The average enzyme principle

Abstract

The Michaelis-Menten equation for an irreversible enzymatic reaction depends linearly on the enzyme concentration. Even if the enzyme concentration changes in time, this linearity implies that the amount of substrate depleted during a given time interval depends only on the average enzyme concentration. Here, we use a time re-scaling approach to generalize this result to a broad category of multi-reaction systems, whose constituent enzymes have the same dependence on time, e.g. they belong to the same regulon. This “average enzyme principle” provides a natural methodology for jointly studying metabolism and its regulation.

Introduction

Biochemical reactions are embedded in complex metabolic networks. The dynamic features of these networks, such as the time-dependent regulation of proteins, underlie the capacity of the cell to cope with variable environmental conditions [1–3], to allocate resources efficiently [4], and to achieve complex adaptive strategies for survival [5]. Several classical studies have addressed the effect of changing enzyme levels on substrate kinetics [6–10]. Most recent efforts in systems biology, however, tend to focus either on understanding genome-scale metabolic networks under steady-state conditions [11,12], or on understanding transcriptional regulation irrespective of the underlying metabolism [13]. While this separation of biological complexity into metabolic and regulatory layers has given rise to extremely insightful techniques and analyses, the search for new approaches to merge these two layers in a unified manner is recognized as a fundamental, albeit difficult challenge.

Here, we revisit the classical Michaelis-Menten equation under the assumption of a time-dependent enzyme concentration. We first summarize our results from an earlier study [14], showing that, for an isolated reaction obeying Michaelis-Menten kinetics, the final substrate concentration depends not on the time-dependent details of enzyme concentration, but simply on its average. Indeed, any two enzyme profiles with the same average concentration and identical kinetic parameters yield the same final substrate concentration.

Next, we show how this “average enzyme principle” can be extended to more complex metabolic networks. This is best illustrated through the example of a linear metabolic pathway, which can be simulated numerically, and which helps formulate the problem in a way that is amenable to analytical proof. We prove that if all enzymes in the pathway follow the same dynamics, the final concentration of metabolites in the pathway depends only on the average enzyme level during the elapsed time. Notably, the invariance to enzyme trajectories remains valid for a much broader category of metabolic networks whose constituent enzymes follow synchronous time-courses.

Results

Consider the following problem: a substrate S is degraded by an enzyme E. How much substrate is left after a given time ΔT? Under appropriate conditions and assumptions [15,16], the answer can be computed using the hundred-year-old Michaelis-Menten equation

$$\frac{dS}{dt}=\frac{-k_{\mathit{cat}}ES}{K_M+S}$$ (1)

As in most descriptions of enzymatic catalysis, the total amount of enzyme is assumed to be constant. Hence, an expression for the final concentration of substrate can classically be obtained by integrating:

$${\int }_{S_0}^{S_f}\frac{(K_M+S)dS}{S}={\int }_0^{\mathrm{\Delta }T}-k_{\mathit{cat}}Edt$$ (2)

These integrals can be evaluated explicitly, yielding the result

$$K_{m}ln\frac{S_f}{S_0}+S_f-S_0=-k_{\mathit{cat}}E\mathrm{\Delta }T$$ (3)

Thus, we can express the final concentration of the substrate as a function of its initial concentration, the time interval of interest, and the relevant enzymatic parameters. Note that while Eq. (3) constitutes an implicit function of S_f, an explicit expression can be obtained using the Lambert W function [17].

What if we pose the same problem, but assume that the enzyme concentration is instead a time-dependent quantity E(t). Can we still easily compute the amount of substrate left after a given time ΔT? Behind the apparently open-ended challenge of addressing this question lies a simple answer, potentially rich of biological implications.

Under appropriate conditions [14], and in line with studies of hybrid metabolic-genetic systems [18,19], we can write the Michaelis-Menten equation in which the enzyme concentration is a time-dependent variable:

$$\frac{dS}{dt}=\frac{-k_{\mathit{cat}}E(t)S}{K_M+S}$$ (4)

Equation (4) is still separable; in other words, enzyme concentration can be still isolated from the terms corresponding to substrate, yielding upon integration:

$$K_{m}ln\frac{S_f}{S_0}+S_f-S_0=-k_{\mathit{cat}}{\int }_0^{\mathrm{\Delta }T}E(t)dt=-k_{\mathit{cat}}\mathrm{\Delta }T{E}_{\mathit{avg}}$$ (5)

where E_avg is the average enzyme level during the time interval [0, ΔT]. Equation (5) states that the final concentration of substrate depends only on the average concentration of enzyme in a time interval, rather than its kinetic details.

The potential implications of this simple result may best be seen by illustrating it in the following alternative way: if two enzyme time-course profiles E₁ and E₂ exhibit the same average enzyme concentration at time ΔT (i.e. ${\int }_0^{\mathrm{\Delta }T}{E}_1(t)dt={\int }_0^{\mathrm{\Delta }T}{E}_2(t)dt$), the metabolic effects of these two enzymes are indistinguishable from each other at time ΔT. In other words, in order to degrade a certain amount of substrate S, any one of an infinite number of equivalent enzyme trajectories with identical averages may be used. While the validity of this statement should be apparent from the analytical nature of this result, a simple experimental assay reported earlier should relieve any further doubts [14]. As we explore in detail in [14], one consequence of this multiplicity of solutions is the possibility of identifying a trajectory that minimizes the cost a cell incurs when sequestering cellular resources for the production/degradation of enzyme.

Here we wish to show that this average enzyme principle holds not only for a single enzyme-catalyzed reaction, but also for a broad category of multi-step metabolic pathways, under certain assumptions on the time-dependence of the enzymes in the pathway. It is not obvious that the principle should extend: because the dynamics of each metabolite will depend on other metabolites in the network, it is not possible to simply integrate over time, as was done in the single-reaction case above.

The simplest case, which we examine in detail, is the one of a linear metabolic pathway of n metabolites in which the product of one reaction is the substrate for the next one. The dynamics of such a system are described by the following differential equations:

$$\frac{{dS}_1}{dt}=-E_1(t)\left(\frac{k_1{S}_1}{K_{M,1}+S_1}\right)$$ (6)

$$\frac{{dS}_i}{dt}=E_{i-1}(t)\left(\frac{k_{i-1}{S}_{i-1}}{K_{M,i-1}+S_{i-1}}\right)-E_i(t)\left(\frac{k_i{S}_i}{K_{M,i}+S_i}\right),i=2\dots n$$ (7)

where S_i corresponds to the i^th substrate in the pathway, k_i is the rate constant of the i^th reaction, and K_M,I is the half-saturation constant of the i^th reaction. We assume that all enzymes vary synchronously in time (c₁E₁(t) = c₂E₂(t)=… =c_nE_n(t), where each c_i is a positive constant). This assumption is reasonable for metabolic pathways (such as many linear ones) whose enzymes are co-regulated and exhibit similar time-dependent expression profiles [20,21]. A numerical solution of these equations shows that, for different enzyme time-courses with identical averages at time ΔT, metabolite concentrations at time ΔT are precisely identical (Figure 1), exactly as in the case of a single enzyme. For simplicity, but without loss of generality, we have assumed that c_i = 1 for all i (i.e. that all enzyme profiles are exactly the same).

Figure 1. Illustration of the average enzyme principle for a linear metabolic pathway.

(A) A linear metabolic pathway of five reactions and five metabolites is assumed to obey the differential equations in Eqs. (6)–(7). (B),(C),(D) The enzyme levels for the reactions in A are assumed to be regulated identically across the five reactions, in one of these three possible trajectories. The three enzyme time-courses shown here have the same average enzyme concentrations during the assigned time interval. (E),(F),(G) Substrate dynamics for the enzyme time courses (B), (C), and (D), respectively. Note that, although the time-dependent details of each metabolite differ across the three panels, all metabolites reach the same final concentration at the end of the time course. This is a direct outcome of the fact that all three enzyme profiles have the same average concentration during the time interval considered. The equations were solved using standard Matlab ODE functions. Parameter used are: k₁=1 sec⁻¹, k₂=2 sec⁻¹, k₃=1.5 sec⁻¹, k₄=1.2 sec⁻¹, k₅=1.9 sec⁻¹, K_M,1=1.5 mM, K_M,2=3 mM, K_M,3=0.9 mM, K_M,4=2mM, K_M,5=1 mM.

The numerical result above is just a special case of a much more general analytical result, which we derive next. We start by writing the differential equations for a metabolic network comprised of n metabolites, denoted by the vector (S₁, S₂, S₃, …S_n) = S. We assume that each reaction in the network can be effectively described by irreversible Michaelis-Menten equations (as in Eqs. (6)–(7)). We further assume that all enzymes in the network have the same dependence on time (again, that c₁E₁(t) = c₂E₂(t)=… c_nE_n(t)).

Then, using vector notation, we can write the differential equations for such as a system as

$$\frac{d\mathit{S}}{dt}=E(t)\mathit{f}(\mathit{S})$$ (8)

where the bold lettering denotes a vector. Note that f is a vector-valued function, which takes as input the vector of substrate concentrations, and outputs a vector of the same dimension. Because each individual c_i (the parameter corresponding to the scaling of each enzyme) is a fixed constant, it can be directly absorbed into the expression for f. Now, let us consider the dynamics of the system above for two different enzyme time-courses, E_A(t) and E_B(t), with the same average concentration at time ΔT (i.e. ${\int }_0^{\mathrm{\Delta }T}{E}_A(t)dt={\int }_0^{\mathrm{\Delta }T}{E}_B(t)dt$). We will show that the final concentration of each metabolite S_i at time ΔT is identical for the two enzyme time courses. To see this, let us make a change of variables by letting

$${\tau }_A={\int }_0^t{E}_A(x)dx$$ (9)

$${\tau }_B={\int }_0^t{E}_B(x)dx$$ (10)

Note that at t = ΔT, τ_A and τ_B are equal (by our assumption of equivalent average enzyme levels). Then, we can use the chain rule and the Fundamental Theorem of Calculus to rewrite the system in Eqs. (9) and (10) as

$$\frac{d\mathit{S}}{d{\tau }_A}=\frac{dt}{d{\tau }_A}\frac{d\mathit{S}}{dt}=\frac{1}{E_A(t)}{E}_A(t)\mathit{f}(\mathit{S})=\mathit{f}(\mathit{S})$$ (11)

$$\frac{d\mathit{S}}{d{\tau }_B}=\frac{dt}{d{\tau }_B}\frac{d\mathit{S}}{dt}=\frac{1}{E_B(t)}{E}_B(t)\mathit{f}(\mathit{S})=\mathit{f}(\mathit{S})$$ (12)

Since the form of Eqs. (11) and (12) are identical, they have identical solutions, except in differently scaled time variables. However, at time ΔT, ${\tau }_A={\int }_0^{\mathrm{\Delta }T}{E}_A(x)dx={\int }_0^{\mathrm{\Delta }T}{E}_B(x)dx={\tau }_B$ and the solutions (i.e. the concentrations of substrate) are identical. Notably, an identical rescaling argument applies for any metabolite network that can be written in the form of Eq. (8). For any such metabolic network, an analogue of the average enzyme principle applies, and final substrate concentrations are invariant to enzyme time courses with identical averages.

Discussion

Distilling the principles by which metabolism is regulated is a fundamental challenge for experimental, computational, and theoretical biologists alike. Here, we have shown that for a broad class of biochemical networks, a simple invariance principle ultimately determines the final concentration of substrate in the system. Our proof of this principle for networks does not follow directly from the proof for isolated reactions; instead, it hinges on a novel time re-scaling argument that takes advantage of the separability of enzyme terms in each differential equation.

Several important challenges lie ahead in extending and generalizing our work. Our proof of the average enzyme principle for networks of reactions depends on the assumption that all enzymes in the system exhibit synchronous dynamics (i.e. their time-dependent functional forms are scalar multiples of each other). Although this assumption often reflects biological reality, recent work has demonstrated the existence of other patterns of regulation in metabolic pathways (e.g. just-in-time transcription [4]). We suspect that analytical approaches similar to our own, perhaps making use of ideas from the field of delay differential equations, may yield more general results for such systems. In addition, while the current proof of the average enzyme principle requires enzyme separability, it will be interesting to consider whether a similar – perhaps approximate – result holds in cases when enzyme concentration is high enough that the enzyme concentration is no longer separable in the rate equation [22].

Finally, our findings here offer a new opportunity to understand how metabolic networks may be optimally regulated. In line with our work in [14], the existence of an invariance principle for a metabolic network makes it possible to search for the dynamic enzyme profile which minimizes some cost (e.g. energetic cost associated with enzyme production/degradation) to the cell while still achieving some metabolic goal. Indeed, the notion of applying optimal control principles to metabolism has been studied before (see [19,23]). It is possible that our work may help uncover further simplifying principles of metabolic regulation.

Highlights.

• In Michaelis-Menten kinetics with time-dependent enzyme, the average enzyme is what matters

• This “average enzyme principle” holds also for co-regulated multi-reaction pathways

• This approach offers new avenues for integrating metabolism and regulation