Sharing of Phosphatases Promotes Response Plasticity in Phosphorylation Cascades

Abstract

Sharing of positive or negative regulators between multiple targets is frequently observed in cellular signaling cascades. For instance, phosphatase sharing between multiple kinases is ubiquitous within the MAPK pathway. Here we investigate how such phosphatase sharing could shape robustness and evolvability of the phosphorylation cascade. Through modeling and evolutionary simulations, we demonstrate that 1) phosphatase sharing dramatically increases robustness of a bistable MAPK response, and 2) phosphatase-sharing cascades evolve faster than nonsharing cascades. This faster evolution is particularly pronounced when evolving from a monostable toward a bistable phenotype, whereas the transition speed of a population from a bistable to monostable response is not affected by phosphatase sharing. This property may enable the phosphatase-sharing design to adapt better in a changing environment. Analysis of the respective mutational landscapes reveal that phosphatase sharing reduces the number of limiting mutations required for transition from monostable to bistable responses, hence facilitating a faster transition to such response types. Taken together, using MAPK cascade as an example, our study offers a general theoretical framework to explore robustness and evolutionary plasticity of signal transduction cascades.

Introduction

In cell signaling cascades, sharing of pathway components is frequently observed at different levels of signal transduction, from the receptor level where signal is initiated, through intermediate levels of signal processing, to the output level where transcription factors are activated (1, 2, 3, 4). Individual pathways also have shared components that connect different signaling intermediates, a typical example being the phosphatase sharing among different layers of the MAPK cascade. In this case, signaling is initiated by the stimulus-dependent phosphorylation of the uppermost kinase (MAPKKK), which then acts as a kinase doubly phosphorylating the intermediate MAPKK; MAPKK in turn doubly phosphorylates MAPK, which mediates output response (5). The sequence of phosphorylation events from MAPKKK to MAPK is well conserved (6), whereas the topology of dephosphorylation reactions is flexible: some MAPK pathways utilize a common phosphatase to dephosphorylate multiple kinases, or they possess kinase-specific phosphatases. For instance, PP2A and PP2C-α act as shared phosphatases at the MAPKK and MAPK levels in the JNK and p38 pathways, respectively (7, 8, 9). In contrast, MAPK phosphatase (MKP) 1 and 3 specifically act at the MAPK level and completely dephosphorylate the double-phosphorylated p38 MAPK and ERK-1/2, respectively (10). Finally, there are also phosphatases controlling individual phosphorylation sites in a single protein, such as PTP1B or CD45 (11). Fig. 1, A–C shows three potential modes of dephosphorylation with identical phosphorylation steps. Physiological significance of different phosphatase-sharing topologies also remains less understood.

Figure 1.

The degree of phosphatase sharing enhances robustness of bistability. (A–C) Schematic representation of the studied topologies of phosphorylation cascades. In the unshared case, four phosphatases (P_1, P_2, P_3, P₄) dephosphorylate the single- (MAPKK-P, MAPK-P) and double-phosphorylated (MAPKK-PP, MAPK-P) forms of the kinases (A). In the partially shared case (B), two specific phosphatases, P1dual and P2dual, respectively dephosphorylate MAPKK and MAPK layers completely. In the fully shared phosphatase case (C), the phosphatase (shown as P1shared) dephosphorylates the MAPKK and the MAPK. Light and dark ellipses represent monostable and bistable cascades with unshared phosphatases (blue), partially shared phosphatases (red), and fully shared phosphatase (cyan). (D) The percentages of bistable responses are shown for fully shared (25%), partially shared (10%), and unshared (5%) cascades obtained from 10,000 independently sampled parameters for each phosphatase-sharing design. (E) The mutational robustness (MR) for monostable (left panel) and bistable (right panel) responses as a function of the number of mutations are shown. To see this figure in color, go online.

Generally, the MAPK cascade can respond to external stimuli in either monostable or bistable fashion, with both behaviors controlling diverse cell fates. For instance, bistable MAPK cascades control inducible production of lineage-specific differentiation factors (12), oocyte maturation (13), long-term survival of epithelial cells (14), or responses to pathological stress (15). In contrast, a graded/monostable MAPK response is observed in growth-factor stimulated HeLa cells and fibroblasts (16, 17), or in the yeast Saccharomyces cerevisiae stimulated by pheromone (18, 19, 20). A hybrid response is observed for the mammalian ERK, where individual sub populations of genetically identical gonadotrope cells exhibit either monostable or bistable responses to gonadotropin-releasing hormone (21). Plasticity in the response of the cascade is also observed in PC-12 cells, which respond to EGF and NGF stimuli by either monostable or bistable ERK phosphorylation and could be explained by the effects of negative and positive feedback loops, respectively (22). The significance of MKP in imparting response plasticity was also shown earlier in a combined experimental and computational analysis that demonstrated that the expression level of MKP determines if the cascade exhibits bistable or monostable responses (23).

The MAPK cascade structure (conserved across a spectrum of lifeforms (5)) arguably also underwent evolutionary selection pressure. Although several experiments suggest that in response to external cues, the cascade response would be mostly bistable or monostable (1, 5, 14, 19, 20), how robust these response phenotypes are when subjected to selection pressure, and how evolvable (capable to evolve novel functions (24)) the cascade would be, has not been well explored. Here, in the context of the observed phosphatase-sharing designs (shown schematically in Fig. 1, A–C), we specifically asked 1) how different modes of the dephosphorylation due to phosphatase sharing differentially impact robustness of the MAPK cascade in generating graded or bistable responses, and 2) how evolutionary transition between these two response types is affected by such phosphatase sharing.

To address these questions, we adopted computational modeling and performed in silico evolution of the MAPK cascade. Usefulness of the in silico evolutionary approach to investigate cellular design principles is shown for various biological systems (25, 26, 27, 28, 29). Here, by parameter sampling and mutational robustness (MR) analysis, we first show that phosphatase sharing dramatically enhances the frequency of occurrence as well as the robustness of bistable responses. Next, by performing in silico evolution of a population of competing cascades with different degree of phosphatase sharing, we show that the speed of evolutionary transition (the “evolutionary transition rate,” i.e., a quantity that reflects the capacity of a population comprising a given cascade design to evolve from one response phenotype to other) from a monostable to a bistable response type is a function of the extent of phosphatase sharing. In addition, phosphatase sharing facilitated better adaptation to rapidly fluctuating environments that demand either bistable or monostable response.

To understand the mechanism of this accelerated adaptation, we further constructed an analytically solvable phenomenological model (30, 31) of the evolutionary process. This model captured the fundamental observations of the evolutionary algorithm and hinted that such distinct adaptation speed may originate from differences in the mutational landscape of the respective cascades. We thus analyzed mutational properties of the parameter space by calculating the information content of each evolved parameter. Our analysis reveals that for monostable to bistable transition, the cascades with full phosphatase sharing require the least number of critical mutations. However, bistable to monostable transition is independent of degree of phosphatase sharing, as this transition is not limited to a few critical parameters. Finally, by varying the mutation rate of most sensitive parameters for the different sharing designs, we validated the significance of critical mutations. The modeling and analysis framework developed here can be easily adapted to study the evolutionary plasticity and robustness of different signaling pathways.

Methods

Robustness by parameter sampling

The pathway schemes in Fig. 1, A and C can be converted to a set of ordinary differential equations (Supporting Material). The models do not incorporate any production or degradation of the kinases or phosphatases, thus the total kinase and phosphatase concentrations are assumed to remain constant throughout the simulations. Also no explicit feedback loops were considered in our study. We assumed a quasi-steady-state approximation for the enzyme-substrate complexes in the reactions (32, 33), so that differential equations were solved only for the unphosphorylated and phosphorylated forms of the kinases. The total phosphatase concentrations were considered as modifiers (32, 33) and were assumed constant throughout the simulations. The two-layer cascade contained six differential equations and 16 kinetic parameters. In the topology with full phosphatase sharing (Fig. 1 C), three initial concentrations were assigned: the two kinases MAPKK and MAPK and a common phosphatase, P1shared. In the topologies with partial and no phosphatase sharing (Fig. 1, A and B, respectively) there are two (P1dual and P2dual) and four (P1, P2, P3, and P4) phosphatases, respectively. The input-output (I/O) curve describes the fraction of the double-phosphorylated MAPK as a function of the input. We checked hysteresis to evaluate if the responses were monostable or bistable. To do this, we performed the simulation by changing the input both in the forward and backward directions (15, 34); a cascade with saturated output was considered to be active if at least 20% of the total MAPK concentration was doubly phosphorylated, and if the saturated value was below the 20% level the respective cascades were considered as nonresponders and were subsequently removed from further analysis. Specifically, the 19 (fully shared), 20 (partially shared), and 22 (fully unshared) dimensional parameter space was generated by Monte Carlo sampling from the hypercube, with 5-fold range around the central values of the parameters (central values and the stimulus range are provided in Supporting Material). It can be noted that we generated the 16 kinetic parameters and the two initial kinase concentrations (MAPKK and MAPK) commonly for the three cascade types, so for one given simulation, all three cascades will have identical kinetic parameters and kinase concentration, hence differing only in their phosphatase interaction designs and phosphatase concentrations. This implementation allowed us to closely compare bistable and monostable occurrences as a function of phosphatase-sharing designs. The ordinary differential equations (ODEs) were solved by CVode ODE-solver, interfaced with MATLAB for each parameter set, enabling the I/O curve to be classified. A total of 10,000 parameter sets were sampled around the central value.

MR

The MR of a particular phenotype is defined as the fraction of the phenotype in the mutational neighborhood of that phenotype (35). For this analysis the genotype of the cascade was defined as one set of parameters. We assumed the one-neighborhood of that genotype as the parameter set where only one parameter value is different, and thus the rest of the parameter values are the same between the one-mutational neighbors. Similarly, when two parameters are different, then they are within the two-mutational neighborhood. To calculate the MR of bistability, we choose a bistable phenotype and randomly select one parameter and mutate it within a five-fold range. The new genotype is a one-mutational neighbor of the starting bistable phenotype. The dose-response curve is generated after the mutation. The procedure is carried out for 500 different mutations and the number of bistable systems is then counted. The number of bistable systems divided by 500 gives the one-MR for that particular bistability. This calculation is repeated for 100 different parameter sets displaying bistability. The MR is the average over the 100 different bistable states. Then, we select two parameters randomly and calculate the MR, following the same procedure. This gives us the two-MR. This procedure is repeated for up to 10 mutations at a time, and generates MR as a function of number of mutations.

Response coefficient

The response coefficients are determined by calculating the Jacobian of the set of differential equations evaluated at the steady state. From the equations in the Supporting Material, the response coefficients from MAPK-PP to MAPK-P can be calculated as

$$R_{\text{MAPK}\text{-}\text{PP}\to \text{MAPKK}\text{-}\text{PP}}={\left(\frac{\partial F_3}{\partial x_6}\right)}_{\text{steady}\text{state}},$$

where $F_3$ represents temporal derivative of MAPKK_PP and $x_6$ = MAPK_PP; this is described in the Supporting Material, Model equations.

The steady state is evaluated at each dose for a particular phosphatase concentration by solving the set of ODEs as mentioned before, and the Jacobian is evaluated at that steady state using the Jacobian function of the MATLAB software.

Evolutionary algorithm

The evolutionary simulation is performed by starting with a homogeneous population of 200 individual cascades. In each generation, the mutation is carried out with a mutation rate of 0.2; this is achieved in the simulations by drawing one random number in the range 0–1 for a given cascade in a population, and the mutation in that cascade is allowed if the random number is ≤0.2. In all individual cascades, one of the 19 (fully shared)/20 (partially shared)/22 (fully unshared) model parameters is mutated by assigning a random value from a 5-fold range around the biologically observed central value (36). The I/O curve is generated for each cascade by solving the ODEs, and selection pressure was imposed based on the qualitative behavior of the I/O curves. For example, when a population is selected for bistability, the bistable I/O curve is assigned a fitness twice that of the monostable I/O curve, i.e., a bistable I/O curve has fitness = 2 if a monostable I/O curve has fitness = 1. Notably, we also observed occasional oscillatory outputs as reported earlier (36), but as dose-response behavior is the focus of this study, we designed our algorithm to remove such cascades from further analysis. Also, we did not differentiate the bistable curve on the basis of the width of the bistable region during evolutionary simulations. This means that the same fitness value is assigned irrespective of the region of bistability, as long as the I/O curve is bistable. From a parent population of 200 cascades, the next generation of cascades was selected randomly with a replacement, following the probability proportional to the fitness. To obtain statistically significant results, we ran such simulations (bistable to monostable or vice versa) with 1000 independent starting genotypes/parameters (with each genotype comprising 200 identical cascades that are subjected to evolution), ensuring that evolution starts from different points in the genotypic space. Finally, the median values of fitness were taken as a measure of the time trace of the evolution. However, we also kept track of the distribution of the population over time. For the evolution in competitive setting, the same procedure was followed, starting with a population of 300 cascades comprised of a mixture of 100 shared, 100 partially shared, and 100 unshared cascades.

We analyzed the evolved parameter space after 50 (shared), 100(partially shared), or 150 (fully unshared) generations of selection for bistability or monostability. As a control, the evolutionary simulation was performed without any selection, where all the mutations are neutral and the parameters can take all possible values in the specified range. However, when evolving under selection pressure, the sensitive parameters will have some constraint on the values that they can take to achieve the assigned goal. We thus argue that the more sensitive parameters will eventually gain more information about the new environment, where bistability is fitter, as the population adapts gradually to the bistable condition. The gain in information content is defined as the reduction in entropy for each parameter with selection for bistability, compared with the control. For a certain parameter, K, the entropy is defined as

$$E_K=-\sum _{i}P\left(K_i\right)\text{log}P\left(K_i\right),$$

where K takes possible values in the assigned range. The information content in the parameter K after selection is the difference in entropy given by

$$I\left(K\right)=E_K\left(\text{without}\text{selection}\right)-E_K\left(\text{with}\text{selection}\right).$$

When calculating the entropy, the number of bins used to discretize the distribution of the parameter values is eight. Thus, the maximum possible information content of a particular parameter is three bits. The information content was performed on population size of 1000 cells evolved with high mutation rate, to gain sufficiently significant statistics.

The analytical solution of the phenomenological selection-mutation model

$N_1$ and $N_2$ denote the number of cells of phenotype 1 and phenotype 2, respectively, in a population and $f_1$ and $f_2$ are the fitnesses of the phenotype 1 and 2, respectively. The cascades switch with intrinsic transition probabilities α and β between monostable and bistable states, and are assigned a fitness depending on their phenotypic behavior (f₁ or f₂). The intrinsic transition rate α (β) represents the probability that a bistable (monostable) phenotype would spontaneously switch to monostability (bistability) by one mutation. The time evolution equations for growth of the population are as follows:

$$\frac{d{N}_1}{dt}=\left(1-\alpha \right)f_1{N}_1+\beta f_2{N}_2,$$ (1)

$$\frac{d{N}_2}{dt}=\beta f_1{N}_1+\left(1-\alpha \right)f_2{N}_2,$$ (2)

with initial condition, $N_1\left(t=0\right)=0$, so that the population starts with a pure population of only phenotype 2 and evolves toward phenotype 1 (phenotypes 1 and 2 refer to bistability and monostability, respectively, in cases of transition from monostability to bistability). We get the fraction of phenotype 1 as

$$\begin{array}{l}{x}_1=\frac{N_1}{N_1+N_2}\hfill \\ x_1\left(t\right)=\frac{2b\left[\text{exp}\left(\gamma t/2\right)-\text{exp}\left(-\gamma t/2\right)\right]}{2b\left[\text{exp}\left(\gamma t/2\right)-\text{exp}\left(-\gamma t/2\right)\right]+\left(a-d+\gamma \right)\text{exp}\left(-\gamma t/2\right)+\left(d-a+\gamma \right)\text{exp}\left(\gamma t/2\right)},\hfill \end{array}$$ (3)

where

$$a=\left(1-\alpha \right)f_1;b=\beta f_2\text{;}c=\alpha f_1\text{;}d=\left(1-\beta \right)f_2\text{;}\gamma =\sqrt{\left[{\left(a+d\right)}^2-4\left(ad-bc\right)\right]}.$$ (4)

At steady state,

$$x_1\left(t\to \infty \right)=\frac{2b}{2b+d-a+\gamma }.$$ (5)

For selection of phenotype 1, $f_1>f_2$. Particularly, $f_1=2$ and $f_2=1$ for our case, and the calculations of$\alpha$ and $\beta$ are described in Quantification of Evolutionary Transition Rates.

For a mixture of phosphatase sharing and nonsharing cascades, time evolution of the fraction of cells with a given degree of phosphatase sharing can be calculated in large time limit (here, symbols 1 and 2 refer to unshared and shared, respectively; hence, ${\alpha }_1$ and ${\beta }_1$ would indicate the switching rates for an unshared cascade, as shown in Fig. 4 A, whereas ${\alpha }_2$ and ${\beta }_2$ would indicate the switching rates for a shared cascade). Thus, the fraction of shared fraction in transition from monostability to bistability is given by

$$x_2^{\text{mix}}\left(t\right)=\frac{{\varphi }_2/{\gamma }_2}{\left({\varphi }_1/{\gamma }_1\right)\text{exp}\left[\left(\left(a_1+d_1+{\gamma }_1\right)-\left(a_2+d_2+{\gamma }_2\right)\right)t/2\right]+{\varphi }_2/{\gamma }_2},$$ (6)

$$\begin{array}{l}{a}_1=\left(1-{\alpha }_1\right)f_1\text{;}{b}_1={\beta }_1{f}_2\text{;}{c}_1={\alpha }_1{f}_1\text{;}{d}_1=\left(1-{\beta }_1\right)f_2\text{;}{\gamma }_1=\sqrt{\left[{\left(a_1+d_1\right)}^2-4\left(a_1{d}_1-b_1{c}_1\right)\right]}\hfill \\ a_2=\left(1-{\alpha }_2\right)f_1\text{;}{b}_2={\beta }_2{f}_2\text{;}{c}_2={\alpha }_2{f}_1\text{;}{d}_2=\left(1-{\beta }_2\right)f_2\text{;}{\gamma }_2=\sqrt{\left[{\left(a_2+d_2\right)}^2-4\left(a_2{d}_2-b_2{c}_2\right)\right]}\hfill \\ {\varphi }_1=2b_1+d_1-a_1+{\gamma }_1\hfill \\ {\varphi }_2=2b_2+d_2-a_2+{\gamma }_2\hfill \end{array}$$

in the limit

$$t\to \infty x_1^{\text{mix}}=0\mathrm{when}\left(a_1+d_1+{\gamma }_1\right)>\left(a_2+d_2+{\gamma }_2\right)$$

$$t\to \infty x_1^{\text{mix}}=1\mathrm{when}\left(a_1+d_1+{\gamma }_1\right)<\left(a_2+d_2+{\gamma }_2\right).$$

This suggests that one of the two topologies would outcompete the other in the long run, even when the switching rates are slightly different. However, time to takeover is dependent on the magnitude of their differences.

Figure 4.

Phosphatase sharing facilitates transition from the monostable to bistable response (phenotype). (A) The time course of the median fitness of bistable or monostable phenotype over 1000 simulation trajectories with different starting genotypes (parameter values). The top panel shows the median evolutionary trajectories for transition from monostable to bistable and the bottom panel shows the same for the bistable to monostable transition. (B) shows corresponding distributions of the fitness for monostable to bistable transitions with the generation time for the three cases. The results reveal that the transition/adaptation speed toward bistability is a function of degree of phosphatase sharing, with fully shared > partially shared > unshared. The transition speeds are comparable for bistability to monostability transition (bottom panel in A). (C) The median evolutionary trajectories with a starting mixture of the three cascades, each with a population size of 100, evolving toward bistability (left panel) or monostability (right panel) for 100 different genotypes, indicate that the fully shared cascade quickly outcompetes the other two topologies as it transitions to bistability. To see this figure in color, go online.

Results

Phosphatase sharing enhances robustness of bistable response

To analyze effects of phosphatase sharing, we used ordinary differential equations to construct a model of the two-layer MAPK cascade comprised of MAPKK and MAPK. We assumed that the top-layer MAPKKK becomes rapidly activated upon external stimulation and remains active during the course of simulation, thus acting as the input signal for the MAPKK layer. We used this modeling approach to explore three cases: 1) specific phosphatases for each reaction (unshared, Fig. 1 A); 2) phosphatases that are specific for MAPKK and MAPK but shared between single- and double-phosphorylated substrates (partially shared, Fig. 1 B); and 3) one phosphatase that is shared among all reactions (fully shared, Fig. 1 C).

We initially adopted parameter values that were previously used to simulate the MAPK cascade of Xenopus oocytes (6), and systematically explored the parameter space of all three models by randomly sampling all kinetic parameters and initial concentrations (see Methods; Supporting Material). By solving the ODEs at different input levels, we generated steady-state I/O curves for each parameter combination and classified them into monostable and bistable response types. The system was defined as bistable if it displayed hysteresis (15), i.e., if the dose-response curves differ when starting the simulations from a low or high initial activation. After extensive simulations, we observed that the frequency of bistable responses was significantly enhanced by phosphatase sharing between two layers of the cascade (Fig. 1 D): ∼25% of the parameter sets showed bistability when the phosphatase was fully shared, whereas only ∼10% showed bistability when the phosphatase was only shared within layers, and in absence of the phosphatase sharing, the fraction of bistable occurrences reduces to ∼5%, which is likely due to the kinase sharing (37). Conceptually this result is in accordance with earlier studies showing that enzyme sharing can result in a positive feedback loop of regulation and bistability (36, 38), but our study (Fig. 1 D) is, to our knowledge, the first exhaustive comparison of frequency of bistable occurrences in cascades with different degrees of phosphatase sharing. Although the exact percentages of monostable and bistable behaviors were dependent on the central parameter and concentration values, we observed that bistability in the fully shared topology was consistently more frequent when several sets of independent central values were tested (data not shown).

To systematically investigate the robustness of bistable behavior, we next calculated a property called the MR, which reflects the robustness of a phenotype in sustaining its original response upon mutations of parameters (35, 39). Here, starting from a reference “genotype” (parameter set) corresponding to a phenotype (bistable or monostable response), we randomly selected one reference parameter and mutated it within a ±5-fold range (see Methods for details). The new genotype was defined as a member of one-mutational neighborhood of the starting genotype. The fraction of mono/bistable phenotypes in the one-mutational neighborhood of a mono/bistable phenotype was calculated by randomly repeating the selection of one parameter for 500 times; this means that for the partially shared cascade with total 20 parameters, each parameter would undergo 25 mutations on average. The average fraction of mono/bistable phenotypes in the neighborhood of a mono/bistable phenotype was then used to estimate the one-MR of that particular phenotypic state. We repeated the same calculations with two (or more) random mutations at a time and estimated the respective two (or higher order) mutational neighborhoods. For each of single or multimutations, calculations were repeated for 100 different starting genotypes showing mono/bistable phenotype, and the mean MR was calculated. The procedure is performed for up to the 10-mutational neighborhood (Fig. 1 E). This analysis shows that the genotypes corresponding to the monostable response have higher MR for cascades with no sharing or partial sharing compared with the cascade with full phosphatase sharing; this was confirmed for 100 independent genotypes exhibiting monostable response (Fig. 1 E, left panel). On the other hand, when genotypes corresponding to a bistable phenotype were subjected to random mutations, the MR was proportional to the degree of phosphatase sharing (Fig. 1 E, right panel). This supports our hypothesis that the cascade with full phosphatase sharing shows a more pronounced robustness of bistable behavior.

Robustness of bistability is achieved through the emergence of intrinsic positive feedback due to phosphatase sharing

Previous studies have shown that the competition of a phosphatase to dephosphorylate MAPK-PP and MAPK-P in one-layer phosphorylation cascade can give rise to bistability and hysteresis (37, 40, 41). Similarly, when the same phosphatase is shared between two layers of the cascade, an increase in the downstream MAPK-PP level could sequester the phosphatase and reduce the available phosphatase for the upstream MAPKK-PP. As a result, the dephosphorylation rate of MAPKK-PP is reduced, effectively increasing the level of MAPK-PP. This competition between MAPK-PP and MAPKK-PP for the same phosphatase results in a positive feedback loop from MAPK-PP to MAPKK-PP. To delineate the effect of the feedback directly, we determined the local response coefficient quantifying the effect of changing MAPK-PP concentrations on the MAPKK-PP level by calculating the Jacobian of the set of differential equations (40) (see Methods for details). The corresponding response coefficient is zero when the phosphatase is not shared between layers, whereas positive values in the shared case confirm the existence of a hidden feedback loop from MAPK-PP to MAPKK-PP. To test the effect of phosphatase concentration, we calculated the response coefficient by varying the shared phosphatase concentration at different signal doses (Fig. 2 A, signal doses are explained in the legend) for the phosphatase fully shared cascade. The same response coefficients are identically zero for the other two cascade types. These calculations suggest that the feedback is most effective at intermediate phosphatase concentration (Fig. 2 A). At low phosphatase concentrations the pathway is very sensitive to the input, so despite the sequestration of phosphatase by MAPK-PP, it does not significantly affect the MAPKK-PP level because it is already highly active. In contrast, the pathway becomes insensitive to input at very high phosphatase concentration and the MAPK-PP is unable to sequester the phosphatase due to stoichiometric limitations. At intermediate phosphatase concentrations the competition between MAPK-PP and MAPKK-PP for the phosphatase is most effective. To exclude the possibility that this result is limited to a certain parameter combination, we next chose 200 different parameter sets (each from fully shared, partially shared, and fully unshared cases topologies) from our sampling results that exhibited bistable response and varied their respective phosphatase concentrations. Namely, we selected a parameter set that displays hysteresis in the original sampling results and checked whether the output still shows hysteresis if the shared phosphatase concentration is systematically varied from low to high values. Fig. 2, B–D shows the effect of phosphatase concentration variation in the three phosphatase sharing designs, the x axis is the phosphatase concentration and the y axis shows the fraction of cases that remain bistable when the original cases are bistable. It can be noted that “fraction of bistability” (y axis) never reaches 1 because of the inherent differences in the original phosphatase concentrations in the 100 different parameter sets. Indeed, the fraction of bistable responses is highest at intermediate phosphatase concentrations, and it decreases as the phosphatase concentration is increased, demonstrating the effect of the competition (Fig. 2 B). For the partially shared cascade, where the phosphatases are only shared within one layer of the cascade, the phosphatase concentrations have a relatively limited effect on the fraction of bistable responses and the influence is not as pronounced as in the case of fully shared cascade (Fig. 2 C). For the fully unshared case, the individual phosphatase level has a less significant influence on the fraction of bistable responses, although this fraction becomes very low at the highest phosphatase concentrations due to the inability of the pathway to become activated (Fig. 2 D). The analysis thus comparatively shows the relative sensitivity of individual phosphatases, particularly for partially and fully unshared cascades on the retention of their original bistable responses. Following the previously developed approach (41), a more detailed theoretical analysis of a simpler one-cycle case was carried out to understand the necessary conditions for achieving bistability when phosphatase is shared between two different layers (Supporting Material).

Figure 2.

Robustness of bistability arises due to emergence of intrinsic positive feedback by phosphatase sharing. (A) The response coefficient from MAPK-PP to MAPKK-PP as a function of the shared phosphatase concentration at different input doses for the fully shared cascade (input doses are indicated as signal). (B) The dependence of the fraction of bistability on the shared phosphatase concentration for 100 different bistable parameter sets for the fully shared cascade. (C) The dependence of the fraction of bistability on the two shared phosphatase concentrations (indicated as P1dual and P2dual) for 100 different bistable parameter sets for the partially shared cascade. (D) The dependence of the fraction of bistability on the four phosphatase concentrations (indicated as P1, P2, P3, and P4) for 100 different bistable parameter sets for the unshared cascade. To see this figure in color, go online.

Evolutionary plasticity is improved by phosphatase sharing

The above observations raise the question of how a population of MAPK cascades with different degrees of phosphatase sharing would respond to selection pressure during evolution. To understand this, we performed in silico evolution to characterize the ability of a population of cascades to adapt the different phenotypes. As schematically shown in Fig. 3 A, we started with an initially homogeneous population of a genotype that yielded a monostable phenotype and tracked evolution of this population until it dominantly adapted genotypes corresponding to a bistable phenotype. Alternatively, we performed the opposite selection by assigning a starting condition of bistability and allowing the population to evolve to a monostable phenotype. The population evolves through mutation (42, 43) of only one randomly chosen parameter in 20% of the individuals per generation. Selection pressure is imposed in each generation by assuming a higher probability of survival for cascades showing the desired response type (see Methods for details). We repeated these evolutionary simulations for 1000 different independent genotypes to confirm the consistency of the results. Initially, we analyzed the evolution of pure populations (with only one particular phosphatase-sharing design) of monostable cascades toward bistability, which shows that transition speed (i.e., amount of generations needed to achieve the evolutionary goal) is proportional to the degree of phosphatase sharing (Fig. 4 A, top panel). Qualitatively, the more rapid evolution of the population with fully shared cascades is due to a combination of two effects: 1) a faster onset in the first appearance of bistable cascades, and 2) a more efficient conversion into a pure population of bistable cascades (Fig. 4 B). Monostable to bistable transition dynamics for 200 representative genotypes is depicted (Fig. S1) for the three cascade types. With increasing degrees of phosphatase sharing, a faster onset of bistability and a more efficient conversion to bistability can be clearly seen in Fig. S1. Transition speed from bistable to monostable phenotypes, however, does not depend on the degree of phosphatase sharing (Fig. 4 A, bottom panel).

Figure 3.

Schematic representation of the evolutionary simulation pipeline with starting condition monostable signal response and desired evolutionary goal as bistability. (A) Pure population of monostable cascades and (B) equal mixture of fully shared, partially shared, and unshared designs and their transitions to bistability. First, a population of monostable or bistable cascades is created through sampling. The oval with gradient fill represents monostable cascade and oval with complete fill represents bistable cascade. Light blue, red, and dark blue color represent phenotypes corresponding to fully shared, partially shared, and unshared cascades. Evolutionary transition of a fully shared cascade population from monostable to bistable phenotype is shown schematically in (A). After assigning an evolutionary goal the cascades are mutated, the evolutionary processes are represented by the box. After several generations the desired phenotype is obtained (or not). In (B) a mixture of cascade, populations compete across generations for a common goal, to achieve bistability from monostability. To see this figure in color, go online.

We next tested a condition where the cascade types compete with each other in a mixed population (cartoon in Fig. 3 B). Consistent with the first set of simulations (Fig. 4 A), the subpopulation of fully shared cascades outcompetes the other two subpopulations by adapting to bistability within ∼20 generations (Fig. 4 C, left panel). The population remains a mixture of cascades with different sharing for a longer duration (∼50 generations) when selecting for monostability (Fig. 4 C, right panel), although a consistent loss of the fully shared cascade can be noted. Thus we next investigated how the plasticity corresponding to different degrees of phosphatase sharing would be utilized in the changing environment.

To this end, we performed a set of simulations with a mixture of all three cascades in a periodically changing environment, where selection criterion changes every 80 generations from monostability to bistability and vice versa. Over 400 generations, the fully shared cascade outcompetes the other two cascades in the mixture of all three cascade populations (Fig. 5 A) or of two sharing cascades (Fig. 5 B). The cascade with partial sharing, on the other hand, can outcompete the unshared cascade in the absence of the fully shared cascade (Fig. 5 C). Hence the multilayer, phosphatase-shared architecture can most rapidly adapt to a changing environment (assuming a simple scenario where the functional requirements dynamically change only between monostability or bistability) and perhaps is reused throughout evolution.

Figure 5.

Adaptation in changing environment reveals the improvement in plasticity by phosphatase sharing. The evolution is performed for 400 generations starting with an initial equal mixture of population with the three cascade designs. The starting population is composed of 100 of each cascade type. The condition for selection is changed every 80 generations from bistability (shown in the top bars as B) to monostability (M) and vice versa. (A) The evolution of the number of fully shared population for 400 generations averaged over 50 independent genotypes. This shows that the fully shared cascade wins over the other two cascades in a changing environment, indicating its enhanced performance in evolutionary plasticity. (B) The number of fully shared cascades in an evolving population containing a mixture of fully shared and partially shared cascades with 150 of each cascade type. (C) The number of partially shared cascades is shown when a population of an equal mixture of partially shared and unshared cascades evolves. The partially shared cascade has better evolutionary plasticity compared with the unshared cascade.

Quantification of evolutionary transition rates

We next asked why the cascade with phosphatase sharing exhibits higher evolutionary plasticity when compared with an unshared cascade. The simplest explanation would be that the enhanced region of bistability in the model with sharing (Fig. 1 D) makes it more likely to transit from monostability to bistability by a random mutation. We therefore derived a two-step phenomenological model of a selection-mutation process (Fig. 6 A) (30, 31) to investigate whether this mechanism is sufficient to qualitatively explain the evolutionary dynamics observed in Fig. 4 (see Supporting Material for details).

Figure 6.

The two-step phenomenological model for the population evolving from bistability to monostability and vice versa. (A) Schematic diagram represents the models where the intrinsic transitions occur with probabilities of α and β, where B stands for bistable and M for monostable. Bistable and monostable responses have fitness f₁ and f_2, respectively. (B) The diagram indicates the procedure by which the intrinsic transition rates (α,β) are calculated. A sequence of transitions is generated through one mutation at each step. The transition rates are calculated from the sequence as described in Results and Methods. (C) Transition rates α and β for the fully shared, partially shared, and unshared topologies, calculated from the simulation as in (B) for 100 different initial states. To see this figure in color, go online.

In the phenomenological model, cascades switch with intrinsic transition probabilities α and β between mono- and bistable states, and are assigned a fitness depending on their phenotypic behavior (f₁ or f₂). The intrinsic transition rate α (β) represents the probability that a bistable (monostable) phenotype would spontaneously switch to monostability (bistability) by one mutation. Using analytical calculations (see Supporting Material for detail), we obtained expressions for the temporal evolution of the mono- and bistable fractions in the population under selection pressure (Eq. 3 in Methods) and for the corresponding fraction of bistable cascades $\left(x_1\right)$ at a steady state, when the population evolves from monostable to bistable phenotype (see Methods):

$$x_1\left(t\to \infty \right)=\frac{2b}{2b+d-a+\gamma },$$ (7)

where $a=\left(1-\alpha \right)f_1\text{;}b=\beta f_2\text{;}c=\alpha f_1\text{;}d=\left(1-\beta \right)f_2\text{;}\gamma =\sqrt{\left[{\left(a+d\right)}^2-4\left(ad-bc\right)\right]}.$

The transition rate α (β) is expected to be low if robustness of bistability (monostability) to parameter changes is high. We argue that the ratio of α and β would follow the ratio of robustness of monostability and bistability. To estimate α and β, the evolutionary simulations were performed without any selection for 1000 generations by mutating one parameter per generation (Fig. 6 B). Then, the occurrence of transitions between mono- and bistable (and vice versa) response over the 1000 steps was counted separately for the three sharing cases. The transition rates averaged over 100 different starting states reveal that mono- to bistability (β) transition is much slower for the unshared case (β = 0.010) compared to the two sharing cases (β = 0.015 for partially shared case and 0.044 for fully shared case) (Fig. 6 C, right panel). Nevertheless, if sufficient time is allowed, eventually both sharing and nonsharing populations attain a very similar distribution of phenotypes (Fig. 6 C, left panel). The calibrated phenomenological models also qualitatively reproduce the fundamental observations of the evolutionary simulations of the MAPK cascade (Fig. S2). If the transition rates are a mere reflection of the robustness of mono- and bistable behavior in parameter space, then the ratio of transition rates (α/β) would be close to the ratio of mono- and bistable frequencies (Fig. 1 D). In contrast, we found that the values of α/β for the fully shared (α = 0.217, β = 0.02, α/β = 10.5), partially shared (α = 0.31, β = 0.015, α/β = 20.66), and unshared (α =0.41, β = 0.01, α/β = 41) cases are far greater than the corresponding ratios of mono- and bistable regions observed in the MAPK cascade (fully shared: 75/25 = 3.0; partially shared: 90/10 = 9.0; unshared: 95/5 = 19.0). These observations suggest that the accelerated evolution toward bistability for the cascade with fully sharing phosphatases cannot be solely explained by the higher frequency of bistable occurrences, and additional major factors must play a role during the selection process.

To further test whether the region of bistability is the critical factor affecting the evolutionary transition speed of the MAPK cascades with different sharing designs (Fig. 4 A), we simulated a situation where both fully shared and partially shared cascades show comparable bistable frequencies (Fig. S3, top left panel, compare “Full_share_high_phosphatase” and “partial_share”). This was achieved by increasing the shared phosphatase concentration, which reduces the competition between MAPKK and MAPK (thus diluting the strength of the implicit positive feedback). Our algorithm made sure that the MAPK_PP reaches at least 20% of the total MAPK concentration in each acceptable simulation, to avoid nonresponding cascades resulting from high phosphatase concentration. We observed that the rate of evolution from mono- to bistability, although affected due to change in region of bistability, remains much higher for the fully shared cascade compared with the partially shared cascade (Fig. S3, top right panel). Further MR for bistability in the fully shared system with a smaller region of bistability remains closely comparable to its counterpart with ∼2.5-fold larger region of bistability (compare Fig. 1 E, right panel, “Fully_shared” with Fig. S3, bottom panel, “Fully_shared”). This clearly indicates that region of bistability is not the only factor determining the speed of evolutionary transition or the MR.

A shared cascade needs less limiting mutations to adapt bistability

Given that the range of bistability in the parameter space is not sufficient to explain the difference in evolutionary transition rates, we investigated if shared and unshared models exhibit different sensitivity to parameter changes. We reasoned that parameter changes that are required to make a transition to bistability should be enriched in the evolved parameter set. We therefore analyzed the evolved parameter set after evolution for 100 generations. As a control, the evolutionary simulations were also performed without any selection (see Methods for details) where the fitness of the population is independent of the parameter values and therefore all mutations are neutral. Consequently, in the control scenario the parameters can randomly assume any possible values in the specified range, whereas in the case of selection, the evolutionary selected parameter sets will be constrained. We used an information-theoretical approach to identify which parameter changes are enriched during evolution (see Methods). Here, the information content as a measure of enrichment attains a value of zero if a parameter is not selected for, and a maximum value of three bits if all values lie within a single bin after selection because the distribution of each parameter value is discretized into eight bins to calculate the entropy (see Methods). We calculated the information content of each parameter and observed distinct behavior dependent on whether sharing or nonsharing cascades were selected for bistability (Fig. 7 A). For the model with full phosphatase sharing, the concentrations of the shared phosphatase (P1shared) and the MAPK and the parameter Km7 (Km of MAPK_PP dephosphorylation) gain the most significant information (Fig. 7 A) in the sense that the evolved values of these parameters are conserved. For the partially shared case, the MAPKK concentration gains significant information together with the phosphorylation and dephosphorylation rates at the MAPKK level (k1-k3 in Fig. 7 A). We also noticed that the number of critical parameters progressively increase from the case of fully shared to fully unshared cascades. From these results, we speculated that the conservation of the parameter values may emerge due to the functional requirement (which is evolutionary pressure for transition to bistability). Thus, in a fully shared cascade very few parameter mutations may be sufficient to convert a monostable response to the bistable response, whereas more parameter mutations are necessary to achieve a bistable phenotype if the degree of phosphatase sharing is reduced.

Figure 7.

The mutational landscape of the evolving population and sensitivity analysis highlights critical parameters for monostable to bistable transitions. (A) The information contents of all the parameters for the three cascades when selected for monostability and bistability. The meaning of the values ki, i = 1…8 and kmi, i = 1…8 are explained in Supporting Material, Model equations. (B) Scatter plot shows mutational information of a parameter (x axis) and its respective sensitivity(y axis). The Spearman correlation coefficient (ρ) and p-value shows high correlation between mutational information of a parameter and its sensitivity to transition rate. (C) The median evolutionary time courses over 200 independent trajectories keeping either MAPK only or MAPK and P1shared constant are plotted, and show dramatic change in the transition rates compared with the control. (D) The median evolutionary time courses of 200 independent trajectories for monostable to bistable transition are shown for the fully shared cascade, where only P1shared and MAPK were allowed to mutate and the rest of the parameters were kept constant. To see this figure in color, go online.

To test whether the parameters with high information content also contribute to the bistable to monostable transition, we kept one of the parameters fixed but allowed all other parameters to mutate, and calculated the mono- to bistable transition rates. The sensitivity of a given parameter was then determined by taking the ratio of the transition rate when all the mutations are allowed, and the resultant transition rate when the parameter is not mutated. The sensitivities of the individual parameters display a significant correlation (Spearman correlation coefficient = 0.71, p < 10⁻¹¹) to their respective information contents (Fig. 7 B). In fact, by simultaneously blocking the mutation of multiple parameters with high information content, we observed strong reduction in the transition speeds. For example, in the full phosphatase-sharing cascade when both MAPK and the shared phosphatase (P1shared) were not allowed to mutate during the evolutionary simulations, the transition from monostable to bistable response changed from ∼15 to >60 generations (Fig. 7 C). For partially shared (unshared) models, mono- to bistable transition time increases by ∼50 (∼100) generations when MAPKK and MAPK (MAPKK and k2) are fixed (Fig. S4 A). These results imply that a combination of mutations (or correlated mutations) of the most significant parameters is critical for such evolutionary transition.

To check whether the set of parameters with maximum information content is already sufficient to convey a rapid of transition to bistability, we next performed simulations by mutating only top sensitive parameters and keeping other parameters fixed. For the case of full sharing, when top two parameters were mutated (MAPK, P1shared), the adaptation speed was almost the same as for when all parameters are allowed to change (Fig. 7 D). However, for partially shared and unshared cases, mutations of even the top three parameters (MAPKK, k2, and k3) were not enough to reach the transition speed of the control simulations (Fig. S4 B, top and bottom right panel) suggesting such transitions are dependent on the mutation of multiple critical parameters. However, when we allowed mutation of multiple sensitive parameters (only mutating MAPKK, MAPK, P1dual, k2, and k4, and keeping the rest of the parameters constant) in the partially shared cascade, the mono-to-bistable transition speed becomes comparable to its control case (Fig S4 C). Alternately, mutations of only noncritical parameters failed to achieve a mono- to bistable transition (shown for the partially shared cascade in Fig. S4 D). These results reemphasize that in the fully shared cascade, fewer critical parameters facilitate mono- to bistable transitions with a higher probability. The transition to monostability, however, is not contingent on any particular mutation, making the transition much faster irrespective of the choice of parameters or degree of phosphatase sharing (data not shown).

Discussion

A large volume of experimental and computational findings have provided a generally good understanding of the working principles of phosphorylation/dephosphorylation cascades and of their connection to different cell fate decisions (44). For the MAPK phosphorylation cascade, mechanistic studies showed significance of its conserved, multilayer structure in conferring robust and flexible output (45, 46), as well as in optimizing response time and signal duration (38, 40). Furthermore, different motifs where kinases or phosphatases are shared between several reactions were analyzed for their ability to trigger monostable or bistable responses (40, 41, 47). However, how signaling pathways with similar structural designs and with comparable potential to deliver a specific output type would deviate from (and probably outcompete) each other in the process of evolution remains poorly understood. In this study, we investigated the significance of an enzyme sharing motif and asked whether there exists a plausible evolutionary advantage when a phosphatase dephosphorylates multiple kinases of a cascade compared with when the kinase-phosphatase interactions are more specific. We took the two-layered MAPK cascade comprising MAPKK and MAPK and studied how the extent of phosphatase sharing influences the occurrences of monostable and bistable responses, which are the two most commonly observed output responses of the cascade (12, 13, 14, 16, 17, 18, 19, 20, 21).

We first showed that a greater extent of phosphatase sharing results in proportionally more frequent occurrences of a bistable response. Subsequently, by MR analysis we showed that phosphatase sharing also enhances the robustness of a genotype (parameter space) that exhibits the bistable response. The evolutionary adaptability (the ability of a system to adapt to changes in environment) (24, 48) is also different for different phosphatase-sharing designs. The cascade with maximum degree of phosphatase sharing showed the maximum evolutionary adaptability (or better evolvable under selection pressure), specifically during evolution of a monostable population to predominantly bistable population. At the mechanistic level, a positive feedback loop resulting from the phosphatase sharing (40, 41) led to enhanced occurrence of bistable responses. However, even if the frequency of bistable responses is lowered by increasing the phosphatase concentration, the MR of bistability as well as the adaptability were always higher for the shared case, indicating that additional factors contribute to the observed faster transition speed of monostable-to-bistable responses. To study the mutational landscape of the parameters, we next used an information theory (49) based approach and analyzed the evolved mutational landscape comprising biochemical rates and concentrations of kinases and phosphatases.

We evaluated the information content of individual parameters for all three phosphatase-sharing designs, highlighting the parameters with maximum information for both mono- to bistable transitions and vice versa. We systematically evaluated the relation between information content of a parameter and its impact on the mono- to bistable transition rate in the three cascade types. We also performed evolutionary simulations by either blocking or only mutating the parameters with maximum information in each cascade type, showing dramatic effects on the mono- to bistable transition rates. The analysis also highlights that degree of phosphatase sharing in the MAPK cascade can uniquely shape the mutational landscape and the information content of a parameter. For instance, the cascade with maximum degree of phosphatase sharing had the least number of critical parameters, and by mutating only the two top parameters with maximum information, we could achieve the transition speed when all parameters were varied. On the contrary, when multiple parameters (shown by mutating 10 parameters with least information in the partially shared cascade) with low information were only subjected to mutation, a mono- to bistable transition was not even possible. These results suggest that the overall improved plasticity in displaying both monostable and bistable responses in the phosphatase fully shared design (as compared to the other two cascade types) is due to a significant reduction in the number of critical mutations.

We argue that the improved plasticity of the phosphatase-sharing design may be the reason for its frequent use in nature, enabling the cell to generate different response types depending on the requirements. MAPK cascades in mammalian cells often share phosphatases between the MAPK and MAPKK layers (7, 8, 9, 23, 50), and in the output level, display both graded and bistable responses. On the other hand, phosphatase sharing between different layers of the MAPK cascades has not yet been identified in yeast, where interestingly, the output of the MAPK cascade frequently shows a monostable response (18, 19, 20); is it possible that introduction of (synthetically designed) shared phosphatase can trigger hysteresis in yeast MAPK cascades? Further experimental studies may shed light on this possibility.

Although the most commonly observed response of the MAPK cascade to external stimuli is either monostable or bistable, there are reports of more specialized I/O responses. In the temporal regime the cascade is able to produce oscillation and adaptation in response to constant external stimulus (32, 33, 36, 51, 52). The core cascade design can also cross talk with other pathways (47), and it can be wired with explicit positive and negative feedback loops (53, 54). In our study, we neglected such additionally complex I/O behaviors and feedback designs as we wanted to compare cascades differing only in their phosphatase interactions. Additional control and fine tuning of signal propagation emerging due to spatial localization of the enzymes (55, 56) were also neglected in our study, as they fall beyond the scope of ODE-based modeling adopted for this study, which required spatially explicit models; future studies may couple spatial localization of signaling components to the topology of signaling pathways, and ask the evolutionary consequences of such coupling.

The improved flexibility and robustness of bistable behavior may make the cascade less (or more) evolvable in a new environment, where some of the other phenotypes are desired. Generally, evolvability of a system often emerges as a trade-off between specialization and flexibility (24, 42, 48). A network that is insensitive to mutation (or less evolvable) will consistently perform the same specialized function. On the contrary, its evolutionary adaptation would require flexible design(s) that can be utilized to perform different functions. Notably, the system requires some degree of MR to allow for genotypic heterogeneity in a population, and to make a new phenotype easily accessible in the neighborhood of the genotypic space (42, 43). Evolvability and its connection to MR have already been demonstrated in transcriptional networks, RNA, and protein structures (42, 43, 57). In the future, such interplay between flexibility and specialization in the emergence of evolvability could also be explored in the context of multilayer, phosphorylation-dephosphorylation cascades and in more complex signal transduction systems with feedback loops and cross talks.

Author Contributions

B.G. and U.S. conceived the study. B.G. and U.S. built the analysis pipeline and performed all the simulations. B.G., U.S., and S.L. analyzed the data. B.G., U.S., V.S., and S.L. wrote the manuscript.

Editor: Stanislav Shvartsman.