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# Structural conditions on complex networks for the Michaelis–Menten input–output response

Edited by Curtis G. Callan Jr., Princeton University, Princeton, NJ, and approved August 14, 2018 (received for review May 9, 2018)

## Significance

The Michaelis–Menten (MM) formula arose to explain simple enzyme behavior. It has since been found to describe input–output responses in several other biological contexts. Its ubiquity has been surprising and poorly understood. Here, we use the graph-based “linear framework” to show how the MM formula arises whenever appropriate structural conditions are satisfied, both at thermodynamic equilibrium and when energy is being dissipated. These conditions are based on separating the graph into two parts and constraining how and where the input variable appears. The conditions do not depend on parameter values and allow many of the details to be arbitrary. This explains the ubiquity of the MM formula and substantially generalizes previous results.

## Abstract

The Michaelis–Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input–output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a “coarse graining” into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula’s ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.

The Michaelis–Menten (MM) formula may be expressed as**1** in their foundational work on enzyme kinetics (1, 2). They derived the formula for a network of three reactions (Fig. 1*A*) in which an enzyme, E, catalyzes the conversion of substrate, S, to product, P. Their derivation relied on assuming a timescale separation in which the intermediate enzyme–substrate complex, **1** is sometimes referred to as the “Langmuir isotherm.”

It has been appreciated gradually that Eq. **1** holds in far greater generality than the simple contexts considered by Michaelis and Menten and Langmuir. Table 1 summarizes the broad range of contexts known to us in which Eq. **1** has been found. These examples suggest a ubiquity in the MM formula, which transcends the variety of contexts, molecular components, and mechanisms under which it arises. It is this ubiquity which the present paper seeks to explain. Similar ubiquity has been explored for the empirical observation of sigmoidal input–output responses (4). Here, we seek answers in the architecture of the underlying molecular mechanisms, rather than the statistics of measurement and information. Our results encompass, in substantial generality, all of the examples in Table 1.

Several theoretical studies have attempted to identify appropriate regimes in which the MM formula appears (7⇓⇓⇓⇓⇓–13, 30⇓⇓⇓⇓⇓–36). Such studies have largely relied on simplifying assumptions, such as symmetric or regular networks (7, 11⇓–13, 30, 35, 36) or restriction to networks in which the steady state is one of thermodynamic equilibrium (12, 13). Numerical studies have suggested that the MM formula generally does not arise away from equilibrium (37⇓⇓–40).

We approach the problem using the graph-based “linear framework” for timescale separation. The framework is described in refs. 41 and 42, applied to biological problems in refs. 43⇓⇓–46, and reviewed in ref. 47. We briefly describe its salient features here, with more details below.

In the linear framework, both macroscopic, deterministic systems, such as enzymes in well-mixed compartments, and microscopic, stochastic systems, such as individual molecular motors, can be described in the same way, by a graph (Fig. 1*B*). The vertices usually represent fast components or states, which are assumed to reach steady state under a timescale separation. The directed edges represent reactions between components or transitions between states. The influence of the slow components is incorporated into the edge labels, which specify the reaction rates.

Such a graph yields a linear differential equation for the concentrations or probabilities of the vertices. In the stochastic context, this is the master equation of the underlying Markov process. The linear dynamics gives the framework its name and is common to all applications but the slow components in the labels, which introduce nonlinearities, are dealt with in ways that depend on the context. Here, we assume that slow components are effectively constant over the timescale of the fast dynamics. Under this timescale separation, the dynamical equation reaches a steady state, which can be expressed algebraically in terms of the edge labels, without having to know in advance the numerical values of parameters or eigenvalues. This holds in generality for any graph (42) although we are concerned here with graphs for which the steady state is essentially unique (below). If the steady state is at thermodynamic equilibrium, the resulting expressions are equivalent to those of equilibrium statistical mechanics. Importantly, they remain valid away from equilibrium (48, 49), which permits a unified approach. The steady-state output response can be described in terms of these expressions as a rational function of the input variable (Eq. **2**). The results below give conditions under which this rational function assumes the simple form in Eq. **1**.

The conditions we find relate to whether the graph can be “split” into two subgraphs and where the input variable occurs with respect to this partition. They leave many other details unspecified, which explains the ubiquity of the MM formula and its independence from underlying mechanistic details.

## Results

### Linear Framework.

We introduce here some notation and terminology; for full details and more background, see *SI Appendix*, section S1 and refs. 41, 42, 44, and 45. We consider finite directed graphs with labeled edges and no self-loops (hereafter, “graphs”). As noted above, vertices typically represent fast components or states, edges represent reactions or transitions, and labels represent rates, with units of (time)^{−1}. The labels may include contributions from slow components that are not represented by vertices but interact with them (Fig. 1*B*). We assume that all graphs are connected, so that they cannot be decomposed into parts between which there are no edges. Vertices are denoted by indexes

The graphs considered here are strongly connected. This means that, given any two distinct vertices, *B*), then

A steady state, **2** is then seen to correspond to the prescription of equilibrium statistical mechanics, with the denominator being the partition function. As noted above, however, Eq. **2** continues to hold away from equilibrium, providing thereby a form of nonequilibrium statistical mechanics.

If a reversible graph forms a tree, with no proper cycles of reversible edges (i.e., cycles with at least three vertices), then it always satisfies detailed balance. Free energy may be dissipated in some reactions but there is no entropy production because that arises only from proper cycles (49).

For the results below, we consider the input–output response of a graph to be the steady-state concentration or probability of a specific vertex, **2**. In some applications the output is a property like an inverse mean first passage time of a Markov process, which can be identified with a steady-state output flux from an appropriate vertex, **1** is the same for all j. This will emerge from the arguments below and we leave it to the reader to draw the appropriate conclusion for the particular context.

### Partitions and Splitting.

We introduce here the concepts needed for the main results. Given a graph G, with vertices forming the set *B*, with

In the examples summarized in Table 1, edge labels are usually simple expressions, such as

### MM at Thermodynamic Equilibrium.

We first consider the case when a graph G is at thermodynamic equilibrium, so that it is reversible. We say that G has monomial ratios if

### Proposition 1.

*Let* G *be a reversible graph with monomial ratios at thermodynamic equilibrium. If* G *is partitioned by* *and* *so that* *for all edges splitting* *from* *and* *for all nonsplitting edges*, *then all vertices in* *satisfy the MM formula*. *Conversely*, *if some vertex satisfies the MM formula and* *consists of all such vertices*, *then* *and* *form a partition of* G *for which* *has the same properties*. *In either direction*, *all vertices with the MM formula have the same denominator*.

Proofs of *Proposition 1* and other results below are in *SI Appendix*, sections S2–S6. The partition shown in Fig. 2 falls under *Proposition 1* provided the graph is reversible and satisfies detailed balance and the only label ratios with x are on the splitting edges, with the magenta edges having degree 1 and the blue edges correspondingly having degree −1. The coarse-graining analogy described above becomes closer here, with *B*.

Sequence graphs, which consist of a series of vertices with nearest neighbors joined by reversible edges, as in Fig. 3*A*, have been used to model enzyme kinetics (5, 8⇓–10, 52), gene regulation (16), and molecular motors (8, 25, 26). Since sequence graphs are trees, they satisfy detailed balance, as noted above, and can be considered to be at thermodynamic equilibrium. In the cited applications, the partitions break the sequence into two adjoining parts (Fig. 3*A*) and the requirements on the degrees of the label ratios conform to *Proposition 1*, from which the MM formula arises.

Several authors have independently shown that the MM formula arises for two sequence graphs connected in parallel (Fig. 3*B*), in the regime in which thermodynamic equilibrium holds (7, 11⇓–13, 27). The application has mainly been to single-molecule enzyme kinetics, with the vertices along the sequences representing different enzyme and enzyme–substrate conformations (Fig. 1*B*). In this case, the input variable is found only on the magenta edges. Accordingly, these derivations of the MM formula at equilibrium follow from *Proposition 1*, with

### MM Away from Equilibrium, with Reversible Edges.

We now consider a reversible graph, G, which may not be at thermodynamic equilibrium. We say that G has monomial labels if, for each edge *Proposition 1*. We say that G is x-acyclic if no label in any proper cycle of edges depends on x, so that *A*, in which the edges with x in their labels (blue or magenta) link subgraphs through a subtree, with the subgraphs being arbitrarily complicated. As mentioned above, entropy production takes place on cycles, so being x-acyclic implies that the variable x does not contribute to entropy production within the graph. With these restrictions, the MM formula arises under the same conditions as in *Proposition 1*.

### Proposition 2.

*Let* G *be a reversible graph with monomial labels that is* x-*acyclic*. *If* G *is partitioned by* *and* *so that* *for all edges splitting* *from* *and* *for all nonsplitting edges*, *then all vertices in* *satisfy the MM formula*. *Conversely*, *if some vertex satisfies the MM formula and* *consists of all such vertices*, *then* *and* *form a partition of* G *for which* *has the same properties*. *In either direction*, *all vertices with the MM formula have the same denominator*.

As noted above, the parallel sequence graph in Fig. 3*B* has been used to model enzyme catalysis at the single-molecule level. If the graph is at thermodynamic equilibrium and the input variable occurs only on the magenta edges, then the graph satisfies the conditions of *Proposition 1*. When the graph is away from equilibrium, it cannot satisfy the conditions of *Proposition 2* because it is not x-acyclic. However, parametric regimes have been identified in which the MM formula does emerge away from equilibrium (7, 11, 14). In these regimes, the bound, or the unbound, enzyme changes conformations very slowly, giving rise to the graphs in Fig. 3 *C* and *D*, respectively (*SI Appendix*, Table S1 and Fig. S2). These graphs are both x-acyclic and the emergence of the MM formula follows from *Proposition 2*.

### MM Away from Equilibrium, with Irreversible Edges.

In many applications in which a system is away from thermodynamic equilibrium, certain reactions or transitions are treated as effectively irreversible. If the corresponding graph acquires an irreversible edge, it would not fall under the scope of either *Proposition 1* or *Proposition 2*. The emergence of the MM formula now becomes much more delicate and, in contrast to the results above, we can identify only certain sufficient conditions. A restrictive case, which nonetheless applies to some examples (8, 15, 21, 29), is relegated to *SI Appendix*, *Proposition S1*.

### Proposition 3.

*Let* G *be a strongly connected graph with monomial labels*. *If* G *is partitioned into* *and* *so that* *for all edges with* *and* *for all edges with* *then all vertices in* *satisfy the MM formula with the same denominator*.

Fig. 4*B* shows the kind of graph structure to which *Proposition 3* applies, with the x-containing labels only on the magenta edges. In contrast to *Proposition 2*, the x-containing labels may be on proper cycles but are now required to be present on, and only on, the outgoing edges from any vertex in

In single-molecule enzyme studies, one of the nonequilibrium parametric regimes in which the MM formula emerges gives rise to a graph like that in Fig. 3*E*, which satisfies the conditions of *Proposition 3* with *SI Appendix*, Table S1 and Fig. S2) (7, 11, 14). Graphs satisfying the conditions of *Proposition 3* have also arisen in four other applications. First, modified versions of the sequence graph in Fig. 3*A* have been used to model enzyme kinetics both macroscopically (6) and at the single-molecule level (8⇓–10). Here, *Proposition 3*, although none of them have exploited the generality it offers, as suggested in Fig. 4*B*.

Our final result requires one more concept: A vertex j is a sink if there exists an incoming edge

### Proposition 4.

*Let* G *be a strongly connected graph that is partitioned by* *and* *with* *containing only a single vertex*, j, *which is a sink*. *Suppose that all edges to and from* j *have monomial labels for which* *and* *and that*, *furthermore*, *does not depend on* i. *Then* *satisfies the MM formula*.

No restriction is placed on the labels of edges other than those to and from the sink vertex. Fig. 4*C* shows the kind of graph structure that satisfies *Proposition 4*. In studies of single-molecule enzyme kinetics, another parameter regime was identified in which the MM formula arises (7, 11, 14). Here, the conformational fluctuations of the bound enzyme are assumed to be extremely fast, so that there is effectively only a single conformation, and the on-rates for substrate binding are assumed to be independent of the unbound conformation, giving rise to the graph in Fig. 3*F* (*SI Appendix*, Table S1 and Fig. S2). These are exactly the assumptions required for *Proposition 4*, with the sink vertex being the unique bound conformation.

## Discussion

The MM formula has been found to arise under timescale separation in a surprising variety of biological contexts (Table 1). Previous attempts to explain its ubiquity have largely focused on regular networks, such as the parallel sequence network in Fig. 3*B*, and identification of specific parameter regimes. In contrast, our approach considers general graphs and gives conditions in terms of graph structure and the location of the input variable (*Propositions 1*–*4* and *SI Appendix*, *Proposition S1*). Most other details—the sizes and topologies of subgraphs, the arrangements of vertices and edges, and the numerical values of labels—prove to be irrelevant to the algebraic nature of the MM formula (Fig. 4). Instead, these details influence the quantities A and B in Eq. **1**.

Our results provide a series of rules for determining whether or not the MM formula arises. If a graph is at thermodynamic equilibrium, then the MM formula holds, if, and only if, there is a partition of the graph for which the input variable, x, is found in the label ratios only on splitting edges in one direction and only to degree 1, as described in *Proposition 1*. This is analogous to a coarse graining into the original MM graph in Fig. 1*B*. If the graph is not at thermodynamic equilibrium but is still reversible, then it is important to check whether x occurs in a label on any proper cycle. If not, then the MM formula holds if, and only if, the same coarse graining can be found as at equilibrium, as described in *Proposition 2*. If the graph is not x-acyclic or if it has irreversible edges, then the situation is more subtle. The MM formula holds if one of the special structural conditions in *Proposition 3*, *Proposition 4*, or *SI Appendix*, *Proposition S1* is satisfied.

Eq. **2** offers a way to understand how the MM formula arises. Recall that the denominator of this rational function generalizes the partition function of equilibrium statistical mechanics. In *Proposition 1* and *SI Appendix*, *Proposition S1*, the partition function takes the restricted form *SI Appendix*, *Proposition S1* and sections S2 and S6), from which MM formula follows easily. In *Propositions 2* and *3*, the partition function takes the form *SI Appendix*, sections S3 and S4). Here, n is related to the structure of the graph. The MM formula follows by canceling *Proposition 4*, the partition function factorizes as *SI Appendix*, section S5). The MM formula arises by canceling *Proposition 4* is the most difficult of our results, despite the restrictive assumption of a single sink vertex. *SI Appendix*, section S7 and Fig. S1 gives further examples in which nontrivial factorizations of the partition function occur, which are not covered by *Propositions 1–4* and *SI Appendix*, *Proposition S1*. It remains an interesting open problem to fully characterize those graphs which give rise to the MM formula.

Eq. **2** tells us that the input–output response of a graph-based system is a rational function of x. *Propositions 1*–*4* and *SI Appendix*, *Proposition S1* provide conditions under which this rational function takes its simplest form. Conditions can also be found which more broadly constrain the algebraic complexity of a response function (45, 46). A general advantage of the graph-based linear framework that we have exploited here is that it does not depend on numerical calculations or simulations, which require all details of the system to be specified and parameter values to be estimated. It focuses instead on the topological features of the graph, thereby rising above many of the mechanistic details. This capability is particularly well suited to analyzing biomolecular systems, in which many of the details may be obscure or unavailable, and it seems likely that more results of this kind remain to be uncovered.

## Materials and Methods

Detailed proofs of *Propositions 1*–*4* and *SI Appendix*, *Proposition S1* and examples falling outside the scope of these results are given in *SI Appendix*, sections S2–S7 and Fig. S1.

## Acknowledgments

We thank the editor and Rob Phillips for very helpful suggestions. F.W. was supported by the US National Science Foundation (NSF) Graduate Research Fellowship under Grant DGE1144152. A.D. was supported by the Indian Ministry of Human Resource Development under the Senior Research Fellowship Scheme. D.C. was supported by the Prof. S. Sampath Chair at Indian Institute of Technology, Kanpur and a J. C. Bose National Fellowship. J.G. was supported by the NSF under Grant 1462629.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: jeremy{at}hms.harvard.edu.

Author contributions: F.W., D.C., and J.G. designed research; F.W., A.D., D.C., and J.G. performed research; F.W. and J.G. wrote the paper; and all authors discussed the results and commented on the manuscript at all stages.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1808053115/-/DCSupplemental.

Published under the PNAS license.

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