Introduction

Enzymes are biocatalysts that enable numerous reactions in living organisms. The Michaelis-Menten mechanism describes their underlying dynamics1. A free enzyme reversibly binds a substrate molecule to form an enzyme-substrate complex that can either lead to the formation of a product via an irreversible catalytic pathway or can simply revert back to the free state without forming a product. Based on this mechanism, the Michaelis-Menten equation predicts a linear relation between the reciprocals of the turnover rate and the substrate concentration. From this linear relation, one can deduce two important kinetic parameters: the maximal rate of the enzymatic reaction and the Michaelis-Menten constant, i.e., the substrate concentration at which half the maximal rate is achieved2.

Significant technological advancements accomplished in the past two decades have now made it possible to follow the stochastic motion and activity of single particles and molecules3,4,5,6,7,8,9,10,11, including individual enzymes12,13,14,15,16,17,18,19. Specifically, under certain assumptions, the Michaelis-Menten equation was shown to hold at the single-molecule level where it predicts a linear relation between the mean turnover time and the reciprocal of the substrate concentration20,21,22,23,24,25. The turnover time probability distribution can also be measured by tracking single enzymes12,13,21. Yet, in contrast to the mean reaction time, higher moments often exhibit complex non-monotonic dependencies on the reciprocal of the substrate concentration20,21,26,27, which complicates the extraction of kinetic information and the physical interpretation of experimental results.

Some progress can still be made for enzymes whose kinetics are well described by nearest-neighbor models27. Expressions for the mean reaction time and the inverse of the squared coefficient of variation have been attained for such enzymes by adopting a Markovian kinetic description in which internal states are linearly connected in a reversible manner28,29,30. Using these, information on the number of interim states involved in substrate binding, the total number of states present in the catalytic network, and the number of non-substrate binding states, can be extracted by measuring the mean and second moment of the catalysis time distribution as a function of substrate concentration27,29,31,32. Yet, if the conformational dynamics is non-Markovian14,33,34,35, or has parallel branching pathways or non-sequential motifs, this modeling approach fails27. One is then left to wonder how to systematically extract useful information from higher moments of enzymatic turnover times in a general way and without making restricting assumptions on the underlying kinetics.

Addressing this research problem, several studies have made distinctive progress within the renewal approach to Michaelis-Menten enzyme kinetics36,37,38,39,40. In this approach, one adopts the classical reaction scheme of Michaelis & Menten, but allows for non-Markovian transitions between three coarse-grained enzymatic states: free enzyme and substrate \(\left(E+S\right),\) enzyme-substrate complex \(\left({ES}\right),\) free enzyme and product \(\left(E+P\right).\)

The coarse-grained description effectively accounts for multiple intermediate kinetic states that are often part of the reaction, even when information about them is partially or completely missing due to experimental limitations. It can be built by retaining the same state space as in the classical Michaelis-Menten model, while replacing the familiar transition rates with generally distributed (non-Markovian) transition times41. Thus, by allowing for arbitrary transition time distributions, one can effectively account for hidden degrees of freedom, e.g., different conformational states and branching pathways, that are lost in the coarse-graining procedure.

A key insight coming from the renewal approach is that the Michaelis-Menten equation is universal38. Namely, regardless of the details of the underlying distributions of binding, unbinding, and catalysis times, the mean turnover time always shows a linear dependence on the reciprocal of the substrate concentration. Yet, extensions of this fundamental result to higher moments of the turnover time have so far remained elusive.

Generalizing the Michaelis-Menten equation to moments of any order will open the door to systematic analysis of turnover time data, thus allowing for characterization of enzyme catalysis beyond the classical, yet incomplete, description of the maximal turnover rate and the Michaelis-Menten constant. Here, we do so by leveraging the renewal approach to identify unique moment combinations that exhibit universal linear dependencies on the reciprocal of the substrate concentration. We call these relations: high-order Michaelis-Menten equations and demonstrate how they can be used to extract kinetic information previously deemed inaccessible, e.g., the binding rate, the mean and variance of the time spent in the bound enzymatic state, and the probability that catalysis occurs before substrate unbinding.

Below, we advance on the renewal framework to derive results for the turnover time distribution and continue to show how these yield high-order Michaelis-Menten equations. The benefits coming from these equations, and how to apply them in practice to infer hidden kinetic parameters, are also discussed in detail.

Results

Renewal approach to enzyme catalysis

Any enzymatic reaction includes three kinetic processes: binding and unbinding of a substrate molecule to and from the enzyme, and catalysis which leads to product formation. These events can be characterized by the random times of binding,\(\,{T}_{{on}}\,,\) unbinding, \({T}_{{off}},\) and of catalysis, \({T}_{{cat}}\,\)(Fig. 1a). Note that at this point we make no assumptions on the distributions of these times, which in turn determine the experimentally observable turnover time: the time it takes a single enzyme to produce a single molecule of product.

Fig. 1: Renewal approach to enzyme catalysis.
figure 1

a A schematic representation of the Michaelis-Menten model, where transitions are characterized in terms of a general binding time \({T}_{{on}}\), unbinding time \({T}_{{off}}\), and catalysis time \({T}_{{cat}}\). The free enzyme, \(E\), binds with a substrate molecule, \(S\), to form a bound-enzymatic state, \({ES}\). The formed enzyme-substrate complex undergoes catalysis to form a product, \(P\); or goes back to the free enzyme state via unbinding. b Turnover time traces representing two consecutive product formation events. Following binding, the enzyme is in the \({ES}\) state. If catalysis happens prior to unbinding, a product is formed. In this case, the waiting time in the \({ES}\) state has the statistics of \({W}_{{ES}}^{{cat}}\) from Eq. (2), i.e., of the catalysis time \({T}_{{cat}}\) given that it was smaller than the unbinding time \({T}_{{off}}\). Else, unbinding occurs and the waiting time in the \({ES}\) state has the statistics of \({W}_{{ES}}^{{off}}\) from Eq. (3), i.e., of the unbinding time \({T}_{{off}}\) given that it was smaller than the catalysis time \({T}_{{cat}}\). In this paper, we show how to extract hidden information on the statistics of the hidden times \({T}_{{on}},{W}_{{ES}}^{{cat}}\), \({W}_{{ES}}^{{off}}\) by analyzing the distribution of the turnover time \({T}_{{turn}}\) that is observed in experiments.

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The turnover time can be calculated considering two possible routes of enzyme-substrate complex breakup:

$${T}_{{turn}}={T}_{{on}}+\left\{\begin{array}{c}{T}_{{cat}} \qquad \, \, \qquad {if} \, {T}_{{cat}} \, < \, {T}_{{off}}\hfill\\ {T}_{{off}}+{T}_{{turn}}^{{\prime} } \quad {if} \, {T}_{{cat}} \, > \, {T}_{{off}}\end{array}\right\},$$
(1)

where \({T}_{{turn}}^{{\prime} }\) is an identical and independent copy of \({T}_{{turn}}.\) Namely, if \({T}_{{cat}}\) is smaller than \({T}_{{off}}\): catalysis follows binding and the enzyme-substrate complex breaks into a free enzyme and product, thus completing the turnover cycle (top row of Eq. (1)). Conversely, if \({T}_{{cat}}\, > \,{T}_{{off}}\): substrate binding is followed by unbinding and the enzyme-substrate complex breaks into a free enzyme and substrate (bottom row of Eq. (1)). In this case, the turnover cycle starts anew.

To move forward, it is convenient to introduce conditional random variables \({W}_{{ES}}^{{cat}}\) and \({W}_{{ES}}^{{off}}\) that stand for the waiting times in the enzyme-substrate complex, provided that either catalysis or unbinding occurs first, respectively. If catalysis occurs first, the waiting time in the enzyme-substrate complex is given by:

$${W}_{{ES}}^{{cat}}=\left.{T}_{{cat}}\right|{T}_{{cat}} \, < \, {T}_{{off}}\,,$$
(2)

and if unbinding occurs first, the waiting time in the enzyme-substrate complex is given by:

$${W}_{{ES}}^{{off}}=\left.{T}_{{off}}\right|{T}_{{cat}}\, > \,{T}_{{off}}.$$
(3)

The waiting times in Eqs. (2) and (3) have a clear physical meaning, which allows for an insightful interpretation of the underlying turnover kinetics. In Fig. 1b, we give two possible turnover paths for example. In the first path, binding is directly followed by catalysis and product formation. In the second path, binding is followed by unbinding which restarts the turnover cycle. This is followed by a second binding event leading to catalysis and product formation.

We note that single-molecule experiments are designed to track product formation events, but in most cases cannot follow state changes within the catalytic trajectory. Thus, the statistics of \({{T}_{{on}},{W}_{{ES}}^{{cat}},W}_{{ES}}^{{off}}\) is hidden and the challenge is to infer them from the observed statistics of the turnover times. Figure 2 illustrates this concept showing the waiting time histograms associated with the probability density functions (PDFs) of the observable turnover time \({T}_{{turn}}\) and the hidden times \({T}_{{on}},\,{W}_{{ES}}^{{cat}}\) and \({W}_{{ES}}^{{off}}\). Three different examples are considered: (1) a classical single-molecule Michaelis-Menten model, where all kinetic processes are Markovian (characterized by exponential PDFs, blue bars); and (2) two non-Markovian kinetic schemes, where binding times are exponentially distributed but catalysis times follow a Gamma-distribution. The non-Markovian schemes considered differ in the distribution of unbinding times, which is assumed to be exponential in one of them (orange bars) and Gamma-distributed in the other (green bars). Our main goal is to develop an approach for extracting information about the hidden PDFs from single-molecule measurements of the turnover time. Next, we develop the theory and practical tools that allow us to do just that.

Fig. 2: Statistics of times characteristic of enzyme kinetics.
figure 2

Several examples of probability distribution functions (PDFs) of a the observable turnover times, \({f}_{{T}_{{turn}}}\left(t\right)\) and the hidden (b) binding times, \({f}_{{T}_{{on}}}\left(t\right)\), c the conditional catalysis times, \({f}_{{W}_{{ES}}^{{cat}}}\left(t\right)\), and d the conditional unbinding times, \({f}_{{W}_{{ES}}^{{off}}}\left(t\right)\). Histograms were obtained from Monte-Carlo simulations of \({10}^{6}\) turnover cycles of three different illustrative models: (i) the classical single-molecule Michaelis-Menten model, where all kinetic processes are characterized by exponential PDFs (\(\exp\): blue bars); (ii) a non-Markovian kinetic scheme, where binding and unbinding times are exponentially distributed but catalysis times follow Gamma-distributions (\({\gamma }_{{cat}}:\) orange bars), and (iii) a non-Markovian scheme where binding times are exponentially distributed but unbinding and catalysis times follow a Gamma-distribution (\({\gamma }_{{cat}},\,{\gamma }_{{off}}\): green bars). Details of the kinetic parameters and probability distribution functions used are provided in the “Methods” section. Codes employed for generating these PDFs have been supplied as Supplementary Software.

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The probability density functions of the conditional random variables \({W}_{{ES}}^{{cat}}\) and \({W}_{{ES}}^{{off}}\) can be written in terms of probability density functions of the catalysis and unbinding times, \({f}_{{T}_{{cat}}}\left(t\right)\) and \({f}_{{T}_{{off}}}\left(t\right),\) as:

$${f}_{{W}_{{ES}}^{{cat}}}\left(t\right)=\frac{\Pr \left[\left.{T}_{{cat}}=t\right|{T}_{{cat}}\, < \,{T}_{{off}}\right]}{\Pr \left[{T}_{{cat}}\, < \,{T}_{{off}}\right]}=\frac{{f}_{{T}_{{cat}}}\left(t\right){\bar{F}}_{{T}_{{off}}}\left(t\right)}{{\phi }_{{cat}}},$$
(4)

and:

$$\,{f}_{{W}_{{ES}}^{{off}}}\left(t\right)=\frac{\Pr \left[\left.{T}_{{off}}=t\right|{T}_{{cat}}\, > \,{T}_{{off}}\right]}{\Pr \left[{T}_{{cat}}\, > \,{T}_{{off}}\right]}=\frac{{f}_{{T}_{{off}}}\left(t\right){\bar{F}}_{{T}_{{cat}}}\left(t\right)}{{\phi }_{{off}}}.$$
(5)

Here \({\bar{F}}_{{T}_{{cat}}}\left(t\right)=\Pr \left[{T}_{{cat}}\, > \,t\right]\) and \({\bar{F}}_{{T}_{{off}}}\left(t\right)=\Pr \left[{T}_{{off}}\, > \,t\right]\) are the survival probability functions of the catalysis and unbinding times respectively, and \({\phi }_{{cat}}={\int }_{0}^{\infty }{f}_{{T}_{{cat}}}\left(t\right){\bar{F}}_{{T}_{{off}}}\left(t\right){dt}\) and \({\phi }_{{off}}=1-{\phi }_{{cat}}\) are respectively the probabilities for catalysis to occur before unbinding and vice versa. Naturally, the sum of the probabilities characterizing the splitting of the catalytic and unbinding pathways is equal to unity. The probability density of the unconditional waiting time at the \({ES}\) state, \({W}_{{ES}}\), is thus given by:

$${f}_{{W}_{{ES}}}\left(t\right)={\phi }_{{cat}}\,{f}_{{W}_{{ES}}^{{cat}}}\left(t\right)+\,\left(1-{\phi }_{{cat}}\right){f}_{{W}_{{ES}}^{{off}}}\left(t\right).$$
(6)

Using the above definitions and assuming that binding times are exponentially distributed with a rate \({k}_{{on}}\left[S\right]\), the Laplace transform of the turnover time, \({T}_{{turn}}\) in Eq. (1), can be written as (Supplementary Discussion 2):

$${\hat{f}}_{{T}_{{turn}}}\left(k\right)={\int }_{0}^{\infty }{e}^{-kt}{f}_{{T}_{{turn}}}\left(t\right){dt}=\frac{{\phi }_{{cat}}{\hat{f}}_{{W}_{{ES}}^{{cat}}}\left(k\right)}{1+\frac{k}{{k}_{{on}}\left[S\right]}\,+\,{\phi }_{{cat}}{\hat{f}}_{{W}_{{ES}}^{{cat}}}\left(k\right)-{\hat{f}}_{{W}_{{ES}}}\left(k\right)}.$$
(7)

Equation (7) expresses the statistics of turnover times in terms of physically meaningful quantities: the substrate concentration \(\left[S\right]\), the binding rate \({k}_{{on}}\), the Laplace transforms \({\hat{f}}_{{W}_{{ES}}^{{cat}}}\left(k\right)\) and \({\hat{f}}_{{W}_{{ES}}}\left(k\right)\) of the conditional and unconditional waiting times in the \({ES}\) state, and the splitting probability \({\phi }_{{cat}}\). Next, we show how moments of the turnover time can be used to infer \({k}_{{on}}\), \({\phi }_{{cat}}\), and moments of the unknown waiting times \({W}_{{ES}}^{{cat}}\) and \({W}_{{ES}}.\)

The mean turnover time

The mean turnover time can be computed directly from Eq. (7). It is given by:

$$\left\langle {T}_{{turn}}\right\rangle=-{\left.\frac{{\partial \hat{f}}_{{T}_{{turn}}}\left(k\right)}{\partial k}\right|}_{k=0}=\overbrace{\left(\frac{1}{{\phi }_{{cat}}\,{k}_{{on}}}\right)}^{{A}_{1}}\frac{1}{\left[S\right]}+\overbrace{\frac{\left\langle {W}_{{ES}}\right\rangle }{{\phi }_{{cat}} }}^{{B}_{1}},$$
(8)

where \(\left\langle {W}_{{ES}}\right\rangle\) is the mean waiting time in the \({ES}\) state, and all other parameters were previously defined. Note that Eq. (8) gives a linear relation between the mean turnover time and the reciprocal of the substrate concentration. To this end, we define \({A}_{1}=1/({\phi }_{{cat}}\,{k}_{{on}})\) and \({B}_{1}=\left\langle {W}_{{ES}}\right\rangle /{\phi }_{{cat}}\) as the slope and intercept of the corresponding line, respectively.

When binding, unbinding, and catalysis times are all exponentially distributed with rates \({k}_{{on}},\,{k}_{{off}}\) and \({k}_{{cat}},\) respectively: the mean turnover time in Eq. (8) reduces to the known Michaelis-Menten formula \(\left\langle {T}_{{turn}}\right\rangle=\frac{{K}_{M}}{{v}_{\max }\,}\frac{1}{\left[S\right]}+\frac{1}{{v}_{\max }\,}.\,\)Note that in contrast to bulk measurements where \({v}_{\max }={k}_{{cat}}\left[{E}_{0}\right]\) with \(\left[{E}_{0}\right]\) standing for the total enzyme concentration, at the single enzyme level we have \({v}_{\max }={k}_{{cat}}\) for the maximal reaction rate. Also, \({K}_{M}=({k}_{{cat}}+{k}_{{off}})/{k}_{{on}}\) is the Michaelis-Menten constant, which corresponds to the substrate concentration at which half the maximal reaction rate, \({v}_{\max }\), is achieved.

In Fig. 3a, we plot \(\left\langle {T}_{{turn}}\right\rangle\) vs. the reciprocal of the substrate concentration for the Markovian case described above (blue solid line). Along with this classical case, we have also chosen two other enzymatic systems where kinetic processes are non-Markovian. The orange solid line in Fig. 3a represents the case where \({T}_{{cat}}\) follows a Gamma distribution. Similarly, the green solid line represents the case where both \({T}_{{cat}}\) and \({T}_{{off}}\) follow Gamma distributions (see “Methods” section for details). The corresponding symbols represent simulation data points, which are in accord with theory (solid lines). These results illustrate that the microscopic details of the kinetic transitions do not change the linear dependence on the reciprocal of the substrate concentration as predicted by Eq. (8).

Fig. 3: Moments and squared coefficient of variation of the turnover time distribution.
figure 3

a The mean turnover time, \(\left\langle {T}_{{turn}}\right\rangle,\) as a function of the reciprocal of the substrate concentration, \(\left[S\right]\) (measured in units of the Michaelis-Menten constant, \({K}_{M}\)). We assume a Markovian substrate binding process and consider three different cases for catalysis and unbinding: (i) exponential catalysis and unbinding times (\(\exp :\) blue), (ii) Gamma distributed catalysis time and exponential unbinding time (\({\gamma }_{{cat}}:\) orange), and (iii) Gamma distributed catalysis and unbinding times (\({\gamma }_{{cat}},\,{\gamma }_{{off}}:\) green). Solid lines come from Eq. (8) which is corroborated by numerical simulations (symbols). In all cases the linear relation predicted by Eq. (8) is shown to hold. b Same as a, but for the second moment \(\left\langle {T}_{{turn}}^{2}\right\rangle\) of the turnover time. c Same as a, but for the third moment \(\left\langle {T}_{{turn}}^{3}\right\rangle\) of the turnover time. d Same as a, but for the squared coefficient of variation, \({{CV}}^{2}=\frac{\left\langle {T}_{{turn}}^{2}\right\rangle -{\left\langle {T}_{{turn}}\right\rangle }^{2}}{{\left\langle {T}_{{turn}}\right\rangle }^{2}}\). It can be appreciated that higher moments, and the squared coefficient of variation, are highly non-linear with respect to the reciprocal of the substrate concentration. Source data are provided as a Source Data file. Codes employed for generating these data sets have been supplied as Supplementary Software.

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Summarizing, Eq. (8) extends the basic Michaelis-Menten formula to situations where catalysis and unbinding times come from general distributions. We then have \({v}_{\max }=1/{B}_{1}={\phi }_{{cat}}/\left\langle {W}_{{ES}}\right\rangle\) and \({K}_{M}={A}_{1}/{B}_{1}=1/\left({k}_{{on}}\left\langle {W}_{{ES}}\right\rangle \right)\,.\) We thus see that the maximal turnover rate is determined by the ratio of the probability, \({\phi }_{{cat}},\) for a catalytic event to occur before unbinding and the mean waiting time, \(\left\langle {W}_{{ES}}\right\rangle\), in the \({ES}\) state. Similarly, the Michaelis-Menten constant is determined by the reciprocal of the product between the binding rate and \(\left\langle {W}_{{ES}}\right\rangle\).

Michaelis-Menten equations for high turnover time moments

As shown in the previous section, the linear relation between the mean turnover time and the reciprocal of the substrate concentration (also known as the Lineweaver–Burk plot) allows us to obtain valuable information on enzymatic activity. This is done by fitting a straight line to the data, extracting the slope \({A}_{1}\), and intercept \({B}_{1}\), of Eq. (8); and determining \({v}_{\max }=1/{B}_{1}\) and \({K}_{M}={A}_{1}/{B}_{1}.\) However, essential microscopic characteristics, such as \({k}_{{on}},\,{\phi }_{{cat}},\left\langle {W}_{{ES}}^{{cat}}\right\rangle,\) and \(\left\langle {W}_{{ES}}\right\rangle\) still remain unknown.

Higher moments of the turnover time distribution can also be analyzed to try and extract kinetic information. Yet, to date, there was no way of doing so in a manner similar to that in which kinetic information is extracted from the Michaelis-Menten equation. Indeed, one thing that quickly becomes clear is that higher moments display nonlinear dependencies on \(1/\left[S\right].\) These are illustrated in Fig. 3b, c for the second and third moments of the turnover time, calculated for the different kinetic models that were considered in Fig. 2. Complicated non-linear behaviors are also displayed by other statistical measures of enzymatic activity such as the randomness parameter, Poisson indicator and the Fano factor20,26,27,30,42. As an example, we present in Fig. 3d the squared coefficient of variation (randomness parameter), \({{CV}}^{2}=\frac{\langle {T}_{{turn}}^{2}\rangle -{\left\langle {T}_{{turn}}\right\rangle }^{2}}{{\left\langle {T}_{{turn}}\right\rangle }^{2}},\) which indeed shows a nonlinear behavior for all considered model systems. Notably, even for the classical Michaelis-Menten model, which is characterized by exponential distributions of binding, unbinding and catalytic times, the randomness parameter exhibits a nonmonotonic dependence on \(1/\left[S\right]\) (blue line and symbols in Fig. 3d).

Next, we build on Eq. (7) to reveal distinguished combinations of higher-order moments that depend linearly on the reciprocal of the substrate concentration. We show that these newly discovered relations open the door for large-scale characterization of hidden parameters in enzyme kinetics. We start by taking the reciprocal of Eq. (7) to arrive at the following relation:

$$\frac{1}{{\hat{f}}_{{T}_{{turn}}}\left(k\right)}=\left(\frac{k}{{\phi }_{{cat}}{\hat{f}}_{{W}_{{ES}}^{{cat}}}\left(k\right){k}_{{on}}}\right)\frac{1}{\left[S\right]}+\frac{1-{\hat{f}}_{{W}_{{ES}}}\left(k\right)}{{\phi }_{{cat}}{\hat{f}}_{{W}_{{ES}}^{{cat}}}\left(k\right)}+1.$$
(9)

Analogous to the Michaelis-Menten equation reported in Eq. (8), the above equation also shows a linear dependence of \(1/{\hat{f}}_{{T}_{{turn}}}\left(k\right)\) on the reciprocal of the substrate concentration. Expanding both sides of Eq. (9) in a Taylor series with respect to the Laplace variable \(k\), and equating equal order terms, we get (Supplementary Discussion 3):

$${\Omega }_{n}=\frac{{A}_{n}}{\left[S\right]}+{B}_{n}+{\delta }_{n,0},$$
(10)

where

$${\Omega }_{n}={\left.\frac{{\partial }^{n}}{\partial {k}^{n}}\left(\frac{1}{{\hat{f}}_{{T}_{{turn}}}\left(k\right)}\right)\right|}_{k=0}\frac{1}{n!}\,,$$

and

$${A}_{n}=\left(\frac{1}{{\phi }_{{cat}}{k}_{{on}}\,}\right){\left.\frac{{\partial }^{n}}{\partial {k}^{n}}\left(\frac{k}{{\hat{f}}_{{W}_{{ES}}^{{cat}}}\left(k\right)}\right)\right|}_{k=0}\frac{1}{n!}\,,$$
$${B}_{n}=\left(\frac{1}{{\phi }_{{cat}}\,}\right){\left.\frac{{\partial }^{n}}{\partial {k}^{n}}\left(\frac{1-{\hat{f}}_{{W}_{{ES}}}\left(k\right)}{{\hat{f}}_{{W}_{{ES}}^{{cat}}}\left(k\right)}\right)\right|}_{k=0}\frac{1}{n!},$$

and where \({\delta }_{n,0}\) is the Kronecker delta.

For \(n=0\) both the left-hand side and right-hand side of Eq. (10) are equal to unity, indicating that all PDFs composing this equation are properly normalized. Also, for \(n=1\), Eq. (10) reduces to Eq. (8) for the first moment. For \(n\, > \,1\) the left-hand side of Eq. (10) is a combination of turnover time moments of order \(\left\{1,\ldots,n\right\}\) whereas its right-hand side is written in terms of kinetic parameters of the enzymatic process (see Table 1 for \(n=1,\,2,\) and \(3\) and Supplementary Discussion 4 for further details). Thus, Eq. (10) provides the desired linear relationship between the turnover time moments and microscopic parameters of the enzyme. Using an analog of the Lineweaver–Burk plot, the parameters \({A}_{n}\) and \({B}_{n}\) can be found as the slope and intercept of \({\Omega }_{n}\) when plotted vs. \(1/\left[S\right]\), respectively. Next, we present and discuss these linear relations for the most important cases of \(n=2\) and \(3\), which include combinations of second and third-order moments. We then proceed to demonstrate how these relations can be used to infer key parameters of enzymatic catalysis in practice.

Table 1 Explicit expressions for \({\Omega }_{n},\,{A}_{n},{B}_{n}\) from Eq. (10), for \(n={{\mathrm{1,2,3}}}\)
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Second and third-order Michaelis-Menten equations

Considering second-order terms in Eq. (10), we get the following equation:

$${\Omega }_{2}=\overbrace{\left(\frac{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }{{\phi }_{{cat}}\,{k}_{{on}}}\right)}^{{A}_{2}}\frac{1}{\left[S\right]}+\overbrace{\frac{2\left\langle {W}_{{ES}}^{{cat}}\right\rangle \left\langle {W}_{{ES}}\right\rangle -\left\langle {W}_{{ES}}^{2}\right\rangle }{{2\phi }_{{cat}}}}^{{B}_{2}},$$
(11)

where \({\Omega }_{2}={\left\langle {T}_{{turn}}\right\rangle }^{2}-\frac{\left\langle {T}_{{turn}}^{2}\right\rangle }{2}\) is the specific combination of \(n\le 2\) turnover time moments, which shows a linear dependence on \(1/\left[S\right]\) (see Fig. 4a). For the classical Michaelis-Menten model, characterized by exponential time distributions, the various PDFs that determine the form of Eq. (10), \({f}_{{W}_{{ES}}^{{cat}}}\left(t\right),\,{f}_{{W}_{{ES}}^{{off}}}\left(t\right)\) and \({f}_{{W}_{{ES}}}\left(t\right)\), have exactly the same functional form: \(({k}_{{cat}}+{k}_{{off}}){e}^{-\left({k}_{{cat}}+{k}_{{off}}\right)t}\) (Supplementary Discussion 5). Thus, in this case \(\left\langle {W}_{{ES}}\right\rangle=\left\langle {W}_{{ES}}^{{cat}}\right\rangle\) and we have \(2\left\langle {W}_{{ES}}^{{cat}}\right\rangle \left\langle {W}_{{ES}}\right\rangle -\left\langle {W}_{{ES}}^{2}\right\rangle=2{\left\langle {W}_{{ES}}\right\rangle }^{2}-\left\langle {W}_{{ES}}^{2}\right\rangle=0\), where the last equality follows from basic properties of the exponential distribution. The parameter \({B}_{2}\) which gives the intersection of the line \({\Omega }_{2}\) with the vertical axis, is then equal to zero (blue line in Fig. 4a). However, this is not true in general. For non-exponential distributions of catalytic and unbinding times, \({B}_{2}\) can be nonzero (orange and green lines in Fig. 4a). Thus, a non-zero value of \({B}_{2}\) can serve as an indicator of the non-classical nature of the enzyme being studied.

Fig. 4: High-order Michaelis-Menten equations.
figure 4

a Second order Michaelis-Menten equation. The moments combination, \({\Omega }_{2}={\left\langle {T}_{{turn}}\right\rangle }^{2}-\frac{\left\langle {T}_{{turn}}^{2}\right\rangle }{2}\), as a function of the reciprocal of the substrate concentration, \(\left[S\right]\) (measured in units of the Michaelis constant, \({K}_{M}\)). Solid lines come from Eq. (11) which is corroborated by numerical simulations (symbols). In all cases the linear relation predicted by Eq. (11) is shown to hold. b Third order Michaelis-Menten equation. The moments combination, \({\Omega }_{3}=\frac{\left\langle {T}_{{turn}}^{3}\right\rangle }{6}-\left\langle {T}_{{turn}}\right\rangle \left\langle {T}_{{turn}}^{2}\right\rangle+{\left\langle {T}_{{turn}}\right\rangle }^{3}\), as a function of the reciprocal of the substrate concentration (measured in units of the Michaelis constant). Solid lines come from Eq. (12) which is corroborated by numerical simulations (symbols). In all cases the linear relation predicted by Eq. (12) is shown to hold. For both (a) and (b), we assume a Markovian substrate binding process and consider three different cases for catalysis and unbinding: (i) exponential catalysis and unbinding times (\(\exp :\) blue), (ii) Gamma distributed catalysis time and exponential unbinding time (\({\gamma }_{{cat}}:\) orange), and (iii) Gamma distributed catalysis and unbinding times (\({\gamma }_{{cat}},\,{\gamma }_{{off}}:\) green) [see “Methods” section for details]. Source data are provided as a Source Data file. Codes employed for generating these data sets have been supplied as Supplementary Software.

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Considering third-order terms in Eq. (10) we get the following equation:

$${\Omega }_{3} =\overbrace{\left(\frac{{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }^{2}-\frac{\left\langle {\left({W}_{{ES}}^{{cat}}\right)}^{2}\right\rangle }{2}}{{\phi }_{{cat}}\,{k}_{{on}}}\right)}^{{A}_{3}}\frac{1}{\left[S\right]}\, \\ \quad+\overbrace{\frac{\left(\frac{\left\langle {W}_{{ES}}^{3}\right\rangle }{6}-\frac{\left\langle {W}_{{ES}}^{{cat}}\right\rangle \left\langle {W}_{{ES}}^{2}\right\rangle }{2}+\left\langle {W}_{{ES}}\right\rangle {\left\langle {W}_{{ES}}^{{cat}}\right\rangle }^{2}-\frac{\left\langle {W}_{{ES}}\right\rangle \left\langle {\left({W}_{{ES}}^{{cat}}\right)}^{2}\right\rangle }{2}\right)}{{\phi }_{{cat}}}}^{{B}_{3}},$$
(12)

where \({\Omega }_{3}=\frac{\left\langle {T}_{{turn}}^{3}\right\rangle }{6}-\left\langle {T}_{{turn}}\right\rangle \left\langle {T}_{{turn}}^{2}\right\rangle+{\left\langle {T}_{{turn}}\right\rangle }^{3}\) is the specific combination of \(n\le 3\) turnover time moments, which shows a linear dependence on \(1/\left[S\right]\) (see Fig. 4b). Here too, we see a clear difference between the classical Michaelis-Menten model which is characterized by exponential time distributions, and non-Markovian scenarios. Specifically, in the Markovian case we find that \({{{{\rm{A}}}}}_{3}={{{{\rm{B}}}}}_{3}=0\) (Supplementary Discussion 6), i.e., the slope and the intercept in Eq. (12) vanish (blue line in Fig. 4b). Once again, note that this is not true in general. For non-exponential distributions of catalysis and unbinding times, \({A}_{3}\) and \({B}_{3}\) can be nonzero (orange and green lines in Fig. 4b).

Inferring hidden kinetic parameters from high-order Michaelis-Menten equations

Considering Eqs. (8) and (11) for \({\Omega }_{1}\) and \({\Omega }_{2}\), respectively, we observe that they include four measurable quantities, \({A}_{1},\,{B}_{1},\,{A}_{2},\,{B}_{2}\), and five unknown kinetic parameters, \({\phi }_{{cat}},\,{k}_{{on}},\left\langle {W}_{{ES}}\right\rangle,\,\left\langle {W}_{{ES}}^{{cat}}\right\rangle,\,\left\langle {W}_{{ES}}^{2}\right\rangle\), that characterize enzymatic activity (see Table 1). Since an additional equation is missing, these kinetic parameters cannot be uniquely determined only by analyzing the dependence of \({\Omega }_{1}\) and \({\Omega }_{2}\) on the reciprocal of the substrate concentration. This problem cannot be resolved by considering combinations of higher turnover time moments, \({\Omega }_{n}\), with \(n\, > \,2\). Indeed, for any \(M\), the set of \({\Omega }_{n}\), with \(n\le M\) allows to obtain \(2M\) quantities, \({A}_{n}\) and \({B}_{n}\), the values of which are determined by \(2M+1\) unknown kinetic parameters. In particular, plotting \({\Omega }_{1}\), \({\Omega }_{2}\) and \({\Omega }_{3}\) vs. \(1/[S]\) one can find six quantities, \({A}_{n}\) and \({B}_{n}\) for \(n\le 3\), which are expressed in terms of seven unknown kinetic parameters, \({\phi }_{{cat}},\,{k}_{{on}},\left\langle {W}_{{ES}}\right\rangle,\,\left\langle {W}_{{ES}}^{{cat}}\right\rangle,\,\left\langle {W}_{{ES}}^{2}\right\rangle\), \(\left\langle {({W}_{{ES}}^{{cat}})}^{2}\right\rangle\) and \(\left\langle {W}_{{ES}}^{3}\right\rangle\) (see Table 1).

From here, one can proceed in multiple ways. First, observe that even if just one of the unknown kinetic parameters can be measured independently, this will provide a closure relation for the set of equations and allow for unique determination of all the other unknown parameters. However, this entails direct measurements of e.g., the splitting probability \({\phi }_{{cat}}\) or the mean time spent in the \({ES}\) state, i.e., \(\left\langle {W}_{{ES}}\right\rangle\). While such direct measurements may be hard to perform, it is noteworthy that when they are feasible, they can be synergistically combined with the framework developed herein to infer hidden kinetic parameters.

Interestingly, missing information can also come from carefully examining other features of the turnover time distribution. In this paper, we focused on moments of the turnover time, but it has recently been shown that short-time behavior also holds valuable kinetic information43,44,45. Specifically, we have recently shown that the short time behavior of the enzymatic turnover time holds information that allows one to uniquely determine the binding rate\(\,{k}_{{on}}\), provided some mild assumptions hold46. Plugging in the value of the inferred binding rate into the higher-order Michaelis-Menten equations we have developed herein, provides the required closure and allows one to uniquely determine the values of all the other kinetic parameters.

Finally, we show that one can proceed even in the absence of additional information to obtain approximate estimates of the unknown kinetic parameters. In what follows, we show that these approximations are exact for Markovian enzymes, and then proceed to investigate the role of non-Markovianity. For this purpose, using Table 1, it is useful to present the desired quantities, such as the binding rate \({k}_{{on}},\) and the conditional probability for catalysis to happen before unbinding \({\phi }_{{cat}}\), in the following form:

$${k}_{{on}}=\left(\frac{{B}_{1}}{{A}_{2}}\right){\left(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\right)}^{-1},$$
(13)

and

$${\phi }_{{cat}}=\left(\frac{{A}_{2}}{{A}_{1}{B}_{1}}\right)\left(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\right).$$
(14)

From Table 1, we also see that the mean conditional catalysis time \(\left\langle {W}_{{ES}}^{{cat}}\right\rangle\) can be calculated using the measured slopes of \({\Omega }_{1}\), and \({\Omega }_{2}\) vs. \(1/[S]\) without any additional assumptions and written as:

$$\left\langle {W}_{{ES}}^{{cat}}\right\rangle=\left(\frac{{A}_{2}}{{A}_{1}}\right).$$
(15)

Multiplying both sides of Eq. (15) by \(\left\langle {W}_{{ES}}\right\rangle /\left\langle {W}_{{ES}}^{{cat}}\right\rangle\), we can write the mean time in the bound enzymatic state as:

$$\,\left\langle {W}_{{ES}}\right\rangle=\left(\frac{{A}_{2}}{{A}_{1}}\right)\,\left(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\right),$$
(16)

which has a similar form to Eqs. (13) and (14).

Equations (13), (14), and (16) still do not allow us to determine \({k}_{{on}}\), \({\phi }_{{cat}}\) and \(\left\langle {W}_{{ES}}\right\rangle\), since in addition to the measurable quantities, \(\left(\frac{{B}_{1}}{{A}_{2}}\right)\), \(\left(\frac{{A}_{2}}{{A}_{1}{B}_{1}}\right)\) and \(\left(\frac{{A}_{2}}{{A}_{1}}\right)\) they also include the unknown ratio \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\). Yet, recall that we have just shown that for classically behaving enzymes, where all the kinetic processes are Markovian, we have \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=1\). In this case, \({k}_{{on}}\) and \({\phi }_{{cat}}\) can be directly determined from the slopes (\({A}_{1}\), and \({A}_{2}\)) and intercept (\({B}_{1}\)) of the \({\Omega }_{1}\) and \({\Omega }_{2}\) lines, as follows:

$${k}_{{on}}=\left(\frac{{B}_{1}}{{A}_{2}}\right),\,{\phi }_{{cat}}=\left(\frac{{A}_{2}}{{A}_{1}{B}_{1}}\right).$$
(17)

In addition, the rates of unbinding and catalysis can also be inferred from the slopes (\({A}_{1}\), and \({A}_{2}\)) and intercept (\({B}_{1}\)) as:

$${k}_{{off}}=\frac{{A}_{1}}{{A}_{2}}-\frac{1}{{B}_{1}},{{{\rm{and}}}}\,{k}_{{cat}}=\frac{1}{{B}_{1}}.$$
(18)

It should be noted that traditional analysis of enzymatic kinetics, which is based only on measurements of the mean turnover times, cannot provide this important information.

Equations (17) and (18) instruct us on how to infer all the characteristic rates, and the fertility fraction (splitting probability) of catalysis, for Markovian enzymes. However, single molecule measurements are intrinsically stochastic and therefore the accuracy of the inferred kinetic parameters depends on the amount of available data. To address this important issue, we present in Fig. 5 boxplots, characterizing spread and skewness of the inferred kinetic parameters as a function of the number of turnover events per substrate concentration, N. Data were obtained by simulating the classical Michaelis-Menten model with the same parameters as in Fig. 3a (blue line and symbols, see also “Methods” section).

Fig. 5: Accuracy of the inferred kinetic parameters as a function of the sample size for the Markovian model.
figure 5

Boxplots for the inferred a binding rate \(\left({k}_{{on}}^{\inf }\right)\), b splitting probability of catalysis \(\left({\phi }_{{cat}}^{\inf }\right)\), c unbinding rate \(\left({k}_{{off}}^{\inf }\right)\), and d catalysis rate \(\left({k}_{{cat}}^{\inf }\right)\) normalized by the corresponding true values \(\left({k}_{{on}},\,{\phi }_{{cat}},\,{k}_{{off}},{k}_{{cat}}\right)\) and plotted as a function of the number of turnover events, \(N\), per substrate concentration. For each \(N,\) \({10}^{3}\) independent repetitions were performed. The bottom and top of each box are the 25th and 75th percentiles of the sample, respectively, red lines mark the median value, and whiskers extend to 1.5 times the interquartile range. Source data are provided as a Source Data file. Codes employed for generating these data sets have been supplied as Supplementary Software.

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The inference procedure was implemented as follows. The quantities \({\Omega }_{1}\) and \({\Omega }_{2}\) were estimated based on their relation to the turnover time moments (First column of Table 1). The results were plotted as a function of the reciprocal of the substrate concentration and fitted to obtain estimates for \({A}_{1},\,{B}_{1},{A}_{2},\) and \({B}_{2}.\) Eqs. (17) and (18) were then used to obtain estimates of \({k}_{{on}},\,{\phi }_{{cat}},\,{k}_{{off}}\) and \({k}_{{cat}}.\) To assess the fidelity of this procedure, it was repeated \({10}^{3}\) times. The presented results show that \({10}^{3}\) turnover events per substrate concentration were already enough to produce reasonable estimates, and that \({10}^{4}\) turnover events per substrate concentration gave good estimates. More data leads to an even better accuracy, such that \({10}^{6}\) turnover events provide near-perfect estimates. We quantify this improving trend in Supplementary Fig. 2, as part of Supplementary Discussion 7. There we show that the normalized standard deviations of our estimates obey the typical \(\sim 1/\sqrt{N}\) scaling law.

Kinetic inference for non-Markovian enzymes

For non-Markovian enzymes, one can estimate \({k}_{{on}}\), \({\phi }_{{cat}},\) and \(\left\langle {W}_{{ES}}\right\rangle\) by plugging in the closure relation \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=1\) in Eqs. (13), (14) and (16). Yet, the quality of this approximation depends on whether the unknown ratio does indeed remain relatively close to unity. In other words, any deviation of \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\) from unity indicates non-Markovian behavior, and the exact value of this ratio provides a measure for the magnitude of the error made when the closure relation is applied. For example, if \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=2\), applying the closure relation \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=1\) in Eq. (13) will overestimate the true values of \({k}_{{on}}\) by a factor of two, and similar logic carries to the error made when using the closure relation to evaluate \({\phi }_{{cat}}\) and \(\left\langle {W}_{{ES}}\right\rangle\) via Eqs. (14) and (16), respectively.

It is thus important to understand in which cases \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\approx 1\) provides a fair approximation for non-Markovian enzymes. To answer this question, we now analyze two prevalent, yet distinctly different, types of non-Markovian behavior in enzymes. Namely, non-Markovian behavior that arises due to sequential catalytic steps and non-Markovian behavior that arises due to parallel catalytic pathways. Non-exponential catalysis times are common to both scenarios. Yet, for sequential catalytic steps we have catalysis time distributions that are narrower than the exponential and \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\, < \,1\), whereas for parallel catalytic pathways we have catalysis time distributions that are broader than the exponential and \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\, > \,1\).

Let us first consider a model involving two sequential catalytic steps (Fig. 6a). In this model: (1) a free-enzyme, \(E\), can reversibly bind a substrate \(S\) with rate \({k}_{{on}}[S]\), to form an enzyme-substrate complex, \({{ES}}_{1}\); (2) the complex \({{ES}}_{1}\) can then be irreversibly transformed into another complex, \({{ES}}_{2}\), with rate \({k}_{{cat}}^{1}\) or, alternatively, unbind with rate \({k}_{{off}}\); and finally (3) the complex \({{ES}}_{2}\) can be converted by the enzyme to form a product, \(P\), with rate \({k}_{{cat}}^{2}\) or unbind with rate \({k}_{{off}}\). Note that the kinetic scheme in Fig. 6a can be mapped onto the renewal kinetic scheme in Fig. 1a by coarse graining \({{ES}}_{1}\) and \({{ES}}_{2}\) into a single \({ES}\) state and defining \({T}_{{off}}\) and \({T}_{{cat}}\) accordingly (Supplementary Discussion 8). This fact allows us to use the results derived in the previous sections.

Fig. 6: Sequential reaction mechanism.
figure 6

a A schematic representation of a model involving two sequential catalytic steps. The free enzyme, \(E\) undergoes binding with a substrate to form the first substrate-bound conformer \(E{S}_{1}\) that sequentially transitions to the second bound-enzymatic state, \(E{S}_{2}.\) This state can lead to the formation of the product, \(P,\) through an irreversible catalytic event. From the bound states \(E{S}_{1}\) and \(E{S}_{2}\), unbinding events can occur—leading back to the free enzyme. The described kinetic processes are characterized by the rates of binding \(\left({k}_{{on}}\left[S\right]\right)\) where \(\left[S\right]\) is the substrate concentration, unbinding \(\left({k}_{{off}}\right)\), the first catalytic transition \(\left({k}_{{cat}}^{1}\right)\), and the second catalytic transition that forms the product \(\left({k}_{{cat}}^{2}\right)\). b 2D color map of the error factor \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\,\) in the space of the normalized catalytic rates \(\frac{{k}_{{cat}}^{1}}{{k}_{{off}}}\) and \(\frac{{k}_{{cat}}^{2}}{{k}_{{off}}}.\,\)Here \(\left\langle {W}_{{ES}}\right\rangle\) and \(\left\langle {W}_{{ES}}^{{cat}}\right\rangle\) are the mean waiting times of the bound-enzymatic state and the conditional catalysis times, respectively.

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For the sequential scheme in Fig. 6a, the ratio \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\) can be calculated analytically, and we find it is given by (Supplementary Discussion 9):

$$\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=\frac{{k}_{{cat}}^{1}+{k}_{{cat}}^{2}+{k}_{{off}}}{{k}_{{cat}}^{1}+{k}_{{cat}}^{2}+2{k}_{{off}}}\cdot$$
(19)

Importantly, Eq. (19) asserts that for any choice of kinetic parameters the ratio lies in the range of \(1/2\le \frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\, < \,1,\) which provides a tight bound on the error coming from the use of the closure relation. We visually illustrate this in Fig. 6b which presents a 2D color map of \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle },\) calculated in the space of the normalized catalytic rates \(\frac{{k}_{{cat}}^{1}}{{k}_{{off}}}\) and \(\frac{{k}_{{cat}}^{2}}{{k}_{{off}}}\). It can be seen that the largest deviations from the classical Markovian behavior, \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=1\), occur when the unbinding rate is much larger than both catalytic rates, \({k}_{{off}}\,{\gg \, k}_{{cat}}^{1}+{k}_{{cat}}^{2}\).

Summarizing, for enzymes that follow the successive catalytic scheme illustrated in Fig. 6a, the desired kinetic parameters \({k}_{{on}}\) and \({\phi }_{{cat}}\) can be well estimated by setting \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=1\) in Eqs. (13) and (14). Also, the mean waiting time in the coarse grained \({ES}\) state can be estimated via Eq. (16) since under the above closure relation, \(\left\langle {W}_{{ES}}\right\rangle=\left\langle {W}_{{ES}}^{{cat}}\right\rangle\). Using the closure relation \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=1\), results in an error factor of no more than two. For example, the binding rate is underestimated compared to its true value, but in no more than a factor of two. In Supplementary Discussion 10, we test the sensitivity of our inference procedure to the amount of available data. For the sequential reaction mechanism described above, we reach similar conclusions as those obtained for the Markovian case where all kinetic processes were exponentially distributed (Supplementary Fig. 3).

We also extended the above analysis to kinetic models with \(n\) identical sequential steps (Supplementary Discussion 11). We show that the maximal error factor is then given by \(n\). Since for real enzymes the value of \(n\) is rather limited, the closure relation \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=1\) can be applied to sequential enzymes yielding estimates that are well within an order of magnitude to the true parameter values.

Next, we consider a reaction scheme involving two parallel catalytic pathways (Fig. 7a). In this model: (1) a free-enzyme, \(E\), reversibly binds a substrate, \(S\), to form either the enzyme-substrate complex \(E{S}_{1}\) or the enzyme-substrate complex \(E{S}_{2}\) with rates \(p{k}_{{on}}[S]\) and \(\left(1-p\right){k}_{{on}}[S]\), respectively, where \(p\) is the splitting probability between pathways; (2) both complexes can then be converted to a product, \(P\), with rates \({k}_{{cat}}^{1}\) or \({k}_{{cat}}^{2}\,\), respectively; or unbind with a common rate \({k}_{{off}}\). Unlike the sequential scheme, here the formation of a product can occur from both enzyme-substrate complexes, \(E{S}_{1}\) and \(E{S}_{2}\). We once again note that the kinetic scheme in Fig. 7a can also be described in the language of Fig. 1a by coarse-graining \({{ES}}_{1}\) and \({{ES}}_{2}\) into a single \({ES}\) state and defining \({T}_{{off}}\) and \({T}_{{cat}}\) accordingly (Supplementary Discussion 12). This fact allows us to use the results derived in the previous sections.

Fig. 7: Parallel reaction mechanism.
figure 7

a A schematic representation of a model involving two parallel catalytic steps. The free enzyme, \(E\) can bind with a substrate molecule via two parallel processes to form the first substrate-bound conformer, \(E{S}_{1}\) or the second bound-enzymatic state, \(E{S}_{2}.\) These branched substrate-binding channels have splitting probabilities \(p\) and \(1-p\), respectively. Both bound states can form the product, \(P\) through irreversible catalytic events. From \(E{S}_{1}\) and \(E{S}_{2}\), unbinding events can occur that lead back to the free enzyme. The described kinetic processes are characterized by the rates of binding \(\left({k}_{{on}}\left[S\right]\right)\) where \(\left[S\right]\) is the substrate concentration, unbinding \(\left({k}_{{off}}\right)\), the first \(\left({k}_{{cat}}^{1}\right)\) and second \(\left({k}_{{cat}}^{2}\right)\) catalytic transitions, respectively. b 2D color map of the error factor \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\) in the space of the normalized catalytic rates \(\frac{{k}_{{cat}}^{1}}{{k}_{{off}}}\) and \(\frac{{k}_{{cat}}^{2}}{{k}_{{off}}}\), for the splitting probability \(p=0.5\). Here \(\left\langle {W}_{{ES}}\right\rangle\) and \(\left\langle {W}_{{ES}}^{{cat}}\right\rangle\) are the mean waiting times of the bound-enzymatic state and the conditional catalysis times, respectively. Note that for \(\frac{{k}_{{cat}}^{1}}{{k}_{{off}}}\) and \(\frac{{k}_{{cat}}^{2}}{{k}_{{off}}}\) that span four orders of magnitude, the error factor is always <30, and in most cases significantly smaller than this upper bound.

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For the parallel kinetic scheme, the ratio \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\) can be calculated analytically, and we find it is given by (Supplementary Discussion 13):

$$\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=\frac{\left({k}_{{cat}}^{1}+{k}_{{off}}+p\left({k}_{{cat}}^{2}-{k}_{{cat}}^{1}\right)\right)\left(\left(1-p\right){k}_{{cat}}^{2}{k}_{{off}}+{k}_{{cat}}^{1}\left({k}_{{cat}}^{2}+p{k}_{{off}}\right)\right)}{p{k}_{{cat}}^{1}{\left({k}_{{cat}}^{2}+{k}_{{off}}\right)}^{2}+\left(1-p\right){k}_{{cat}}^{2}{\left({k}_{{cat}}^{1}+{k}_{{off}}\right)}^{2}}.$$
(20)

Analysis of this equation asserts that the ratio \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\) exhibits the strongest deviations from unity for the splitting probability \(p=0.5\), i.e., when the two enzyme-substrate complexes are formed with equal probability. We thus focus on this worst-case scenario.

The corresponding 2D color map of \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\,\), calculated as a function of the normalized catalytic rates\(\,\frac{{k}_{{cat}}^{1}}{{k}_{{off}}}\) and \(\frac{{k}_{{cat}}^{2}}{{k}_{{off}}}\) is presented in Fig. 7b. This map clearly demonstrates that in most of the parameter space the ratio \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }\) is close to unity, albeit in small corner regions where significant deviations from unity are observed. This happens under extreme conditions where the normalized catalytic rates,\(\,\frac{{k}_{{cat}}^{1}}{{k}_{{off}}}\) and \(\frac{{k}_{{cat}}^{2}}{{k}_{{off}}}\), differ from each other by more than three orders of magnitude; and note that in all other cases applying the closure relation \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=1\) in Eqs. (13), (14), and (16) will lead to estimates that are withing an order of magnitude from true values. Similar analysis applies to asymmetric cases where one of the catalytic routes is preferentially chosen over the other (\(p\ne 0.5\)), albeit systematic estimation errors being even smaller (Supplementary Discussion 14). Thus, similar to the sequential scheme, setting \(\frac{\left\langle {W}_{{ES}}\right\rangle }{\left\langle {W}_{{ES}}^{{cat}}\right\rangle }=1\) as a closure relation in the parallel kinetic scheme leads to very good approximations for the values of kinetic parameters that are otherwise very hard to estimate.

Discussion

In this paper, we presented a pioneering approach to enzyme kinetics at the single-molecule level. Building on a renewal approach to enzyme kinetics, we derived a set of high-order Michaelis-Menten equations that reveal universal linear dependencies between unique combinations of turnover time moments and the reciprocal of the substrate concentration. These relations effectively generalize the single-molecule Michaelis-Menten equation from the mean turnover time to moments of arbitrary order.

We demonstrated the capacity of our approach to infer concealed kinetic parameters, such as the lifetime of the enzyme-substrate complex, the rate of binding, and the probability of successful product formation. With this, we pave the way for systematic characterization of enzyme kinetics from single-molecule data, going well beyond the limited information provided by the all-familiar Michaelis-Menten constant and maximal reaction velocity.

The results apply to Markovian and non-Markovian enzymes alike. For Markovian enzymes, which adhere to memoryless stochastic transitions between states, the framework provides a way to accurately estimate kinetic parameters such as the binding, unbinding and catalysis rates. By leveraging the high-order Michaelis-Menten equations, which capture universal linear dependencies, precise determination of these parameters is achieved.

For non-Markovian enzymes, where kinetic processes exhibit memory effects, our framework is also exact but incomplete. Namely, for inference using our framework to be exact, one of the unknown kinetic parameters, e.g., the binding rate, must be measured or inferred independently. This is not a serious limitation as the binding rate can be inferred based on information held in the short-time tail of the turnover time distribution, provided some mild assumptions hold46. Combining such additional information with the results derived herein gives a closed and complete inference framework for non-Markovian enzymes.

Moreover, our framework can be used on non-Markovian enzymes even in lieu of external information on the binding rate or other kinetic parameters. To this end, one assumes that the mean time spent in the enzyme-substrate complex is identical to the (conditional) mean time spent in this complex provided product formation is the direct result of the enzyme-substrate encounter. While this approximation introduces some error, we find that this is typically limited. We demonstrated this by analyzing kinetic models with sequential and parallel product formation pathways. These models display canonical, yet completely orthogonal, types of non-Markovian behavior. Nevertheless, in all cases considered, we showed that applying the above assumption provides estimates that are within an order of magnitude from the true value of kinetic parameters that would otherwise remain completely unknown.

In summary, through the derivation of high-order Michaelis-Menten equations, we have shown that traditional analysis methods in enzymology can be extended to unveil central kinetic parameters that have so far remained hidden. We thus offer a systematic and straightforward approach to characterize enzyme kinetics far and beyond the current state of the art.

Methods

Kinetic parameters used in Figs. 2–4

The results presented in Figs. 2–4 were obtained considering three different catalytic models: (1) exponential catalysis and unbinding times (blue), (2) Gamma distributed catalysis times and exponential unbinding times (orange), and (3) Gamma distributed catalysis and unbinding times (green). Data coming from numerical simulations, histograms in Fig. 2 and symbols in Figs. 3 and 4, were obtained by taking \({10}^{6}\) turnover events at each substrate concentration, for each one of the cases discussed below.

The substrate binding event is common in all three enzymatic systems. It is governed by an exponential distribution with a mean binding time of \(\left\langle {T}_{{on}}\right\rangle={\left({k}_{{on}}\left[S\right]\right)}^{-1}\equiv {\left(5*\left[S\right]\right)}^{-1}\,[{{{\rm{s}}}}]\). In the first model, exponential catalysis and unbinding times, the mean unbinding and catalysis times were taken to be \(\left\langle {T}_{{off}}\right\rangle={\left({k}_{{off}}\right)}^{-1}=1/10\,[{{{\rm{s}}}}]\) and \(\left\langle {T}_{{cat}}\right\rangle={\left({k}_{{cat}}\right)}^{-1}=1/15\,[{{{\rm{s}}}}]\), respectively. Thus, following the definition of the Michaelis-Menten constant, we have \({K}_{M}=\frac{{k}_{{cat}}+{k}_{{off}}}{{k}_{{on}}}=5\,[{{{\rm{\mu }}}}{{{\rm{M}}}}].\) This concentration was also used in Fig. 2 where we have set \(\left[S\right]=5\,[{{{\rm{\mu }}}}{{{\rm{M}}}}]\) throughout. The substrate concentrations used in Figs. 3 and 4 lie in the range: \(0.2{K}_{M}-4{K}_{M}\), where the reciprocal of the substrate concentrations are regularly spaced to ensure uniform coverage of the horizontal axis. All concentrations were normalized by the Michaelis-Menten constant above, i.e., \({K}_{M}=5\,[{{{\rm{\mu }}}}{{{\rm{M}}}}]\).

In the second model, Gamma distributed catalysis times and exponential unbinding times, the unbinding time distribution was taken to be the same as in the first case, and catalysis times were taken to follow a Gamma distribution of the form, \({f}_{{T}_{{cat}}}\left(t\right)=\frac{{e}^{\frac{-t}{\theta }}{t}^{k-1}}{{\theta }^{k}\Gamma \left[k\right]},\) with shape factor \(k=6\frac{2}{3}\) and scale factor \(\theta=\frac{1}{100}\,[{{{\rm{s}}}}].\) The parameters of the Gamma distribution were chosen such that the mean catalysis time for this system, \(\left\langle {T}_{{cat}}\right\rangle=k\theta=1/15\,[{{{\rm{s}}}}],\) would be identical to the mean catalysis time of the exponential case, \(\left\langle {T}_{{cat}}\right\rangle=\frac{1}{{k}_{{cat}}}=1/15\,[{{{\rm{s}}}}]\), thereby allowing for a fair comparison.

In the third model, both the catalytic and unbinding times follow Gamma distributions. The parameters for the catalytic time distribution are the same as in the second case, while the unbinding time distribution has a form, \({f}_{{T}_{{off}}}\left(t\right)=\) \(\frac{{e}^{\frac{-t}{\theta }}{t}^{k-1}}{{\theta }^{k}\Gamma \left[k\right]},\) with shape factor \(k=10\) and scale factor \(\theta=\frac{1}{100}\,[{{{\rm{s}}}}]\). Note that these kinetic parameters were intentionally chosen such that the mean unbinding time, \(\left\langle {T}_{{off}}\right\rangle=k\theta=\frac{1}{10}[{{{\rm{s}}}}]\), is exactly equal to the inverse of the unbinding rate constant used for the exponential case, \(\left\langle {T}_{{off}}\right\rangle=\frac{1}{{k}_{{off}}}=\frac{1}{10}[{{{\rm{s}}}}].\,\)

In Supplementary Fig. 1, we give histograms of the catalysis and unbinding times that were used in the three models described above. It can be seen that these underlying distributions are qualitatively different from one another, which emphasizes our main point: the set of high-order Michaelis-Menten equations is valid regardless of the underlying kinetics.

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