequality given in (21) translates to Through scaling analysis, Segel [25] went on to show that the RSA determines single-handedly the validity of the asymptotic solutions (17) and (18).
Introducing the dimensionless variablesŝ 1 = s 1 /s 0 1 andĉ 1 = c 1 /c 1 , Segel and Slemrod [27] demonstrated that, with respect to the dimensionless timescale where κ 1 = k −1 /k 2 and ε = e 0 1 /(k M1 + s 0 1 ). In contrast, under the timescale T = t/t s1 , (11a)-(11a) become: Thus, it is apparent from the dimensionless equations (23) then not only will the RSA hold, but the QSSA (which assumes that the c 1nullcline is a good approximation to M ε ) also holds. In fact the RSA (i.e., 195 ε 1) is more restrictive than separation of timescales. After some algebraic calculations, the separation of timescales (t c1 /t s1 1) can be written as: where K S1 = k −1 /k 1 , and K 1 = k 2 /k 1 . For the RSA to be valid, the condition must be satisfied, which is more stringent than condition (25) of s 2 and c 2 . We will start by trying to estimate a slow timescale for the indicator reaction. An accurate slow timescale should give us a reasonable estimation of the completion time for the indicator reaction. In the case of the zymogen activation coupled to an enzyme catalyzed reaction, the completion of the indicator reaction can be faster, as fast, or slower than the non-observable 210 reaction. For the non-observable reaction, the slow timescale is expressed in terms of the initial quantities s 0 1 and e 0 1 , and the Michaelis constant K M1 : The quantity e 0 1 is the total amount of enzyme for the non-observable reaction. The construction of a homologous slow timescale for the indicator reaction is problematic in that the total amount of available enzyme e A 2 , is a time-dependent quantity. If we start by assuming the QSSA is valid, then the mass action equations reduce to where V 2 (t) ≡ k 4 e A 2 (t). The general solution to (29b) is given in terms of a Lambert-W function where "s" has been employed as a dummy variable and σ 2 ≡ s 0 2 /K M2 . We will employ a mean-field approach to derive a slow (depletion) timescale for the indicator reaction. Let us first assume that we know the slow timescale for the 220 indicator reaction, and denote this timescale as T s2 . Then, the mean available enzyme over the time course of the indicator reaction, which we will denote as e A 2 , is given by If the completion of the indicator reaction occurs long before the completion of the non-observable reaction, then we expect that e A 2 s 0 1 . In contrast, if 225 the completion of the indicator reaction occurs long after the completion of the non-observable reaction, then we expect e A 2 ≈ s 0 1 . In any case, we can define the slow timescale as which should yield a reasonable estimate for the slow timescale if the depletion of s 2 is influenced by a slow manifold. Note that K M2 = (k −3 + k 4 )/k 3 is the 230 Michaelis constant of the indicator reaction.
Next, we want to scale the mass action equations that model the indicator reaction with respect to the quantities T = t/t, s 0 2 , and max(e A 2 ), where max(e A 2 ) is the maximum amount of e A 2 over the course of the indicator reaction: Utilizing max(e A 2 ) as an upper bound on the available enzyme dictates a natural 235 scaling of c 2 Scaling the mass action equations with respect to the following dimensionless variables,s wheret denotes an arbitrary timescale. Substitution of these quantities into the mass action equation yields In the above expressions, the dimensionless quantities σ 2 , κ 2 and α are: The parameter λ, defined as is unique in that if it is sufficiently small, then it mathematically characterizes the indicator reaction as a singularly perturbed differential equation for which model reduction is possible through means of projecting onto the slow manifold "M λ ".
where δ S is the ratio of the substrate depletion timescales, δ S = T s2 /t s1 , and T = t/t s1 . Based on the scaling given in (39a) and (39b, we will derive an estimate for T s2 as well as solutions for three particular cases, which are defined by the scale of δ S : (i) Case 1: the indicator reaction is faster than the non-250 observable reaction (δ S 1), Case 2: the indicator reaction is roughly the same speed as the non-observable reaction (δ S ≈ 1), and Case 3: the indicator reaction is much slower than the non-observable reaction (δ S 1).

Case 1: The indicator reaction is faster than the non-observable reaction
If the indicator reaction is fast, and δ S 1, then the dominant slow timescale is t s1 , and thus the completion of the non-observable reaction will occur long after the completion of the indicator reaction. To start the analysis, we will rescale the mass action equations that govern the non-observable reaction with respect toT = t/T s2 : By inspection of (40a), if δ S 1, then s 1 will be a slow variable over the T s2 timescale, and thus we will expect s 1 to be essentially constant over the time course of the indicator reaction. In addition, let us assume that T s2 t c1 , in which case c 1 will be on the order of its maximum value on the T s2 timescale.
Combining these observations leads to the approximation for the non-observable reaction over the timescale T s2 . Equations (41a) and (41b) seem to suggest that e A 2 1 over the T s2 timescale. Furthermore, since the changes in s 1 and c 1 are comparatively minimal when t c1 ≤ t ≤ T s2 , the production of e A 2 is effectively constant over the T s2 timescalė Integration of (42) yields the following approximation of e A 2 on the T s2 timescale where u is a dummy variable. The approximate average value e A 2 on T s2 is easily obtainable through straightforward integration and inserting (44) into (32) yields the following estimate for T s2 : We can write (45) in a slightly more convenient form. Defining the limiting slow allows us to express T s2 as Note that V 2 = k 4 s 0 1 is defined as the limiting rate of the indicator reaction. T * s2 should provide an accurate estimate for total completion time of the indicator reaction as long as the non-observable reaction is comparatively slow.

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For a generic (and linear) dynamical system of the forṁ the depletion or characteristic timescale is 1/a, and thus we look for a timescale that is indicative of the time it takes for the initial quantity (i.e., x 0 in the context of (48)) to deplete to an amount that is less than or equal to x 0 /e.
Following suit from the linear theory, we will consider the timescale T * s2 to be a 275 sufficient depletion timescale as long as Numerical solutions of the mass action equations confirm the validity of the timescale T * s2 when the indicator reaction is much faster than the non-observable reaction provided t c1 T * s2 (see Figures 2a and 2b). Next, we develop an asymptotic solution to the mass action equations that 280 will be valid when: (1) T * s2 is an accurate and precise depletion timescale, (2) the concentrations s 1 and c 1 remain on the order of their maximum values (s 0 1 and εs 0 1 respectively) for the duration of the indicator reaction, and (3) the fast timescale t c1 is negligibly short. To begin, let us assume that the initial concentration s 0 2 is large enough so that in which case we can assume λ 1. Then, from Tikhonov's theorem, and due to the existence of the slow manifold M λ , we have as a leading order approximation. Insertion of this approximation into the mass Substitution of e A 2 ≈ t into (52) gives us and provides an accurate approximation to the mass action model (see Figures 3a and 3b).
we can formulate a nonlinear algebraic equation that will allow us to compute an estimate for the depletion timescale when the reactions are equivalent in speed. First, and thus we see that T s2 should satisfy 305 Second, under the RSA, the concentration c 1 is expressible (algebraically) in terms of s 1 , and therefore where ∆s 1 = s 0 1 − s 1 (the timescale t c1 has been assumed to be negligibly small and hence left out of the integrand, although it is straightforward to include this term). Third, the definite integral on the right hand side of (58) is straight-310 forward to compute analytically; evaluating it will yield a nonlinear equation in terms of the variable T s2 , and the solution to (57) can be approximated numerically. Using the average e A 2 provides an accurate estimate of the slow (depletion) timescale (see Figure 4).
From a practical point of view, the utility in estimating T s2 numerically is rather minimal. The objective here will be to construct a criteria from which a reduced model can be be extracted from the mass action equations that will be valid without any a priori knowledge of the intrinsic timescales of the indicator reaction (or the non-observable reaction). To do this, let us first revisit the generic scaling introduced in the previous section: Bearing in mind the assumption δ S ≈ 1, it is sufficient (but not necessary) to 315 bound λ in order to assemble a dynamical model that can be reduced (asymptotically) through slow manifold projection. The upper bound on λ, which we denote as λ max , is The parameter λ max is the natural small parameter when the indicator is very slow. Furthermore, if the non-observable reaction completes very quickly rel-320 ative to the non-observable reaction, and δ S 1, then the average available enzyme should be on the order of s 0 1 : Thus, if s 0 2 s 0 1 , then the approximatioṅ will be valid regardless of the relative speeds of the reactions when λ max 1.
Furthermore, (62) admits a closed-form solution using separation of variables 325 that consists of composite Lambert-W functions (we do not present this expression here, although we remark that it is straightforward, albeit somewhat tedious to derive). Under the RSA, we obtaiṅ as the final form of our reduced differential equation forṡ 2 .

Case 3:
The indicator reaction is much slower than the non-observable 330 reaction (δ S 1) We now consider the case when δ S 1, and the completion of the nonobservable occurs much sooner than the completion of the indicator reaction.
As mentioned in the previous subsection, a very slow indicator reaction suggests that s 2 will be slow over the timescale t s1 . Consequently, we can approximate 335 s 2 as Furthermore, because the non-observable reaction has effectively completed when t = t s1 , we can approximate ∆s 1 = s 0 1 when t ≥ t s1 , in which casė The validity of the approximate solution (64) can be established by the mathematical formulation of the RSA for the indicator reaction. If s 2 ≈ s 0 2 over the interval [0, t s1 ], then max t≤ts 1 The inequality given in (67) translates to with maxṡ 2 = k 3 s 0 1 s 0 2 . In the case of a slow indicator reaction, we expect that T s2 = t * s2 . Thus, we have a RSA that is pertinent to the indicator reaction and establishes a region of validity for the solution to the mass action equations during the initial build-up of c 2 when t ≤ t s1 . Equation (69) is analogous to the term used to measure the strength of fully competitive enzyme reactions with 350 alternative substrates [29,30]. Numerical simulations (see Figure 5) confirm the validity of t * s2 and (65).

Interpretation of fast timescales of the indicator reaction
Up until this point, we have not mentioned the equivalent of a fast timescale that is pertinent to the indicator reaction. In the case of the non-observable reaction, the fast timescale corresponds to the time it takes the reaction to reach QSS. However, based on our scaling analysis, we have demonstrated that a QSSA can be imposed over the T s2 timescale reaction as long as λ 1.
At first glance, it would seem that the kinetics of the indicator reaction omit the influence of a fast timescale. This is however false. To derive the fast timescale, we will assume that initial conditions are not experimental, and that the indicator reaction is equipped with a non-trivial amount of complex c 0 2 at the start of the reaction. Furthermore, we will assume that the substrate, the fast timescale as t c2 , the mass action equation can be linearized: The solution to the linear equation (70) is given by Duhamel's Principle from which the naturally occurring characteristic timescale is 1/µ: Since typical experimental initial conditions start on the c 2 -nullcline, we turn to scaling to provide a biochemical interpretation of the timescale t c2 . Defining T * = t/t c2 , we obtain: We see from the scaling that t c2 defines a stagnation timescale when experimental initial conditions are prescribed. If the timescale t c2 is short, then we expect the indicator reaction to be effectively stationary over t c2 . This is because s 2 scales as a slow variable over t c2 and the phase space trajectory should stay near the c 2 -nullcline over short timescales, and thusċ 2 ≈ 0 when t ≤ t c2 . Thus,

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if t c2 is small (i.e., t c2 min{t s2 , t s1 }), then the fast timescale of the indicator reaction defines translates to a scale over which the indicator reaction exhibits a "slow response". In fact, any timescale t * such that defines a "fast, stagnation timescale".
The relationship between λ, t c2 and T s2 is now evident. The ratio of fast and 365 slow timescales is bounded above by λ t c2 The strict inequality follows from the fact that t c2 whereλ is given byλ Furthermore, since e A 2 ≤ max(e A 2 ), we have that from which (75) follows.
To explore the relationship between the QSSA and the RSA, we note that the parameter λ max is easily derived using Segel's heuristic approach [25]: Since it is clear thatλ it follows that the RSA (i.e., λ max 1) ensures separation of fast and slow timescales. Consequently, the RSA for the indicator reaction implies the QSSA 375 and is thus a universal qualifier for the validity of the reduced model over the timescale T s2 .
In addition, it is important to note that c 2 may or may not accumulate in QSS. We rescale the mass action equation for c 2 with respect to T = t/t s1 to determine how c 2 accumulates: From the previous section we have that and consequently we see that c 2 accumulates in QSS when t c2 /t s1 1. We remark this is equivalent to demanding that maxė A 2 · t c2 s 0 1 , where maxė A 2 ≡ k 2 εs 0 1 .

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Under the QSSA, enzyme catalyzed reactions usually express a lag time. The lag time is normally defined as the time is takes for the rate of product generation to reach its maximum (steady-state) value. This coincides with the time it takes for c 2 to reach its maximum value, and is straightforward to calculate under the limiting circumstances. In this section, we calculate the lag time under the 390 assumption that the indicator reaction is extremely fast and extremely slow.

Estimation of the lag time for fast indicator reactions
Let us start by considering the case when the indicator reaction is very fast for which s 2 is given by Next, notice that under the QSSA we have If we differentiate both sides of (85), then we see thatċ 2 vanishes whens 2 vanishes: Inserting (84) into the right hand side of (86) and setting the left hand side to zero yields an expression for t: The timescale t c2 is identically the lag time when the indicating reaction is fast and σ 2 1 and the indicator reaction is fast (see Figure 6).

Estimation of the lag time for slow indicator reactions
For slow indicator reactions will can employ the RSA max t≤ts 1 and thus we obtain Furthermore, we will assume that max(c 2 ) is given by when the indicator reaction is slow. The timescale t s1 will serve as a good approximation to the lag time when σ 1 very large. However, when σ 1 is small, the asymptotic solution to the MM equation reduces to and consequently the timescale t s1 is characteristic, which means roughly 1/3 410 of s 0 1 still needs to be converted to product when t = t s1 . Thus, we need an estimate for the time it takes for the non-observable reaction to complete when σ 1 is small. To do this we set and solve for t. This yields that is a much better estimate of the lag time when σ 1 is small. A similar 415 analysis can be carried out when σ 1 is of order unity, but we will not dive into the detail of this calculation here. Numerical results confirm the lag time estimates t s1 and t * s1 when the indicator reaction is slow (see Figures 7a-7b).

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The primary contribution of this paper is to introduce methods for the appropriate scaling and timescale estimates of coupled enzyme catalyzed reactions.
As a case study, we present the analysis of a zymogen activation coupled to an enzyme catalyzed reaction. The identification of specific parameters through scaling has yielded necessary and sufficient conditions for the QSSA, whereas the slow timescales t s1 and T s2 : When these parameters are small, and the timescales t c2 and t s1 are adequately separated, the indicator reaction can be assumed to be in a QSS for the duration of the reaction (i.e, for t ≥ 0). There is a twofold reasoning to this assumption.
First, if λ max 1, then λ ≡ max(e A 2 ) K M2 + s 0 2 1, and model reduction from slow manifold projection is valid regardless of which reaction finishes first (non-observable or indicator). Since it is not generally possible to determine which reaction is faster in the typical experiment a priori, the condition that λ max 1 serves as a sufficient qualifier to ensure the validity of the reduced model for the reaction rate of depletion of the indicator substrate: Second, as long as t c2 t s1 , a QSSA will effectively hold for all time since experimental initial conditions lie on the c 2 -nullcline. From the theory of singular perturbations, the slow manifold, M λ , is well-approximated by the c 2 -nullcline when λ max 1. Because experimental initial conditions lie on the c 2 -nullcline, the phase space trajectory is already extremely close to the slow manifold, and 440 therefore there is no need for an initial fast transient in order for the trajectory to reach the slow manifold. As we have pointed out, the slow manifold is a geometrical representation of the steady-state rate equation for the reaction.
Note that this is very different from the non-observable reaction, since a fast transient (the duration of the fast transient is approximated by the timescale 445 t c1 ) must elapse before the QSSA is justifiable.
In addition, simple asymptotic solutions to the mass action equations were derived that are valid when the indicator reaction is very fast or very slow in comparison to the non-observable reaction. If the indicator reaction is fast, then the time course of the indicator substrate s 2 is accurately model by where W denotes the Lambert-W function. In contrast, if the indicator reaction is very slow, then the time course of s 2 can be modeled by Note that the above two expressions are analogous to the Schnell-Mendoza equation [21].
It should be pointed out that the condition λ max 1, which can be ensured by requiring an excess of the initial amount of substrate s 2 (i.e., requiring that 450 s 0 2 be large enough so that s 0 1 s 0 2 ), is sufficient but not necessary for the validity of the reduced model presented in (63). In general, it is desirable that s 0 2 be much larger than the maximum of amount of e A 2 over the timescale of the indicator reaction. If the indicator reaction is fast, then the maximum amount of available enzyme, max(e A 2 ), will be small, and thus the requirement that In this simulation k 3 = 1, k 4 = 100, k −3 = 10, s 0 2 = 1, and k 1 = 1, k 2 = 1, k −1 = 1, e 0 1 = 1 and s 0 1 = 100. s 0 1 /s 0 2 = 100 and λ max ≈ 1. However, max e A 2 ≈ 1.543 and therefore λ ≈ 0.014 1.
Finally, three reduced models have been derived that can be utilized in the analysis of the inverse problem. Our analysis seems to suggest that a fast indicator reaction is the most beneficial case for parameter estimation. Under this circumstance, two expressions can be simultaneously utilized to estimate the four unknown parameters V 1 , V 2 , K M1 , and K M2 . Additionally, previous studies [6,1,7,4,2] that have analyzed this reaction mechanism in terms of PFO kinetics can be bridged together with the 460 work here. Since we have treated the first reaction as independent of the second, an additional model could be obtained by imposing PFO kinetics on the non-observable reaction. However, while one can construct many models as a result of our nonlinear analysis, the full understanding of the inverse problem is beyond the scope of this paper. We hope to theoretically investigate the scope 465 of parameter estimation in coupled assays in subsequent future work.