Fig 1.
Signal-induced polarity switching.
A Schematic representation of a rod-shaped cell with polarity marker A shown in yellow. Proteins can either be bound to the poles or diffuse in the cytoplasm. The abundances of the polarity marker at the two poles are denoted by A1 and A2. The release of a signal protein X in the cytoplasm, shown in purple, can lead to a polarity reversal, such that the polarity marker switches from pole 1 to pole 2. B Schematic representation of the molecular interactions of the polarity model. The polarity marker A and its antagonist B inhibit each others binding to the pole. B can cooperatively recruit itself to the pole and promotes binding of the recruitment factor R, which in turn recruits A. Dashed lines indicate exemplary hypothetical interactions of the signal protein X with the polarity proteins. C The switching signal is implemented as a pulse in the total amount of X, parameterized by the signal duration τ and signal amplitude Xmax.
Fig 2.
Schematic representation of the workflow.
A Switching signals are parameterized by the choice of i) a reaction rate it acts on, ii) an inhibitory or enhancing effect and iii) the amplitude Xmax and duration τ of the transient signal. X can act on any of the 11 parameters of the polarity model. B Example of a deterministic and stochastic simulation before, during and after the signal. The signal is applied between t = 0 and t = 3. Thick lines indicate the concentrations of A (yellow), B (red), R (green) and X (purple) at pole 1, and thin lines at pole 2. C Switching is evaluated by comparing the signs of the asymmetry ωA(t) in A before and after the switch. For the stochastic simulation a switching probability is calculated from 100 trajectories. D Switching regimes are plotted in phase space as a function of Xmax and τ for the modification of each model parameter. For the deterministic model, successful switches are shown by the gray regions with a black outline, for the stochastic model switching probabilities are shown in green. E The state of the system during the signal is identified by simulating the deterministic model with the signal applied for the duration of the simulations. The dynamics is classified into three states: symmetric (blue), oscillatory (orange) and polarized (yellow).
Fig 3.
Switching regimes for each of the model parameters.
Regions in which the deterministic model shows switches are indicated by thick black outlines. The green shading shows the switching probability of the stochastic model with N = 103.75. The upper half of the phase diagram shows results for a signal that enhances the reaction rate, and the lower half for a repression of the rate. The colored bars to the right of each panel indicate the class of dynamics when the corresponding amplitude of signal is applied, with yellow for polarized, orange for oscillatory and blue for symmetric polar distribution of A. The red symbols indicate the signal amplitude and duration of the trajectories shown in Fig 4.
Fig 4.
Trajectories of the model during switches, classified as four different switching classes.
Signal parameters Xmax and τ and the parameter modified are indicated by the corresponding symbols in Fig 3. Vertical dashed lines indicate the period during which the signal is present. A Relaxation oscillator. For a short signal (plus-symbol), the polarity switches during the applied signal as shown by the solid lines. For a longer signal (open triangle), the system switches a second time as shown by the dashed lines. B Prime-release switch. During the signal the polarity is unchanged, but switches after the signal is released. C Reset switch. During the signal, the system relaxes to a symmetric distribution of the polarity marker and establishes a reversed polarity after the signal is removed. D Push switch. The system switches while the signal is applied and does not switch back when the signal is applied longer.
Fig 5.
In the presence of the signal, the polarity system can display three qualitatively different phase space topologies, here denoted as ‘polarized’, ‘oscillatory’, and ‘symmetric’.
A For each case, the dynamics of the system is shown in the three-dimensional space (A1 − A2, B1 − B2, R1 − R2), in which the origin corresponds to a completely symmetric protein distribution. A In a polarized state, the system is bistable, with two stable fixed points, marked grey and blue, which correspond to the two polarities of the cell. Depending on the initial condition, the system approaches one or the other stable fixed point, as illustrated by the shown trajectories. B In an oscillatory state, all trajectories of the system run into a stable limit cycle, marked in black. C In a symmetric state, the system is monostable, with a single stable fixed point at the origin, corresponding to an unpolarized cell.
Fig 6.
Nonlinear dynamical behavior of the four different mechanisms of signal-induced polarity switching.
In each case, the system dynamics are shown both during (red) and after (black) a signal pulse, with projections onto the (A1 − A2, R1 − R2)-plane, the (B1 − B2, R1 − R2)-plane, and the (A1 − A2, B1 − B2)-plane. A Transient oscillator switch. B Reset switch. C Prime-release switch. D Push switch.
Fig 7.
Behavior of the model at different noise levels.
A Switching probability as a function of noise strength for different switching mechanisms. Signal parameters are indicated by the corresponding symbols in Figs 3 and 4. Each data point represents the results of 104 stochastic realizations. B Probability of different numbers n of switching events at different noise levels for the prime-release switch. Signal parameters are as for Fig 4B. C States of 200 stochastic realizations (N = 104) of the prime-release switch at t = τ. Dashed line shows an estimate of the position of the separatrix in the absence of the signal (see ‘Methods’). D States of 200 realizations of the relaxation oscillator at t = τ (N = 104). Dotted line shows the deterministic limit cycle of the system during the signal, dashed lines indicate where the limit cycle intersects with the separatrix in the absence of signal. The three different panels in C and D show different two-dimensional projections of the nine dimensional phase space. Point type and color indicate whether the system switches polarity (red) or not (black).
Fig 8.
A Power spectral density of A1(t)−A2(t) in the absence of any X signal for different noise strengths. A peak in the power spectrum at high noise indicates stochastic coherence. B The maximal power density relative to the power at zero frequency shows a non-monotonic dependence on the noise strength. C The mean time between switching events, defined as points when A1 = A2, varies as different signals are applied, at a noise level N = 103.75. Signals that generate deterministic oscillations have been excluded. Times between switches were extracted from stochastic trajectories with the signal applied continuously for 50000 min.
Fig 9.
Illustration of the working principles underlying the four classes of switching mechanisms.
A The different nonlinear dynamical behaviors are schematically represented in a two-dimensional phase space. Red/black symbols indicate the state space and dynamics when the signal is present/absent. Transient oscillator switch. B Reset switch. C Prime-release switch. D Push switch.