Kinetics of diffusion-influenced multisite phosphorylation with enzyme reactivation

Abstract

The simplest way to account for the influence of diffusion on the kinetics of multisite phosphorylation is to modify the rate constants in the conventional rate equations of chemical kinetics. We have previously shown that this is not enough and new transitions between the reactants must also be introduced. Here we extend our results by considering enzymes that are inactive after modifying the substrate and need time to become active again. This generalization leads to a surprising result. The introduction of enzyme reactivation results in a diffusion-modified kinetic scheme with a new transition that has a negative rate constant. The reason for this is that mapping non-Markovian rate equations onto Markovian ones with time-independent rate constants is not a good approximation at short times. We then developed a non-Markovian theory that involves memory kernels instead of rate constants. This theory is now valid at short times, but is more challenging to use. As an example, the diffusion-modified kinetic scheme with new connections was used to calculate kinetics of double phosphorylation and steady-state response in a phosphorylation-dephosphorylation cycle. We have reproduced the loss of bistability in the phosphorylation-dephosphorylation cycle when the enzyme reactivation time decreases, which was obtained by particle-based computer simulations.

Graphical Abstract

1. INTRODUCTION

Post-translational modification by phosphorylation is an important mechanism to regulate and alter protein function.^1–4 Multisite phosphorylation of a protein can lead to unltrasensitivity, switch-like behavior between bistable steady states, and oscillatory behavior.^5–10 Conventional approaches to quantitatively describe the kinetics of multisite phosphorylation are based on the rate equations of chemical kinetics that are only valid for fast diffusive equilibration of local concentrations of the reactants. However, computer simulations of microscopic models showed that translational diffusion even in spatially homogeneous systems cannot be ignored.¹¹

In order to react, the reactants have to approach (diffuse) together and reorient their active sites. The simplest way to account for this effect is to replace the chemical rate constants by the diffusion-influenced ones. However, when two or more sites are modified by an enzyme, this naive theory deviates significantly from the results of particle-based simulations.¹¹ This is due to pseudo-processivity that arises because of diffusion. Specifically, after phosphorylating the first site, the enzyme and product may not completely diffuse apart. Thus the same enzyme molecule can phosphorylate the second site of the substrate before diffusing away. This effect alters kinetics of phosphorylation and can lead to the loss of bistability.

In our previous studies,^12,13 we showed how the pseudo-processivity emerges due to the relative diffusion of the reactants. For example, consider the double phosphorylation of a substrate $S_i$ (index $i=\mathrm{0,1},2$ indicates the number of phosphorylated sites) by an enzyme $E$, which occurs via the Michaelis-Menten mechanism:

$$\begin{gathered} S_0+E\underset{{\kappa }_0^d}{\stackrel{{\kappa }_0^a}{\rightleftharpoons }}{S}_{0}E\stackrel{{\kappa }_0^c}{⟶}{S}_1+E \\ {S}_1+E\underset{{\kappa }_1^d}{\stackrel{{\kappa }_1^a}{\rightleftharpoons }}{S}_{1}E\stackrel{{\kappa }_1^c}{⟶}{S}_2+E \end{gathered}$$ (1)

Here ${\kappa }_i^a$ and ${\kappa }_i^d(i=\mathrm{0,1})$ are the binding (association) and dissociation intrinsic rate constants in the limit of fast diffusion, and ${\kappa }_i^c$ are the catalytic rate constants. In general, the description of the influence of diffusion on this reaction is a challenging problem. However, our theory¹² is remarkably simple when the details of the kinetics at very short times and asymptotically long times are ignored. The effect of diffusion can be described using new rate equations that correspond to a new kinetic scheme. In this scheme (we call it the diffusion-modified kinetic scheme) not only are the intrinsic rate constants rescaled, but also an additional reaction channel appears, which connects the two bound states. The rate constant of the new channel and the scaling coefficients of the rates of the old channels have a clear physical meaning. They are expressed in terms of the capture and escape probabilities of an isolated pair of the enzyme and substrate that can only bind irreversibly. The escape probability is the probability that the enzyme-substrate pair located initially near an active site will eventually diffuse apart. The capture probability is the probability that this pair will eventually bind. We have shown^13,14 that such a theory holds not only for the reactive spheres, but also for more realistic models where the reactive sites have arbitrary shapes.

In this paper we extend our previous results to the case when the reactivation time of the enzyme is finite. This model accounts for the fact that after phosphorylating the substrate, the enzyme becomes inactive. The enzyme in the inactive state $\left(E^{*}\right)$ needs time to release ADT and to bind ATP and thus become active again (state $E$). This model was previously studied in Ref. 11 using numerical simulations of diffusing particles, where the enzyme reactivation process was modelled by a first-order reaction. We adopt the same model but develop an analytically tractable theory for the kinetics. We begin by assuming that double phosphorylation in the limit of fast diffusion is described by the Michaelis-Menten distributive mechanism:

$$\begin{array}{l}{S}_0+E\underset{{\kappa }_0^d}{\stackrel{{\kappa }_0^a}{\rightleftharpoons }}{S}_{0}E\stackrel{{\kappa }_0^c}{\to }{S}_1+E^{*}\\ S_1+E\underset{{\kappa }_1^d}{\stackrel{{\kappa }_1^a}{\rightleftharpoons }}{S}_{1}E\stackrel{{\kappa }_1^c}{\to }{S}_2+E^{*}\\ E^{*}\stackrel{k^{*}}{\to }E\end{array}$$ (2)

where the rate constant $k^{*}$ is the reciprocal of the enzyme reactivation time. This conventional kinetic scheme corresponds to a set of rate equations that allow one to get the time-dependent concentrations. We would like to know how diffusion of the enzymes and substrates influences the above description.

We considered this problem previously¹² in the limit that the reactivation time was sufficiently short so that the concentration of $E^{*}$ was negligible and derived a diffusion-modified kinetic scheme (see Eq. 51) using a many-particle approach. Then Lawley and Keener^15,16 studied the situation where the concentration of $E^{*}$ is arbitrary and presented a different kinetic scheme (see Eq. 52), whose prediction was shown to be in excellent agreement with the results of many-particle simulations of Takahashi et al.¹¹ When we subsequently investigated the general case, we were unable to rederive their results. Rather, as will be shown in this paper, we obtained a diffusion-modified kinetic scheme (see Eq. 50) that unexpectedly contained a negative rate constant. The purpose of this paper is therefore to derive this surprising result in two ways that have a sound theoretical foundation. The first derivation is a step-by-step hopefully pedagogical approach, where we gradually increase the sophistication of the description. The second is a more formal derivation using our general theory of diffusion-influenced coupled reaction networks¹⁴ that has been shown to be accurate for two coupled association-dissociation reactions.¹⁷

The Methods section of this paper contains a detailed description of our theoretical considerations. We have made a special effort not to skip too many steps in the mathematical derivations as is so often done. The diffusion-modified rate equations for double phosphorylation with finite reactivation time of the enzyme are derived in Sec. 2.3. All rate constants in these equations are expressed in terms of the intrinsic rate constants and appropriate capture probabilities. The capture probabilities are considered in detail in Sec. 2.4. In Sec. 2.5, we extend the theory to a non-Markovian description with time-dependent memory kernels. Section 2.6 contains an alternative derivation of the diffusion-modified rate equations. These equations are mapped onto a kinetic scheme with new reaction channels in Sec. 3. This Section contains illustrative calculations for the kinetics of the substrates. Our theory is then applied to the double phosphorylation-dephosphorylation cycle, in which bistability of the steady-state concentrations can emerge as a result of competition of substrates for kinases and phosphotases. It was shown before by numerical simulations that diffusion can lead to the loss of bistability.¹¹ We show that our diffusion-modified kinetic scheme quantitatively reproduces the simulation results to the same degree as the scheme proposed by Lawley and Keener.¹⁵

2. METHODS

2.1. Rate equations of chemical kinetics

Consider double phosphorylation of an enzyme that is described by the conventional kinetics scheme in Eq. 2 in the limit of fast diffusion. Following Ref. 11, we assume that the enzyme and substrate bind when they come in contact irrespective of their orientation. The bound state can dissociate with or without catalytic conversion and form a pair of molecules separated by the contact radius $R$. This model can be readily extended to a more realistic one where the reaction sites are anisotropic and reaction occurs only when the reactants come in contact with the right orientation, as will be shown in Sec. 2.6.

Let us start with the conventional rate equations that correspond to the kinetic scheme in Eq. 2. The equations for the concentrations $\left[S_0\right],\left[S_1\right],\left[S_{1}E\right]$, and $\left[E^{*}\right]$ are:

$$\begin{gathered} \frac{d\left[S_0\right]}{dt}=-{\kappa }_0^a[E]\left[S_0\right]+{\kappa }_0^d\left[S_{0}E\right] \\ \frac{d\left[S_1\right]}{dt}=-{\kappa }_1^a[E]\left[S_1\right]+{\kappa }_1^d\left[S_{1}E\right]+{\kappa }_0^c\left[S_{0}E\right] \\ \frac{d\left[S_{1}E\right]}{dt}={\kappa }_1^a[E]\left[S_1\right]-\left({\kappa }_1^d+{\kappa }_1^c\right)\left[S_{1}E\right] \\ \frac{d\left[E^{*}\right]}{dt}={\kappa }_0^c\left[S_{0}E\right]+{\kappa }_1^c\left[S_{1}E\right]-k^{*}\left[E^{*}\right] \end{gathered}$$ (3)

The equations for $\left[S_2\right],\left[S_{0}E\right]$, and $\left[E\right]$ are similar. The total concentration is conserved, so $\left[S_0\right]+\left[S_1\right]+\left[S_2\right]+\left[S_{0}E\right]+\left[S_{1}E\right]=S_{tot}$ and $\left[E\right]+\left[E^{*}\right]+\left[S_{0}E\right]+\left[S_{1}E\right]=E_{tot}$.

2.2. Pair distribution functions

When the reaction is influenced by diffusion of the reactants, the binding rate is not ${\kappa }_i^a\left[E\right]\left[S_i\right]$, but is related to the pair distribution function ${\rho }_i(r,t)(i=\mathrm{0,1})$ of the substrate $S_i$ and the enzyme $E$ in the active state (since only the active enzyme can bind). The distribution functions depend on the distance between the enzyme and the substrate, $r$. When $r$ is large, the enzyme and substrate are independent, so that ${\rho }_i\to \left[E\right]\left[S_i\right]$ as $r\to \mathrm{\infty }$. At small $r$, the enzyme and substrate are correlated due to the reaction between them, and the binding rate is ${\kappa }_i^a{\rho }_i(R,t)$ when the reaction occurs at the contact radius $R$.

Replacing the binding rates in Eq. 3 by the rates that depend on the pair distribution functions, we have:

$$\begin{gathered} \frac{d\left[S_0\right]}{dt}=-{\kappa }_0^a{\rho }_0(R,t)+{\kappa }_0^d\left[S_{0}E\right] \\ \frac{d\left[S_1\right]}{dt}=-{\kappa }_1^a{\rho }_1(R,t)+{\kappa }_1^d\left[S_{1}E\right]+{\kappa }_0^c\left[S_{0}E\right] \\ \frac{d\left[S_{1}E\right]}{dt}={\kappa }_1^a{\rho }_1(R,t)-\left({\kappa }_1^d+{\kappa }_1^c\right)\left[S_{1}E\right] \\ \frac{d\left[E^{*}\right]}{dt}={\kappa }_0^c\left[S_{0}E\right]+{\kappa }_1^c\left[S_{1}E\right]-k^{*}\left[E^{*}\right] \end{gathered}$$ (4)

and similar equations for other concentrations. The dissociation and catalytic rates do not change and are the same as in Eq. 3. The equation for $\left[E^{*}\right]$ does not involve the binding rates, so it is the same as that in Eq. 3. These equations for the concentrations are formally exact for a homogeneous system, but to make progress, the pair distribution functions must be approximated. Different levels of approximation were considered in detail¹⁸ for the simplest reversible binding reaction $A+B\rightleftharpoons C$, and the resulting predictions were compared (without adjustable parameters) with accurate many-particle simulations for a model, in which association is described by the Collins-Kimball¹⁹ boundary condition. Here we start by deriving equations for the pair distribution functions in the lowest order using a physically intuitive approach.

For the sake of convenience, we first introduce the sink term ${\kappa }_0^a\sigma \left(r\right){\rho }_0$, which describes the binding rate of the unphosphorylated substrate and the active enzyme. ${\kappa }_0^a\sigma \left(r\right)$ is the binding rate constant (with the dimensionality 1/time) when the enzyme-substrate pair is separated by $r$. For the contact reaction, $\sigma \left(r\right)=\delta (r-R)/4\pi R^2$. The rate constant to form an unbound pair of $S_0$ and $E$ separated by $r$ via dissociation is ${\kappa }_0^d\sigma \left(r\right)$. The distribution function for the unphosphorylated substrate, ${\rho }_0$, changes due to diffusion (which is described by $\left.D{\nabla }^2{\rho }_0\right)$, binding $\left({\kappa }_0^a\sigma \left(r\right){\rho }_0\right)$ and dissociation $\left({\kappa }_0^d\sigma \left(r\right)\left[S_{0}E\right]\right)$ of the bound state $S_{0}E$. We assume that after a short transient time comparable to the diffusion time (i.e., the time required to diffuse a distance comparable to the contact radius, $R^2/D$) the pair distribution achieves a steady state, so the distribution changes in time only implicitly due to the slowly changing concentrations. In this case the equation for ${\rho }_0$ is:

$$D{\nabla }^2{\rho }_0-{\kappa }_0^a\sigma (r){\rho }_0+{\kappa }_0^d\sigma (r)\left[S_{0}E\right]=0$$ (5)

with the reflecting boundary condition at contact, $r=R$, and ${\rho }_0(r\to \mathrm{\infty })=\left[E\right]\left[S_0\right]$. Here $D$ is the relative diffusion constant of an enzyme-substrate pair, ${\nabla }^2=r^{-2}d/dr{r}^{2}d/dr$ is the three-dimensional diffusion operator. For a contact reaction, this equation is equivalent to $D{\nabla }^2{\rho }_0=0$ with the additional boundary condition at contact, $D4\pi R^{2}d{\rho }_0/{\left.dr\right|}_{r=R}={\kappa }_0^a{\rho }_0-{\kappa }_0^d\left[S_{0}E\right]$

The pair distribution of the singly phosphorylated substrate and the enzyme in the active state, ${\rho }_1$, changes due to diffusion, binding and dissociation of $S_{1}E$, similar to ${\rho }_0$. In addition, ${\rho }_1$ increases because $E^{*}$ decays to $E$. One would naively account for this increase by including the term $k^{*}{\rho }_1^{*}$, where ${\rho }_1^{*}$ is the pair distribution function of $E^{*}$ and $S_1$. However, at large $r$, where ${\rho }_1$ is equal to $\left[S_1\right]\left[E\right]$, the unimolecular decay of $E^{*}$ is already taken into account by the rate equation for $E$ coupled with $E^{*}$. Thus, instead of $k^{*}{\rho }_1^{*}$, we must use $k^{*}\left({\rho }_1^{*}-\left[E^{*}\right]\left[S_1\right]\right)$, and so the equation for ${\rho }_1$ in the steady-state limit is

$$D{\nabla }^2{\rho }_1-{\kappa }_1^a\sigma (r){\rho }_1+{\kappa }_1^d\sigma (r)\left[S_{1}E\right]+k^{\mathbf{*}}\left({\rho }_1^{\mathbf{*}}-\left[E^{\mathbf{*}}\right]\left[S_1\right]\right)=0$$ (6)

with ${\rho }_1(r\to \mathrm{\infty })=\left[E\right]\left[S_1\right]$.

Finally, the pair correlation function of the singly phosphorylated substrate and enzyme in the inactive state, ${\rho }_1^{*}(r,t)$, changes due to diffusion, the catalytic conversion of $S_{0}E$ to form $S_1$ and $E^{*}$, and conversion to the active enzyme:

$$D{\nabla }^2{\rho }_1^{\mathbf{*}}+{\kappa }_0^c\sigma (r)\left[S_{0}E\right]-k^{\mathbf{*}}\left({\rho }_1^{\mathbf{*}}-\left[E^{\mathbf{*}}\right]\left[S_1\right]\right)=0$$ (7)

with ${\rho }_1^{*}(r\to \mathrm{\infty })=\left[E^{*}\right]\left[S_1\right]$. Note that when there is no contribution from catalysis $\left({\kappa }_0^c=\right.0)$, it follows from the above equation that ${\rho }_1^{*}=\left[E^{*}\right]\left[S_1\right]$, as it must be because there is no correlation between $S_1$ and $E^{*}$ in this case. This confirms our approximation for the unimolecular decay rate as $k^{*}\left({\rho }_1^{*}-\left[E^{*}\right]\left[S_1\right]\right)$.

It is interesting to note that, unlike Eq. 6, Eq. 5 for ${\rho }_0$ does not contain a contribution from the pair correlation function between $E^{*}$ and $S_0,{\rho }_0^{*}$. One might expect that Eq. 5 should contain a term $k^{*}\left({\rho }_0^{*}-\left[E^{*}\right]\left[S_0\right]\right)$ in analogy to Eq. 6. However, this term is actually zero because $E^{*}$ and $S_0$ are not correlated. To verify this, consider the equation for ${\rho }_0^{*}$, which is analogous to Eq. 7 but without the catalytic term:

$$D{\nabla }^2{\rho }_0^{*}-k^{*}\left({\rho }_0^{*}-\left[E^{*}\right]\left[S_0\right]\right)=0$$

with ${\rho }_0^{*}(r\to \mathrm{\infty })=\left[E^{*}\right]\left[S_0\right]$. The solution of this is just ${\rho }_0^{*}=\left[E^{*}\right]\left[S_0\right]$, and so the term $k^{*}\left({\rho }_0^{*}-\left[E^{*}\right]\left[S_0\right]\right)$ vanishes.

2.3. Diffusion-modified rate equations

Equations 5–7 for the pair distribution functions are the key approximations. They can be analytically solved for our microscopic model and expressed in terms of the bulk concentrations. The steady-state equation 5 for ${\rho }_0$ is equivalent to the diffusion equation

$$D{\nabla }^2{\rho }_0=\frac{D}{r^2}\frac{d}{dr}{r}^2\frac{d{\rho }_0}{dr}=0$$ (8)

with the boundary conditions ${\rho }_0\to \left[E\right]\left[S_0\right]$ when $r\to \mathrm{\infty }$ and

$${\left.4\pi R^{2}D\frac{d{\rho }_0}{dr}\right|}_{r=R}={\kappa }_0^a{\rho }_0-{\kappa }_0^d\left[S_{0}E\right]$$ (9)

at contact $r=R$. The solution of Eq. 8 has the form ${\rho }_0-\left[E\right]\left[S_0\right]=A/r$, where $A$ is found from the boundary condition at contact. As a result, we have

$${\rho }_0=[E]\left[S_0\right]+\frac{R}{r}\frac{{\kappa }_0^d\left[S_{0}E\right]-{\kappa }_0^a[E]\left[S_0\right]}{{\kappa }_0^a+k_D}$$ (10)

where $k_D=4\pi DR$ is the diffusion-controlled rate constant.

Now consider Eqs. 6 and 7, which are equivalent to

$$\begin{gathered} \frac{D}{r^2}\frac{d}{dr}{r}^2\frac{d{\rho }_1}{dr}+k^{\mathbf{*}}\left({\rho }_1^{\mathbf{*}}-\left[E^{\mathbf{*}}\right]\left[S_1\right]\right)=0 \\ \frac{D}{r^2}\frac{d}{dr}{r}^2\frac{d{\rho }_1^{\mathbf{*}}}{dr}-k^{\mathbf{*}}\left({\rho }_1^{\mathbf{*}}-\left[E^{\mathbf{*}}\right]\left[S_1\right]\right)=0 \end{gathered}$$ (11)

with the boundary conditions ${\rho }_1\to \left[E\right]\left[S_1\right],{\rho }_1^{*}\to \left[E^{*}\right]\left[S_1\right]$ when $r\to \mathrm{\infty }$ and

$$\begin{gathered} {\left.4\pi R^{2}D\frac{d}{dr}{\rho }_1\right|}_{r=R}={\kappa }_1^a{\rho }_1-{\kappa }_1^d\left[S_{1}E\right] \\ {\left.4\pi R^{2}D\frac{d}{dr}{\rho }_1^{\mathbf{*}}\right|}_{r=R}=-{\kappa }_0^c\left[S_{0}E\right] \end{gathered}$$ (12)

at contact $r=R$. The solutions of these equations have the form

$$\begin{gathered} {\rho }_1\left(r\right)=\left[E\right]\left[S_1\right]+\frac{R}{r}\left(A-B{e}^{-(r-R)\sqrt{k^{\mathbf{*}}/D}}\right) \\ {\rho }_1^{\mathbf{*}}\left(r\right)=\left[E^{\mathbf{*}}\right]\left[S_1\right]+B\frac{R}{r}{e}^{-(r-R)\sqrt{k^{\mathbf{*}}/D}} \end{gathered}$$ (13)

where the constants $A$ and $B$ are found using the boundary condition at contact, Eq. 12:

$$\begin{gathered} A=\frac{{\kappa }_1^d\left[S_{1}E\right]-{\kappa }_1^a\left[E\right]\left[S_1\right]+{\kappa }_0^c\left[S_{0}E\right]+{\kappa }_1^{a}B}{{\kappa }_1^a+k_D} \\ B=\frac{{\kappa }_0^c\left[S_{0}E\right]}{k_D\left(1+\sqrt{k^{\mathbf{*}}{R}^2/D}\right)} \end{gathered}$$ (14)

The binding rates ${\kappa }_0^a{\rho }_0(R,t)$ and ${\kappa }_1^a{\rho }_1(R,t)$ in Eq. 4 are then obtained using Eqs. 10, 13, and 14:

$${\kappa }_0^a{\rho }_0(R,t)={\kappa }_0^a{ϵ}_0\left[E\right]\left[S_0\right]+q_0{\kappa }_0^d\left[S_{0}E\right]$$ (15a)

$$\begin{gathered} {\kappa }_1^a{\rho }_1(R,t)={\kappa }_1^a{ϵ}_1\left[E\right]\left[S_1\right]+q_1{\kappa }_1^d\left[S_{1}E\right] \\ +q^{*}{\kappa }_0^c\left[S_{0}E\right] \end{gathered}$$ (15b)

where

$$q_0=1-{ϵ}_0={\kappa }_0^a/\left(k_D+{\kappa }_0^a\right)$$ (16a)

$$q_1=1-{ϵ}_1={\kappa }_1^a/\left(k_D+{\kappa }_1^a\right)$$ (16b)

$$q^{*}=\frac{{\kappa }_1^a}{\left({\kappa }_1^a+k_D\right)}\frac{\sqrt{k^{*}{R}^2/D}}{\left(1+\sqrt{k^{*}{R}^2/D}\right)}$$ (16c)

The physical meaning of coefficients $q_0,q_1$, and $q^{*}$ are considered in detail in Sec. 2.4. It will be shown there that they are equal to the capture (binding) probabilities defined for an isolated pair of the enzyme and a substrate that can only bind irreversibly (i.e., without possibility of dissociating, see Fig. 1). $q_0$ is the probability that an enzyme in the active state and unphosphorylated substrate initially in contact will eventually bind (be captured) rather than diffuse apart. $q_1$ is the capture probability for a singly phosphorylated substrate, which is initially in contact with an active enzyme. ${ϵ}_i=1-q_i,i=\mathrm{0,1}$, is the corresponding escape probability. $q^{*}$ is the probability that an enzyme in the inactive state initially in contact with singly phosphorylated substrate will eventually convert onto the active state and then bind the substrate rather than diffuse away. For the simplest model of reactive spheres, these capture probabilities can be found analytically (see Eq. 16). In general, for non-contact and non-isotropic reactivity, Eq. 15 is still valid, but the coefficients $q_i$ and $q^{*}$ can be found only numerically.

FIG. 1:

Capture and escape probabilities when the enzyme has a finite reactivation time. (a) An unphosphorylated substrate and the enzyme in the active state can either bind to either of the two sites with probability $q_0$ or diffuse apart with probability ${ϵ}_0=1-q_0$. (b) Left: a singly phosphorylated substrate and an enzyme in the active state located near the unmodified site can bind or diffuse apart with the probabilities $q_1$ and ${ϵ}_1$, respectively. Right: the enzyme in the inactive state located near the modified site of the substrate converts to the active state and then binds to the substrate (with probability $q^{*}$) or diffuses away (with probability ${ϵ}^{*}=1-q^{*}$).

Replacing the binding rates in Eq. 4 by those in Eq. 15, we get the diffusion-modified rate equations for $\left[S_0\right],\left[S_1\right]$, and $\left[S_{1}E\right]$

$$\begin{gathered} \frac{d\left[S_0\right]}{dt}=-{\kappa }_0^a{ϵ}_0[E]\left[S_0\right]+{\kappa }_0^d{ϵ}_0\left[S_{0}E\right] \\ \frac{d\left[S_1\right]}{dt}=-{\kappa }_1^a{ϵ}_1[E]\left[S_1\right]+{\kappa }_1^d{ϵ}_1\left[S_{1}E\right]+{\kappa }_0^c\left(1-q^{*}\right)\left[S_{0}E\right] \\ \frac{d\left[S_{1}E\right]}{dt}={\kappa }_1^a{ϵ}_1[E]\left[S_1\right]-\left({\kappa }_1^d{ϵ}_1+{\kappa }_1^c\right)\left[S_{1}E\right]+{\kappa }_0^c{q}^{*}\left[S_{0}E\right] \end{gathered}$$ (17)

Comparing these equations with Eq. 3, we note that the binding and dissociation rates in the conventional rate equations are scaled by the escape probabilities ${ϵ}_0$ and ${ϵ}_1$. In addition, the new terms (with $q^{*}$) appear.

The diffusion-modified equations for other concentrations are obtained similarly:

$$\begin{gathered} \frac{d\left[S_{0}E\right]}{dt}={\kappa }_0^a{ϵ}_0\left[E\right]\left[S_0\right]-\left({\kappa }_0^d{ϵ}_0+{\kappa }_0^c\right)\left[S_{0}E\right] \\ \frac{d\left[E\right]}{dt}=-{\kappa }_0^a{ϵ}_0\left[E\right]\left[S_0\right]+{\kappa }_0^d{ϵ}_0\left[S_{0}E\right]-{\kappa }_1^a{ϵ}_1\left[E\right]\left[S_1\right]+{\kappa }_1^d{ϵ}_1\left[S_{1}E\right]+k^{\text{*}}\left[E^{\text{*}}\right]-{\kappa }_0^c{q}^{\text{*}}\left[S_{0}E\right] \\ \frac{d\left[E^{\text{*}}\right]}{dt}={\kappa }_0^c\left[S_{0}E\right]+{\kappa }_1^c\left[S_{1}E\right]-k^{\text{*}}\left[E^{\text{*}}\right] \end{gathered}$$ (18)

Note that the equation for $\left[E^{*}\right]$ does not involve the binding terms and, therefore, is the same as the conventional rate equation.

2.4. Capture and escape probabilities

In this section we define the capture and escape probabilities and derive the expressions in Eq. 16 as well as the pair association fluxes, which are required for the memory kernels considered later in Sec. 2.5. All capture and escape probabilities are defined for an isolated enzyme-substrate pair initially in contact, which can only irreversibly bind (i.e., without possibility of dissociating or undergoing catalytic conversion). The capture probability is the probability that the pair eventually binds rather than diffuses apart. Various capture probabilities differ by the initial state of the pair, i.e., whether the substrate is unphosphorylated or singly phosphorylated and whether the enzyme is active or inactive (see Fig. 1).

Let us start with the capture probability for a pair of an unphosphorylated substrate $S_0$ and an active enzyme $E$ (see Fig. 1(a)). This is related to the Green's function $G_0\left(r,t\mid r^{\prime }\right.$), which is the probability density for an isolated pair of $S_0$ and $E$ to be in a distance $r$ apart at time $t$ given initially the pair was at $r^{\prime }$. The equation for this function contains a sink term, which accounts for irreversible binding at distance $r$:

$$\frac{\partial G_0}{\partial t}=D{\nabla }^2{G}_0-{\kappa }_0^a\sigma (r)G_0$$ (19)

The equation is subject to the initial condition $G_0(t=0)=\delta \left(r-r^{\prime }\right)/4\pi r^2$ and the reflecting boundary condition at $r=R$.

The irreversible pair association flux is defined as the probability density that the enzyme-substrate pair initially separated by $R$ irreversibly binds at time $t$. This flux is

$$J_0\left(t\right)={\kappa }_0^a{G}_0(R,t\mid R,0)$$ (20)

The capture probability (i.e., the probability to eventually bind) is the time integral of this quantity:

$$q_0={\int }_0^{\mathbf{\infty }}{J}_0(t)dt={\hat{J}}_0(0)$$ (21)

where ${\hat{J}}_0\left(s\right)={\int }_0^{\mathrm{\infty }}{J}_0\left(t\right)\mathrm{e}\mathrm{x}\mathrm{p}(-st)dt$ is the Laplace transform of $J\left(t\right)$. The probability of the alternative pathway (i.e., diffusing apart) is the escape probability ${ϵ}_0=1-q_0$.

To find the capture probability explicitly, one can directly solve Eq. 19, as was done in Eqs. 8–14. Alternatively, one can express $G_0\left(r,t\mid r^{\prime }\right)$ (with $\left(r\right)=\delta (r-R)/4\pi r^2$) in terms of the free Green's function, $g(r,t\mid R)$, as

$$G_0\left(r,t\mid r^{\prime }\right)=g\left(r,t\mid r^{\prime }\right)-{\kappa }_0^a{\int }_0^{t}g\left(r,t-t^{\prime }\mid R\right)G_0\left(R,t^{\prime }\mid r^{\prime }\right)d{t}^{\prime }$$ (22)

where the free Green's function $g\left(r,t\mid r^{\prime }\right)$ satisfies Eq. 19 without the sink term with the same initial condition and the reflecting boundary condition:

$$\frac{\partial g}{\partial t}=D{\nabla }^{2}g$$ (23)

Eq. 22 can be verified by differentiating it with respect to $t$ and using Eqs. 19 and 23.

Equation 22 can be solved in Laplace space to find the Laplace transform ${\hat{G}}_0(R,s\mid R)$. Using this in Eq. 20, we find the Laplace transform of $J_0\left(t\right)$:

$$\begin{gathered} {\hat{J}}_0\left(s\right)=\frac{{\kappa }_0^a\hat{g}\left(R,s\mid R\right)}{1+{\kappa }_0^a\hat{g}\left(R,s\mid R\right)} \\ =\frac{{\kappa }_0^a}{4\pi DR\left(1+\sqrt{s{R}^2/D}\right)+{\kappa }_0^a} \end{gathered}$$ (24)

where

$$\hat{g}(R,s\mid R)=\frac{1}{4\pi DR\left(1+\sqrt{s{R}^2/D}\right)}$$ (25)

By setting $s=0$, we arrive at the capture probability $q_0={\hat{J}}_0\left(0\right)$ in Eq. 16a.

To find the capture probabilities for the singly phosphorylated substrate in Fig. 1b, one has to introduce three different Green's functions, $G_{AA}\left(r,t\mid r^{\prime }\right),G_{AI}\left(r,t\mid r^{\prime }\right)$, and $G_{II}\left(r,t\mid r^{\prime }\right)$, where the subscripts refer to the state of the enzyme (i.e., active, $A$, or inactive, $I).G_{MN}(M,N=A,I)$ is the probability density for a singly phosphorylated substrate and the enzyme in state $M$ to be a distance $r$ apart at time $t$, given that initially the enzyme was in state $N$ a distance $r^{\prime }$ from the substrate. These functions satisfy

$$\frac{\partial G_{AA}}{\partial t}=D{\nabla }^2{G}_{AA}-{\kappa }_1^a\sigma (r)G_{AA}$$ (26a)

$$\frac{\partial G_{AI}}{\partial t}=D{\nabla }^2{G}_{AI}-{\kappa }_1^a\sigma (r)G_{AI}+k^{*}{G}_{II}$$ (26b)

$$\frac{\partial G_{II}}{\partial t}=D{\nabla }^2{G}_{II}-k^{*}{G}_{II}$$ (26c)

with the initial condition $G_{MN}(t=0)={\delta }_{MN}\delta \left(r-r^{\prime }\right)/4\pi r^2$. Note that the equations for $G_{AA}$ and $G_{AI}$ (but not $G_{II}$) contain a sink term, because the enzyme can bind only in the active state.

The irreversible reaction flux of an isolated pair of an enzyme and singly phosphorylated substrate depends on the initial state of the enzyme. The fluxes for the enzyme initially in the active and inactive state are $J_1\left(t\right)$ and $J^{*}\left(t\right)$, respectively, and are given by

$$J_1\left(t\right)={\kappa }_1^a{G}_{AA}(R,t\mid R,0)$$ (27a)

$$J^{*}\left(t\right)={\kappa }_1^a{G}_{AI}(R,t\mid R,0)$$ (27b)

Consequently, the capture probabilities for the enzyme initially in the active and inactive state are

$$q_1={\int }_0^{\mathbf{\infty }}{J}_1(t)dt={\hat{J}}_1(0)$$ (28a)

$$q^{*}={\int }_0^{\mathrm{\infty }}{J}^{*}(t)dt={\hat{J}}^{*}(0)$$ (28b)

Let us now find the expressions for $J_1$ and $J^{*}$. First note that the equation for $G_{AA}$ in Eq. 26a is Eq. 19 for $G_0$ with ${\kappa }_0^a$ replaced by ${\kappa }_1^a$. Consequently, it follows from Eq. 24 that

$${\hat{J}}_1(s)=\frac{{\kappa }_1^a}{4\pi DR\left(1+\sqrt{s{R}^2/D}\right)+{\kappa }_1^a}$$ (29)

To find $J^{*}\left(t\right)$ and $q^{*}={\hat{J}}^{*}(s=0)$, we solve the equation for $G_{AI}$ by rewriting Eq. 26b in a form similar to that in Eq. 22:

$$\begin{array}{l}{G}_{AI}\left(r,t\mid r^{\prime }\right)=g_{AI}\left(r,t\mid r^{\prime }\right)\\ -{\kappa }_1^a{\int }_0^t{g}_{AA}\left(r,t-t^{\prime }\mid R\right)G_{AI}\left(R,t^{\prime }\mid r^{\prime }\right)d{t}^{\prime }\end{array}$$ (30)

where $g_{AA}\left(r,t\mid r^{\prime }\right)$ and $g_{AI}\left(r,t\mid r^{\prime }\right)$ are the free Green's functions, which satisfy Eq. 26 without sink terms. The equation for $g_{AA}$ is the same as Eq. 23, so that $g_{AA}=g$. The equation for $g_{AI}$ is coupled with that for $g_{II}$:

$$\begin{gathered} \frac{\partial g_{AI}}{\partial t}=D{\nabla }^2{g}_{AI}+k^{\mathbf{*}}{g}_{II} \\ \frac{\partial g_{II}}{\partial t}=D{\nabla }^2{g}_{II}-k^{\mathbf{*}}{g}_{II} \end{gathered}$$ (31)

with the same initial condition $g_{MN}(t=0)={\delta }_{MN}\delta \left(r-r^{\prime }\right)/4\pi r^2,M,N=A,I$. The solution of these equations can be expressed in terms of the free Green's function $g$ by noting that both $g_{AI}+g_{II}$ and $g_{II}\mathrm{e}\mathrm{x}\mathrm{p}\left(k^{*}t\right)$ satisfy Eq. 23; therefore,

$$\begin{gathered} g_{AI}\left(r,t\mid r^{\prime }\right)=g\left(r,t\mid r^{\prime }\right)\left(1-e^{-k^{\mathbf{*}}t}\right) \\ {g}_{II}\left(r,t\mid r^{\prime }\right)=g\left(r,t\mid r^{\prime }\right)e^{-k^{\mathbf{*}}t} \end{gathered}$$ (32)

Laplace transforming $g_{AI}$ in the above equation, we get ${\hat{g}}_{AI}\left(r,s\mid r^{\prime }\right)=\hat{g}\left(r,s\mid r^{\prime }\right)-\hat{g}\left(r,s+k^{*}\mid r^{\prime }\right)$. This is used to solve Eq. 30 in Laplace space to find ${\hat{G}}_{AI}(R,s\mid R)$, which is then used in the Laplace transform of Eq. 27b. As a result we get

$$\begin{gathered} {\hat{J}}^{\text{*}}\left(s\right)=\frac{{\kappa }_1^a\left(\hat{g}\left(R,s\mid R\right)-\hat{g}\left(R,s+k^{\text{*}}\mid R\right)\right)}{1+{\kappa }_1^a\hat{g}\left(R,s\mid R\right)} \\ ={\hat{J}}_1\left(s\right)\frac{\sqrt{\left(s+k^{\text{*}}\right)R^2/D}-\sqrt{s{R}^2/D}}{1+\sqrt{\left(s+k^{\text{*}}\right)R^2/D}} \end{gathered}$$ (33)

It follows from Eqs. 28 and 33 (with $s=0$), that $q^{*}$ is indeed given by Eq. 16c.

2.5. Rate equations with memory kernels

In the above theory we have used the steady-state approximation for the pair distribution functions. A more accurate theory can be developed when the time dependence of the pair distribution functions is explicitly taken into account. To do so, we replace the steady-state Eqs. 5–7 by the equations for the time-dependent distribution functions:

$$\frac{\partial {\rho }_0}{\partial t}=\frac{d\left[E\right]\left[S_0\right]}{dt}+D{\nabla }^2{\rho }_0-{\kappa }_0^a\sigma (r){\rho }_0+{\kappa }_0^d\sigma (r)\left[S_{0}E\right]$$ (34a)

$$\frac{\partial {\rho }_1}{\partial t}=\frac{d\left[E\right]\left[S_1\right]}{dt}+D{\nabla }^2{\rho }_1-{\kappa }_1^a\sigma (r){\rho }_1+{\kappa }_1^d\sigma (r)\left[S_{1}E\right]+k^{*}\left({\rho }_1^{*}-\left[E^{*}\right]\left[S_1\right]\right)$$ (34b)

$$\begin{array}{rr}\frac{\partial {\rho }_1^{*}}{\partial t}=& \frac{d\left[E^{*}\right]\left[S_1\right]}{dt}+D{\nabla }^2{\rho }_1^{*}+{\kappa }_0^c\sigma (r)\left[S_{0}E\right]-k^{*}\left({\rho }_1^{*}-\left[E^{*}\right]\left[S_1\right]\right)\end{array}$$ (34c)

Initially, the molecules are uncorrelated, so ${\rho }_i=\left[E\right]\left[S_i\right],i=\mathrm{0,1}$, and ${\rho }_1^{*}=\left[E^{*}\right]\left[S_1\right]$. The boundary conditions are the same as before. The first term on the right-hand side of each equation describes the change in the distribution function due to the change in the appropriate bulk concentration. The other terms are the same as in the steady-state version of these equations, Eqs. 5–7. These equations together with the equation for the concentrations, Eq. 4, lead to the kinetics that is valid both at short and long times.

To proceed further, we first solve Eqs. 34a with $\sigma \left(r\right)=\delta (r-R)/4\pi R^2$ in terms of the time-dependent Green's function $G_0\left(r,t\mid r^{\prime }\right)$, which is defined in Eq. 19:

$$\begin{gathered} {\rho }_0(r,t)-[E]\left[S_0\right] \\ ={\int }_0^{t}d{t}^{\prime }{G}_0\left(r,t-t^{\prime }\mid R\right)f_0\left(t^{\prime }\right) \\ {f}_0(t)\equiv {\kappa }_0^d\left[S_{0}E\right]-{\kappa }_0^a[E]\left[S_0\right] \end{gathered}$$ (35)

Multiplying the solution by ${\kappa }_0^a$, we find the binding rate:

$${\kappa }_0^a{\rho }_0(R,t)={\kappa }_0^a[E]\left[S_0\right]+{\int }_0^t{J}_0\left(t-t^{\prime }\right)f_0\left(t^{\prime }\right)d{t}^{\prime }$$ (36)

where $J_0\left(t\right)$ is the irreversible association reaction flux for an isolated pair of unphosphorylated substrate and enzyme defined in Eqs. 20. This equation is reduces to Eq. 15a if ${\int }_0^t{J}_0\left(t-t^{\prime }\right)f_0\left(t^{\prime }\right)d{t}^{\prime }$ is approximated by ${\int }_0^{\mathrm{\infty }}{J}_0\left(t^{\prime }\right)d{t}^{\prime }{f}_0\left(t\right)\equiv q_0{f}_0\left(t\right)$. This is the steady-state approximation, which assumes that $J_0\left(t\right)$ changes faster than the concentrations in $f_0\left(t\right)$.

Similarly, Eq. 34b for ${\rho }_1$ can be solved using the Green's functions in Eqs. 26a and 26b:

$$\begin{gathered} {\rho }_1(r,t)-[E]\left[S_1\right] \\ ={\int }_0^t{G}_{AA}\left(r,t-t^{\prime }\mid R\right)f_1\left(t^{\prime }\right)d{t}^{\prime } \\ +{\int }_0^t{G}_{AI}\left(r,t-t^{\prime }\mid R\right)f^{*}\left(t^{\prime }\right)d{t}^{\prime } \\ {f}_1(t)\equiv {\kappa }_1^d\left[S_{1}E\right]-{\kappa }_1^a[E]\left[S_1\right] \\ {f}^{*}(t)\equiv {\kappa }_0^c\left[S_{0}E\right] \end{gathered}$$ (37)

Multiplying this by ${\kappa }_1^a$, we find the binding rate:

$$\begin{gathered} {\kappa }_1^a{\rho }_1(R,t)={\kappa }_1^a[E]\left[S_1\right] \\ +{\int }_0^t\left(J_1\left(t-t^{\prime }\right)f_1\left(t^{\prime }\right)+J^{\mathbf{*}}\left(t-t^{\prime }\right)f^{\mathbf{*}}\left(t^{\prime }\right)\right)d{t}^{\prime } \end{gathered}$$ (38)

where $J_1\left(t\right)$ and $J^{*}\left(t\right)$ are the irreversible association fluxes defined in Eq. 27. $J_1\left(t\right)$ corresponds to the enzyme-substrate pair with the enzyme that is initially in the active state and $J^{*}\left(t\right)$ corresponds to the initially inactive enzyme. This equation is reduced to Eq. 15b by approximations ${\int }_0^t{J}_1\left(t-t^{\prime }\right)f_1\left(t^{\prime }\right)d{t}^{\prime }\approx q_1{f}_1\left(t\right)$ and ${\int }_0^t{J}^{*}\left(t-t^{\prime }\right)f^{*}\left(t^{\prime }\right)d{t}^{\prime }\approx q_1^{*}{f}^{*}\left(t\right)$.

Thus the rate equations with the memory kernel can be obtained from Eqs. 17–18 using the following replacements $(i=\mathrm{0,1})$:

$$\begin{gathered} {\kappa }_i^a{ϵ}_i[E]\left[S_i\right]\to {\kappa }_i^a[E]\left[S_i\right]-{\kappa }_i^a{\int }_0^t{J}_i\left(t-t^{\prime }\right)\left[E\left(t^{\prime }\right)\right]\left[S_i\left(t^{\prime }\right)\right]d{t}^{\prime } \\ {\kappa }_i^d{ϵ}_i\left[S_{i}E\right]\to {\kappa }_i^d\left[S_{i}E\right]-{\kappa }_i^d{\int }_0^t{J}_i\left(t-t^{\prime }\right)\left[S_{i}E\left(t^{\prime }\right)\right]d{t}^{\prime } \\ {\kappa }_0^c{q}_1^{*}\left[S_{0}E\right]\to {\kappa }_0^c{\int }_0^t{J}^{*}\left(t-t^{\prime }\right)\left[S_{0}E\left(t^{\prime }\right)\right]d{t}^{\prime } \end{gathered}$$ (39)

The Laplace transforms of $J_0\left(t\right),J_1\left(t\right)$, and $J^{*}\left(t\right)$ are given in Eqs. 24, 29, and 33.

2.6. Application of the theory of diffusion-influenced reaction networks

In this section, we rederive the rate equations using the general theory of diffusion-influenced reaction networks in Ref. 14. We begin by rewriting the ordinary rate equations, Eq. 3, in terms of the reaction fluxes. There is one unimolecular flux

$$u\left(t\right)=k^{\mathbf{*}}\left[E^{\mathbf{*}}\right]$$ (40)

and four bimolecular net fluxes:

$$\begin{gathered} v_1\left(t\right)={\kappa }_0^a\left[S_0\right]\left[E\right]-{\kappa }_0^d\left[S_{0}E\right] \\ {v}_2\left(t\right)=-{\kappa }_0^c\left[S_{0}E\right] \\ {v}_3\left(t\right)={\kappa }_1^a\left[S_1\right]\left[E\right]-{\kappa }_1^d\left[S_{1}E\right] \\ {v}_4\left(t\right)=-{\kappa }_1^c\left[S_{1}E\right] \end{gathered}$$ (41)

Note that there are no binding terms in fluxes $v_2$ and $v_4$.

The rate equations corresponding to the scheme in Eq. 2 can be written in terms of these fluxes as

$$\begin{gathered} \frac{d\left[S_0\right]}{dt}=-v_1(t) \\ \frac{d\left[S_{0}E\right]}{dt}=v_1(t)+v_2(t) \\ \frac{d\left[S_1\right]}{dt}=-v_2(t)-v_3(t) \\ \frac{d\left[S_{1}E\right]}{dt}=v_3(t)+v_4(t) \\ \frac{d\left[S_2\right]}{dt}=-v_4(t) \\ \frac{d\left[E^{*}\right]}{dt}=-v_2(t)-v_4(t)-u(t) \\ \frac{d\left[E\right]}{dt}=-v_1(t)-v_3(t)+u(t) \end{gathered}$$ (42)

If we introduce a vector of concentrations $\mathit{x}\left(t\right)$ with elements $\left[S_0\right],\left[S_{0}E\right],\left[S_1\right],\left[S_{1}E\right],\left[S_2\right],\left[E^{*}\right],\left[E\right]$ and a vector of bimolecular fluxes $\mathit{v}\left(t\right)$ with elements $v_i\left(t\right),i=1,\dots ,4$, then the rate equations in Eq. 42 can be written in the matrix form as

$$\frac{d\mathit{x}}{dt}={\mathit{S}}_{1}u(t)+{\mathit{S}}_2\mathit{v}(t)$$ (43)

where ${\mathit{S}}_1$ and ${\mathit{S}}_2$ are the unimolecular and bimolecular matrices of stoichiometric coefficients defined as

$${\mathit{S}}_1=\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ -1\\ 1\end{array}\right),{\mathit{S}}_2=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 1& 1& 0& 0\\ 0& -1& -1& 0\\ 0& 0& 1& 1\\ 0& 0& 0& -1\\ 0& -1& 0& -1\\ -1& 0& -1& 0\end{array}\right)$$ (44)

The dimension of ${\mathit{S}}_1$ and ${\mathit{S}}_2$ is defined by the number of species (7) and the number of fluxes for unimolecular (1) and bimolecular (4) reactions.

According to the theory of diffusion-influenced reaction networks,¹⁴ the diffusion-modified rate equations are

$$\frac{d\mathit{x}}{dt}={\mathit{S}}_{1}u(t)+{\mathit{S}}_2\left(\mathit{v}(t)-{\int }_0^t\mathit{J}\left(t-t^{\prime }\right)\mathit{v}\left(t^{\prime }\right)d{t}^{\prime }\right)$$ (45)

where $\mathit{J}\left(t\right)$ is the 4 × 4 matrix of irreversible association fluxes. It has two non-zero diagonal elements corresponding to the first and third fluxes. The diagonal elements $[\mathit{J}{]}_{22}$ and $[\mathit{J}{]}_{44}$ are zero since there are no association terms in these fluxes. The only off-diagonal term is $[\mathit{J}{]}_{32}$.

There are several levels of the theory ranging from the non-Markovian equations with “memory” to the simplest Markovian description with time-independent rate constants. In the lowest level of the non-Markovian theory, which is valid at small concentrations of the reactants, the fluxes are defined for an isolated enzyme-substrate pair, so the flux matrix $\mathit{J}$ is

$$\mathit{J}(t)=\left(\begin{array}{cccc}{J}_0(t)& 0& 0& 0\\ 0& 0& 0& 0\\ 0& J^{\mathbf{*}}(t)& J_1(t)& 0\\ 0& 0& 0& 0\end{array}\right)$$ (46)

For the model adopted in this paper, the Laplace transforms of $J_0\left(t\right),J_1\left(t\right)$, and $J^{*}\left(t\right)$ are given in Eqs. 24, 29, and 33, respectively. The non-Markovian equation 45 is identical to Eq. 4 with the binding rates in Eqs. 36 and 38.

Equation 45 can be simplified assuming that the pair association fluxes change much faster than the bulk concentrations. In this case the equations with memory kernels, Eq. 45, are transformed into the ordinary differential equations:

$$\frac{d\mathit{x}}{dt}={\mathit{S}}_{1}u(t)+{\mathit{S}}_2(\mathit{I}-\mathit{Q})\mathit{v}(t)$$ (47)

where $\mathit{I}$ is the 4 × 4 identity matrix and $\mathit{Q}$ is the matrix of capture probabilities:

$$\mathit{Q}={\int }_0^{\mathbf{\infty }}\mathit{J}(t)dt=\left(\begin{array}{cccc}{q}_0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& q^{\mathbf{*}}& q_1& 0\\ 0& 0& 0& 0\end{array}\right)$$ (48)

where $q_0,q_1$, and $q^{*}$ are given in Eq. 16 for the model adopted in this paper. Equation 47 is identical to Eqs. 17 and 18.

A less drastic approximation to Eq. 45 is

$$\frac{d\mathit{x}}{dt}={\mathit{S}}_{1}u(t)+{\mathit{S}}_2(\mathit{I}-\mathit{Q}(t))\mathit{v}(t)$$ (49)

where $\mathit{Q}\left(t\right)={\int }_0^t\mathit{J}\left(t\right)d{t}^{\prime }$. The matrix elements of $\mathit{Q}\left(t\right)$ are the probabilities to be bound by time t. If $\mathit{Q}\left(t\right)$ is replaced by $\mathit{Q}\left(\mathrm{\infty }\right)\equiv \mathit{Q}$, the steady-state result in Eq. 47 is recovered with the matrix of probabilities to be bound by any time. Equation 49 corresponds to the rate equations in Eqs. 17 and 18 with time-dependent rate coefficients.

The accuracy of Eqs. 45, 47, and 49 was studied for a simple case of single-site reversible binding^18,20 (i.e., $+B\rightleftharpoons C$) and for two coupled reversible binding reactions¹⁷ (i.e., $C_1\rightleftharpoons A+B\rightleftharpoons C_2$). The simplest Markovian equations with time-independent rate constants, Eq. 47, are accurate at times longer than the time required to diffuse a distance comparable to the size of the reactants. For example, when reaction occurs at a contact distance $R$, this time is approximately $R^2/D$. Therefore, for $R=5\mathrm{n}\mathrm{m}$ and $D=1\mu {\mathrm{m}}^2/\mathrm{s}$, the Markovian equations are not satisfactory on the time scale of 0.1 ms and faster.

To describe kinetics on the time scale of the diffusion time and faster, the non-Markovian equations in Eq. 45 or Eq. 49 can be used. The convolution-type equations in Eq. 45 are more accurate when $\left(\left[S_i\right]+\left[E\right]\right){\kappa }_i^a/{\kappa }_i^d$ is comparable to or smaller than 1, whereas the equations with the time-dependent rate coefficients, Eq. 49, are more accurate when $\left(\left[S_i\right]+\left[E\right]\right){\kappa }_i^a/{\kappa }_i^d$ is comparable to or larger than 1.¹⁷ The accuracy of the theory at high concentrations can be increased by accounting for the influence of bulk molecules on the pair association fluxes.^14,17 The most accurate version of the theory is when $\mathit{J}\left(t\right)$ is found self-consistently.^18,20

The convolution-type equations, Eq. 45, can also be used to find how the concentrations approach their steady-state values at long times. The Markovian equations result in the exponential relaxation to the steady state as expected from the ordinary chemical kinetics. However, the reversible binding reactions without unimolecular decay approach equilibrium as a power law because the final approach is determined by the redistribution of the molecules through diffusion.^21,22 For $A+B\rightleftharpoons C$ when $B$ 's are in excess and $A$ and $C$ are static, the change from exponential to power law behavior occurs when $t\approx t^{*}$, where $\mathrm{e}\mathrm{x}\mathrm{p}\left(-\left({\kappa }^a\left[B\right]+\right.\right.\left.\left.{\kappa }^d\right)ϵt^{*}\right)=K_{eq}{\left(1+K_{eq}\left[B\right]\right)}^{-2}{\left(4\pi D{t}^{*}\right)}^{-3/2},K_{eq}={\kappa }^a/{\kappa }^d$ (i.e., when the chemical kinetics and power law relaxation functions are equal).

For a complex dynamics with anisotropic reactivity (orientational constraints), the rate equations stay the same, but the reaction fluxes $J$'s and capture probabilities $q$'s change. They are defined for a pair of reactants that can only irreversibly bind, with various initial states of the enzyme and substrate (see Fig. 1). In general, the capture probabilities can be found only numerically. However, this problem involves only an isolated pair of molecules and many-particle simulations are not required.

In the case of the reactions on a surface of membranes, the kinetics is always non-Markovian since, in two dimensions, the escape probability is zero and the rate coefficients never achieve a steady state. Therefore, Eq. 47 with time-independent rate constants cannot be applied directly to the reactions on a surface (although the approximated Markovian description can be suggested¹⁷). Unlike Eq. 47, equations 45 and 49 are applicable directly in two dimensions.

3. RESULTS AND DISCUSSION

3.1. Diffusion-modified kinetic scheme

The diffusion-modified equations in Eqs. 17 and 18 (or Eq. 47) correspond to the kinetic scheme:

$$\begin{gathered} S_0+E\underset{{\kappa }_0^d{ϵ}_0}{\stackrel{{\kappa }_0^a{ϵ}_0}{\rightleftharpoons }}{S}_{0}E\stackrel{{\kappa }_0^c}{⟶}{S}_1+E^{*} \\ {S}_1+E\stackrel{{\kappa }_1^a{ϵ}_1}{\underset{{\kappa }_1^d{ϵ}_1}{\rightleftharpoons }}{S}_{1}E\stackrel{{\kappa }_1^c}{⟶}{S}_2+E^{*} \\ {S}_{0}E\stackrel{{\kappa }_0^c{q}^{*}}{⟶}{S}_{1}E \\ {E}^{*}\stackrel{k^{*}}{⟶}E \\ {S}_{0}E\stackrel{-{\kappa }_0^c{q}^{*}}{⟶}{S}_1+E \end{gathered}$$ (50)

This scheme, which is shown in Fig. 2(b), is the main result of the paper. It is to be compared with the chemical kinetics scheme in Eq. 2 (shown in Fig. 2(a)). The black arrows in Fig. 2(b) correspond to the channels that are not affected by diffusion, they are the same as those in Fig. 2(a). The rate constants of the channels denoted by the green arrows are the binding and dissociation rate constants scaled by the corresponding escape probabilities, which are described in Fig. 1. Finally, the red arrows in Fig. 2(b) correspond to the new reaction channels. The solid red arrow describes the new transition $S_{0}E\to S_{1}E$ between the two bound states. This transition appears because after phosphorylating the first site and releasing the substrate, the enzyme does not diffuse away but stays around long enough for it to become active and bind to the same substrate to form $S_{1}E$. The rate constant of this process is the product of the catalytic rate, ${\kappa }_0^c$, and the capture probability $q^{*}$ defined in Fig. 1(b).

FIG. 2:

Double phosphorylation with finite enzyme reactivation time. (a) Ordinary kinetic scheme in the limit of fast diffusion (starting on the left and going clock-wise). (b) Diffusion-modified kinetic scheme with the rate constants specified in Eq. 50. New reaction channels have positive (red solid arrows) and negative (red dashed arrows) rate constants. Green arrows correspond to the channels with the rate constants scaled by the escape probabilities. Black arrows correspond to the reaction channels that are not modified.

Another new reaction channel (the last reaction in Eq. 50 and the dashed red arrow in Fig. 2(b)) connects $S_{0}E$ to the active enzyme $E$ and singly phosphorylated substrate $S_1$. This channel is unusual because it has a negative rate constant, so that it describes not the increase in the concentrations of $E$ and $S_1$, but the decrease, which is proportional to $\left[S_{0}E\right]$. To understand this rate, consider the catalytic conversion of the bound state $S_{0}E$. $S_{0}E$ decays with the rate constant ${\kappa }_c^0$ producing $S_1$ and the inactive enzyme $E^{*}$. Since $E^{*}$ cannot react with $S_1$, the rate constant of this reaction channel stays unmodified. The reaction flux $S_{0}E\to S_1+E$ with the negative rate constant compensates the reaction flux $S_{0}E\to S_{1}E$, so that the total decay of $S_{0}E$ is described by the rate constant ${\kappa }_c^0$.

The reaction channel with a negative rate constant is a consequence of the reduction of a complex reaction-diffusion system to a simple kinetic scheme, which corresponds to a set of ordinary differential equations with time-independent rate constants. Except at very short times, this set can be used to accurately find the time dependence of the concentrations of all species. At short times and special initial conditions, the above scheme can even lead to negative concentrations. However, at longer times the diffusion-modified kinetic scheme accurately describes the time dependence of the concentrations. To get both short and long times, one has to abandon the steady-state approximation that has been used to derive the kinetic scheme in Eq. 50 with time-independent rate constants and use the non-Markovian equations, Eq. 45.

When the reactivation time is short, the concentration of $E^{*}$ is very small and can be neglected. Then the above scheme can be reduced to a simpler version, which we considered earlier¹²:

$$\begin{gathered} S_0+E\underset{{\kappa }_0^d{ϵ}_0}{\stackrel{{\kappa }_0^a{ϵ}_0}{\rightleftharpoons }}{S}_{0}E\stackrel{{\kappa }_0^c\left(1-q^{\mathbf{*}}\right)}{⟶}{S}_1+E \\ {S}_1+E\underset{{\kappa }_1^d{ϵ}_1}{\stackrel{{\kappa }_1^a{ϵ}_1}{\rightleftharpoons }}{S}_{1}E\stackrel{{\kappa }_1^c}{⟶}{S}_2+E \\ {S}_{0}E\stackrel{{\kappa }_0^c{q}^{\mathbf{*}}}{⟶}{S}_{1}E \end{gathered}$$ (51)

This scheme does not involve $\left[E^{*}\right]$ and all the rates here are positive.

The kinetic schemes in Eqs. 50 and 51 are closely related to that of Lawley and Keener^15,16:

$$\begin{gathered} S_0+E\underset{{\kappa }_0^d{ϵ}_0}{\stackrel{{\kappa }_0^a{ϵ}_0}{\rightleftharpoons }}{S}_{0}E\stackrel{{\kappa }_0^c\left(1-q^{*}\right)}{⟶}{S}_1+E^{*} \\ {S}_1+E\underset{{\kappa }_1^d{ϵ}_1}{\stackrel{{\kappa }_1^a{ϵ}_1}{\rightleftharpoons }}{S}_{1}E\stackrel{{\kappa }_1^c}{⟶}{S}_2+E^{*} \\ {S}_{0}E\stackrel{{\kappa }_0^c{q}^{*}}{⟶}{S}_{1}E \\ {E}^{*}\stackrel{k^{*}}{⟶}E \end{gathered}$$ (52)

It can be obtained by replacing the last $E$ in Eq. 51 by $E^{*}$ and adding the channel $E^{*}\to E$. Alternatively, this can be obtained by replacing $E$ in the last reaction channel in Eq. 50 by $E^{*}$. The rate equations corresponding to the schemes in Eqs. 52 and 50 coincide except those for $\left[E^{*}\right]$ and $\left[E\right]$. The rate equation for $\left[E^{*}\right]$ in the scheme 52 is

$$\frac{d\left[E^{\mathbf{*}}\right]}{dt}=\left(1-q^{\mathbf{*}}\right){\kappa }_0^c\left[S_{0}E\right]+{\kappa }_1^c\left[S_{1}E\right]-k^{\mathbf{*}}\left[E^{\mathbf{*}}\right]$$ (53)

The exact equation for $\left[E^{*}\right]$ is the above equation with $q^{*}$ set to 0, as shown in Eq. 4, and thus involves only catalytic steps and unimolecular decay. Since these are irreversible unimolecular steps that do not involve diffusion, they cannot depend on the rebinding probability $q^{*}$. The rate equation for $\left[E^{*}\right]$ in the scheme 50 coincides with the exact equation, whereas that in the scheme 52 does not. This is the price that had to be paid to obtain a diffusion-modified kinetic scheme with all positive rate constants. However, for the model and parameters considered in this paper, the difference between the three kinetic schemes is very small.

3.2. Kinetics of double phosphorylation

To illustrate the utility of the diffusion-modified kinetic scheme in Eq. 50, we consider the time dependence of the substrate concentrations for various enzyme reactivation times. Initially, all substrates are unbound and unphosporylated, and the enzymes are in the active state. The kinetics of the substrate concentrations relative to the total substrate concentration, $\left[S_i\right]/S_{tot},i=\mathrm{0,1},2$, are shown in Fig. 3. The concentrations were obtained by numerically solving Eqs. 17–18, which correspond to the scheme in Eq. 50.

FIG. 3:

Kinetics of relative substrate concentration for different enzyme reactivation times ${\tau }^{*}=\mathrm{l}\mathrm{n}2/k^{*}$. Full lines are calculated by numerically solving the rate equations that correspond to the diffusion-modified kinetic scheme in Eq. 50. Dashed lines correspond to the ordinary chemical kinetics scheme with the rescaled (diffusion-influenced) rate constants (i.e., Eq. 50 with $q^{*}=0$). The parameters, taken from Ref. 11, are $k_a^0=0.027{\mathrm{n}\mathrm{M}}^{-1}{\mathrm{s}}^{-1},k_a^1=0.056{\mathrm{n}\mathrm{M}}^{-1}{\mathrm{s}}^{-1},k_d^0=1.35{\mathrm{s}}^{-1}{,k}_d^1=1.73{\mathrm{s}}^{-1},k_c^0=1.5{\mathrm{s}}^{-1},k_c^1=15{\mathrm{s}}^{-1},S_{tot}=200nM,E_{tot}=50nM,R=5\mathrm{n}\mathrm{m},D=0.1\mu {\mathrm{m}}^2/\mathrm{s}$. The capture and escape probabilities in Eq. 50 are defined in Eq. 16 and, with these parameters, are ${ϵ}_0=0.12,{ϵ}_1=0.06,q^{*}$ are 0.94, 0.75, 0.53, and 0.27 for ${\tau }^{*}$ equal to $\mathrm{0,10}\mu s,100\mu s$, and $1\mathrm{m}\mathrm{s}$, respectively.

The new reaction channels in the scheme facilitate phosphorylation of the second site and decrease the population of the intermediate singly phosphorylated substrate. The efficiency of the new reaction channels is proportional to the capture probability $q^{*}$, which depends on the ratio of the reactivation time (defined here as ${\tau }^{*}=\mathrm{l}\mathrm{n}2/k^{*}$) to the diffusion time $R^2/D$ (i.e., the time required to diffuse apart) (see Eq. 16c). When the reactivation time is much longer than the time required to diffuse apart, the chances that the same enzyme molecule phosphorylates both the first and the second site of the substrate are very small. In this case the capture probability $q^{*}$ is small, the efficiency of the new reaction channels is low, and the kinetics of the concentrations is almost the same as predicted by the ordinary kinetic scheme in Eq. 2 with the rescaled association and dissociation rate constants (shown by the dashed lines in Fig. 3). When the enzyme reactivation is instantaneous, the effect of the new reaction channels is the largest. Thus, the ratio of diffusion and reactivation times, $k^{*}{R}^2/D$, regulates the degree of pseudo-processivity resulting from the diffusing motion of the reactants.

3.3. Bistability in double phosphorylation-dephosphorylation cycle

The diffusion-modified scheme can also be used to study steady-state concentrations. As an illustration, consider bistability in the steady state concentration of doubly phosphorylated substrate that arises in a double phosphorylation-dephosphorylation cycle. In this cycle the substrate is phosphorylated by a kinase and dephosphorylated by a phosphotase. The diffusion-modified kinetic scheme in Eq. 50 describes double phosphorylation by kinase $E$. The similar scheme describes diffusion-modified dephosphorylation by a phosphotase $F$:

$$\begin{gathered} S_2+F\underset{{\kappa }_2^d{ϵ}_2}{\stackrel{{\kappa }_2^a{ϵ}_2}{\rightleftharpoons }}{S}_{2}F\stackrel{{\kappa }_2^c}{⟶}{S}_1+F^{*} \\ {S}_1+F\stackrel{{\kappa }_3^a{ϵ}_3}{\underset{{\kappa }_3^d{ϵ}_3}{\rightleftharpoons }}{S}_{1}F\stackrel{{\kappa }_3^c}{⟶}{S}_0+F^{*} \\ {S}_{2}F\stackrel{{\kappa }_2^c{q}^{**}}{⟶}{S}_{1}F \\ {F}^{*}\stackrel{k^{**}}{⟶}F \\ {S}_{2}F\stackrel{-{\kappa }_2^c{q}^{**}}{⟶}{S}_1+F \end{gathered}$$ (54)

where ${\kappa }_i^a,{\kappa }_i^d$, and ${\kappa }_i^c,i=\mathrm{2,3}$ are the association, dissociation, and catalytic rate constants, $k^{**}$ is the rate constant of the phosphotase reactivation. Here the capture and escape probability are defined similar to Eq. 16:

$$\begin{gathered} q_2=1-{ϵ}_2={\kappa }_2^a/\left(k_D+{\kappa }_2^a\right) \\ {q}_3=1-{ϵ}_3={\kappa }_3^a/\left(k_D+{\kappa }_3^a\right) \\ {q}^{\mathbf{*}\mathbf{*}}=\frac{{\kappa }_3^a}{\left(k_D+{\kappa }_3^a\right)}\frac{\sqrt{k^{\mathbf{*}\mathbf{*}}{R}^2/D}}{\left(1+\sqrt{k^{\mathbf{*}\mathbf{*}}{R}^2/D}\right)} \end{gathered}$$ (55)

The steady-state equations corresponding to the scheme in Eqs. 50 and 54 are solved numerically with the parameters used in Ref. 11. With these parameters, the ordinary kinetic scheme with diffusion-influenced rate constants (shown by the black dashed lines in Fig. 4) leads to multiple steady state concentrations. Only two of the three steady states are stable. We assume $k^{*}=k^{**}$ and plot the relative concentration of the doubly phosphorylated substrate as a function of the reactivation time ${\tau }^{*}=\mathrm{l}\mathrm{n}2/k^{*}$ of the enzymes.

FIG. 4:

Steady-state concentration of doubly phosphorylated substrate as a function of the enzyme reactivation time ${\tau }^{*}$. Red lines correspond to the prediction of the diffusion-modified reaction scheme in Eqs. 50 and 54. Black dashed lines are calculated by solving steady-state equations corresponding to the conventional chemical kinetics with the rescaled binding and dissociation rate constants (first three reactions in Eqs. 50 and 54). The dots as well as the parameters of the model are taken from Fig. 6 of Ref. 11. The parameters are ${\kappa }_2^a={\kappa }_0^a,{\kappa }_3^a={\kappa }_1^a,{\kappa }_2^d={\kappa }_0^d,{\kappa }_3^d={\kappa }_1^d,{\kappa }_2^c={\kappa }_0^c{,\kappa }_3^c={\kappa }_1^c,D=2\mu {\mathrm{m}}^2/\mathrm{s},E_{tot}=F_{tot}=50nM,S_{\text{tot}}=500nM$, and other parameters are the same as in Fig. 3.

The plot with steady-state concentrations is superimposed with the results in Fig. 6 in Takahashi et al.¹¹ obtained by numerical simulations of many particles starting from various initial conditions. The plot illustrates that bistability is replaced by a single steady state as reactivation time decreases. In this case the capture probability $q^{*}$, Eq. 16c, increases, as well as the efficiency of the new reaction channels, and the distributive mechanism of phosphorylation at long reactivation times is replaced by a pseudo-processive mechanism at shorter reactivation times.

In addition to the rate equations corresponding to the schemes in Eqs. 50 and 54 we also considered the schemes in Eqs. 51 and 52 with corresponding dephosphorylation channels. However, for the parameters for which simulations were done¹¹, all three schemes agree with simulations equally well. As a consequence, Fig. 4 looks identical to Fig. 6 of Lawley and Keener.¹⁵

4. CONCLUDING REMARKS

We presented the theory that describes how diffusion influences the kinetics of double phosphorylation when the enzyme needs time to be reactivated. In the limit of fast diffusion, this reaction is described by the chemical kinetic scheme in Eq. 2 (Fig. 2(a)). The simplest way to take slow diffusion into account is to scale the association and dissociation intrinsic rate constants by the escape probabilities. These are the probabilities that an active enzyme and substrate initially in contact will eventually diffuse apart rather than bind (see Fig. 1 and Eq. 16). However, when there are more than one phosphorylation site, this is not sufficient, and the kinetics scheme itself should be modified, as shown in Eq. 50 and Fig. 2(b). In this scheme, not only the association and dissociation rate constants are scaled by the escape probabilities, but also two new reaction channels appear. The rate constant of the new channels are the catalytic rate constant, ${\kappa }_0^c$, multiplied by the capture (rebinding) probability $q^{*}$, which is defined in Fig. 1(b). This capture probability is the probability that an inactive enzyme initially in contact with singly phosphorylated substrate will eventually convert to the active state and then bind the substrate rather than diffuse away. For the contact isotropic reactivity model, $q^{*}$ is given in Eq. 16c.

The unusual aspect of the kinetic scheme in Eq. 50 (Fig. 2(b)) is the negative rate of one of the new reaction channels. This is because we approximated the intrinsically non-Markovian rate equations with memory by the ordinary rate equations with constant rate coefficients. Since the appearance of a negative rate constant was so unexpected, we went to great lengths to show that this is indeed the correct lowest order result by presenting a detailed step-by-step derivation. Non-Markovian rate equations in Eq. 45, or their approximated version in Eq. 49, lead to the kinetics that is also accurate at very short times. The same kind of theory can be applied to ligand binding to multiple sites.^17,23

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.