Pattern localization to a domain edge

The formation of protein patterns inside cells is generically described by reaction-diffusion models. The study of such systems goes back to Turing, who showed how patterns can emerge from a homogenous steady state when two reactive components have different diffusivities (e.g., membrane-bound and cytosolic states). However, in nature, systems typically develop in a heterogeneous environment, where upstream protein patterns affect the formation of protein patterns downstream. Examples for this are the polarization of Cdc42 adjacent to the previous bud site in budding yeast and the formation of an actin-recruiter ring that forms around a PIP3 domain in macropinocytosis. This suggests that previously established protein patterns can serve as a template for downstream proteins and that these downstream proteins can “sense” the edge of the template. A mechanism for how this edge sensing may work remains elusive. Here we demonstrate and analyze a generic and robust edge-sensing mechanism, based on a two-component mass-conserving reaction-diffusion (McRD) model. Our analysis is rooted in a recently developed theoretical framework for McRD systems, termed local equilibria theory. We extend this framework to capture the spatially heterogeneous reaction kinetics due to the template. This enables us to graphically construct the stationary patterns in the phase space of the reaction kinetics. Furthermore, we show that the protein template can trigger a regional mass-redistribution instability near the template edge, leading to the accumulation of protein mass, which eventually results in a stationary peak at the template edge. We show that simple geometric criteria on the reactive nullcline's shape predict when this edge-sensing mechanism is operational. Thus, our results provide guidance for future studies of biological systems and for the design of synthetic pattern forming systems.

Figure 1

Illustration of the phase-space geometric analysis for two-component McRD systems. (a) The reactive equilibria (black dots) are given by the intersections between the reactive subspaces $m+c=n$ (gray lines) and the reactive nullcline $f=0$ (black line) in the $(m,c)$-phase space. Hence, the reactive nullcline encodes the qualitative structure of the reactive flow as illustrated by the red arrows. (b) Sketch of the membrane profile of a mesa pattern composed of a high- and low-density domains, $m^{+}$ and $m^{-}$, connected by a diffusive interface around the inflection point $x_0$. (c) Flux-balance construction of the mesa pattern in phase space. The intersections of the flux-balance subspace (FBS) (purple dashed line) and the reactive nullcline yield the concentrations at the plateaus $m^{\pm }$ and the inflection point $m^0$. The balance of net reactive flows in the system (red arrows) determines the FBS-offset ${\eta }_0$. In the regime where the slope of the reactive nullcline is steeper than the slope of the FBS, an homogenous steady state is laterally unstable. (d) Linearization of the dynamics in the vicinity of the homogeneous steady state, yields a dispersion relation for growth rates of the eigenfunctions (Fourier modes indexed by wave number $q$). The fastest growing mode, $q_c$, dominates the length scale of the initial dynamics.

Figure 2

Illustration of a step-like template (a) that acts on the reaction kinetics defines two subdomains, labeled A and B, with respective reaction kinetics $f_{\mathrm{A}}$ and $f_{\mathrm{B}}$. (b) We consider a two-component system describing, for example, a single protein species that cycles between a membrane bound and a cytosolic state. (c) The reactive flow due to the reaction kinetics can be visualized in the $(m,c)$-phase space of concentrations. Due to mass-conservation, the flow points along the reactive phase spaces $m+c=n$ (indicated by gray lines). Along the nullclines $f_{\mathrm{A},\mathrm{B}}$, the flow vanishes. Therefore, the reactive flow in each subdomain is qualitatively captured by the shape of the respective nullcline. (d) With the reactive flow encoded by the nullclines, the phase portraits of the two subdomains can be combined (“overlaid”) into a single phase space. This combined phase space will be used for the construction of stationary states.

Figure 3

Illustration of the construction of monotonic steady states (heterogeneous base states) in $(m,c)$-phase space. (a) Density profile with the inflection point at the template edge $x_{\mathrm{E}}^{}$. (b) Phase space including the reactive nullclines of subdomain A (orange) and B (blue) with the corresponding density distribution (thick orange and blue line). (c) Total turnover as a function of the membrane density at the template edge $m_{\mathrm{E}}^{}$. Total turnover balance determines the steady-state value $m_{\mathrm{E}}^{*}$. (d) Density profile for a case where the inflection point of the profile, $x_0$, lies within subdomain A, that is, when $m_{\mathrm{A}}^0>m_{\mathrm{B}}^{}$ as illustrated in the corresponding phase space representation (e). (f) In this case, the total turnover becomes a nonmonotonic function of $m_{\mathrm{E}}^{}$, such that total turnover balance may have multiple solutions, or no solution at all. These different cases are sketched here for different average global total densities $\overline{n}$. (g) Bifurcation diagram of the base state, showing the saddle-node bifurcation due to breakdown of total-turnover balance. In the region beyond the saddle-node bifurcation (shaded in gray) no monotonic base state exists. Note that monotonicity enforces $m_{\mathrm{E}}^{*}-m_{\mathrm{B}}^{}<0$ here.

Figure 4

Bifurcation structure and phase space construction of stationary patterns. (a) Bifurcation structure of stationary states for the average mass, $\overline{n}$, obtained by numerical continuation, together with spatial profiles for the stable steady states (monotonic base states and nonmonotonic patterns). In the area shaded in gray, no monotonic base states exist. Solid (dashed) lines indicate stable (unstable) branches. The unstable branches in the central region, marked by numbers 1–6 correspond to unstable patterns with multiple inflection points in subdomain A (see Fig. 13 in Appendix pp5). The instability of these patterns is related to coarsening, which is generically exhibited by two-component McRD systems on a homogeneous domain [6, 7, 30, 31]. (b), (c) Sketches of stable, nonmonotonic, stationary patterns together with the corresponding phase space constructions. At the density profile's extremum, marked by the dash dotted vertical line, there are no gradients (${\partial }_x\tilde{m}=0={\partial }_x\tilde{c}$), such that a notional no-flux boundary can be introduced. Thus, the resulting two segments (labeled I and II), can now be studied separately. In the phase portraits, the density distributions of the two segments connect at the extremal concentrations $m_{\mathrm{A}}^{-}$ and $m_{\mathrm{A}}^{+}$, in (a) and (b), respectively. They are shown slightly offset from the FBS (dashed purple line) for visual clarity. The true density distribution must of course be embedded in a single FBS to fulfill diffusive flux-balance. Parameters for the bifurcation diagram: $D_m=0.01$, $D_c=0.5$, ${\widehat{k}}_{\mathrm{fb}}=0.25$, ${\widehat{k}}_{\mathrm{off}}=4$, ${\theta }_{\mathrm{B}}=20$, ${\theta }_{\mathrm{A}}=2$, $L=10$, $x_{\mathrm{E}}^{}=5$.

Figure 5

Regional lateral instability at the base state's saddle-node bifurcation. (a) The numerically calculated eigenfunction, ${\mathrm{\Phi }}_m$, associated to the vanishing eigenvalue at the saddle-node bifurcation is localized at the template edge. (Parameters as in Fig. 4 at the saddle-node bifurcation $\overline{n}\approx 2.65541$). (b) The concentration profile $\tilde{m}\left(x\right)$ of the base state at the saddle-node bifurcation. A spatial region that is fully contained in subdomain A and centered around the profile's inflection point $x_0$ is marked in green. (c) This spatial region corresponds to a phase space region where the dynamics is guided by a section of the nullcline with a negative slope, i.e., where the system is laterally unstable. (The phase space is shown as a sketch for visual clarity. See Fig. 8 in Appendix pp2 for a plot from numerical simulation.)

Figure 6

Nullcline criterion for edge sensing, i.e., the emergence of a stable density peak at the template edge. Edge sensing is possible is the nullclines intersect in a point where one of them has negative slope (a). If they do not intersect (b) or intersect in a point where they have both positive (or both negative) slope, then stable peaks only exist at a system boundary, and hence, edge sensing is not possible.

Figure 7

Time evolution of $m$-profile due to dynamic template interface position. The Supplemental Material contains a movie for each of the four scenarios. (a) Slow pull (the template edge moves away from the peak): This results in pinning of the peak to the template edge (see Supplemental Material Movie 3 [46]). (b) Slow push (the template edge moves toward the peak): This results in pinning of the peak to the template edge (see Supplemental Material Movie 4). (c) Fast pull: This results in depinning as the peak cannot follow the template edge. However, the peak stays where it depins as it is quasistable in region A on the observed timescale (see Supplemental Material Movie 5). (d) Fast push: This results in suppression as the peak is not stable in region B. As the peak dissipates the average mass at the new edge position slowly increases and potentially (if there is enough total mass in the system) leads to to a reentrance of the peak (see Supplemental Material Movie 6). Parameters: $D_m=0.01$, $D_c=10$, $k_{\mathrm{on}}=1$, ${\widehat{k}}_{\mathrm{fb}}=1$, $k_{\mathrm{off}}=2$, ${\widehat{K}}_{\mathrm{d}}=1$, $\overline{n}=5$, ${\theta }_{\mathrm{B}}=20$, ${\theta }_{\mathrm{A}}=0.5$, $L=20$, $x_{\mathrm{E}}^{}=3/5L$, $v_{\mathrm{E}}\left(t\right)=\pm 2.4\times {10}^{-14}{t}^2$ in (a), (b), $v_{\mathrm{E}}\left(t\right)=\pm 8.9\times {10}^{-13}{t}^2$ in (c), (d).

Figure 8

Base states for nullclines shapes that do not facilitate edge-sensing. (a) The low-mass base state with a high-concentration plateau in subdomain A and a low-concentration plateau in subdomain B. (b) At a critical average mass, the base state undergoes a saddle-node bifurcation. The bifurcation arises from the “annihilation” of the FBS-NC intersection point $m_A^{-}$ and $m_A^0$, and not from the break down of turnover balance as discussed in Sec. 2c the main text. In this case, a regional instability is triggered at the system boundary. An adiabatic sweep of $\overline{n}$ through the saddle-node bifurcation of the FBS-NC intersection points is shown in Supplemental Material Movie 7.

Figure 9

An adiabatically changing template triggers peak formation at the template edge. See also Supplemental Material Movie 8. (a) Initial state with a homogeneous template profile ${\theta }_{\mathrm{A}}={\theta }_{\mathrm{B}}$. (b) Base state analogous to Fig. 3. (c) Base state right before the saddle-node bifurcation. The region that becomes laterally unstable in the bifurcation is highlighted in light green in the spatial profile and in phase space (cf. Fig. 5). (d) Peak pattern state after the system transitioned through the bifurcation. This state is qualitatively the same as sketched in Fig. 4. Parameters same as described in the caption of Fig. 7 but with $L=5$, ${\theta }_{\mathrm{B}}=20$, and ${\theta }_{\mathrm{A}}^{\mathrm{f}}=0.5$.

Figure 10

Effect of nonadiabatic mass upregulation on the pattern formation dynamics. (a) Peak formation at the template edge: The density distribution is not embedded in a single FBS, leading to the “zigzag”-shaped density distribution in phase space. The regional instability is still triggered at the template edge, as highlighted by the (green) shaded region. (b) Peak formation at the system boundary: The faster mass-inflow leads to a more pronounced “zigzag”-shaped density distribution in phase space. The regional instability is now first triggered at the system boundary. This results in a peak forming at the system boundary as shown in Fig. 4.

Figure 11

Phase diagram for the timescale of mass upregulation (global cytosolic inflow ${\kappa }_{\mathrm{s}}^{-1}$) against the timescale of mass-redistribution across the entire domain ($L^2/D_c$). Edge sensing, i.e., formation of a single density peak a the template edge, is operational in regime (1), see Supplemental Material Movie 9. In regimes (2)–(6) a peak at the system boundary ($x=0$) or multiple peaks form, see Supplemental Material Movies 10–14. Parameters: $D_m=0.01$, $D_c=10$, $k_{\mathrm{on}}=1$, ${\widehat{k}}_{\mathrm{fb}}=1$, $k_{\mathrm{off}}=2$, ${\widehat{K}}_{\mathrm{d}}=1$, ${\theta }_{\mathrm{A}}=0.5$, ${\theta }_{\mathrm{B}}=20$, $x_E=3/5L$, ${\overline{n}}_{\mathrm{f}}=5$.

Figure 12

One-parameter bifurcation structure in $\overline{n}$ (average mass) connecting the low-mass base state (monotonic, i.e., $m_{\mathrm{E}}^{}<m_{\mathrm{B}}^{}$ and the peak pattern at the template edge (nonmonotonic, i.e., $m_{\mathrm{E}}^{}>m_{\mathrm{B}}^{}$). Solid (dashed) lines indicate stable (unstable) branches from numerical continuation for different domain sizes (increasing from dark to light lines). The corresponding steady-state profiles are shown in Supplemental Material Movies 15 and 16 [46], for domain sizes $L=10,40$. Solutions from the analytic construction of base states in the large domain size limit ($L\to \infty$) are shown as red dots and (position of the saddle-node bifurcation denoted by $n_{\mathrm{bs}}^{\infty }$). Note that for small domain size ($L=5$), the saddle-node bifurcations vanish and the base state smoothly transitions into a stable peak pattern upon increasing $\overline{n}$. The red, dash dotted line indicates the analytically constructed edge-localized pattern (limit $L\to \infty$). The lower bound in average mass for the existence of these patterns is denoted by $n_{\mathrm{pattern}}^{\infty }$. Fixed parameters as described in the caption of Fig. 4.

Figure 13

Spatial profiles representative of the unstable branches numbered (1–6) in the $\overline{n}$-bifurcation diagram Fig. 4.

Figure 14

Demonstration of the edge-sensing criterion for a phenomenological model for Cdc42 pattern formation. (a) Nullclines of the reaction kinetics Eq. (F1) in subdomains A (orange) and B (blue), defined by the reaction rates in Eq. (F4). Multiple B-nullclines are shown for off-rate factors ${\gamma }_{\mathrm{off}}=1,1.5,...,5$ while the on-rate factor is fixed at ${\gamma }_{\mathrm{on}}=8$. The critical values ${\gamma }_{\mathrm{off}}^{\pm }$ between which the A-nullcline is intersected at negative slope (i.e., the edge-sensing criterion is fulfilled) are shown as dashed and dash-dotted lines, respectively. (b) Phase diagram of on- and off-rate factors $({\gamma }_{\mathrm{on}},{\gamma }_{\mathrm{off}})$. The shaded region shows the regime where edge-sensing is observed in numerical simulations with adiabatically increasing average mass [cf. Eq. (D1)]. Dashed red lines show the curves ${\gamma }_{\mathrm{off}}^{\pm }\left({\gamma }_{\mathrm{on}}\right)$, cf. Eq. (F6), between which the edge-sensing criterion is fulfilled. (c) Quasistationary density profiles obtained from numerical simulations with adiabatically increasing average mass $\overline{n}$ illustrating the typical pattern emerging from the base-states's saddle-node bifurcation in the three regimes of the $({\gamma }_{\mathrm{on}},{\gamma }_{\mathrm{off}})$ phase diagram (${\gamma }_{\mathrm{on}}=8$; top: ${\gamma }_{\mathrm{off}}=5.5,\overline{n}=2.9$; ${\gamma }_{\mathrm{off}}=3,\overline{n}=2.35$; bottom: ${\gamma }_{\mathrm{off}}=1.5,\overline{n}=2.24$.). Fixed parameters: $k_{\mathrm{fb}}=1,k_{\mathrm{on}}=0.07,k_{\mathrm{off}}=1,D_m={10}^{-4},D_c=0.1,L=1,x_{\mathrm{E}}^{}=0.5,{\kappa }_{\mathrm{s}}={10}^{-5}$.