Distributive reaction network
A distributive reaction network is one in which a substrate molecule is catalytically modified multiple times by an enzyme that must unbind from the molecule between each catalytic step. Here we consider the simple example of a substrate protein containing two residues that can each be either phosphorylated or unphosphorylated. A kinase protein catalyzes the phosphorylation of each site, and a phosphatase protein catalyzes the dephosphorylation of each site. We denote the state of the substrate protein by distinguishing between three states: no sites phosphorylated (S0
), one of the sites phosphorylated (S1
), or both sites phosphorylated (S2
). Denoting the kinase by E
, the phosphatase by P
, and inactive forms of the enzymes by E*
, the reaction network considered is:
This generic distributive reaction network has been used by Takahashi et al. as a model for a portion of the mitogen-activated protein kinase cascade.21
We choose kinetic parameters to be consistent with their work. We vary the value of ka
, which reflects the time required for a kinase to exchange ADP with ATP, over several orders of magnitude as it is unknown for most protein kinases. We find its value to be qualitatively important when considering the effects of the membrane environment, which highlights the importance of experimentally determining its value for protein kinases.
This network model is sufficiently general to provide insight into signaling pathways that contain opposed distributive modification cycles, provided appropriate kinetic parameters are chosen. For example, the phosphorylation of ITAM regions on the ζ-chain of the T cell receptor by the tyrosine kinase Lck involves multiple kinase modifications. It is unknown whether this reaction occurs by a distributive or the alternative processive mechanism, in which an enzyme does not have to unbind from a substrate protein between catalytic events. Additionally, it is unknown how the membrane environment might affect qualitative cellular-level behavior of distributive reaction networks, so principles gleaned from simulations such as those in this paper may prove useful in designing experiments to distinguish whether a phosphorylation cascade occurs via a distributive mechanism.
It was demonstrated by Markevich et al. that a distributive reaction network can exhibit bistable behavior.26
The physical argument is based on the sequestration of enzymes by the substrate molecule, and a necessary condition for the presence of bistability is that the number of substrate molecules must exceed the number of at least one type of enzyme.27
Here we consider the case in which the substrate molecules outnumber each type of enzyme. When bistability is present, in one steady state, fully phosphorylated proteins (S2
) outnumber the unphosphorylated proteins (S0
), while in the other steady state, the relative numbers are reversed. Consider a system that resides in the steady state with mostly unphosphorylated proteins. Most kinases will be bound and sequestered by S0
proteins. Since there are relatively few S2
proteins, there will be an excess of free phosphatase. Hence, when a kinase creates a singly phosphorylated substrate protein, the new S1
protein is much more likely to bind to a phosphatase than a kinase in a well-mixed system. This provides a driving force to return the protein to the unphosphorylated state, thereby stabilizing the S0
state. An analogous argument holds for the other steady state.
In three dimensions, Takahashi et al. recently showed that a smaller diffusion coefficient can change the characteristics of the distributive reaction network due to enhanced rebinding of substrate protein and enzyme after the first catalytic reaction.21
Rebinding of enzyme and substrate can result in a second catalytic event quickly following the first, leading to an effective processive-like step taking S0
directly to S2
. Processive enzyme catalysis cannot exhibit bistable behavior. It was also noted that increasing ka
served to weaken features associated with the distributive reaction network. Recent experimental results provide evidence that molecular crowding in mammalian cells, which presumably increases the likelihood of rebinding between proteins, converts the double phosphorylation of MAP kinase proteins from a distributive to a processive mechanism.28
Our focus is on the effects of confining the distributive reaction network to a 2-dimensional membrane-like environment. Features of the membrane include increased concentration and smaller distances between proteins, lower protein mobility, and differences in the properties of random walks in two and three dimensions. Our simulation results parse the consequences of the interplay between these effects, which are difficult to intuit.
In our simulations, we use a typical membrane protein concentration of 100 molecules/μm2 for the substrate protein, using initial conditions of NS2 = NS0 = 50 and NE = NP = 25, with all other species and complexes initially unpopulated. We simulate many independent dynamic trajectories in which the position and state of all proteins are tracked in time. Initially, the molecules are placed uniformly at random in space. Each trajectory corresponds to an experiment with an isolated patch of membrane. Starting simulations with NS2 = NS0 and all enzymes unbound gives an unbiased starting point: for a bistable system, half of the trajectories (on average) should go to each of the steady states.
Confining the distributive network promotes bistability through increased concentration
Results in show the effects of decreasing the confinement length. For each simulation, we hold constant the number of proteins, the diffusion coefficient, and the rate of enzyme activation. The diffusion coefficient is characteristic of cytoplasmic diffusion and the value of ka
is of the same order of magnitude as that estimated by recent experiments measuring the half-life of ADP release after phosphorylation for a serine-threonine protein kinase.29
At large l
, the distribution of the number of S2
molecules at steady state is unimodal, and as the system is confined, the distribution at first broadens. At sufficiently small l
, the distribution becomes bimodal, which implies that a population of isolated experiments under these conditions would partition into two subpopulations, one containing cells with most substrate proteins fully phosphorylated and one containing cells with most substrate proteins unphosphorylated. This trend continues for values of l
smaller than those considered in , as is demonstrated in the Supporting Information
. Stochastic fluctuations at short times determine which state a trajectory reaches.
Decreasing the confinement length promotes the emergence of bistability in the distributive reaction network
A dominant feature as the system becomes confined is that the concentration of molecules increases. This favors reactions involving the binding of enzyme and substrate proteins, leading to a higher fraction of bound enzyme-substrate pairs. This enhances the sequestration effect that is necessary for bistability in distributive reaction networks. This results in the emergence of bistability as the system is confined, with more confinement resulting in less frequent fluctuation-driven transitions between states.
Decreasing protein mobility and/or increasing the rate of enzyme activation suppresses bistability
The membrane environment can also suppress bistability because of smaller diffusion coefficients. A transition from bistable to monostable behavior can be seen by decreasing the diffusion coefficient while holding the confinement length (l
) and other parameters fixed. At sufficiently small values of D
, the system is monostable, with a unimodal distribution of NS2
. Increasing D
, when l
is sufficiently small, eventually leads to bistable behavior, with larger D
giving rise to more separated states that are less likely to undergo fluctuation-driven switching. The overall dependence of the bistability on D
, with ka
= 0.7 s−
1, is summarized in the “bistability diagram” in . The effect of increasing ka
, and hence reducing the refractory time of enzymes, is shown in . Experimental studies measuring the rate of ADP release for the protein kinase PKA estimate a rate 30 to 50 times larger than ka
= 0.7 s−
1, suggesting that enzyme turnover rates may vary over orders of magnitude, depending on the kinase.30
In general, as ka
increases, greater confinement and higher mobility are necessary to support bistability.
Bistability diagram for the distributive reaction network
Approximate dividing lines between monostable and bistable regions for various values of ka
Protein mobility affects features of bistability in part because the transport of molecules influences the reaction kinetics.17
As mobility decreases, effective diffusion-influenced kinetic rates are reduced, leading to a smaller fraction of bound enzyme-substrate pairs. This weakens the sequestration effect necessary for bistability. At sufficiently high mobility, effective kinetic rates approach their well-mixed values, yielding the maximum sequestration effect.
Both protein mobility and the rate of enzyme reactivation (or refractory time of the enzyme) can affect features of bistability through their effect on the rebinding between enzyme and substrate molecules. The half-life of an inactive enzyme associated with ka
= 0.7 s−
1 is approximately 1 s. In this time, the characteristic distance traveled by the enzyme in an unbounded 2-dimensional domain is (4Dt
= 2 μ
m. This distance is of the same order as the system size simulated, which implies that the location of the enzyme upon reactivation will be weakly correlated with its position immediately after catalysis. Thus, a processive-like step is unlikely to occur as spatial correlation between the enzyme and substrate molecule does not persist on time scales associated with enzyme reactivation. For ka
= 0.7 s−
1, rebinding plays a minimal role compared with the influence of transport on the effective kinetic rates for the values of D
considered. The effects of rebinding become more prominent as ka
increases and D
decreases, since spatial correlation between the enzyme and substrate molecule is more likely to persist for times associated with the reactivation of the enzyme. For a given value of l
, the effects of diffusion on the transport of molecules are expected to be similar. This suggests that differences in the line of conditions separating bistable and monostable responses as ka
is varied () are due primarily to rebinding effects. Re-binding plays a more prominent role with increasing ka
, converting the distributive mechanism to an effectively processive one (which does not support bistability). A larger value of ka
, at a given value of l
, requires higher protein mobility to minimize rebinding and support bistability, which is similar to the results in Ref. 21
Decreasing protein mobility enhances rebinding for multiple reasons: (i) the enzyme is more likely to become active before hopping occurs; (ii) once an enzyme is active and in contact with S1
, a binding event is more likely to occur; and (iii) if hopping occurs before enzyme activation, the enzyme and substrate protein are closer together on average when the enzyme becomes active. To gain insight into the importance of rebinding in the distributive reaction network, we developed a quantitative estimate of the probability of rebinding and catalysis between one E*
and one S1
protein, given they just reacted and occupy the same lattice site (see the Appendix
for details). shows that the effects of rebinding are much more pronounced with ka
= 700 s−
1, in both two and three dimensions, and that rebinding is more significant in two dimensions.
Rebinding is more pronounced in two dimensions and for smaller enzyme refractory times (larger ka)
At fixed concentration, shape of the signaling volume influences bistability
To emphasize that the emergence of bistability upon confinement (change in geometry) is not simply a concentration effect, we examined the steady state distribution of NS2 in two slab-shaped systems that have equal volume and contain equal numbers of molecules. However, they differ in shape, with dimensions of 2μm × 2μm × 0.01μm and 1μm × 1μm × 0.04μm. Although both systems have the same concentration of proteins, differences in shape affect the distribution of distances between proteins, with the average distance between neighboring molecules smaller in the second case. This holds since, in general, the expected distance from a molecule to its nearest neighbor is well approximated by setting ρV(r) ~ 1, where V (r) is the volume of the region contained within a sphere of radius r, ρ is the concentration of molecules, and ρV(r) gives the expected number of molecules within the region. For the two systems considered in this section, the average distance between neighboring molecules is larger than the slab thickness, so we can approximate the volume within radius r of a point at the center of the slab as V(r) ~ πr2λ, where λ denotes the slab thickness. Using ρV(r) ~ 1 and solving for r, it follows that the expected distance to a nearest neighbor scales as λ−1/2. Thus, at constant concentration, as the slab thickness increases, the average distance between neighboring molecules decreases. This result holds for confined regions in which the slab thickness is smaller than the average distance between neighboring molecules. At sufficiently large values of λ, the average distance between nearest neighbors approaches a constant value.
For the two systems considered here, the average distance between neighboring molecules in the less confined system is roughly half the distance between molecules in the more confined system. A more complete analysis, presented in the Supporting Information
, shows that the average distance from a molecule to its kth
nearest neighbor is smaller in the less confined system. The smaller average distance between molecules reduces the effect of diffusion on the effective kinetic rates and reduces the likelihood of rebinding, since molecules must diffuse a smaller distance before they become effectively well mixed. This is reflected in the steady state distributions shown in : while bistability is present in both systems, the stable states are more separated when λ
= 0.04 μ
m, with less frequent fluctuation-induced switching between steady states. Notice that, in the case in which concentration is fixed, a difference in shape (less confinement) promotes bistability. This result shows that differences in the time (or distance) required for a protein to encounter other proteins after its last signaling act can result in qualitative differences in the input-output characteristics of signaling modules on cell membranes. In the limit of very high mobility, the systems are well mixed and shape does not affect the dynamics.
The shape of the confining region, at fixed volume and concentration, affects features of the bistability
Positive feedback regulation of Ras activation
Ras is a small G protein (GTPase) that plays an important role in many signaling pathways (e.g., T cell receptor and growth factor receptor signaling).31
Ras is inactive when bound to GDP and active when bound to GTP. In its active state it can enzymatically modify numerous substrate proteins involved in signal transduction, particularly those involved in cell proliferation.32
Ras activity is regulated by the GTPase-activating protein RasGAP and the guanine nucleotide exchange factors RasGRP and SOS. RasGAP, when bound to Ras, enhances the rate of conversion of GTP to GDP, thus promoting the inactivation of Ras. RasGRP and SOS both catalyze the exchange of GDP with GTP, thus activating Ras. SOS has been shown to have both an allosteric and a catalytic pocket that bind to Ras. The activation of Ras leads to more SOS molecules with RasGTP bound to the allosteric pocket, and an enhancement in the rate of catalysis.33
Thus, there is a SOS-mediated positive feedback loop in the Ras activation network, with RasGTP enhancing the production of additional RasGTP. This positive feedback loop can lead to bistability, hysteresis, and digital signaling.34
Ras is a membrane-anchored protein, and RasGRP and SOS are recruited to the membrane by diacylglycerol (DAG) and adaptor proteins, respectively. The degree to which they are recruited to the membrane is regulated by upstream signaling pathways. The importance of the membrane environment in Ras activation by SOS is illustrated by recent experiments in which the rate of Ras activation is greatly enhanced when membrane-bound Ras is used instead of Ras in solution.35
We consider the effects of membrane confinement and diffusion on the following simple representation of the Ras activation network (adapted from Das et al.34
). We denote RasGDP by D and RasGTP by T, and we assume the allosteric pocket of SOS must be occupied for catalytic activity.
This network exhibits bistability in appropriate parameter regimes. When the network is bistable, the steady states can be characterized by the number of active Ras molecules. One state is “active” and contains a relatively large number of RasGTP, while the other state is “inactive” and contains relatively few RasGTP. To test for the presence of bistability, trajectories are started from two different sets of initial conditions. In each case, all molecules begin unbound and are uniformly distributed in space. In one case all Ras are in the active form and in the other case all Ras are in the inactive form. If the system is bistable, trajectories starting from the all-RasGDP state will likely fall into the basin of attraction for the inactive state, while trajectories starting from the all-RasGTP state will likely fall into the basin for the active state. If the system does not exhibit bistability, then all trajectories will evolve in time toward a common state. In bistable systems, it is common for one steady state to be more stable than the other. In our stochastic simulations, this is reflected by stochastic switching from the less stable state to the more stable state on the time scale of the simulations. As such, in simulations with steady states of disparate stability, the less stable steady state is only transiently populated by trajectories starting in its basin of attraction.
Membrane confinement of the Ras network and/or decreasing protein mobility promotes Ras activation
As before, we simulate the Ras network in a slab geometry which is confined in one spatial dimension, varying the diffusion coefficient as well as the number of DAG and RasGRP molecules. The simulations are performed with 150 Ras molecules, 80 SOS molecules, 8 RasGAPs, and a variable number of DAG and RasGRP molecules (NDAG = NGRP = 0; NDAG = 6, NGRP = 20; and NDAG = 24, NGRP = 80). The three most confined systems all exhibit bistable behavior (), but with l = 0.01μm, trajectories from the inactive state spontaneously switch to the active state, and with l = 0.03μm, trajectories from the active state spontaneously switch to the inactive state. This gives an indication of the relative stability of each steady state. At l = 0.04μm, the system is at the limit of bistability, and most trajectories from the active regime quickly populate the inactive state.
Confining the Ras activation network affects properties of the bistability at fixed protein mobility
These results can be understood in terms of concentration effects: as the system is confined, the concentration of proteins increases, promoting binding between proteins. This is the same principle that resulted in enhanced binding between substrate proteins and enzymes in the distributive reaction network. Here, more concentrated conditions favor the binding of Ras molecules to the allosteric and catalytic pockets of SOS, which enhances the production of RasGTP. At sufficiently high levels, RasGTP can initiate and sustain positive feedback, which is necessary for bistability.
Along with the results for D = 1μm2/s (), shows that reducing the mobility of molecules decreases the stability of the inactive state relative to the active state. At high mobility, the system is bistable, with the active state more stable than the inactive state, while at low mobility, the system is monostable and resides in the active state. Lower mobility allows more cells to achieve the highly active state in the bistable system. Two effects promote stability of the active state as the mobility of proteins decreases. The first is that RasGAPs, which have the highest binding rate (k11) to Ras, experience effects of lowering diffusion at higher mobility than other binding reactions; i.e., the fastest reaction becomes diffusion-influenced first. This serves to decrease the effective binding rate between RasGAP and RasGTP, thus increasing the survival time of an active Ras protein. This promotes positive feedback through SOS and can enhance the further production of RasGTP. This result highlights that different reactions are influenced differently upon reduction in protein mobility, and predicting the effects of diffusion on the network requires knowledge of rates of the biochemical reactions. Additionally, as the rate of diffusion decreases, rebinding between recently bound proteins becomes more likely. For example, RasGTP bound to the allosteric site of SOS becomes more likely to rebind, increasing its effective lifetime.
Decreasing the diffusion coefficient stabilizes the active Ras state relative to the inactive Ras state
The average time that RasGTP is bound to the SOS allosteric pocket, per binding event, is
. Two effects increase the effective lifetime: multiple bindings before hopping and multiple diffusive excursions and returns before the molecules become mixed. We treat the probability of returning as in the distributive reaction network and assume that if RasGTP and SOS diffuse beyond a characteristic distance apart, they are effectively mixed. This distance is set roughly by the expected distance to the nearest RasGDP molecule, which competes for the allosteric pocket of SOS. The number of bindings before hopping is geometrically distributed, as is the number of returns before diffusion beyond a cutoff distance. The expected number of each type of event is
Note that here pon is the probability that RasGTP binds SOS before either molecule hops away. Putting these results together, the effective lifetime is
With D = 0.01μm2/s, the effective time (including rebinding) that RasGTP is bound to the allosteric pocket of SOS, compared to the case without rebinding, is 1.9 times longer in two dimensions and 1.3 times longer in three dimensions. Additionally, as mobility decreases, it is more likely for a recently produced RasGTP to stay in the proximity of the SOS molecule for long enough to enhance its binding to the allosteric pocket when RasGDP unbinds. Both of these effects serve to enhance the effects of positive feedback.
An additional feature of interest is that within the monostable regime, the expected time to reach the active steady state from the all-RasGDP initial condition varies non-monotonically with the diffusion coefficient, as is demonstrated by the response times reported in . As the diffusion coefficient decreases from 0.1μ
/s, the average time to reach the active state first decreases (faster response time); however, for sufficiently small D
, the average time increases (slower response time). Above the crossover diffusivity, decreasing mobility decreases the response time by reducing the effectiveness of RasGAPs and enhancing positive feedback through rebinding effects. Below the crossover diffusivity, the response time is dominated by the time it takes for molecules to diffuse into contact. Hence, reducing D
in this regime increases the overall response time. The non-monotonic response time is similar to a result from Ref. 21
, in which lowering mobility first decreased response times due to enhanced rebinding but eventually increased response times due to longer molecular encounter times.
The bistability diagram for the Ras activation network is shown in and summarizes the influence of confinement, the diffusion coefficient, and the number of DAG and RasGRP on bistability (previous studies have explored the influence of RasGRP on Ras activation in well-mixed systems34,36
). There are three dominant effects at play in the results: (i) confinement increases the concentration of molecules, which enhances protein-protein binding and the effects of positive feedback; (ii) decreasing D
enhances production of RasGTP through reduced effective RasGAP activity and enhanced positive feedback effects due to rebinding; and (iii) increasing NGRP
enhances the activation of Ras, thus biasing the system toward an active state.
Bistability diagrams for the Ras-SOS reaction network