Including Rebinding Reactions in Well-Mixed Models of Distributive Biochemical Reactions

Abstract

The behavior of biochemical reactions requiring repeated enzymatic substrate modification depends critically on whether the enzymes act processively or distributively. Whereas processive enzymes bind only once to a substrate before carrying out a sequence of modifications, distributive enzymes release the substrate after each modification and thus require repeated bindings. Recent experimental and computational studies have revealed that distributive enzymes can act processively due to rapid rebindings (so-called quasi-processivity). In this study, we derive an analytical estimate of the probability of rapid rebinding and show that well-mixed ordinary differential equation models can use this probability to quantitatively replicate the behavior of spatial models. Importantly, rebinding requires that connections be added to the well-mixed reaction network; merely modifying rate constants is insufficient. We then use these well-mixed models to suggest experiments to 1) detect quasi-processivity and 2) test the theory. Finally, we show that rapid rebindings drastically alter the reaction’s Michaelis-Menten rate equations.

Introduction

Modification of cellular proteins by phosphorylation, methylation, ubiquitination, lipidation, acetylation, hydroxylation, and glycosylation are fundamental mechanisms for cell signaling and regulation. It is well known that the qualitative behavior of these systems depends critically on whether or not the modifying molecule acts on the protein in a processive or distributive manner. Whereas processive enzymes bind only once to a substrate before carrying out a sequence of modifications, distributive enzymes release the substrate after each modification and thus require repeated bindings. Many groups have studied the effect that processive and distributive enzymes can have on reaction network dynamics. For example, various experimental and mathematical studies have implicated processivity versus distributivity as a factor underlying ultrasensitivity (1, 2, 3), thresholding (4), bistability (5, 6), and oscillations (7, 8) in the mitogen-activated protein kinase cascade (MAPK).

Recent in vitro experimental results by Aoki et al. show that rapid enzyme-substrate rebindings can drastically alter the dynamics of the MAPK pathway (9). Using particle-based simulations in space and time, Takahashi et al. similarly demonstrated the dramatic effect that rebindings can have on the MAPK pathway (10). In essence, the experimental results of (9) and the numerical computations of (10) show that rapid enzyme-substrate rebindings can turn a distributive mechanism into a processive mechanism (so-called quasi-processive). Based on these results, the authors of (10) argue for the importance of spatial models of such systems, despite the computational expense. That is, because ordinary differential equation (ODE) models (models based on mean-field chemical rate equations) ignore spatiotemporal correlations and thus ignore rebindings, it is argued that one must resort to models that explicitly include space to properly account for the effects of rebinding.

Complementing the experimental results of (9) and the computational results of (10), in this study we use mathematical analysis to study these rebindings. In particular, we derive an analytical formula (first derived in (11)) for the probability of rapid enzyme-substrate rebinding that depends on only two dimensionless parameters. This formula therefore explicitly shows how different biophysical parameters contribute to quasi-processivity and allows one to predict if and to what degree a given biochemical reaction will be quasi-processive.

Importantly, we show that if an ODE model incorporates this formula, then it accurately mimics the dynamics of a spatial model. These precise fits are highly significant as they negate some of the impetus for spatial models and therefore allow for a drastic decrease in computational expense and complexity.

In addition to being computationally cheaper than spatial models, ODE models are amenable to analysis. We exploit this feature and analyze the ODE model to suggest experiments to 1) detect quasi-processivity and 2) test the theory. We further analyze the ODE model to determine how rebindings alter the Michaelis-Menten reduced rate equations. We find that the effect is dramatic.

We now outline the article and describe how it relates to previous work. In Materials and Methods, we show how to modify a reaction network to include the probability of rapid rebinding, and we derive an analytical formula for this probability. Unbeknownst to the current authors, the essential results of this section were first found in (11). In contrast to (11), we show in Results and Discussion that ODE models that incorporate this probability can quantitatively replicate spatial models. Related investigations into the consequences of quasi-processivity were made in (12, 13, 14). In our study, we take advantage of the analytic tractability of ODE models. Our analysis extends the methods developed in (15) to analyze processivity. We determine how rebindings alter the Michaelis-Menten reduced rate equations, and use these equations to investigate the effect of rebindings on chemical reactions with many steps. We conclude by highlighting several other biochemical systems where rebinding may play a significant role.

Materials and Methods

Review of Smoluchowski-Collins-Kimball theory

To set up the analysis of rebindings, we first review the Smoluchowski-Collins-Kimball theory of diffusion-controlled reaction rates (16, 17). Consider the following simple reaction of a substrate binding to an enzyme to form a complex:

$$S+E\stackrel{a}{\to }SE,$$

where a is the intrinsic association rate that is the association rate for two molecules in contact. Incorporating diffusion essentially entails renormalizing the association rate to include the time it takes for molecules to find each other. The probability density, $p\left(r,t\right)$, that two diffusing spherical molecules are distance r from each other at time t satisfies the following:

$$\frac{\partial p}{\partial t}=D\Delta p,r>\epsilon ,$$

where $D>0$ and $\epsilon >0$ are the sum of their diffusion coefficients and radii, respectively. The following partially absorbing (Robin) boundary condition is imposed to model binding:

$$4\pi {\epsilon }^{2}D\frac{d}{dr}p\left(\epsilon ,t\right)=ap\left(\epsilon ,t\right),$$

and the probability density is taken to be fixed at spatial infinity, ${\text{lim}}_{r\to \infty }p\left(r,t\right)=1$. The diffusion-influenced association rate α is then defined to be the flux of p at $r=\epsilon$ at large time, which is (after some calculation) the following:

$$\alpha \left(\epsilon ,D,a\right):=\underset{t\to \infty }{\text{lim}}4\pi {\epsilon }^{2}D\frac{d}{dr}p\left(\epsilon ,t\right)=\frac{4\pi \epsilon Da}{4\pi \epsilon D+a}.$$ (1.1)

Throughout this article we use α to denote diffusion-influenced association rates and a to denote intrinsic association rates.

Including rebinding in well-mixed models

Now, consider the following reaction network:

$$\begin{array}{l}{S}_0+E\underset{b_1}{\stackrel{a_1}{\rightleftharpoons }}{S}_{0}E\stackrel{c_1}{\to }{S}_1+E^{\ast }\hfill \\ S_1+E\underset{b_2}{\stackrel{a_2}{\rightleftharpoons }}{S}_{1}E\stackrel{c_2}{\to }{S}_2+E^{\ast }\hfill \\ E^{\ast }\stackrel{\rho }{\to }E.\hfill \end{array}$$ (1.2)

In this reaction, substrates exist in three states, $S_0$, $S_1$, and $S_2$, and enzymes have an active form, E, and an inactive form, $E^{\ast }$. Active enzymes convert the substrate from $S_0$ to $S_1$ and from $S_1$ to $S_2$. After such a modification, the enzyme becomes inactive. Inactive enzymes become active at a “recharge” rate $\rho >0$. The parameters $a_i$, $b_i$, and $c_i$ are the intrinsic rates for association, dissociation (“breakup”), and catalysis, respectively.

It was shown in (10, 18) that the behavior of a network similar to Eq. 1.2 depends critically on whether or not one explicitly models the diffusion of molecules. In particular, merely renormalizing the association rates, $a_1$ and $a_2$, in an ODE model is not enough to replicate particle-based simulations in space and time. The reason for this is simple. If the enzyme recharge rate ρ is large, then it is possible that a single enzyme molecule can rapidly modify the same substrate molecule twice. That is, after first converting a substrate from $S_0$ to $S_1$, the enzyme can become active again and rebind to the $S_1$ substrate, rather than diffusing away, and convert it to $S_2$ since the enzyme is still physically near the substrate after the first modification. Such rebindings effectively transform a distributive mechanism into a processive one.

A natural way to incorporate such a rebinding is to add an arrow to the reaction diagram from the complex $S_{0}E$ to the complex $S_{1}E$ as shown in Fig. 1. To assign a rate to this reaction, we calculate the probability that an enzyme will rapidly rebind to a substrate after modifying it. We propose the following idealized model to approximate this probability.

Figure 1.

Modified reaction network that incorporates rebinding. A new connection has been added involving the probability of rebinding, $\pi \left(\rho ,\epsilon ,D,a_2\right)$, using Eq. 1.5, and the association/dissociation rates have been renormalized using Eqs. 1.1 and 1.6. To see this figure in color, go online.

Consider two spherical particles diffusing in three-dimensional space. We assume that the particles are initially in contact, but that no reaction can occur during an initial transitory time modeled as an exponentially distributed random variable with rate $\rho >0$. After this exponential time, the particles react immediately upon contact.

Let $D>0$ and $\epsilon >0$ be the sum of their respective diffusion coefficients and radii. By a direct application of Theorem 3 in (19), it follows that the probability that they react before becoming distance $R>\epsilon$ apart, given that they are initially distance $r\in \left(\epsilon ,R\right)$ apart, is $p_1\left(r;R\right)$ that satisfies the following:

$$\begin{array}{c}D\Delta p_1+\rho \left(p_0-p_1\right)=0,r\in \left(\epsilon ,R\right)\\ \frac{d}{dr}{p}_1\left(\epsilon ;R\right)=0,\\ p_1\left(R;R\right)=0,\end{array}$$ (1.3)

where $p_0\left(r;R\right)$ is the solution to the following:

$$\begin{array}{l}\Delta p_0=0,r\in \left(\epsilon ,R\right)\hfill \\ p_0\left(\epsilon ;R\right)=1,\hfill \\ p_0\left(R;R\right)=0.\hfill \end{array}$$ (1.4)

A formal derivation of Eqs. 1.3 and 1.4 is in the Appendix. To see intuitively the origin of Eqs. 1.3 and 1.4, first note that $p_0\left(r;R\right)$ is the probability that a diffusing particle starting at $r\in \left[\epsilon ,R\right]$ is absorbed at ε before R, which is well known to satisfy the elliptic problem in Eq. 1.4. However, because we assume that the particle cannot be absorbed during an initial transitory time, we impose a no flux boundary condition at $r=\epsilon$ in Eq. 1.3. Since this transitory time is exponential with rate $\rho >0$, and since after this transitory time the probability that the particle is absorbed at ε is identical to $p_0$, we add the second term in the equation for $p_1$.

Now suppose the particles do not react immediately upon contact, but rather react at some rate $a>0$ when in contact. The effect of this on the probability can be modeled by changing the boundary condition in Eq. 1.4 to the following:

$$p_0\left(\epsilon ;R\right)-\left(\frac{4\pi {\epsilon }^{2}D}{a}\right)\frac{d}{dr}{p}_0\left(\epsilon ;R\right)=1.$$

With this new definition of $p_0$, it is straightforward to solve for $p_1$ and find an explicit formula for the rebinding probability, $p_1\left(\epsilon ;R\right)$. Taking the limit $R\to \infty$, we arrive at the following approximation for the probability of rapid rebinding:

$$\pi \left(\rho ,\epsilon ,D,a\right):=\underset{R\to \infty }{\text{lim}}{p}_1\left(\epsilon ;R\right)=\frac{\gamma }{\left(\gamma +1\right)\left(\delta +1\right)},$$ (1.5)

where

$$\gamma :=\sqrt{{\epsilon }^2\rho /D}\text{and}\delta :=4\pi \epsilon D/a$$

are dimensionless parameters. This formula was first derived in (11) by considering a different model that is similar to the model presented here. In contrast, we have appealed to recent rigorous results on diffusion in the presence of randomly switching boundaries that depend on the generalized Ito formula to find Eq. 1.5.

In Fig. 2, we show a comparison of Eq. 1.5 (the curves) and the computational estimates of this probability (the squares) that were found by Monte Carlo simulations of a many-body stochastic spatial model in (10). The close fit in Fig. 2 bolsters confidence in the theory.

Figure 2.

Rebinding probability as a function of diffusion coefficient. To see this figure in color, go online.

We remark that we have assumed that rebinding is only possible when the enzyme has reactivated, that the two molecules are spherical, and that the surface of each molecule is uniformly reactive. Violating this third assumption would break the symmetry in the problem and would change the ODEs in Eqs. 1. 3 and 1.4 to complicated partial differential equations (see (12) for a consideration of this case).

Renormalizing dissociation rates

Finally, the discussion of rebindings suggests that the dissociation rates, $b_1$ and $b_2$, in Eq. 1.2 should also be renormalized. That is, if the substrate and enzyme dissociate, then they are still physically close to each other and thus might rapidly rebind. We thus multiply $b_i$ by the probability of not rapidly rebinding. Since the enzyme does not need to recharge in this case, we renormalize $b_i$ to be the following:

$${\beta }_i\left(\epsilon ,D,a_i,b_i\right)=b_i\left(1-\underset{\rho \to \infty }{\text{lim}}\pi \left(\rho ,\epsilon ,D,a_i\right)\right)=b_i\frac{{\delta }_i}{1+{\delta }_i},$$ (1.6)

where ${\delta }_i=4\pi \epsilon D/a_i$ for $i=\mathrm{1,2}$. Throughout this article we use β to denote diffusion-influenced breakup rates and b to denote intrinsic breakup rates. We note that Eq. 1.6 recovers a well-established result (20, 21).

Results and Discussion

Our results were obtained by mathematical analysis and numerical computation. A list of parameters and their default values used in our numerical results is provided in Table 1.

Table 1.

A List of Parameters and Their Default Values Used in Numerical Results Unless Indicated Otherwise

	Parameter	Value
$a_1$	intrinsic association rate	$0.027{\text{nM}}^{-1}{\text{s}}^{-1}$
$a_2$	intrinsic association rate	$0.056{\text{nM}}^{-1}{\text{s}}^{-1}$
${\alpha }_i$, $i=\mathrm{1,2}$	diffusion-influenced association rate	Eq. 1.1
$b_1$	intrinsic dissociation rate	$1.35{\text{s}}^{-1}$
$b_2$	intrinsic dissociation rate	$1.73{\text{s}}^{-1}$
${\beta }_i$, $i=\mathrm{1,2}$	diffusion-influenced dissociation rate	Eq. 1.6
$c_1$	catalysis rate	$1.5{\text{s}}^{-1}$
$c_2$	catalysis rate	$15{\text{s}}^{-1}$
D	sum of diffusion coefficients	$2\mu {\text{m}}^2{\text{s}}^{-1}$
ε	sum of radii	$5\text{nm}$
$\left[E_{\text{tot}}\right]$	total E enzyme concentration	$50\text{nM}$
$\left[P_{\text{tot}}\right]$	total P enzyme concentration	$50\text{nM}$
$\left[S_{\text{tot}}\right]$	total substrate concentration	$200\text{nM}$

Numerical values were taken from (10).

Well-mixed models can quantitatively replicate spatial models

Consider the following double phosphorylation reaction network:

$$\begin{array}{c}{S}_0+E\underset{b_1}{\stackrel{a_1}{\rightleftharpoons }}{S}_{0}E\stackrel{c_1}{\to }{S}_1+E^{\ast }\\ S_1+E\underset{b_2}{\stackrel{a_2}{\rightleftharpoons }}{S}_{1}E\stackrel{c_2}{\to }{S}_2+E^{\ast }\\ S_2+P\underset{b_1}{\stackrel{a_1}{\rightleftharpoons }}{S}_{2}P\stackrel{c_1}{\to }{S}_1+P^{\ast }\\ S_1+P\underset{b_2}{\stackrel{a_2}{\rightleftharpoons }}{S}_{1}P\stackrel{c_2}{\to }{S}_0+P^{\ast }\\ E^{\ast }\stackrel{\rho }{\to }E,P^{\ast }\stackrel{\rho }{\to }P.\end{array}$$ (2.1)

This network is the same as the network considered above, except that there are now phosphatase enzymes, P, that can take the substrate down the phosphorylation cycle. Due to its biological ubiquity, this network has been well studied (4, 8, 10, 15, 18).

In (10), the authors identified three ways in which the behavior of Eq. 2.1 depends on whether or not one explicitly includes spatial effects. In particular, by comparing a well-mixed ODE model with Monte Carlo simulations of a stochastic spatial model, the authors found that rapid enzyme-substrate rebindings can 1) speed up the network response, 2) weaken the sharpness of the response, and 3) lead to a loss of bistability. Based on these findings, the authors argue against the use of well-mixed ODE models in favor of stochastic spatial models, despite their computational expensive and analytic intractability.

In this section, we show that by incorporating the rebinding probability found in Materials and Methods into well-mixed ODE models, these simpler nonspatial models can quantitatively replicate each of these three effects. Thus, we show that one can circumvent spatial models by incorporating this rebinding probability into nonspatial models.

To use a well-mixed model, one first renormalizes the association and dissociation rates according to Eqs. 1.1 and 1.6. Then, we propose to incorporate the probability of rapid rebinding to change the reaction network Eq. 2.1 into the network in Fig. 3, with the following:

$$\pi =\pi \left(\rho ,\epsilon ,D,a_2\right),$$

where $\pi \left(\rho ,\epsilon ,D,a_2\right)$ is the probability of rapid rebinding defined above in Eq. 1.5. Note that the topology of the network in Fig. 3 reduces to Eq. 2.1 in the case of zero rebinding probability, $\pi =0$.

Figure 3.

Modified reaction network that incorporates rebinding. New connections have been added involving the probability of rebinding, $\pi \left(\rho ,\epsilon ,D,a_2\right)$, using Eq. 1.5, and the association/dissociation rates have been renormalized using Eqs. 1.1 and 1.6. To see this figure in color, go online.

Modeling the network in Fig. 3 by ODEs with mass-action kinetics yields the following system:

$$\begin{array}{ll}\frac{d\left[S_1\right]}{dt}\hfill & ={\beta }_2\left[S_{1}E\right]+{\beta }_2\left[S_{1}P\right]+\left(1-\pi \right)c_1\left[S_{0}E\right]+\left(1-\pi \right)c_1\left[S_{2}P\right]-{\alpha }_2\left[S_1\right]\left[E\right]-{\alpha }_2\left[S_1\right]\left[P\right]\hfill \\ \frac{d\left[S_2\right]}{dt}\hfill & ={\beta }_1\left[S_{2}P\right]+c_2\left[S_{1}E\right]-{\alpha }_1\left[S_2\right]\left[P\right]\hfill \\ \frac{d\left[E^{\ast }\right]}{dt}\hfill & =-\rho \left[E^{\ast }\right]+\left(1-\pi \right)c_1\left[S_{0}E\right]+c_2\left[S_{1}E\right]\hfill \\ \frac{d\left[P^{\ast }\right]}{dt}\hfill & =-\rho \left[P^{\ast }\right]+\left(1-\pi \right)c_1\left[S_{2}P\right]+c_2\left[S_{1}P\right]\hfill \\ \frac{d\left[S_{0}E\right]}{\left[dt\right]}\hfill & ={\alpha }_1\left[S_0\right]\left[E\right]-\left({\beta }_1+c_1\right)\left[S_{0}E\right]\hfill \\ \frac{d\left[S_{1}E\right]}{\left[dt\right]}\hfill & ={\alpha }_2\left[S_1\right]\left[E\right]+\pi c_1\left[S_{0}E\right]-\left({\beta }_2+c_2\right)\left[S_{1}E\right]\hfill \\ \frac{d\left[S_{2}P\right]}{\left[dt\right]}\hfill & ={\alpha }_1\left[S_2\right]\left[P\right]-\left({\beta }_1+c_1\right)\left[S_{2}P\right]\hfill \\ \frac{d\left[S_{1}P\right]}{\left[dt\right]}\hfill & ={\alpha }_2\left[S_1\right]\left[P\right]+\pi c_1\left[S_{2}P\right]-\left({\beta }_2+c_2\right)\left[S_{1}P\right].\hfill \end{array}$$ (2.2)

We have not written the superfluous equations for $\left[S_0\right]$, $\left[E\right]$, or $\left[P\right]$ since by the following conservation:

$$\left[S_0\right]=\left[S_{\text{tot}}\right]-\left[S_1\right]-\left[S_2\right]-\left[S_{0}E\right]-\left[S_{1}E\right]-\left[S_{2}P\right]-\left[S_{1}P\right],$$

$$\left[E\right]=\left[E_{\text{tot}}\right]-\left[S_{0}E\right]-\left[S_{1}E\right]-\left[E^{\ast }\right],$$

$$\left[P\right]=\left[P_{\text{tot}}\right]-\left[S_{2}P\right]-\left[S_{1}P\right]-\left[P^{\ast }\right],$$

where $\left[S_{\text{tot}}\right]$, $\left[E_{\text{tot}}\right]$, and $\left[P_{\text{tot}}\right]$ are the total respective concentrations of substrate, E enzyme, and P enzyme.

In the following three subsections and in Figs. 4, 5, and 6, we show that this system of ODEs quantitatively replicates the Monte Carlo simulations of the stochastic spatial model of (10).

Figure 4.

Response time as a function of diffusion coefficient. To see this figure in color, go online.

Figure 5.

Steady state $\left[S_2\right]/\left[S_{\text{tot}}\right]$ as a function of enzyme ratio, $\left[E_{\text{tot}}\right]/\left[P_{\text{tot}}\right]$, for $\rho =\text{log}\left(2\right)\times {10}^6{\text{s}}^{-1}$ in the top plot and $\rho =\text{log}\left(2\right)\times {10}^2{\text{s}}^{-1}$ in the bottom plot. To see this figure in color, go online.

Figure 6.

Bifurcation diagram showing possible steady states of $\left[S_2\right]/\left[S_{\text{tot}}\right]$ as a function of the expected recharge time. The squares are from simulations of a spatial model in (10) (different colored squares correspond to different initial conditions), the red curves are for the ODE system with rebinding (Eq. 2.2), and the black curves are for the ODE system without rebinding (Eq. 2.2 with $\pi =0$). Solid curves represent stable steady states, and the dashed curve is an unstable steady state. The ODE system without rebinding has an unstable steady state for all values of ρ considered, but we omit it for simplicity. We set $\left[S_{\text{tot}}\right]=500\text{nM}$. To see this figure in color, go online.

Response time

Fig. 4 compares the response time of Eq. 2.1 when the network is modeled by 1) ODEs without the probability of rapid rebinding, 2) ODEs with the probability of rapid rebinding, and 3) stochastic simulations of a full spatial model (spatial simulation data taken from (10)). The response time is defined to be the amount of time the system takes to reach one half of the steady-state value of $\left[S_2\right]$ given that all of the substrate is initially unphosphorylated.

Several features of this figure are noteworthy. First, although modeling the network by ODEs without rebinding does not replicate the spatial model, incorporating rebinding into the ODEs replicates the spatial model to a very high degree of accuracy. Second, the effect of rebinding is most significant at large values of the enzyme recharge rate and small values of the diffusion coefficient. This phenomenon is readily explained by the formula for rebinding Eq. 1.5 wherein the probability of rapid rebinding increases as ρ increases and D decreases. Third, incorporating rebinding predicts that there is an optimal diffusion coefficient that minimizes the response time. This optimum results from the fact that increasing the diffusion coefficient has the competing effects of increasing the rate of the first binding and decreasing the rebinding probability.

Sharpness of the response

Fig. 5 plots the steady-state proportion of doubly phosphorylated substrate against the ratio of kinase to phosphatase (the so-called input-output relation) when Eq. 2.1 is modeled by 1) ODEs with the probability of rapid rebinding and 2) stochastic simulations of a spatial model. The curves in the figure are for our ODEs in Eq. 2.2, and the squares are spatial simulation data taken from (10). As in the previous subsection, we see that ODEs that incorporate rebinding replicate the spatial model. Further, the effect of rebinding is most significant for a large enzyme recharge rate and a small diffusion coefficient, and again this is immediately explained by the formula Eq. 1.5 for the rebinding probability.

Moreover, Fig. 5 suggests that one can use Eq. 1.5 to predict the degree to which a system exhibits a sigmoidal, cooperative response. That is, as the rebinding probability Eq. 1.5 decreases, the input-output relation shows a sharper, more sensitive response. One can understand this phenomenon by noting that the rebinding probability Eq. 1.5 is independent of the enzyme concentration. Thus, if the rebinding probability is large, then a substrate requires only a single enzyme molecule in order to be phosphorylated twice and hence the steady-state twice-phosphorylated substrate concentration depends less on the enzyme concentration.

Loss of bistability

Fig. 6 is a bifurcation diagram comparing the possible steady-state concentrations of doubly phosphorylated substrate in Eq. 2.1 when the network is modeled by 1) ODEs without the probability of rapid rebinding, 2) ODEs with the probability of rapid rebinding, and 3) stochastic simulations of a full spatial model (spatial simulation data taken from (10)). Significantly, ODEs that incorporate rebinding show the same loss of bistability as seen in the full spatial model for large values of the enzyme recharge rate and hence large values of the rebinding probability. That is, we see in this plot that for small recharge rate, there are multiple stable steady states (bistability) whereas at large recharge rate, there is only one stable steady state if the network is modeled by ODEs with rebinding or a spatial system. Further, since ODE models are amenable to analysis, we are able to discover that this loss of bistability is a pitchfork bifurcation (i.e., not a hysteretic switch), as depicted in Fig. 6.

Quasi-processivity can be identified through steady-state invariants

In addition to reducing computational expense, ODE models have the benefit of being amenable to analysis. In this and the following subsection, we exploit this feature.

In (15), the author showed how one might be able to distinguish between distributivity and processivity in two-step phosphorylation/dephosphorylation reactions through steady-state invariants. That is, the author showed that in a well-mixed mass-action ODE model of Eq. 2.1 where both enzymes act distributively and are always active $\left(\rho =+\infty \right)$, the steady-state value of

$$\frac{\left[S_0\right]\left[S_2\right]}{{\left[S_1\right]}^2}$$ (2.3)

is always the same regardless of the conditions under which the system is initiated, such as the total amounts of enzymes or substrate. However, if one or both of the enzymes acts processively, then the quantity Eq. 2.3 changes as a function of the steady-state ratio $\left[S_2\right]/\left[S_1\right]$. Whereas previous techniques for experimental determination of processivity or distributivity were based on time course data, this result allows one to make this determination based solely on steady-state measurements.

In this subsection, we extend these results to distinguish between distributive and quasi-processive two-step phosphorylation/dephosphorylation reactions. This extension requires some care since the topology of our quasi-processive reaction in Fig. 3 differs from the topology of the processive reaction in (15). Specifically, our reaction contains inactive states for the enzymes, and our reaction contains arrows from $S_{0}E$ to $S_{1}E$ and from $S_{2}P$ to $S_{1}P$, whereas the reaction in (15) contains arrows from $S_{0}E$ to $S_2+E$ and from $S_{2}P$ to $S_0+P$.

In this subsection, we allow the rate parameters in Fig. 3, ${\alpha }_i,{\beta }_i,c_i$, to depend on the enzyme involved in their reaction, and so we give them the corresponding superscripts (${\alpha }_i^E,{\beta }_i^E,c_i^E$ or ${\alpha }_i^P,{\beta }_i^P,c_i^P$). At steady state, the fluxes of $\left[S_{0}E\right]$ must balance so that

$$\left[S_{0}E\right]=\frac{\left[S_0\right]\left[E\right]}{K_1^E}\text{where}{K}_1^E=\frac{{\beta }_1^E+c_1^E}{{\alpha }_1^E}.$$ (2.4)

Combining this with the fact that the fluxes of $\left[S_{1}E\right]$ must also balance at steady state, we have the following:

$$\left[S_{1}E\right]=\frac{\left[S_1\right]\left[E\right]}{K_2^E}+q^E\frac{\left[S_0\right]\left[E\right]}{K_1^E},\text{where}{K}_2^E=\frac{{\beta }_2^E+c_2^E}{{\alpha }_2^E}\text{and}{q}^E=\frac{{\pi }^E{c}_1^E}{{\beta }_2^E+c_2^E}.$$

By symmetry, at steady state, we have the following:

$$\left[S_{2}P\right]=\frac{\left[S_2\right]\left[P\right]}{K_1^P}\text{where}{K}_1^P=\frac{{\beta }_1^P+c_1^P}{{\alpha }_1^P},$$

$$\left[S_{1}P\right]=\frac{\left[S_1\right]\left[P\right]}{K_2^P}+q^P\frac{\left[S_2\right]\left[P\right]}{K_1^P},\text{where}{K}_2^P=\frac{{\beta }_2^P+c_2^P}{{\alpha }_2^P}\text{and}{q}^P=\frac{{\pi }^P{c}_1^P}{{\beta }_2^P+c_2^P}.$$ (2.5)

Balancing the fluxes of $\left[S_0\right]$ at steady state, we have the following:

$${\alpha }_1^E\left[S_0\right]\left[E\right]-{\beta }_1^E\left[S_{0}E\right]=c_2^P\left[S_{1}P\right].$$ (2.6)

Combining Eq. 2.6 with Eqs. 2.4 and 2.5 at steady state, we have the following:

$$\frac{c_1^E}{K_1^E}\left[S_0\right]\left[E\right]=\frac{c_2^P}{K_2^P}\left[S_1\right]\left[P\right]+\frac{c_2^P{q}^p}{K_1^P}\left[S_2\right]\left[P\right].$$ (2.7)

And by symmetry at steady state, we have the following:

$$\frac{c_1^P}{K_1^P}\left[S_2\right]\left[P\right]=\frac{c_2^E}{K_2^E}\left[S_1\right]\left[E\right]+\frac{c_2^E{q}^E}{K_1^E}\left[S_0\right]\left[E\right].$$ (2.8)

If we let

$$\begin{array}{cc}{\lambda }_0& =\left(\frac{c_1^E}{K_1^E}\right)\left(\frac{K_2^P}{c_2^P}\right){\lambda }_{+}=\left(\frac{c_1^E}{K_1^E}\right)\left(\frac{K_1^P}{q^P{c}_2^P}\right)\\ {\lambda }_1& =\left(\frac{c_2^E}{K_2^E}\right)\left(\frac{K_1^P}{c_1^P}\right){\lambda }_{-}=\left(\frac{q^E{c}_2^E}{K_1^E}\right)\left(\frac{K_1^P}{c_1^P}\right),\end{array}$$ (2.9)

then we can rearrange Eqs. 2.7 and 2.8 and write

$$\begin{array}{cc}\frac{\left[S_0\right]}{\left[S_1\right]}& =\frac{1}{{\lambda }_0}\frac{\left[P\right]}{\left[E\right]}+\frac{1}{{\lambda }_{+}}\frac{\left[P\right]}{\left[E\right]}\frac{\left[S_2\right]}{\left[S_1\right]}\\ \frac{\left[S_2\right]}{\left[S_1\right]}& ={\lambda }_1\frac{\left[E\right]}{\left[P\right]}+{\lambda }_{-}\frac{\left[E\right]}{\left[P\right]}\frac{\left[S_0\right]}{\left[S_1\right]}.\end{array}$$ (2.10)

Given our parameter definitions in Eq. 2.9, the analysis now becomes identical with that in (15). In particular, it follows from Eq. 2.10 that at steady state we have the following:

$$\frac{\left[S_0\right]\left[S_2\right]}{{\left[S_1\right]}^2}=\omega \nu \frac{\left[S_2\right]}{\left[S_1\right]}+\frac{\kappa \eta }{\nu \frac{\left[S_2\right]}{\left[S_1\right]}-\kappa }+\eta \left(2\nu -1\right),$$ (2.11)

where

$$\nu =1-\frac{{\lambda }_{-}}{{\lambda }_{+}},\kappa =\frac{{\lambda }_{-}}{{\lambda }_0},\omega =\frac{{\lambda }_1}{{\lambda }_{+}},\eta =\frac{{\lambda }_1}{{\lambda }_0}.$$

Note that if ${\pi }^E=0$, then $\kappa =0$ (since ${\pi }^E=0$ implies $q^E=0$, which implies ${\lambda }_{-}=0$) and if ${\pi }^P=0$, then $\omega =0$ (since ${\pi }^P=0$ implies $q^P=0$, which implies ${\lambda }_{+}=\infty$). Thus, by performing multiple experiments with various initial conditions and plotting the steady-state value of $\left[S_0\right]\left[S_2\right]/{\left[S_1\right]}^2$ against the steady-state value of $\left[S_2\right]/\left[S_1\right]$, one can identify quasi-processivity. That is, such a plot will be flat if ${\pi }^E\approx {\pi }^P\approx 0$, and hence the system is not quasi-processive. Otherwise, the system is quasi-processive, and the plot will increase linearly if ${\pi }^E\approx 0$ and ${\pi }^P\not\approx 0$, decrease hyperbolically if ${\pi }^P\approx 0$ and ${\pi }^E\not\approx 0$, or be biphasic if ${\pi }^E\not\approx 0$ and ${\pi }^P\not\approx 0$.

We emphasize that the steady-state substrate concentrations satisfy Eq. 2.11 regardless of initial conditions. Thus, even though the system may have multiple stable steady states (see discussion above), each of these steady states must obey Eq. 2.11.

There are several points to take away from this analysis. The first is simply that the method for identifying processivity developed in (15) applies to quasi-processivity. The second is that previous studies that have identified processivity may have actually mistaken processivity for quasi-processivity because of the possibility of enzyme rebinding.

Finally, Eq. 2.11 provides a way to test the predictions of our analysis. As just noted, if the plot of the steady-state value of $\left[S_0\right]\left[S_2\right]/{\left[S_1\right]}^2$ against the steady-state value of $\left[S_2\right]/\left[S_1\right]$ is not flat, then the system is quasi-processive. Our analysis then predicts that experimentally increasing the diffusion coefficient will cause the rebinding probability to decrease (see Eq. 1.5) and this plot will become flat.

Michaelis-Menten (quasi-steady-state) reduction

As biochemical modelers routinely use reduced Michaelis-Menten rate equations in lieu of full substrate-complex systems, it is important to determine how rebindings alter these reduced equations. In this subsection, we find that rebindings can affect them dramatically.

We first find the Michaelis-Menten approximation to the reaction network in Fig. 1 involving only phosphorylations (we have chosen this network for simplicity; extension to the phosphorylation/dephosphorylation network in Fig. 3 is immediate). Under the assumption that the total substrate concentration is much greater than the total enzyme concentration (the usual Michaelis-Menten assumption), the reactions involving substrate-enzyme complexes are much faster than other reactions. Thus, we suppose that the complexes are in quasi-steady state (see Chapter 1 in (22) for an introduction to these techniques). This quasi-steady-state assumption amounts to setting the time derivative of each complex equal to zero, and so the complexes satisfy the following:

$${\alpha }_1\left[S_0\right]\left(\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]-\left[S_{0}E\right]-\left[S_{1}E\right]\right)=\left({\beta }_1+c_1\right)\left[S_{0}E\right],$$

$${\alpha }_2\left[S_1\right]\left(\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]-\left[S_{0}E\right]-\left[S_{1}E\right]\right)+\pi c_1\left[S_{0}E\right]=\left({\beta }_2+c_2\right)\left[S_{1}E\right].$$

Solving this linear system, we find that

$$\begin{array}{c}{c}_1\left[S_{0}E\right]=V_0=V_0\left(\left[S_0\right],\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right)=c_1\frac{\left(\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right)K_2\left[S_0\right]}{K_1{K}_2+\left(K_2+r\pi \right)\left[S_0\right]+K_1\left[S_1\right]}\\ c_2\left[S_{1}E\right]=V_1=V_1\left(\left[S_0\right],\left[S_1\right],\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right)=c_2\frac{\left(\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right)\left(r\pi \left[S_0\right]+K_1\left[S_1\right]\right)}{K_1{K}_2+\left(K_2+r\pi \right)\left[S_0\right]+K_1\left[S_1\right]},\end{array}$$ (2.12)

where

$$K_i=\left({\beta }_i+c_i\right)/{\alpha }_i\text{and}r=c_1/{\alpha }_2.$$

(To get Eq. 2.12 in terms of intrinsic rates, $a_i,b_i$, and physical parameters, $D,\epsilon$, recall Eqs. 1.1, 1.5, and 1.6.) Thus, the mass-action system for the network in Fig. 1 is reduced to the following:

$$\frac{d\left[S_1\right]}{dt}=V_0\left(\left[S_0\right],\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right)-V_1\left(\left[S_0\right],\left[S_1\right],\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right),$$ (2.13)

$$\frac{d\left[S_2\right]}{dt}=V_1\left(\left[S_0\right],\left[S_1\right],\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right),$$ (2.14)

$$\frac{d\left[E^{\ast }\right]}{dt}=-\rho \left[E^{\ast }\right]+\left(1-\pi \right)V_0\left(\left[S_0\right],\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right)+V_1\left(\left[S_0\right],\left[S_1\right],\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right).$$

In the case that the probability of rapid rebinding is significant, we can further assume that enzymes are always active $\left(\rho \gg 1\right)$ so that $\left[E^{\ast }\right]=0$. In this limit, the system reduces to the pair of ODEs, Eqs. 2.13 and 2.14.

We have carried out this analysis in the case of a two-step phosphorylation reaction, but it is straightforward to extend it to an n-step reaction. We note that if $n=1$, then rebinding does not affect the system. For n > 1, our procedure reduces a system of $3n+2$ ODEs to $n+2$ ODEs and provides an explicit formula for the production rate of each substrate form as a function of the other n substrate form concentrations.

To illustrate, consider the following n-step reaction:

$$S_0\underset{P}{\stackrel{E}{\rightleftharpoons }}{S}_1\underset{P}{\stackrel{E}{\rightleftharpoons }}\dots \underset{P}{\stackrel{E}{\rightleftharpoons }}{S}_n,$$

where the enzymes act according to the following:

$$\begin{array}{l}{S}_i+E\underset{\beta }{\stackrel{\alpha }{\rightleftharpoons }}{S}_{i}E\stackrel{c}{\to }{S}_{i+1}+E^{\ast },0\le i<n\hfill \\ S_i+P\underset{\beta }{\stackrel{\alpha }{\rightleftharpoons }}{S}_{i}P\stackrel{c}{\to }{S}_{i-1}+P^{\ast },0<i\le n\hfill \\ E^{\ast }\stackrel{\rho }{\to }E,P^{\ast }\stackrel{\rho }{\to }P.\hfill \end{array}$$ (2.15)

For ease, we have taken the rate constants, $\left(\alpha ,\beta ,c\right)$, to be the same for every reaction. This assumption is likely to be satisfied in the case of polyubiquitination.

Because of the possibility of rebindings, this reaction network needs to be modified according to Fig. 7. As above, one can show that if the complexes are in quasi-steady state, then

$$\frac{d\left[S_m\right]}{dt}=-V_m^E-V_m^P+V_{m-1}^E+V_{m+1}^P,0\le m\le n,$$ (2.16)

with $V_0^P=V_n^E=V_{n+1}^P=V_{-1}^E=0$, and

$$V_m^E=c\left[S_{m}E\right]=\frac{c\left(\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right)\left({\sum }_{j=0}^m{\left(\pi p\right)}^j\left[S_{m-j}\right]\right)}{\left(\beta +c\right)/\alpha +{\sum }_{j=0}^{n-1}\left({\left(\pi p\right)}^j{\sum }_{i=0}^{n-j-1}\left[S_i\right]\right)},0\le m\le n-1,$$ (2.17)

where $p=c/\left(\beta +c\right)$, and $V_m^P$ is defined analogously. In contrast, if one neglects rebindings, then the terms involving π drop out of Eq. 2.17 and we are left with the following:

$$V_m^E=c\left[S_{m}E\right]=\frac{c\left(\left[E_{\text{tot}}\right]-\left[E^{\ast }\right]\right)\left[S_m\right]}{\left(\beta +c\right)/\alpha +{\sum }_{i=0}^{n-1}\left[S_i\right]},0\le m\le n-1.$$ (2.18)

The numerator of Eq. 2.17 has a simple probabilistic interpretation. Rebinding permits a molecule in state $\left[S_{i}E\right]$ to jump directly to $\left[S_{i+1}E\right]$, and the probability of such a jump is $\pi p$ (see Fig. 7). For a molecule to go directly from the $m-j$ phospho-form to the m phospho-form, it must rebind j times. This can happen for any $1\le j<m$, hence the following extra term:

$$\stackrel{m}{\sum _{j=1}}{\left(\pi p\right)}^j\left[S_{m-j}\right]$$

in the numerator of Eq. 2.17 compared with the numerator of Eq. 2.18.

Figure 7.

Modified reaction network that incorporates rebinding for 0 ≤ i < n − 1. A new connection has been added involving the probability of rebinding, $\pi \left(\rho ,\epsilon ,D,a_2\right)$, using Eq. 1.5, and the association/dissociation rates have been renormalized using Eqs. 1.1 and 1.6. To see this figure in color, go online.

The results of this subsection carry out the necessary step of determining how rebinding affects Michaelis-Menten rate equations, which are used extensively by biochemical modelers. Comparing Eqs. 2.17 and 2.18 shows that rebinding drastically changes the form of these equations. This change in equations translates into a change in behavior of the dynamical system. Indeed, one can show that including rebinding in the Michaelis-Menten system produces the effects seen above for the full substrate-complex system (namely, rebinding can decrease the response time, decrease the sharpness of response, and lead to a loss of bistability).

Furthermore, rebinding has an even greater impact on systems with many steps (large n). To illustrate, Fig. 8 plots the time course of fully phosphorylated substrate for the Michaelis-Menten system with and without rebinding for different numbers of phosphorylation steps. The importance of rebinding for systems with many steps is straightforward to understand: rebinding allows a substrate to jump directly from $S_0$ to $S_n$, but in the absence of rebinding the substrate must transition through the $n-1$ intermediate states, $S_1,\dots ,S_{n-1}$. This may be particularly relevant for eukaryotic proteins, as they can be phosphorylated many times (23). For example, p53, which integrates a cell’s response to DNA damage, has 16 phosphorylation sites (24). Indeed, some eukaryotic proteins have over 150 phosphorylation sites (25).

Figure 8.

Time course of fully phosphorylated substrate for the Michaelis-Menten system (Eq. 2.16), with and without rebinding for different numbers of phosphorylation steps, n, in the system. Here, $D=.25\mu {\text{m}}^2{\text{s}}^{-1}$, $\rho =\infty$, and the intrinsic rates are in Table 1: $a=a_2$, $b=b_2$, $c=c_2$. To see this figure in color, go online.

Conclusions

Stated broadly, our work continues the tradition of using mathematical analysis to incorporate spatial effects into nonspatial models, a research program with a long and fruitful history in biochemistry, dating back at least to Smoluchowski’s work on diffusion-limited reaction rates (16). In this vein, since we use many of the same assumptions, our work can be viewed as a Smoluchowski theory for diffusion-limited rebinding.

Though our work has been primarily motivated by phosphorylation/dephosphorylation reactions, our results apply to any diffusion-limited reaction in which a substrate can be modified repeatedly by a single molecule. Biological examples abound. Polyubiquitination is one such example that is particularly important. Another example is the human body’s detoxification mechanism for ingested arsenic, in which inorganic arsenic is methylated twice in the liver. Our results can find immediate application to mathematical models of this pathway (26, 27).

Author Contributions

S.D.L. and J.P.K. jointly designed and performed the research and wrote the manuscript.

Editor: Alexander Berezhkovskii.

Appendix

The purpose of this appendix is to give a self-contained derivation of Eqs. 1.3 and 1.4. A detailed proof can be found in (19), and similar derivations of exit statistics of diffusing particles in the presence of randomly switching boundary conditions can be found in (28, 29, 30).

Let $r\left(t\right)\in \left[\epsilon ,R\right]$ denote the radial position at time t of a diffusing particle with diffusion coefficient $D>0$ in the region $\left\{x\in {\mathrm{ℝ}}^3:0<\epsilon \le \left|x\right|\le R\right\}$. Suppose the outer boundary, $r=R$, is always absorbing. Suppose the inner boundary, $r=\epsilon$, is reflecting at time t if $n\left(t\right)=1$ and absorbing if $n\left(t\right)=0$, where $n\left(t\right)\in \left\{\mathrm{0,1}\right\}$ is a Markov jump process that jumps from state 1 to state 0 at rate $\rho >0$ and never leaves state 0. The backward Kolmogorov equation for the process $\left(r\left(t\right),n\left(t\right)\right)\in \left[\epsilon ,R\right]\times \left\{\mathrm{0,1}\right\}$ is as follows:

$$\begin{array}{l}{\partial }_t{q}_0=D\Delta q_0\hfill \\ {\partial }_t{q}_1=D\Delta q_1+\rho q_0-\rho q_1,\hfill \end{array}$$ (2.19)

with

$$q_0\left(R,t\right)=q_1\left(R,t\right)=0,q_0\left(\epsilon ,t\right)=0,{\partial }_r{q}_1\left(\epsilon ,t\right)=0.$$

Here $q_n\left(r,t\right)=p\left(r^{\prime },n^{\prime },t|r,n,0\right)$ is the probability density that the process is at $\left(r^{\prime },n^{\prime }\right)$ at time $t>0$, given that it started at $\left(r,n\right)$ at time 0.

Let ${\varphi }_n\left(r,t\right)$ be the probability that the particle is absorbed at the inner boundary after time t given that the process started at $\left(r,n\right)$. That is, ${\varphi }_n$ is the following probability flux at the inner boundary:

$${\varphi }_n\left(r,t\right)=-4\pi {\epsilon }^{2}D{\int }_t^{\infty }{\partial }_{r}p\left(\epsilon ,0,t^{\prime }|r,n,0\right)d{t}^{\prime }.$$

Differentiating Eq. 2.19 with respect to r and integrating with respect to t, we obtain that ${\varphi }_0$ and ${\varphi }_1$ satisfy Eq. 2.19.

Observe that $p_n\left(r;R\right):={\varphi }_n\left(r,0\right)$ is the probability that the particle is absorbed at the inner boundary, given that it started at position r and the boundary was initially in state n. Setting $t=0$ in the PDE that ${\varphi }_n$ satisfies and observing that ${\partial }_t{\varphi }_n\left(r,0\right)=0$, if $r\ne \epsilon$ yields the PDEs in Eqs. 1.3 and 1.4. The boundary conditions in Eqs. 1.3 and 1.4 are immediate.