Enhancement of Tunability of MAPK Cascade Due to Coexistence of Processive and Distributive Phosphorylation Mechanisms

Abstract

The processive phosphorylation mechanism becomes important when there is macromolecular crowding in the cytoplasm. Integrating the processive phosphorylation mechanism with the traditional distributive one, we propose a mixed dual-site phosphorylation (MDP) mechanism in a single-layer phosphorylation cycle. Further, we build a degree model by applying the MDP mechanism to a three-layer mitogen-activated protein kinase (MAPK) cascade. By bifurcation analysis, our study suggests that the crowded-environment-induced pseudoprocessive mechanism can qualitatively change the response of this biological network. By adjusting the degree of processivity in our model, we find that the MAPK cascade is able to switch between the ultrasensitivity, bistability, and oscillatory dynamical states. Sensitivity analysis shows that the theoretical results remain unchanged within a reasonably chosen variation of parameter perturbation. By scaling the reaction rates and also introducing new connections into the kinetic scheme, we further construct a proportion model of the MAPK cascade to validate our findings. Finally, it is illustrated that the spatial propagation of the activated MAPK signal can be improved (or attenuated) by increasing the degree of processivity of kinase (or phosphatase). Our research implies that the MDP mechanism makes the MAPK cascade become a flexible signal module, and the coexistence of processive and distributive phosphorylation mechanisms enhances the tunability of the MAPK cascade.

Introduction

Cellular proteins are often subject to posttranslational modifications like phosphorylation (1), acetylation (2), hydroxylation (3), and ubiquitinylation (4). Protein posttranslational modifications are chemical modifications that play a key role in functional proteomics, because they regulate activity, localization, and interaction with other cellular molecules such as proteins, nucleic acids, lipids, and cofactors.

These covalent modifications are made on specific amino acid residues of the target protein by modifying enzymes (kinases, methylases, etc.). Usually, the modifications are removed by other types of enzyme working in the opposite direction (phosphatases, demethylases, etc.). Often, the same type of modification happens at more than one amino acid of the target protein, a phenomenon called multisite modification.

The most characteristic example of reversible multisite protein modification is phosphorylation and dephosphorylation by protein kinases and phosphatases. Multisite protein phosphorylation and dephosphorylation are key cellular regulatory mechanisms (5) that have been studied extensively (6).

Like any other multisite modification, multisite phosphorylation can happen via a processive or distributive mechanism (7,8). During processive phosphorylation, the kinase phosphorylates more than one amino acid residue on its substrate during a single binding event. Therefore the kinetics of a processive mechanism is not essentially different from a single phosphorylation event. In contrast, under the distributive rule, only one phosphorylation takes place during a single enzyme-substrate binding event. After the substrate is phosphorylated, the kinase dissociates from its substrate and then must rebind at a different site of phosphorylation.

A typical example of multisite phosphorylation is the mitogen-activated protein kinase (MAPK) cascade with dual phosphorylation and dephosphorylation. MAPK cascades mediate decision-making processes throughout eukaryotic life (9–11). Changes in cell fate leading to differentiation, proliferation, and apoptosis in response to hormones, as well as metabolic changes in response to stress, involve MAPK pathways.

The work of Ferrell et al. proposed that in Xenopus oocytes the MAPK pathway responds to increasing levels of progesterone in an all-or-none or switchlike manner (12). The authors showed that the switchlike responses can arise from dual phosphorylation steps of the MAPK cascade, i.e., a distributive phosphorylation model (13). However, a linear graded response of MAPK was observed in mammalian cells (14). It was suggested that the kinase is phosphorylated in a processive manner.

As stated in Salazar and Höfer (5), the processive and distributive mechanisms are just two extremes of a continuous spectrum. Molecular crowding can make enzyme-substrate rebindings more likely due to subdiffusion (14,15); hence, crowding plays a vital role in changing the nature of MAPK phosphorylation from distributive to processive. For a reaction place with a low number of crowding molecules, distributive phosphorylation is preferred, whereas in crowded environments, the processive mechanism dominates over the distributive one (16). Microscopically, the processive and distributive mechanisms should coexist under cellular conditions. Recently, phosphorylation reactions in the p38 MAPK cascade were explored by mass spectrometry (17). Both distributive and processive reactions are observed with precision experimentally.

Considerable efforts have been made to establish a theoretical model of the phosphorylation network with only the distributive or only the processive phosphorylation mechanism (18–20). However, the coexistence mechanisms play a nontrivial role in the multisite phosphorylation reactions and cannot be neglected. To our knowledge, a hybrid model of the phosphorylation reaction mechanisms is still absent. Therefore, to explore the effects of a pseudoprocessivity mechanism on the phosphorylation network, a mixed model combining the two kinds of phosphorylation mechanisms is required. In addition, the underlying dynamical behavior and the possible biological signaling feature for the mixed multisite phosphorylation mechanisms are unclear. The related theoretical research on these issues is of great interest.

To gain insight into the above issues, we focus our attention on the MAPK cascade. In this article, we set up a degree model and a proportion model of three-layer MAPK cascade with dual phosphorylation and dephosphorylation, in which the processive and the distributive mechanisms are involved. This combination of two kinds of phosphorylation/dephosphorylation mechanisms in MAPK cascade is the main contribution of this work. For simplicity, we refer to this mixed dual-site phosphorylation mechanism as the MDP mechanism. It enables us to reveal more biological significance about the MAPK signaling pathway.

This article is organized as follows. The MDP mechanism in a one-layer phosphorylation cycle is theoretically derived, and this derivation is further applied to a degree model of three-layer MAPK cascade. Next, by bifurcation analysis and sensitivity analysis, the dynamical features of MAPK cascade in homogeneous environments are clarified. The impacts of processive mechanism on spatial propagation of activated MAPK signal are further demonstrated in heterogeneous environments. Finally, a proportion model of the MAPK cascade is established, and the possible biological relevance of our theoretical findings is discussed.

Model and Methods

MDP mechanism in a one-layer cycle

We consider that substrate S, which has two phosphorylation sites, can be phosphorylated by kinase E and dephosphorylated by phosphatase F. There exist three phosphoforms of S, including $S_0$, $S_1$, and $S_2$. The one-layer phosphorylation cycle about S is described in Fig. 1, which is the same as the dual-phosphorylation cycle in the MAPK cascade discussed below. In the absence of $S_1$, the full processive mechanism is achieved, because $S_2$ can only be produced by process 3 and $S_2$ is also dephosphorylated directly through process 6. However, if processes 1 and 2, and 4 and 5, take the places of processes 3 and 6, respectively, it becomes a fully distributive mechanism, because the transformation between $S_0$ and $S_2$ must be completed via $S_1$.

Figure 1.

Schematic of processive and distributive mechanisms of the two-site phosphorylation and dephosphorylation cycle. Reactions 3 and 6 represent the corresponding processive mechanisms of kinase and phosphatase, respectively, in this single-layer cycle.

Recent experiments have illustrated that the distributive phosphorylation mechanism can be converted into the processive one under certain conditions (14,15,17). Hence, it is reasonable to infer that both mechanisms exist in this one-layer cycle of dual phosphorylation and dephosphorylation. The detailed reaction processes in Fig. 1 are displayed in Figs. 2 and 3, where $a_i^E$ and $a_i^F$ are association rates, $b_i^E$ and $b_i^F$ are dissociation rates, and $c_i^E$ and $c_i^F$ are catalysis rates, with subscript i corresponding to the substrate of the reactions. Applying the law of mass action and the assumption of a quasi-steady state, we can render our MDP model by a set of nonlinear ordinary differential equations that define the rate of change of reactants with t as follows.

$$\frac{d\left[S_1\right]}{dt}=v_1+v_4-v_2-v_5$$ (1)

$$\frac{d\left[S_2\right]}{dt}=v_2+v_3-v_4-v_6$$ (2)

$$\left[S_0\right]=\left[S_{tot}\right]-\left[S_1\right]-\left[S_2\right]$$ (3)

$$v_1=\frac{k_1^{cat}\left[S_0\right]\left[E_{tot}\right]/K_0^E}{\left(1+\left[S_0\right]/K_0^E+\left[S_1\right]/K_1^E\right)}$$ (4)

$$v_2=\frac{k_2^{cat}\left[S_1\right]\left[E_{tot}\right]/K_1^E}{\left(1+\left[S_0\right]/K_0^E+\left[S_1\right]/K_1^E\right)}$$ (5)

$$v_3=\frac{k_3^{cat}\left[S_0\right]\left[E_{tot}\right]/K_0^E}{\left(1+\left[S_0\right]/K_0^E+\left[S_1\right]/K_1^E\right)}$$ (6)

$$v_4=\frac{V_{max4}\left[S_2\right]/K_2^F}{\left(1+\left[S_1\right]/K_1^F+\left[S_2\right]/K_2^F\right)}$$ (7)

$$v_5=\frac{V_{max5}\left[S_1\right]/K_1^F}{\left(1+\left[S_1\right]/K_1^F+\left[S_2\right]/K_2^F\right)}$$ (8)

$$v_6=\frac{V_{max6}\left[S_2\right]/K_2^F}{\left(1+\left[S_1\right]/K_1^F+\left[S_2\right]/K_2^F\right)},$$ (9)

where $\left[E_{tot}\right]$ is the total kinase concentration, $\left[S_{tot}\right]$ is the total substrate concentration, $k_1^{cat}=c_{0,1}^E$, $k_2^{cat}=c_1^E$, $k_3^{cat}=c_{0,2}^E$, $K_0^E=\left(b_0^E+c_{0,1}^E+c_{0,2}^E\right)/a_0^E$, $K_1^E=\left(b_1^E+c_1^E\right)/a_1^E$, $K_2^F=\left(b_2^F+c_{2,1}^F+c_{2,0}^F\right)/a_2^F$, $K_1^F=\left(b_1^F+c_1^F\right)/a_1^F$, $V_{max4}=c_{2,1}^F\left[F_{tot}\right]$, $V_{max5}=c_1^F\left[F_{tot}\right]$, $V_{max6}=c_{2,0}^F\left[F_{tot}\right]$, and $\left[F_{tot}\right]$ is the total phosphatase concentration. For the details of the derivation process of these equations, see the Appendix. Equations 1 and 2 represent the rate equations of this MDP model for a single-layer phosphorylation cycle. We call $v_i\left(i=1,\mathrm{2..},6\right)$ the transition rate of process i in Fig. 1.

Figure 2.

Scheme A. Elementary reactions for the MDP mechanism of kinase phosphorylation in the degree model. The substrate $S_0$ binds reversibly to the kinase, E, forming the enzyme-substrate complex $E{S}_0$. By catalysis reactions, the phosphorylation products of $S_1$ and $S_2$ are released from the intermediate complex with a distributive reaction and a processive reaction, respectively.

Figure 3.

Scheme B. Elementary reactions for the MDP mechanism of phosphatase dephosphorylation in the degree model. The substrate $S_2$ binds reversibly to the phosphatase, F, forming the enzyme-substrate complex $F{S}_2$. By catalysis reactions, the dephosphorylation products of $S_1$ and $S_0$ are released from the intermediate complex with a distributive reaction and a processive reaction, respectively.

To conveniently investigate the effects of the processive mechanism on the transduction dynamics of the MDP model, we need to define a degree of enzyme processivity that can measure the relative strength of the processive mechanism. Since the degree of processivity depends on the relative timescales of enzyme dissociation and catalytic reaction (5), the ratio of the processive reaction rate to the distributive reaction rate can be used as the processivity degree. Specifically, in scheme A (Fig. 2) and scheme B (Fig. 3), $k=c_{0,2}^E/c_{0,1}^E=k_3^{cat}/k_1^{cat}$ is the degree of kinase processivity and $p=c_{2,0}^F/c_{2,1}^F=V_{max6}/V_{max4}$ is the degree of phosphatase processivity. It is noted that other possible measures for degree of processivity do not affect our research below.

Degree model of a three-layer MAPK cascade in homogeneous environments

A typical MAPK cascade consists of three phosphoproteins: MAPK, MAPK kinase (MAP2K), and MAP2K kinase (MAP3K), as shown in Fig. 4. MAP3K in the first layer is initially activated by Ras-GTP and becomes phosphorylated MAP3K (pMAP3K). Then pMAP3K activates MAP2K in the second layer by phosphorylating its two residues sequentially. Bisphosphorylated MAP2K (ppMAP2K) also phosphorylates MAPK in the third layer on two conserved threonine and tyrosine residues. It is in this stage that bisphosphorylated MAPK (ppMAPK) is generated. In addition, different kinds of phosphatases reverse these phosphorylation reactions. Note that a similar model was analyzed by Markevich et al. (21). However, the processive mechanisms (Fig. 4, red arrows) are not involved in their original model. The integration of two kinds of phosphorylation/dephosphorylation mechanisms is the key point in our article.

Figure 4.

Schematic of the MDP model of three-layer MAPK cascade. The processive mechanisms of kinase (reactions 12 and 11) and phosphatase (reactions 14 and 13) are integrated into the traditional distribution mechanism in the MAP2K and MAPK layers, respectively. The dash-dotted line indicates the potential feedback of ppMAPK on MAP3K layer. To see this figure in color, go online.

For a homogeneous (well-stirred) environment, all the molecules are distributed evenly and the effect of diffusion can be ignored. Thus, the rate equations are employed to describe this MAPK signaling pathway. We consider that the dual-site phosphorylation and dephosphorylation in the second and third layers both adopt the MDP mechanism. Based on the rate expressions derived above for a one-layer phosphorylation cycle, the rate equations that describe the MAPK cascade in the homogeneous environment can be written easily as

$$\frac{d\left[\text{pMAP}3\text{K}\right]}{dt}=v_1-v_2$$ (10a)

$$\frac{d\left[\text{pMAP}2\text{K}\right]}{dt}=v_3-v_4+v_5-v_6$$ (10b)

$$\frac{d\left[\text{ppMAP}2\text{K}\right]}{dt}=v_4-v_5+v_{12}-v_{14}$$ (10c)

$$\frac{d\left[\text{pMAPK}\right]}{dt}=v_7-v_8+v_9-v_{10}$$ (10d)

$$\frac{d\left[\text{ppMAPK}\right]}{dt}=v_8-v_9+v_{11}-v_{13}.$$ (10e)

The expressions of transition rate $v_i\left(i=1,2,\dots 14\right)$ are given in Table S1. The values of parameters are listed in Table S2. For further details about the definitions and estimations of the parameters, see Markevich et al. (21) and Kholodenko (22). It is worth noting that in Table S1, $F_a$ and $F_a^{\prime }$ can be regarded as the intensities of positive and negative feedback, respectively. According to the values of $F_a$ and $F_a^{\prime }$, which are estimated empirically, the MDP mode of the MAPK cascade can be classified into three categories.

• Mode I, where $F_a=F_a^{\prime }=1$, $\left(1+F_a{\left(\left[\text{ppMAPK}\right]/K_a\right)}^2/1+F_a^{\text{'}}{\left(\left[\text{ppMAPK}\right]/K_a\right)}^2\right)=1$, and there is no feedback of ppMAPK on the upstream protein.

• Mode II, where $F_a=5$, $F_a^{\prime }=1$, and $1+F_a{\left(\left[\text{ppMAPK}\right]/K_a\right)}^2/1+F_a^{\text{'}}{\left(\left[\text{ppMAPK}\right]/K_a\right)}^2$ increases with increasing ppMAPK level. Hence, there is positive feedback of ppMAPK on the upstream protein.

• Mode III, where $F_a=1$, $F_a^{\prime }=5$, and $1+F_a{\left(\left[\text{ppMAPK}\right]/K_a\right)}^2/1+F_a^{\text{'}}{\left(\left[\text{ppMAPK}\right]/K_a\right)}^2$ increases as the ppMAPK level decreases. Thus, there is negative feedback of ppMAPK on the upstream protein.

In the next two sections, the degrees of processivity k (for kinase) and p (for phosphatase) are treated as control parameters. We explore the impacts of different p and k on dynamical behaviors under the three modes via bifurcation analysis. Bifurcation curves are generated numerically using XPPAUT. Furthermore, we assume that all processivity degrees for kinase are equal (i.e., $k_{12}^{cat}/k_3^{cat}=k_{11}^{cat}/k_7^{cat}=k$) and all the processivity degrees for phosphatase are equal (i.e., $V_{max14}/V_{max5}=V_{max13}/V_{max9}=p$). For simplicity, we refer to the above model as a degree model of the MAPK cascade.

Results

Dynamical analysis for the degree model of the MAPK cascade in homogeneous environments

From Fig. 4, it is clear that when the processivity degrees are $k\ne 0$ and $p\ne 0$, the MAPK cascade adopts the MDP mechanism to transmit signals. However, if 1), $k=0$ and $p\ne 0$, reactions 11 and 12 are absent, because $v_{11}=v_{12}=0$, and the phosphorylation processes of the MAPK cascade only employ the distributive mechanism; 2), $k\ne 0$ and $p=0$, $v_{13}=v_{14}=0$, and the dephosphorylation processes of the MAPK cascade are distributive; 3), $k=0$ and $p=0$, then each dual-phosphorylation layer in the MAPK cascade performs its function via a purely distributive mechanism (denoted by Dis).

In this section, starting from the distributive mechanism, we increase the degree of processivity of kinase or phosphatase alone, and then investigate the dynamical properties of each mode by bifurcation analysis. In contrast, the results for the Dis case are supplied simultaneously.

Our numerical results for the three modes of the MAPK cascade are displayed in Figs. 5–7, respectively. It can be seen that the different modes exhibit diverse behaviors, such as monostability, bistability, and limit cycle, simply by changing the value of k or p. That is to say, compared with the distributive mechanism, the tunability of the MAPK cascade is largely enhanced by introducing the processivity mechanism of kinase or phosphatase. The detailed results are clarified as follows.

Figure 5.

Dose-response curves of the degree model for nonfeedback mode with different degree of kinase processivity, k (a) and phosphatase processivity, p (b).

Figure 6.

Dose-response curves of the degree model for positive feedback mode with different degrees of kinase processivity, k (a) and phosphatase processivity, p (b).

Figure 7.

Dose-response curves of the degree model for negative feedback mode with different degrees of kinase processivity, k (a) and phosphatase processivity, p (b). To see this figure in color, go online.

First, we investigate the effects of the degree (k and p) of enzyme processivity on the dynamics of mode I. As observed in Fig. 5 a, with $k=0$ and $p=0$ (Dis), in the absence of any imposed feedback regulation, bistability can arise solely from a distributive kinetic mechanism of the two-site MAPK phosphorylation and dephosphorylation. When k is increased (e.g., $k=0.5$), the bistability is still maintained, but its range becomes narrower. For $k=1$, the system becomes monostable, with a single steady state, and exhibits ultrasensitivity. If k is further increased (e.g., $k=5,10$), the cascade’s response is converted from a switch form to a graded form, becoming more and more gradual.

It can also be noted that for any given k, with increasing doses of Ras-GTP, the steady-state values of ppMAPK rise to nearly the same saturation level. In addition, it is worth mentioning that the threshold value of Ras-GTP, at which the level of ppMAPK achieves saturation, decreases as k increases. This means that the processive mechanism of kinase enhances the sensitivity of response to the external signal.

Similar to Fig. 5 a, Fig.5 b shows that as p increases, the cascade also changes from bistability (e.g., $p=0,0.03$) to ultrasensitivity $\left(p=0.1\right)$ and then to general monostability $\left(p=0.3\right)$. It is clear that for $p\ge 0.1$, increasing the dose of Ras-GTP causes the steady-state level of ppMAPK to rise monotonically to a saturation value, whereas the saturation value decreases as p increases. Moreover, the critical dose of Ras-GTP, beyond which the ppMAPK level reaches saturation, is increased as p increases. This implies that the role played by the processive mechanism of phosphatase is different from that played by kinase processivity; it prefers to reduce the efficiency and sensitivity of signal propagation, presenting a brakelike function.

From a qualitative comparison of Fig. 5, a and b, it is obvious that for increasing k and p, the bistability regions both become narrower, and finally the bistable behavior disappears. However, with the increase of k or p, the dose-response curves shift to the left or right, respectively.

We next characterize the impacts of enzyme processivity on the signaling dynamics in mode II. As shown in Fig. 6, due to the presence of positive feedback, bistability is created more easily in mode II than in mode I under the same values of k or p. In addition, with increasing k or p, the region of bistability is decreased. This suggests that a processive mechanism with high degree is able to make the bistable switch adapt to fluctuating environments (23). Moreover, the difference between the high and low steady-state values of ppMAPK is almost unchanged as k increases, whereas it decreases as p increases. Hence, the processive mechanism of kinase presents flexibility as well as robustness in the dynamics of the MAPK cascade. It is known that the selection of cell fate in response to internal and external stimuli is essential to a cell’s life (24). Since the occurrence of bistability defines the switch between two different cell fates (25), these rich dynamical behaviors contribute to the specification and maintenance of cell fate.

By comparing Fig. 6, a and b, similar to the case for the nonfeedback mode, it is found that bistability ranges also become narrower for both increasing k and increasing p, whereas the bistable curve shifts to the left (or right) with increasing k (or p).

Finally, in Fig. 7, the effects of enzyme processivity in mode III are explored. In contrast to positive feedback, negative feedback has the potential to evoke oscillations in the cascade. As shown in Fig. 7 a, with $k=0$ and $p=0$ (Dis), this cascade exhibits limit-cycle oscillations in the level of ppMAPK. As k increases to 0.3, this limit cycle is sustained, although it becomes smaller. Compared with the distributive mechanism, the oscillation amplitude of ppMAPK is weaker, and the oscillatory region of the Ras-GTP level is also decreased. When $k\ge 5$, the oscillation behavior disappears, and this cascade becomes a monostable system. As we know, the time delay introduced by negative feedback is crucial for limit-cycle dynamics. However, when k increases, the productive rate of the ppMAPK increases so that the time delay is shortened equivalently. Hence, very large k destroys the system’s oscillation behavior.

Similar to the case of k, with increasing p (Fig. 7 b), the limit cycle (e.g., $p=0.05$) shrinks gradually until it disappears $\left(p=0.1,0.3\right)$, although the mechanism underlying the disappearance of oscillation is different from that observed for k. With increasing p, the output of ppMAPK is decreased so that the intensity of negative feedback becomes very weak.

By comparing the effects of increasing k and p (Fig. 7, a and b, respectively), it is clear that the overall trends of dose-response are the same, i.e., the oscillation amplitudes gradually become smaller until finally they disappear.

Sensitivity analysis of dynamical behaviors in the degree model

Due to the lack of experimental data available to determine all the parameters, it is necessary to analyze the sensitivity of our results to changes in system parameters. Since the aim of this study is to explore transduction dynamics with the processive mechanism, we further study the effects of parameter changes on bifurcation behaviors (especially the bistability and limit-cycle ranges). The method of sensitivity analysis is described in the Supporting Material.

First, each parameter is perturbed with 10% variations (i.e., the perturbations of each parameter are 10%). The dependence of the difference between high and low steady-state values of ppMAPK on Ras-GTP level are plotted for the nonfeedback and positive-feedback modes in Figs. S1 and S2, respectively. It is shown that at the same value of k or p, different curves emerge for different parameter sets due to the parameter fluctuations, implying different bistability ranges (i.e., the region where the difference between high and low steady-state values of ppMAPK is >0). Thus, the bistability range is just dependent on parameters chosen with the fixed k or p. However, the bistability regions of all curves (i.e., different disturbed parameters) generally become narrower when k (Figs. S1 and S2, left columns) or p (Figs. S1 and S2, right columns) is increased. Especially, for $k=5$ or $p=0.3$ in nonfeedback mode, the bistability range almost disappears. Moreover, it is found that the bistability range moves toward the direction of Ras-GTP = 0 with increasing k. Conversely, as p increases, the bistability region moves away from this direction. If the perturbations of our original parameters are 30% (see Figs. S4 and S5), the overall trends of the bistability range, with left shift (for increasing k) or right shift (for increasing p), are also nearly unchanged. Therefore, our results are independent of the parameters chosen.

In a similar way, the results of sensitivity analysis for the negative-feedback mode are provided in Figs. S3 and S6. The dependences of the oscillation amplitude of ppMAPK on Ras-GTP level are presented. It is shown that the Ras-GTP ranges corresponding to oscillation dynamics are dependent on the parameters chosen with fixed k or p, although with increasing k (Figs. S3 and S6, left columns) or p (Figs. S3 and S6, right columns), the oscillation amplitudes are generally decreased and finally shrink. Thus, the change trends of oscillation amplitude are also robust to parameter fluctuations.

Degree model of the MAPK cascade in heterogeneous environments

Reaction-diffusion model for the MAPK cascade

The MAPK cascade transmits extracellular stimuli from the plasma membrane to the periphery of the cell nucleus by a series of phosphorylation and dephosphorylation reactions (see Fig. 4). Actually, the associated kinases in the MAPK cascade are localized in space so that this cascade is activated spatially (26). For example, MAP3K is phosphorylated and dephosphorylated at the cell membrane, whereas MAPK is phosphorylated and dephosphorylated in the cytoplasm. The stimulus signal from extracellular ligands must propagate into the nucleus by diffusion. In addition, the distribution of some large crowded molecules is not uniform in space. Therefore, for such a heterogeneous system, these spatial effects should be considered in our original MDP model of the MAPK cascade (i.e., Eq. 10). To extend the above theoretical model to the homogeneous case, a reaction-diffusion MDP model for heterogeneous environments is built.

First, we suppose that the cell and its nucleus are two homocentric spheres and the environments of the points that are on a spherical surface are the same. Noting that MAP3K is only at the cell membrane, we formulate the reaction-diffusion model for the MAPK cascade in the heterogeneous system as follows:

$$\frac{d\left[\text{pMAP}3\text{K}\right]}{dt}=v_1-v_2$$ (11a)

$$\frac{\partial \left[\text{pMAP}2\text{K}\right]}{\partial t}=\left(\frac{D}{R^2}\right)\frac{1}{x^2}\frac{\partial }{\partial x}\left(x^2\frac{\partial \left[\text{pMAP}2\text{K}\right]}{\partial x}\right)+v_5-v_6$$ (11b)

$$\frac{\partial \left[\text{ppMAP}2\text{K}\right]}{\partial t}=\left(\frac{D}{R^2}\right)\frac{1}{x^2}\frac{\partial }{\partial x}\left(x^2\frac{\partial \left[\text{ppMAP}2\text{K}\right]}{\partial x}\right)-v_5-v_{14}$$ (11c)

$$\frac{\partial \left[\text{pMAPK}\right]}{\partial t}=\left(\frac{D}{R^2}\right)\frac{1}{x^2}\frac{\partial }{\partial x}\left(x^2\frac{\partial \left[\text{pMAPK}\right]}{\partial x}\right)+v_7-v_8+v_9-v_{10}$$ (11d)

$$\frac{\partial \left[\text{ppMAPK}\right]}{\partial t}=\left(\frac{D}{R^2}\right)\frac{1}{x^2}\frac{\partial }{\partial x}\left(x^2\frac{\partial \left[\text{ppMAPK}\right]}{\partial x}\right)+v_8-v_9+v_{11}-v_{13}.$$ (11e)

The boundary conditions for protein components are set as

$$\frac{\partial \left[\text{pMAP}2\text{K}\right]}{\partial x}{|}_{x=R}=\frac{R^2}{3D}\left(v_3-v_4+v_5^m-v_6^m\right)$$ (12a)

$$\frac{\partial \left[\text{ppMAP}2\text{K}\right]}{\partial x}{|}_{x=R}=\frac{R^2}{3D}\left(v_4-v_5^m+v_{12}-v_{14}^m\right)$$ (12b)

$$\frac{\partial \left[\text{pMAP}2\text{K}\right]}{\partial x}{|}_{x=r}=\frac{\partial \left[\text{ppMAP}2\text{K}\right]}{\partial x}{|}_{x=r}=0$$ (12c)

$$\frac{\partial \left[\text{pMAPK}\right]}{\partial x}{|}_{x=R}=\frac{\partial \left[\text{pMAPK}\right]}{\partial x}{|}_{x=r}=0$$ (12d)

$$\frac{\partial \left[\text{ppMAPK}\right]}{\partial x}{|}_{x=R}=\frac{\partial \left[\text{ppMAPK}\right]}{\partial x}{|}_{x=r}=0,$$ (12e)

where R and r are the cell radius and the nuclear radius, respectively. Although the expressions of $v_5^m$, $v_6^m$, and $v_{14}^m$ in Eq. 12 are the same as those of $v_5$, $v_6$, and $v_{14}$ in Eq. 11, the corresponding values of these parameters listed in Table S2 are slightly different due to the different reaction places (e.g., $V_{max5}$ and $V_{max6}$ = 250 nM/s, whereas $V_{max5}^m$ and $V_{max6}^m$ = 100 nM/s ). The diffusion coefficient, D, in the cytoplasm was reported to be in the range $1-10$ μm²/s (27). D is set to 5 μm²/s under crowded cytoplasmic condition (28). For illustrative purposes, R and r are set to 15 μm and 6 μm, respectively. We use d to denote propagation length, which is the distance from a point in the cytoplasm to the cell membrane. Thus, the total propagation length is $L=R-r=9$ μm. The variations in these reported parameter values do not change our main conclusions below.

Numerical simulation of our reaction-diffusion model is implemented in Fortran and Matlab. The Laplacian in 3D has the form $△u=\left({\partial }^{2}u/\partial x^2\right)+\left({\partial }^{2}u/\partial y^2\right)+\left({\partial }^{2}u/\partial z^2\right)$. For the spherical symmetry, $\left({\partial }^{2}u/\partial x^2\right)=\left({\partial }^{2}u/\partial {\rho }^2\right)\left(x^2/{\rho }^2\right)+\left(\partial u/\partial \rho \right)\left({\rho }^2-x^2/{\rho }^3\right)$, y and z have similar forms. Further, we know that the Laplace operator can be written in the form $△$$u=\left(1/{\rho }^2\right)\left(\partial /\partial \rho \right)\left({\rho }^2\left(\partial u/\partial \rho \right)\right)$, where ρ is the distance from the cell center. Using this, our analysis of signaling in three dimensions is nearly equivalent to that in one dimension. The equations are discretized in time and space by the finite-difference method, and the secant method is applied to deal with a nonvanishing boundary condition.

Spatial propagation of the MAPK cascade

As a transcriptional factor, the activated MAPK (i.e., ppMAPK) can be moved into the cell nucleus; hence, a spatial distribution of the ppMAPK level indirectly reflects the signal transmission in the cell.

In this section, based on the MDP model of the MAPK cascade in heterogeneous environments, we explore the effects of processive mechanisms on the spatial distribution of the ppMAPK level. For comparison, the numerical results for three feedback modes are shown using the reaction-diffusion equations (i.e., Eqs. 11 and 12). The concentrations of Ras-GTP are set to 30 nM in all modes. As depicted in Fig. 8, some interesting observations about biological signaling of the MAPK cascade are summarized below.

Figure 8.

Spatial distributions of ppMAPK from cell membrane (left) to nucleus (right) for nonfeedback (a), positive-feedback (b), and negative-feedback (c) modes in the degree model. The distribution curves gradually converge (or diverge) in the direction of propagation when modulating kinase processivity, k (or phosphatase processivity, p) independently.

In all modes, for fixed k (degree of kinase processivity) or p (degree of phosphatase processivity), the concentration of ppMAPK decays gradually as the propagation distance, d, increases. In addition, for a given mode, the ppMAPK level at each point in the cytoplasm increases as k increases, suggesting that the enhancement of degree of kinase processivity improves the ppMAPK level and hence supports signal transmission. Conversely, an increase in the degree of phosphatase processivity, p, attenuates the ppMAPK concentration in cytoplasm so that it inhibits signal transmission.

When only k is adjusted independently starting from a distributive mechanism (we fix $p=0$), both convergence-like and divergence-like propagation features are observed. For the modes of nonfeedback (Fig. 8 a) and positive feedback (Fig. 8 b), the concentrations of ppMAPK for different k $\left(k=0,1,50\right)$ near the cell membrane are nearly the same, whereas with the increase of propagation distance d, the differences in the levels of ppMAPK become larger and larger (for simplicity, we refer to this as the divergence effect). This means that the increase of kinase processivity constrains the decay of the ppMAPK activity for these two modes. Conversely, in the mode of negative feedback (Fig. 8 c), the concentrations of ppMAPK at the cell membrane are very different for $k=0,1,5,50$, however, these differences decrease with increasing d (for simplicity, we call this the convergence effect).

When p is regulated along from distributive mechanism (we fix $k=0$), the convergence-like effects of spatial transduction in the MAPK cascade are identified. For example, in the case of nonfeedback and negative feedback modes, the propagation curves (e.g., $p=0.1,0.2$ in Fig. 8 a and $p=0,0.1,0.5$ in Fig. 8 c) initially are divergent near the cell membrane, but gradually focus with increasing d, because the concentrations of ppMAPK at points near the nucleus for various p are all close to zero. The convergence-like effect is slightly weak for the positive-feedback mode (e.g., $p=0.1,0.2$ in Fig. 8 b).

Discussion

It is noted that the processive mechanism we used is technically pseudoprocessive, as the enzyme still detaches and binds again. Impeded diffusion in the crowded media accelerates this process. Macromolecular crowding is also termed the excluded-volume effect because its most basic characteristic is the mutual impenetrability of all solute molecules. Crowding has a complex effect on the rate of biochemical reactions (e.g., forward reaction and backward reaction) in the cell (29–33). On one hand, under crowding conditions the thermodynamic activity of the reactants increases. On the other hand, crowding reduces diffusion, so that the possibility of two reactants meeting decreases. The overall effect of these opposing factors depends on the nature of each reaction. To simulate macromolecular crowding, crowding agents (e.g., bovine serum albumin, ovalbumin, Ficoll 70, polyethylene glycol, and dextrans) should be added to the solution. The kinetics of an in vitro enzymatic reaction in response to changing the concentration and size of the crowding agents have been studied experimentally (34). This is, to our knowledge, the first reported experiment on the crowding effect in an enzymatic reaction with a mixed inhibition by product. Therefore, by adjusting the concentrations of these crowding agents, it is possible to change experimentally the degree of enzyme processivity.

Furthermore, according to Aoki et al. (14), the effects of macromolecular crowding on ERK MAPK phosphorylation by MEK MAPKK can be reflected by two parameters, crowding factor c and proportion of processivity of phosphorylation, $p_1$, where c and $p_1$ are functions of the ratio of the encounter rate to conversion rate, the crowder occupying fraction, the diffusion coefficient, and so on. Hence, we reconsider our model with these two parameters (i.e., c and $p_1$) and another parameter, $p_2$ (i.e., the proportion of processivity of dephosphorylation). c is used to scale the reaction rates, $p_1$ (or 1 − $p_1$) represents the probability that the phosphorylation product $S_2$ (or $S_1$) is generated by processive (or distributive) reaction from intermediate complex $E{S}_0$, and $p_2$ (or 1 − $p_2$) represents the probability that the dephosphorylation product, $S_0$ (or $S_1$), is generated by processive (or distributive) reaction from intermediate complex $F{S}_2$. Therefore, new connections are introduced into the kinetic scheme. The new model, called the proportion model, is derived in the Supporting Material. It can be seen from Table S1 that changing the processivity degree in the degree model only affects the rates of processive reactions (e.g., Fig. 4, reactions 12 and 14). However, as observed based on Table S3, if modulating the processivity proportion in the proportion model, the rates of distributive and processive reactions are both modified (e.g., Fig. 4, reactions 12 and 3, and 14 and 5). The difference originates from a biophysical constraint in the proportion model (Fig. S7) that two catalysis reaction probabilities for the intermediate complex (e.g., $E{S}_0$) should be normalized (e.g., $c_{01}^E+c_{02}^E=c{c}_0$).

As shown in Table S3, the crowding factor, c, is the common multiplier in each $v_i$; hence, it affects only the timescale, not the steady-state behavior, of this signaling system. The effects of enzyme processivity on the bifurcation diagram are just determined by the processivity proportions of kinase, $p_1$, and phosphatase, $p_2$. Therefore, we further explore, via bifurcation analysis, the impact of different $p_1$ and $p_2$ on the dynamical behavior under three feedback modes. As shown in Figs. S8–S10, with $p_1$ or $p_2$ increasing, the trends in change of bistability range (such as the direction of its movement) and limit cycle (such as the changes of its amplitude) in the proportion model are qualitatively consistent with the results in our degree model (where the degrees of processivity k and p are used).

It is noted, from Table S3, that the differences between the transition rates of processive reactions ($v_{11},v_{12},v_{13}$, and $v_{14}$) and those of the corresponding distributive reactions ($v_7,v_3,v_9$, and $v_5$) are only $p_1$ and $1-p_1$ or $p_2$ and $1-p_2$. Therefore, the relationships between the processivity degrees (i.e., k and p in the degree model) and the processivity proportions (i.e., $p_1$ and $p_2$ in the proportion model) are $k=\left(1-p_1\right)/p_1$ and $p=\left(1-p_2\right)/p_2$, respectively. Since the processivity proportions of phosphorylation, $p_1$, and dephosphorylation, $p_2$, can be experimentally measured, as in Aoki et al. (14), it is reasonable to predict that the processivity degrees for kinase k and phosphatase p, directly adjusted by $p_1$ and $p_2$, are also controllable in the experiments.

We may devise some experiments to validate our theoretical results by time series for different k or p with any fixed Ras-GTP level. For example, according to Fig. 5 b, when Ras-GTP = 20.5 nM, and $k=0$ $\left(p_1=0\right)$ and $p=0$ $\left(p_2=0\right)$, the system is bistable. However, when p (or $p_2$) is increased to 1 (or 0.5), the bistability range disappears. These observations are verified by time series (Fig. S11). When Ras-GTP = 20.5 and $k=0$ $\left(p_1=0\right)$ and $p=0$ $\left(p_2=0\right)$, the time series with high initial value (Fig. S11 a; ppMAPK = 360 nM) converge to a high steady-state value, whereas those with low initial value (ppMAPK = 0) fall toward a low steady-state value. The model exhibits a bistability. In contrast, as p (or $p_2$) increases to 1 (or 0.5) (Fig. S11 b) the time series beginning with high initial value (ppMAPK = 360) and low initial value (ppMAPK = 0) converge to a common steady-state value. The model becomes monostable. Furthermore, the phosphorylated protein at different time points can be detected by Phos-tag Western Blot. Thus, our results from bifurcation analysis could be tested by the time series from experiments.

Enzyme processivity may be enhanced by the presence of scaffold protein. Scaffold proteins have been believed to convert distributive reactions to processive ones (35,36), which implies that the scaffold proteins are closely related to the processivity parameter (k, p, p₁, or p₂). Changes of concentration, localization, dynamics of scaffold proteins, or binding strength of kinases with these scaffold proteins may lead to significant adjustments in the signaling transduction dynamics. These influences of scaffold proteins on the signaling pathway can further lead to changes in the cell-fate decision.

Multisite phosphorylation is also significant in the regulation of activities of some other proteins, such as p53. The effects of sequential phosphorylation of p53 on cell-fate decisions have been investigated (37,38). Since the processive mechanism plays crucial roles in the modulation of phosphorylation dynamics, it may also affect the DNA damage/repair mediated by p53. If the processive mechanism is involved in the sequential phosphorylation, the primary phosphorylation at Ser-15/20 (which can induce cell-cycle arrest) and the further phosphorylation at Ser-46 (which can trigger apoptosis) occur simultaneously. It means that active p53 is directly converted into p53 killer with a certain degree or proportion. To ensure the two-phase dynamics of p53, we expect that the processive degree or proportion should be set to an appropriate region. In addition, the interlinked positive and negative feedback loops existing in the p53 network are expected to elicit further investigation of with the processive mechanism is added.

Finally, we offer a simple discussion of our spatial model. In contrast to the original model provided by Markevich et al. (21), we introduce processive mechanisms in our modified model. By doing so, we extend their work from $k=0$ and $p=0$ to general conditions. The reaction-diffusion equations and boundary conditions are also adjusted accordingly. Using our spatial model, we have quantitatively studied the effects of processive reactions on spatial transmission of activated kinases. It is found that for different feedback mechanisms, the MDP model of the MAPK cascade in the heterogeneous system can exhibit diverse efficiencies of signal transmission by modulation of k and p.

Conclusions

Protein posttranslational modification, including phosphorylation, glycosylation, ubiquitination, nitrosylation, methylation, acetylation, lipidation, and proteolysis, influences almost all aspects of normal cell biology and pathogenesis (39,40). Therefore, identifying and understanding posttranslational modification is critical in the study of cell biology and disease treatment and prevention (41–43).

In this article, we have explored the multisite phosphorylation mechanism in a three-layer MAPK cascade. Most of the existing works about dual phosphorylation reactions have focused on only the distributive or only the processive phosphorylation mechanism (5,11,21,44–48). However, both distributive and processive reactions were clearly observed experimentally (14,15). We note, especially, that an excess of crowding agent can turn a distributive mechanism into a processive one (17), increasing the degree of processivity. Therefore, we developed a mixed mechanism of dual phosphorylation and dephosphorylation (i.e., the MDP mechanism) by combining the processive and distributive mechanisms. Further, the degree model of the three-layer MAPK cascade is presented with the MDP mechanisms applied in each dual-phosphorylation layer. Some interesting dynamical behaviors of MAPK cascade are revealed.

We analyzed the dynamical features of the MAPK cascade under homogeneous conditions and found that the MAPK cascade can transit between the ultrasensitivity, bistability, and oscillatory dynamical regions. Especially for the nonfeedback mode, the bistability induced by two-site phosphorylation can be destroyed by increasing the processivity degree (Fig. 5). For the positive-feedback mode, the bistable region is sensitive to the change of enzyme processivity, whereas the steady-state values are nearly maintained when regulating kinase processivity (Fig. 6). Thus, for the nonfeedback and positive-feedback modes, increases of processivity in either kinase or phosphatase make the bistable curve move to the left or right, respectively. For the negative-feedback mode, large enzyme processivity has a destructive effect on oscillation behaviors (Fig. 7), leading to shrinking of the limit cycle. Hence, the multiple functions of this biological module induced by the processive mechanism are clarified. Sensitivity analysis shows that our theoretical results from the degree model are robust to parameter fluctuations to a certain extent.

Further, to explore signal transduction from the cell membrane to the nucleus, a dynamical model of the MAPK cascade under heterogeneous conditions is established. The impact of the processive mechanism on spatial transmission of activated MAPK is demonstrated (Fig. 8). When adjusting only phosphatase processivity via the distributive mechanism, convergence effects of spatial propagation emerge from all three modes. When regulating only kinase processivity, divergence effects are observed in the nonfeedback and positive-feedback modes.

Finally, a proportion model of the MAPK cascade is constructed taking into consideration some explicit biophysical constraints. The bifurcation analysis illustrates that the qualitative dynamical behaviors in the proportion model are very close to those in the degree model.

All the results confirm that the degree (or proportion) of processivity is an appropriate control parameter that makes the MAPK cascade become a flexible function module, and the MDP mechanism enhances the tunability of the MAPK cascade. Our research provides a theoretical frame in which to consummate the MAPK cascade dynamics and gives a clear description of the relationship between the structure and function of the MAPK cascade.

Some problems should be mentioned. We only investigate a dual-site phosphorylation-and-dephosphorylation signal pathway in this research. For multisite phosphorylation systems with more than three site numbers, it becomes more complicated to obtain detailed mathematical models when the MDP mechanisms are adopted. This will be our next task. Also, our model has too many parameters. It is necessary to optimize these parameters and improve the model with more experimental observations in the future. Finally, the random or sequential model for phosphorylation order in MAPK modules also has not been studied.

Appendix

In this section, we provide the detailed derivation of Eqs. 1–9. In fact, scheme A and scheme B depict a series of biochemical reactions between enzyme and substrate. Applying the law of mass action, we have

$$\frac{d\left[S_0\right]}{dt}=b_0^E\left[E{S}_0\right]-a_0^E\left[E\right]\left[S_0\right]+c_1^F\left[F{S}_1\right]+c_{2,0}^F\left[F{S}_2\right]$$ (13)

$$\frac{d\left[S_1\right]}{dt}=c_{0,1}^E\left[E{S}_0\right]+b_1^E\left[E{S}_1\right]-a_1^E\left[E\right]\left[S_1\right]+b_1^F\left[F{S}_1\right]-a_1^F\left[F\right]\left[S_1\right]+c_{2,1}^F\left[F{S}_2\right]$$ (14)

$$\frac{d\left[S_2\right]}{dt}=c_{0,2}^E\left[E{S}_0\right]+c_1^E\left[E{S}_1\right]+b_2^F\left[F{S}_2\right]-a_2^F\left[F\right]\left[S_2\right]$$ (15)

$$\frac{d\left[E{S}_0\right]}{dt}=a_0^E\left[E\right]\left[S_0\right]-\left(b_0^E+c_{0,1}^E+c_{0,2}^E\right)\left[E{S}_0\right]$$ (16)

$$\frac{d\left[E{S}_1\right]}{dt}=a_1^E\left[E\right]\left[S_1\right]-\left(b_1^E+c_1^E\right)\left[E{S}_1\right]$$ (17)

$$\frac{d\left[F{S}_1\right]}{dt}=a_1^F\left[F\right]\left[S_1\right]-\left(b_1^F+c_1^F\right)\left[F{S}_0\right]$$ (18)

$$\frac{d\left[F{S}_2\right]}{dt}=a_2^F\left[F\right]\left[S_2\right]-\left(b_2^F+c_{2,1}^F+c_{2,0}^F\right)\left[F{S}_2\right].$$ (19)

We assume that the concentration of the intermediate complex does not change on the timescale of product formation, i.e., the quasi-steady-state assumption. The equations below are derived:

$$\frac{d\left[E{S}_0\right]}{dt}=0,\frac{d\left[E{S}_1\right]}{dt}=0,\frac{d\left[F{S}_1\right]}{dt}=0,\frac{d\left[F{S}_2\right]}{dt}=0.$$ (20)

Noting that $E{S}_0+E{S}_1+E=E_{tot}$ and $F{S}_0+F{S}_1+F=F_{tot}$, and combining Eqs. 16–19 and Eq. 20, we obtain

$$\left[E{S}_0\right]=\frac{\left[E_{tot}\right]\left[S_0\right]/K_0^E}{1+\frac{S_0}{K_0^E}+\frac{S_1}{K_1^E}}$$ (21)

$$\left[E{S}_1\right]=\frac{\left[E_{tot}\right]\left[S_1\right]/K_1^E}{1+\frac{S_0}{K_0^E}+\frac{S_1}{K_1^E}}$$ (22)

$$\left[F{S}_2\right]=\frac{\left[F_{tot}\right]\left[S_2\right]/K_2^F}{1+\frac{S_2}{K_2^F}+\frac{S_1}{K_1^F}}$$ (23)

$$\left[F{S}_1\right]=\frac{\left[F_{tot}\right]\left[S_1\right]/K_1^F}{1+\frac{S_2}{K_2^E}+\frac{S_1}{K_1^F}},$$ (24)

where $K_0^E=b_0^E+c_{0,1}^E+c_{0,2}^E/a_0^E$, $K_1^E=b_1^E+c_1^E/a_1^E,$ $K_2^F=b_2^F+c_{2,1}^F+c_{2,0}^F/a_2^F,$ and $K_1^F=b_1^F+c_1^F/a_1^F$.

Thus, combining Eqs. 13–15 and Eqs. 21–24, we get

$$\frac{d\left[S_0\right]}{dt}=v_5+v_6-v_1-v_3$$ (25)

$$\frac{d\left[S_1\right]}{dt}=v_1+v_4-v_2-v_5$$ (26)

$$\frac{d\left[S_2\right]}{dt}=v_2+v_3-v_4-v_6,$$ (27)

where

$$\begin{array}{l}{v}_1=\frac{c_{0,1}^E\left[E_{tot}\right]\left[S_0\right]/K_0^E}{1+\frac{S_0}{K_0^E}+\frac{S_1}{K_1^E}},\hfill \\ v_2=\frac{c_1^E\left[E_{tot}\right]\left[S_1\right]/K_1^E}{1+\frac{S_0}{K_0^E}+\frac{S_1}{K_1^E}},\hfill \\ v_3=\frac{c_{0,2}^E\left[E_{tot}\right]\left[S_0\right]/K_0^E}{1+\frac{S_0}{K_0^E}+\frac{S_1}{K_1^E}},\hfill \\ v_4=\frac{c_{2,1}^F\left[F_{tot}\right]\left[S_1\right]/K_1^F}{1+\frac{S_1}{K_1^F}+\frac{S_2}{K_2^F}},\hfill \\ v_5=\frac{c_1^F\left[F_{tot}\right]\left[S_1\right]/K_1^F}{1+\frac{S_1}{K_1^F}+\frac{S_2}{K_2^F}},\hfill \\ v_6=\frac{c_{2,0}^F\left[F_{tot}\right]\left[S_1\right]/K_1^F}{1+\frac{S_1}{K_1^F}+\frac{S_2}{K_2^F}}.\hfill \end{array}$$