Diffusive coupling of two well-mixed compartments elucidates elementary principles of protein-based pattern formation

Spatial organization of proteins in cells is important for many biological functions. In general, the nonlinear, spatially coupled models for protein-pattern formation are only accessible to numerical simulations, which has limited insight into the general underlying principles. To overcome this limitation, we adopt the setting of two diffusively coupled, well-mixed compartments that represents the elementary feature of any pattern—an interface. For intracellular systems, the total numbers of proteins are conserved on the relevant timescale of pattern formation. Thus the essential dynamics is the redistribution of the globally conserved mass densities between the two compartments. We present a phase-portrait analysis in the phase-space of the redistributed masses that provides insights on the physical mechanisms underlying pattern formation. We demonstrate this approach for several paradigmatic model systems. In particular, we show that the pole-to-pole Min oscillations in Escherichia coli are relaxation oscillations of the MinD polarity orientation. This reveals a close relation between cell polarity oscillatory patterns in cells. Critically, our findings suggest that the design principles of intracellular pattern formation are found in characteristic features in these phase portraits (nullclines and fixed points). These features are not uniquely determined by the topology of the protein-interaction network but depend on parameters (kinetic rates, diffusion constants) and distinct networks can give rise to equivalent phase portrait features.

Figure 1

Reduction from the full dynamics in cell-geometry to the phase portrait of mass-redistribution dynamics. (a) Spherocylinder geometry of a rod-shaped E. coli cell (top) and reduced two-compartment setting representing the two cell halves (bottom). Purple arrows illustrate diffusive mass transport. (b) Protein-interaction network of the Min system of E. coli (see main text Sec. 3 for details. (c) Time traces of the protein mass in the compartments relative to the mean, $\mathrm{\Delta }{n}_{\mathrm{D},\mathrm{E}}$ defined via $n_{\mathrm{D},\mathrm{E}}^{(1,2)}={\overline{n}}_{\mathrm{D},\mathrm{E}}\pm \mathrm{\Delta }{n}_{\mathrm{D},\mathrm{E}}$, showing the pole-to-pole oscillations in three-dimensional cell geometry. (d) Oscillations persist in the two-compartment setting, with diffusive exchange rates set to a slow timescale. (e) Phase portrait of the mass-redistribution dynamics showing the flow field (gray arrows) and the nullclines of MinD and MinE redistribution (blue and red lines). The origin (0,0) corresponds to the homogeneous state which is unstable against perturbations redistributing mass. The limit cycle trajectory (black) corresponds to pole-to-pole oscillations. The cartoons in the four corners illustrate the two-compartments (separated by a vertical dashed line) where the color intensity indicates the mass distribution of MinD (blue) and MinE (red) in the respective quadrant of the phase portrait.

Figure 2

Phase space structure of a two-compartment two-component MCRD system, with reaction and diffusion on the same timescale (a) and with diffusion set to a slower timescale (b). The concentrations $(m_i,c_i)$ in the two compartments are marked by blue dots, labeled 1 and 2, respectively. The local phase spaces corresponding to the masses in the two compartments $n_{1,2}=\overline{n}\pm \mathrm{\Delta }n$ are shown as gray lines. Gray arrows indicate the reactive flow towards the reactive nullcline $f=0$ (solid black line). Black dots mark the local equilibria (intersection points between reactive nullcline and local phase spaces) and red arrows indicate the reactive flow towards these local equilibria. (b) When diffusion is set to a slower timescale, the local concentrations adiabatically follow the reactive nullcline. Thus the only remaining degree of freedom is the mass difference $\mathrm{\Delta }n$, whose dynamics is governed by the concentration differences $\mathrm{\Delta }{m}^{*}\left(\mathrm{\Delta }n\right)$ and $\mathrm{\Delta }{c}^{*}\left(\mathrm{\Delta }n\right)$ (see Eq. (2)).

Figure 3

Graphical construction of the control-space dynamics for two-component MCRD systems. (a) Reactive nullcline (line of reactive equilibria) $c^{*}\left(n\right)$. (b) For $\overline{n}<n_{\mathrm{crit}}$, the lines $c^{*}(\overline{n}+\mathrm{\Delta }n)$ (black solid line) and $c^{*}(\overline{n}-\mathrm{\Delta }n)$ (gray dashed line) only intersect once at $\mathrm{\Delta }n=0$, corresponding to the homogeneous steady state. The flow direction in control space (indicated by blue arrows) is determined by the sign of the difference between $c^{*}(\overline{n}+\mathrm{\Delta }n)$ and $c^{*}(\overline{n}+\mathrm{\Delta }n)$, as indicated by the green and purple shading; cf. Eq. (7). (c) For $\overline{n}>n_{\mathrm{crit}}$, there are two additional intersection points between $c^{*}(\overline{n}+\mathrm{\Delta }n)$ and $c^{*}(\overline{n}-\mathrm{\Delta }n)$, corresponding polarized steady states. The flow in control space is directed away from the homogeneous state $\mathrm{\Delta }n=0$, which is therefore unstable, and drives the system towards one of the stable polarized states.

Figure 4

Graphical construction of the dynamics in control space from the reactive nullcline surfaces. [(a) and (b)] Reactive nullcline surfaces showing MinD and MinE cytosol concentration (shaded blue and red, respectively) as a function of the mass differences $\mathrm{\Delta }{n}_{\mathrm{D}},\mathrm{\Delta }{n}_{\mathrm{E}}$. The intersection of each surface with its point reflection (shaded in gray with dashed outlines) determine the mass-redistribution nullclines (see text for details). These nullclines are a generalization of the fixed points shown in Figs. 3 and 3. (c) Phase portrait of the dynamics Eq. (14) with the MinD and MinE mass-redistribution nullclines obtained by the construction in (a) and (b) and the limit cycle trajectory (black) corresponding to pole-to-pole oscillations [cf. Fig. 1]. (d) Setting MinE diffusion to a slower timescale transforms the limit cycle trajectory to the shape characteristic for relaxations oscillations. For parameters, see Table 1.

Figure 5

Phase diagrams of Min protein dynamics. In each panel, the points and shaded background indicate the results from numerical simulations, distinguishing no patterns (gray), oscillations (purple) and stationary polarity (green). (a) Phase diagram for the LQSSA dynamics Eq. (12). The solid purple and green line indicate the Hopf and pitchfork bifurcations found by linear stability analysis of the LQSSA dynamics. Along the dashed red line, the MinE-redistribution nullcline intersects MinD-redistribution nullcline at the latter's extrema. In the limit ${\mathcal{D}}_{\mathrm{E}}^{},{\mathcal{D}}_{\mathrm{de}}^{}\ll {\mathcal{D}}_{\mathrm{D}}^{}$, this marks the transition between relaxation oscillations and stationary polarity. The gray line indicates the line ${\mathcal{D}}_{\mathrm{E}}^{}={\mathcal{D}}_{\mathrm{de}}^{}$. (b) Example trajectories in the $(\mathrm{\Delta }{n}_{\mathrm{D}},\mathrm{\Delta }{n}_{\mathrm{E}})$-phase plane [cf. Fig. 4]: (i) no instability, (ii) pole-to-pole oscillations, (iii) stationary polarity for ${\mathcal{D}}_{\mathrm{E}}^{}>{\mathcal{D}}_{\mathrm{de}}^{}$, and (iv) stationary polarity for ${\mathcal{D}}_{\mathrm{E}}^{}<{\mathcal{D}}_{\mathrm{de}}^{}$ (note the opposite slope of the MinE-redistribution nullcline). In (iii) and (iv), the dashed gray line shows the separatrix, that separates the basins of attraction of the two polarized states. (c) Phase diagram from numerical simulations of the full two-compartment dynamics, Eq. (1). Note the excellent agreement with LQSSA (a). (d) Phase diagram from numerical simulations of the PDEs on a line (varying $D_{\mathrm{E}}$ in full three-dimensional cell geometry affects also the vertical gradients rather than just lateral diffusion). For parameters, see Table 1

Figure 6

Reaction networks, reactive nullcline surfaces and control-space phase portraits for the PAR system of C. elegans [(a)–(c)] and the Cdc42 system of S. Cerevisiae [(d)–(f)]. (a) Cartoon of a C. elegans embryo showing the segregated aPAR and pPAR domains which as a result of mutual detachment of aPAR and pPAR proteins from the membrane. (b) Reactive nullcline surfaces of aPARs (blue, left) and pPARs (red, right). Note the symmetry under the exchange $\mathrm{A}\leftrightarrow \mathrm{P}$. (c) Control-space phase portrait showing the mass-redistribution dynamics and the mass-redistribution nullclines of aPARs (blue) and pPARs (red). Both mass-redistribution nullclines intersect the lines $\mathrm{\Delta }{n}_{\mathrm{A},\mathrm{P}}=0$ only once, indicating that pattern formation requires redistribution of both protein species. For parameters, see Table 2. (d) Cartoon of a budding yeast cell showing a polar cluster of co-localized Cdc42 and Bem1-GEF complexes. In WT cells, Cdc42 and Bem1-GEF complexes (homologous to Scd1-Scd2 complexes in fission yeast) mutually recruit each other to the membrane and are therefore co-localized in the resulting pattern. (e) Reactive nullcline surfaces of Cdc42 (blue, left) and Bem1-GEF (red, right). For parameters, see Table 3. (f) Control-space phase portrait showing the mass-redistribution dynamics and the mass-redistribution nullclines of Cdc42 (blue) and Bem1-GEF (red). The N-shaped Cdc42-redistribution nullcline intersects the line $\mathrm{\Delta }{n}_{\mathrm{B}}=0$ three times, indicating that redistribution of Bem1-GEF complexes is not required for pattern formation. In contrast to MinE in the Min system, where cytosolic MinE redistribution drives oscillations, the cytosolic redistribution of Bem1-GEF complexes has a stabilizing effect on stationary patterns.

Figure 7

Comparison of the Min-protein dynamics in the full 3D geometry of an E. coli cell (a) to the two-compartment system [(b) and (c)] representing the two cell-halves (poles); see Sec. 3 for details. Nonlinear reactions ($f$, red arrows) account for cycling between membrane-bound and cytosolic states (concentrations $m$ and $c$). Diffusive exchange is indicated by purple arrows. Time traces (center) and phase-space trajectories (right) of the redistributed masses $\mathrm{\Delta }{n}_{\mathrm{D},\mathrm{E}}$ between the two cell-halves/compartments show good qualitative agreement between the full 3D simulation and the two-compartment system. Importantly, setting the diffusive exchange rates to a much slower timescale ($\mathcal{D}\to \varepsilon \mathcal{D}$, here $\varepsilon ={10}^{-2}$) does not qualitatively alter the pole-to-pole oscillation (c).

Figure 8

Illustration of an in vitro setup using a flat microchamber with two membrane surfaces (gray planes) on top and bottom of a bulk volume [26]. An individual column of that system, comprising two membrane patches and the bulk volume in-between them can be pictured as an analog to the cell geometry, where the two membrane patches correspond to the cell poles. The analogous approximation by two compartments, as shown on the right, is valid as long as the vertical bulk gradient is approximately linear. Comparing to Fig. 1, the analogy between pole-to-pole oscillations in cells and vertical membrane-to-membrane oscillations in microchambers becomes immediately evident.

Figure 9

Qualitative equivalence of the Min dynamics at physiological protein densities and with scaled down densities. (a) Parameter space of the protein densities $n_{\mathrm{D}},n_{\mathrm{E}}$ showing the regime of bistable reaction kinetics in gray and trajectories from simulations of the two compartment model. The inset on the top right shows a curve of reactive equilibria (blue) in a slice through the bistable region at constant $n_{\mathrm{E}}$. The inset on the bottom right shows a blow-up of the trajectory in the low density regime. Purple dots mark the average masses. [(b) and (c)] Time traces of the mass differences corresponding to the two trajectories in (a). Note the differently scaled time axes.

Figure 10

Reactive equilibrium surfaces showing the membrane concentrations $m_{\mathrm{d}}$ and $m_{\mathrm{de}}$.