The Michaelis–Menten Reaction at Low Substrate Concentrations: Pseudo-First-Order Kinetics and Conditions for Timescale Separation

We demonstrate that the Michaelis–Menten reaction mechanism can be accurately approximated by a linear system when the initial substrate concentration is low. This leads to pseudo-first-order kinetics, simplifying mathematical calculations and experimental analysis. Our proof utilizes a monotonicity property of the system and Kamke’s comparison theorem. This linear approximation yields a closed-form solution, enabling accurate modeling and estimation of reaction rate constants even without timescale separation. Building on prior work, we establish that the sufficient condition for the validity of this approximation is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \ll K$$\end{document}s0≪K, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=k_2/k_1$$\end{document}K=k2/k1 is the Van Slyke–Cullen constant. This condition is independent of the initial enzyme concentration. Further, we investigate timescale separation within the linear system, identifying necessary and sufficient conditions and deriving the corresponding reduced one-dimensional equations.


Introduction and overview of results
The irreversible Michaelis-Menten reaction mechanism describes the mechanism of action of a fundamental reaction for biochemistry.Its governing equations are: with positive parameters e 0 , s 0 , k 1 , k −1 , k 2 , and typical initial conditions s(0) = s 0 , c(0) = 0. Understanding the behavior of this system continues to be crucial for both theoretical and experimental purposes, such as identifying the rate constants k 1 , k −1 and k 2 .However, the Michaelis-Menten system (2) cannot be solved via elementary functions, hence approximate simplifications have been proposed since the early 1900's.
In 1913, Michaelis and Menten [12] introduced the partial equilibrium assumption for enzyme-substrate complex formation, effectively requiring a small value for the catalytic rate constant k 2 .Later, for systems with low initial enzyme concentration (e 0 ), Briggs and Haldane [3] derived the now-standard quasi-steadystate reduction (see Segel and Slemrod [16] for a systematic analysis).Both these approaches yield an asymptotic reduction to a one-dimensional equation.A mathematical justification is obtained from singular perturbation theory.Indeed, most literature on the Michaelis-Menten system focuses on dimensionality reduction, with extensive discussions on parameter combinations ensuring this outcome.
We consider a different scenario: low initial substrate concentration (s 0 ).This presents a distinct case from both the low-enzyme and partial-equilibrium scenarios, which yield one-dimensional reductions in appropriate parameter ranges.Instead, with low substrate, we obtain a different simplification: This linear Michaelis-Menten system (2) was first studied by Kasserra and Laidler [11], who proposed the condition of excess initial enzyme concentration (e 0 ≫ s 0 ) for the validity of the linearization of (1) to (2).This aligns with the concept of pseudo-first-order kinetics in chemistry, where the concentration of one reactant is so abundant that it remains essentially constant [17].
Pettersson [14] further investigated the linearization, adding the assumption that any complex concentration accumulated during the transient period remains too small to significantly impact enzyme or product concentrations.Later, Schnell and Mendoza [15] refined the validity condition s 0 ≪ K M (where K M = (k −1 + k 2 )/k 1 is the Michaelis constant) to be sufficient for the linearization (alias dictus the application of pseudo-first-order kinetics to the Michaelis-Menten reaction).However, Pedersen and Bersani [13] found this condition overly conservative.They introduced the total substrate concentration s = s + c (used in the total quasi-steady state approximation [2]) and proposed the condition s 0 ≪ K M + e 0 , reconciling the Schnell-Mendoza and Kasserra-Laidler conditions.
In this paper, we demonstrate that the solutions of the Michaelis-Menten system (1) admit a global approximation by solutions of the linear differential equation (2) whenever the intial substrate concentration is small.Our proof utilizes Kamke's comparison theorem for cooperative differential equations. 1This theorem allows us to establish upper and lower estimates for the solution components of (1) via suitable linear systems.Importantly, these bounds are valid whenever s 0 < K, where K = k 2 /k 1 is the Van Slyke-Cullen constant, and the estimates are tight whenever s 0 ≪ K.This effectively serves as sufficient condition for the validity of the linear approximation Michaelis-Menten system (2).We further prove that solutions of the original Michaelis-Menten system (1) converge to solutions of the linear system (2) as s 0 → 0, uniformly for all positive times.Moreover, the asymptotic rate of convergence is of order s 2 0 .
Additionally, we explore the problem of of timescale separation within the linear Michaelis-Menten system (2).While a linear approximation simplifies the system, it does not guarantee a clear separation of timescales.In fact, there are cases where no universal accurate one-dimensional approximation exists.Since both eigenvalues of the matrix in (2) are real and negative, the slower timescale will dominate the solution's behavior.True timescale separation occurs if and only if these eigenvalues differ significantly in magnitude.Importantly, we find that this separation is not always guaranteed.There exist parameter combinations where a one-dimensional approximation would lack sufficient accuracy.Building upon results from [5], we establish necessary and sufficient conditions for timescale separation and derive the corresponding reduced one-dimensional equations.

Low substrate: Approximation by a linear system
To set the stage, we recall some facts from the theory of cooperative differential equation systems.For a comprehensive account of the theory, we refer to the H.L. Smith's monograph [18,Chapter 3] on monotone dynamical systems.Consider the standard ordering, here denoted by ≺, on R n , given by be given, with f continuously differentiable.We denote by F (t, y) the solution of the initial value problem (3) with x(0) = y, and call F the local flow of the system.Given a convex subset D ⊆ U , we say that (3) is cooperative on D if all non-diagonal elements of the Jacobian are nonnegative; symbolically As shown by M.W. Hirsch ([8, and subsequent papers]) and other researchers (see [18, and the references therein]), cooperative systems have special qualitative features.Qualitative properties of certain monotone chemical reaction networks were investigated by De Leenheer et al. [4].In the present paper, we consider a different feature of such systems.Our interest lies in the following simplified version of Kamke's comparison theorem [9]. 2roposition 1.Let D ⊆ U be a convex positively invariant set for (3), and assume that this system is cooperative on D. Moreover let ẋ = g(x) be defined on U , with continuously differentiable right hand side, and local flow G.
• If f (x) ≺ g(x) for all x ∈ D, and y ≺ z, then F (t, y) ≺ G(t, z) for all t ≥ 0 such that both solutions exist.
• If g(x) ≺ f (x) for all x ∈ D, and z ≺ y, then G(t, z) ≺ F (t, y) for all t ≥ 0 such that both solutions exist.
2.1 Application to the Michaelis-Menten systems (1) and (2) For the following discussions, we recall the relevant derived parameters where K S is the complex equilibrium constant, K M is the Michaelis constant, and K is the van Slyke-Cullen constant.
The proof of the following facts is obvious.
The off-diagonal entries are ≥ 0 in the physically relevant subset of the phase space, and D is positively invariant and convex.Thus system (1) is cooperative in D.
hence by Kamke's comparison theorem the solution of the linear system with matrix and initial value in D provides component-wise lower estimates for the solution of (1) with the same initial value.
hence by Kamke's comparison theorem the solution of the linear system with matrix and initial value in D provides component-wise upper estimates for the solution of (1) with the same initial value.
Remark 1.The upper estimates become useless in the case k 1 s 0 > k 2 , because then one eigenvalue of H becomes positive.For viable upper and lower bounds one requires that s The right hand sides of both linear comparison systems from parts (b) and (c) of the Lemma converge to as s 0 → 0. We will use this observation to show that a solution of (1) with initial value in D converges to the solution of (2) with the same initial value on the time interval [0, ∞), as s 0 → 0.

Comparison Michaelis-Menten systems (1) and (2)
Both G and H, as well as Df (0), are of the type with all entries > 0, and γ > β (to ensure usable estimates).The trace equals −(α + γ), the determinant equals α(γ − β), and the discriminant is The eigenvalues are The following is now obtained in a straightforward (if slightly tedious) manner.We specialize the initial value to the typical case, with no complex present at t = 0.
Lemma 2. The solution of the linear differential equation with matrix (6) and initial value s 0 0 is equal to This enables us to write down explicitly the solution of (2), as well as upper and lower estimates, in terms of familiar constants.We use Lemma 2 The solution to the Michaelis-Menten system (1) converges to the solution of the linear Michaelis-Menten system (2) as s 0 → 0. In all panels, the solid black curve is the numerical solution to the Michaelis-Menten system (1).The thick yellow curve is the numerical solution to linear Michaelis-Menten system (2).The dashed/dotted curve is the numerical solution to the linear system defined by matrix G.The dotted curve is the numerical solution to the linear system defined by matrix H.All numerical simulations where carried out with the following parameters (in arbitrary units): In all panels, the solutions have been numerically-integrated over the domain t ∈ [0, T ], where T is selected to be long enough to ensure that the long-time dynamics are sufficiently captured.For illustrative purposes, the horizontal axis (in all four panels) has been scaled by T so that the scaled time, t/T , assumes values in the unit interval: Observe that the solution components of (2) become increasingly accurate approximations to the solution components of (1) as s 0 decreases.
as the solution of (2), after some simplifications.
Proposition 2. To summarize: (a) The solution s * c * of the linear differential equation (2) with initial value s 0 0 is given by (7).
(b) The solution s low c low of the linear differential equation with matrix G and initial value s 0 0 is obtained from replacing K M by K M + s 0 in (7).
(c) The solution s up c up of the linear differential equation with matrix H and initial value s 0 0 is obtained from replacing K S by K S + s 0 in (7).
(d) Given that s 0 < K, for all t ≥ 0 one has the inequalities denotes the solution of (1) with initial value s 0 0 .
(e) Let M be a compact subset of the open positive orthant R 4 >0 , abbreviate π := (e 0 , k 1 , k −1 , k 2 ) tr and K * := min{k 2 /k 1 ; π ∈ M}.Then, there exists a dimensional constant C (with dimension concentration −1 ), depending only on M, such that for all π ∈ M and for all s 0 with 0 < s 0 ≤ K * /2 the estimates hold for all t ∈ [0, ∞).Thus, informally speaking, the approximation errors of s by s * , and of c by c * , are of order s 2 0 .
Proof.The first three items follow by straightforward calculations.As for part (d), the first column of ( 8) is just a restatement of Lemma 1, while the second follows from the observation that (mutatis mutandis) the right-hand side of (2) also obeys the estimates in parts (b) and (c) of this Lemma.
There remains part (e).In view of part (d) it suffices to show that for some constant and, in turn, it suffices to show such estimates for both s up − s * , s * − s low , and c up − c * , c * − c low .We will outline the relevant steps, pars pro toto, for the upper estimates.By (7) and part (c), one may write with the B i and It suffices to show that the maximum norm ∥ • ∥) From compactness of [0, K * /2] × M, continuous differentiability and the explicit form of A i and B i one obtains estimates, with the parameters and variables in M: • There is B * > 0 so that ∥B i (s 0 , π)∥ ≤ B * .
• By the mean value theorem there exists σ between 0 and s 0 so that So, with the constant A * * satisfying A * * ≥ | ∂A i ∂s 0 | for all arguments in M one gets and this proves the assertion.
For illustrative purposes, a numerical example is given in Figure 1.
Remark 2. Our result should not be mistaken for the familiar fact that in the vicinity of the stationary point 0 the Michaelis-Menten system (1) is (smoothly) equivalent to its linearization (2); see, e.g.Sternberg [19,Theorem 2].The point of the Proposition is that the estimate, and the convergence statement, hold globally.Remark 3. Some readers may prefer a dimensionless parameter (and symbols like ≪) to gauge the accuracy of the approximation.In view of Lemma 1 and equation (5) it seems natural to choose ε := s 0 /K S < s 0 /K M , which estimates the convergence of G and H to Df (0), and set up the criterion s 0 /K S ≪ 1.But still the limit s 0 → 0 has to be considered.Remark 4. Since explicit expressions are obtainable from (7) for the A i and B i in the proof of part (e), one could refine it to obtain quantitative estimates.This will not be pursued further here.

Timescales for the linear Michaelis-Menten system (2)
In contrast to familiar reduction scenarios for the Michaelis-Menten system (1), low initial substrate linearization (2) does not automatically imply a separation of timescales.Timescale separation depends on further conditions that we discuss next.
Recall the constants introduced in (4) and note In this notation, the eigenvalues of the matrix in the linearized system (2) are This matrix is the Jacobian at the stationary point 0, thus it is possible to use the results from earlier work [5].
Remark 5. Since timescales are inverse absolute eigenvalues, by [5, subsection 3.3], (9) shows: (a) The timescales are about equal whenever which is the case, notably, when K S ≈ 0 (so K M ≈ K) and K M ≈ e 0 .
(b) On the other hand, one sees from (9) that a significant timescale separation exists whenever (loosely speaking, employing a widely used symbol) Notably this is the case when e 0 /K M is small, or when K/K M is small.Proposition 3. We take a closer look at the scenario with significant timescale separation, stating the results in a somewhat loose language.
(a) Given significant timescale separation as in (10), the slow eigenvalue is approximated by and the reduced equation is d dt (b) Given significant timescale separation as in (10), the eigenspace for the slow eigenvalue is asymptotic to the subspace spanned by Proof.Given (10) one has by Taylor approximation, from which part (a) follows.
Moreover, according to (7), the direction of the slow eigenspace is given by By the same token, we have the approximation which shows part (b).Remark 6.One should perhaps emphasize that Proposition 3 indeed describes a singular perturbation reduction when λ 1 → 0. This can be verified from the general reduction formula in Goeke and Walcher [7] (note that the Michaelis-Menten system (2) is not in standard form with separated slow and fast variables), with elementary computations.
Finally we note that Proposition 3 is applicable to the discussion of the linearized total quasi-steadystate approximation.Details are discussed in [6].

Conclusion
Our comprehensive analysis of the Michaelis-Menten system under low initial substrate concentration addresses an important gap in the literature.We obtain the sufficient3 condition, s 0 ≪ K for the global approximation of the Michaelis-Menten system (1) by the linear Michaelis-Menten system (2).This simplification, known as the pseudo-first-order approximation, is widely used in transient kinetics experiments to measure enzymatic reaction rates.Moreover, given the biochemical significance of this reaction mechanism and the renewed interest in the application of the total quasi-steady state approximation in pharmacokinetics experiments under low initial substrate concentration [1,20], the linearization analysis holds clear practical relevance.
From a mathematical perspective, it is intriguing to observe a simplification that arises independently of timescale separation or invariant manifolds.The natural next step is to explore this approach with other familiar enzyme reaction mechanism (e.g., those involving competitive and uncompetitive inhibition, or cooperativity [see, for example, Keener and Sneyd [10]]).However, a direct application of Kamke's comparison theorem is not always feasible.Enzyme catalyzed reaction mechanism with competitive or uncompetitive inhibition do not yield cooperative differential equations, and standard cooperative networks do so only within specific parameter ranges.Investigating these systems under low initial substrate conditions will require the development of further mathematical tools.

Figure 1 :
Figure 1: Illustration of Proposition 2.The solution to the Michaelis-Menten system (1) converges to the solution of the linear Michaelis-Menten system (2) as s 0 → 0. In all panels, the solid black curve is the numerical solution to the Michaelis-Menten system (1).The thick yellow curve is the numerical solution to linear Michaelis-Menten system (2).The dashed/dotted curve is the numerical solution to the linear system defined by matrix G.The dotted curve is the numerical solution to the linear system defined by matrix H.All numerical simulations where carried out with the following parameters (in arbitrary units):k 1 = k 2 = k −1 = e 0 = 1.In all panels, the solutions have been numerically-integrated over the domain t ∈ [0, T ], where T is selected to be long enough to ensure that the long-time dynamics are sufficiently captured.For illustrative purposes, the horizontal axis (in all four panels) has been scaled by T so that the scaled time, t/T , assumes values in the unit interval: t T ∈ [0, 1].Top Left: The numericallyobtained time course of s with s 0 = 0.5 and c(0) = 0.0.Top Right: The numerically-obtained time course of c with s 0 = 0.5 and c(0) = 0.0.Bottom Left: The numerically-obtained time course of s with s 0 = 0.1 and c(0) = 0.0.Bottom Right: The numerically-obtained time course of c with s 0 = 0.1 and c(0) = 0.0.Observe that the solution components of (2) become increasingly accurate approximations to the solution components of (1) as s 0 decreases.
Figure 1: Illustration of Proposition 2.The solution to the Michaelis-Menten system (1) converges to the solution of the linear Michaelis-Menten system (2) as s 0 → 0. In all panels, the solid black curve is the numerical solution to the Michaelis-Menten system (1).The thick yellow curve is the numerical solution to linear Michaelis-Menten system (2).The dashed/dotted curve is the numerical solution to the linear system defined by matrix G.The dotted curve is the numerical solution to the linear system defined by matrix H.All numerical simulations where carried out with the following parameters (in arbitrary units):k 1 = k 2 = k −1 = e 0 = 1.In all panels, the solutions have been numerically-integrated over the domain t ∈ [0, T ], where T is selected to be long enough to ensure that the long-time dynamics are sufficiently captured.For illustrative purposes, the horizontal axis (in all four panels) has been scaled by T so that the scaled time, t/T , assumes values in the unit interval: t T ∈ [0, 1].Top Left: The numericallyobtained time course of s with s 0 = 0.5 and c(0) = 0.0.Top Right: The numerically-obtained time course of c with s 0 = 0.5 and c(0) = 0.0.Bottom Left: The numerically-obtained time course of s with s 0 = 0.1 and c(0) = 0.0.Bottom Right: The numerically-obtained time course of c with s 0 = 0.1 and c(0) = 0.0.Observe that the solution components of (2) become increasingly accurate approximations to the solution components of (1) as s 0 decreases.
(c) Moreover, condition (10) is satisfied in a large part of parameter space (in a well-defined sense) as shown in Eilertsen et al. [5, subsection 3.3, in particular Figure 2].