The membrane environment can promote or suppress bistability in cell signaling networks

Abstract

Many key biochemical reactions that mediate signal transduction in cells occur at the cell membrane, yet how the two-dimensional membrane environment influences the collective behavior of signaling networks is poorly understood. We study models of two topologically different signaling pathways that exhibit bistability, examining the effects of reduced protein mobility and increased concentration at the membrane, as well as effects due to differences in spatiotemporal correlations between the membrane environment and three-dimensional cytoplasm. The two model networks represent the distributive enzymatic modification of a protein at multiple sites and the positive feedback-mediated activation of a protein. In both cases, we find that confining proteins to a membrane-like environment can markedly alter the emergent dynamics. For the distributive protein modification network, increased concentration promotes bistability through enhanced protein-protein binding, while lower mobility and membrane-enhanced spatiotemporal correlations suppress bistability. For the positive feedback-mediated activation network, confinement to a membrane environment enhances protein activation, which can induce bistability or stabilize a monostable, active state. Importantly, the influence of the membrane environment on signaling dynamics can be qualitatively different for signaling modules with different network topologies.

Figure 1. Decreasing the confinement length promotes the emergence of bistability in the distributive reaction network

(A–E) Each histogram summarizes the distribution of the final number of S₂ molecules from 500 independent trajectories (1000 s each). The diffusion coefficient is D = 1 μm²/s, the rate of enzyme activation is k_a = 0.7 s⁻¹, and the initial conditions are the same for each trajectory. Bimodality, which reflects an underlying bistability, emerges as the system becomes more confined. Within the bistable regime, additional confinement reduces the frequency of spontaneous state switching. (F) The number of S₂ molecules as a function of time for two stochastic trajectories which reach different steady states (l = 0.01 μm).

Figure 2. Bistability diagram for the distributive reaction network

At fixed number of molecules and with k_a = 0.7 s⁻¹, the confinement length (l) and diffusion coefficient (D) together determine whether the network exhibits bistability. The dashed line is simply intended to guide the eye in distinguishing between the bistable (circles) and monostable (diamonds) regions. The color scheme indicates the degree of separation between the two modes in the bistable region, with hotter colors indicating greater separation between modes. The degree of separation is measured by the standard deviation of the N_S2 distribution (see Supporting Information for details). There exists a confinement length l^*, such that for l > l^*, the system is monostable for all D. Analysis in the well-mixed limit (D → ∞) gives l^* to be approximately 0.6 μm.

Figure 3. Approximate dividing lines between monostable and bistable regions for various values of k_a

For a given choice of D and l, productive rebinding between an enzyme and substrate protein is more likely for larger values of k_a. As a result, the transition from bistable to monostable behavior occurs at larger values of the diffusion coefficient for larger values of k_a, which takes on values of 0.7 s⁻¹, 7 s⁻¹, 70 s⁻¹, and 700 s⁻¹. The full bistability diagrams are presented in the Supporting Information.

Figure 4. Rebinding is more pronounced in two dimensions and for smaller enzyme refractory times (larger k_a)

The color scheme indicates the approximate probability for an enzyme and substrate protein to rebind and react given a first catalytic event just occurred. The molecules are assumed to be unable to rebind if they diffuse farther apart than the mean distance separating nearest neighbor molecules in the system.

Figure 5. The shape of the confining region, at fixed volume and concentration, affects features of the bistability

The average distance between particles is larger in the system corresponding to the top figures, which leads to slower effective kinetic rates and a greater likelihood of rebinding between an enzyme and substrate protein after the first catalytic step.

Figure 6. Confining the Ras activation network affects properties of the bistability at fixed protein mobility

Here, D = 1 μm²/s and N_GRP = 0. Each figure displays the time dependence of numerous trajectories. Trajectories in red start with all Ras inactive and trajectories in blue start with all Ras active. Twenty five trajectories from each initial condition are displayed for each value of l. Increasing l decreases the relative stability of the active Ras state.

Figure 7. Decreasing the diffusion coefficient stabilizes the active Ras state relative to the inactive Ras state

Here the system is 2-dimensional (l = 0.01 μm) with no RasGRP. The value of the response time (τ_resp) reported in each figure gives the average time for trajectories starting from the all-RasGDP state (red) to first reach a state in which N_RasGTP is half the value of its steady state average. An intermediate value of D minimizes the average time to transition from all RasGDP to the long-time value of RasGTP.

Figure 8. Bistability diagrams for the Ras-SOS reaction network

The active state is promoted by increased confinement, decreased mobility, and increased numbers of N_GRP.