Geometry-induced protein pattern formation

The Impact of Cell Geometry on Protein Gradients in Elongated Cells.

We performed a numerical analysis of this reaction module, paying particular attention to the effect of varying the cell geometry and the degree of cooperativity in membrane binding (Fig. 1B). Our simulations show that, in elongated cells, the protein density on the membrane is always inhomogeneous and reflects the local cell geometry. Indeed, one can show analytically that the homogeneous steady state ceases to exist as one passes from circular to elliptical geometry (cf. SI Appendix). We observe two distinct types of pattern: membrane-bound proteins either accumulate at midcell or form a bipolar pattern with high densities at both cell poles. The polarity of these patterns is quantified by the ratio of the density of membrane-bound proteins located at the cell poles (upole) to that at midcell (umidcell): P=upole/umidcell. First, we investigated the impact of preferential recruitment of either PNTP or PNDP to the membrane, defined as R=(kdD−ktT)/(kdD+ktT), on cell polarity. We find that proteins accumulate at the cell poles (P>1) if there is a preference for cooperative binding of PNDP (R>0). Moreover, the polarity P of this bipolar pattern becomes more pronounced with increasing R. This scenario corresponds to the strongly bipolar pattern of AtMinD observed in mutant E. coli cells lacking EcMinD and EcMinE (18). In contrast, when cooperative binding favors PNTP (R<0), proteins accumulate at midcell (P<1). Thus, the sign of the recruitment preference R for a protein in a particular nucleotide state controls the type, whereas its magnitude determines the amplitude of the pattern. Next, we investigated how cell geometry affects the pattern, while keeping R fixed. Upon varying the length of the long axis, L, while keeping the length of the short axis fixed at l=1 μm, we find that the aspect ratio L/l controls the amplitude of the pattern, but leaves the type of pattern unchanged. With increasing eccentricity of the ellipse, the respective pattern becomes more sharply defined; for a spherical geometry, the pattern vanishes. In summary, cell geometry controls the definition of the pattern, and the preference for membrane recruitment of a certain nucleotide state determines the location on the cell membrane where the proteins accumulate and how pronounced this accumulation becomes.

Why Geometry Influences Patterning.

Our finding that recruitment is a major determinant of cell polarity suggests that there is some underlying intrinsic affinity of the two protein species for either the cell poles or the midzone. This affinity cannot be encoded in the attachment or recruitment rates alone, because these are position independent. Instead, it must emerge from the interplay between these reactions, cell geometry, and diffusion. To uncover the underlying mechanism, we first performed a numerical study where we omitted all cooperative membrane binding processes, such that the dynamics became linear. Interestingly, we observed that, although the overall protein density is homogeneous in the cytosol (see SI Appendix), PNDP and PNTP are nevertheless spatially segregated, accumulating in the vicinity of the cell poles and close to midcell, respectively (Fig. 2A). This observation, the origin of which will be discussed later, explains how patterns of membrane-bound proteins arise: these inhomogeneities in protein densities in the cytosol serve as seeds for the polarization of the protein pattern on the membrane, and their respective impact is regulated by the attachment rates ωD+ and ωT+. The pattern of the protein species with the higher membrane affinity determines the type of the pattern (Fig. 2C). If PNDP has the larger membrane affinity, a bipolar pattern emerges, whereas one observes enrichment of membrane-bound proteins at midcell if attachment of PNTP dominates. Note that the detachment rates have the inverse effect (cf. SI Appendix).

• Download figure
• Open in new tab
• Download powerpoint

Fig. 2.

Membrane affinity controls, and recruitment amplifies geometry adaption. The cells used for the numerical studies have a length of L = 5 μm and a width of l = 1 μm. (A) Even when recruitment is turned off, PNTP and PNDP form inhomogeneous density profiles in the cytosol. PNDP accumulates close to the poles and is depleted at midcell. In contrast, PNTP exhibits high concentration at midcell and a low concentration at the poles. The attachment and detachment rates are set to 1 μm/s and 1 s⁻¹, respectively, which gives a penetration depth lλ≈1.6 μm. (B) Illustration of the source degradation mechanism for the spatial segregation of cytosolic PNDP and PNTP. All proteins that detach from the membrane are in an NDP-bound state and can undergo nucleotide exchange; the range of PNDP in the cytosol is limited to a penetration depth lλ (dashed lines); here, lλ=0.35 μm. At the poles, this reaction volume receives input from opposing faces of the membrane, resulting in an accumulation of cytosolic PNDP (dark red). The magnitude of this accumulation depends on the penetration depth. The polarity PNDP=udpole/udmid−cell of membrane-bound PNDP plotted as a function of lλ shows a maximum at lλ≈0.35 μm and vanishes in the limits of large as well as small penetration depths. (C) Polarity P of membrane-bound proteins as a function of the attachment rates, ωD+ and ωT+, with cooperative binding (recruitment) turned off. While for ωD+>ωT+ membrane-bound proteins form a bipolar pattern (P>1), they accumulate at midcell (P<1) for ωD+<ωT+. (D) Density profiles of membrane-bound proteins in the limit where the attachment rates of the two species are equal, ωD+=ωT+=1 μm/s, and recruitment is switched off. The membrane profile of the total protein density (green) is flat, whereas membrane-bound PNTP (blue) accumulates at midcell and PNDP (red) forms a bipolar pattern. (E) Polarity P of the membrane-bound proteins as a function of kdD for ωD+=ωT+. Increasing the recruitment rate restores polarity. (F) Density profiles of membrane-bound proteins for the same parameter configuration as in E with kdD=0.1 μm²/s. The density of PNDP (red) as well as the overall protein density (green) exhibit strongly bipolar patterns, which are much more pronounced than the corresponding patterns in the absence of cooperative membrane binding. The density of PNTP (blue) is comparatively flat, and there are much less membrane-bound proteins in this nucleotide state than in the PNDP state. The overall protein pattern is strongly dominated by PNDP.

Next, to analyze the additional nonlinear effects of membrane recruitment, we considered a situation, illustrated in Fig. 2D, where both nucleotide states have the same membrane affinity. As a result, the steady-state membrane density becomes uniform (see SI Appendix). Because cooperative membrane binding effectively increases the affinity of a protein species just like an increase in the respective attachment rate, we expected that membrane patterns could be restored by switching the recruitment processes back on. Indeed, we found a strong increase in polarity upon raising the recruitment rate kdD for fixed ktT=0 (Fig. 2E). Moreover, for large recruitment rates, not only does the relative level of the two species on the membrane change, but the pattern of PNDP becomes highly polar (Fig. 2F). The reason is the positive feedback facilitated by cooperative membrane binding: in membrane regions facing a cytosolic region with an enhanced PNDP concentration, binding leads to a locally increased concentration, which in turn increases the net attachment rate. Recruitment strongly amplifies the slight dominance of PNDP already existing at the cell poles in the absence of cooperative membrane binding, and thereby leads to the observed strongly bipolar PNDP membrane pattern.

In summary, the above analysis shows that the mechanism underlying the pattern-forming process is intrinsic to the protein dynamics: an inhomogeneous protein density in the cytosol together with unequal membrane affinities of the two forms leads to a spatially nonuniform accumulation of membrane-bound proteins. Nonlinear dynamics in the form of cooperative membrane binding (recruitment) serves to amplify these weakly nonuniform profiles into pronounced membrane patterns.

Cytosolic Reaction Volume Determines the Pattern.

After investigating the phenomenology of geometry-dependent pattern formation, we were left with the key question: what is the origin of the observed spatial segregation of PNTP and PNDP in the cytosol? Because these patterns form without cooperative membrane binding, the mechanism must be based on the combined effect of membrane attachment and detachment, diffusion, and nucleotide exchange. Moreover, as all chemical processes are spatially uniform, the key to understanding the impact of cell geometry must lie in the diffusive coupling of these biochemical processes.

Consider the situation where the attachment rates for PNDP and PNTP are equal, such that the total protein density on the membrane becomes spatially homogeneous (Fig. 2D). Only PNDP is released from the membrane. Hence, the latter acts as a source of cytosolic PNDP. Because, in addition, cytosolic PNDP is transformed into cytosolic PNTP by nucleotide exchange, we have all of the elements of a source degradation process. The ensuing density profile for PNDP in the cytosol is exponential with the decay length set by lλ=Dc/λ. Due to membrane curvature these reaction volumes overlap close to the cell poles (Fig. 2B, Bottom), which implies an accumulation of PNDP at the cell poles. The effect becomes stronger with increasing membrane curvature. Moreover, there is an optimal value for the penetration depth lλ, roughly equal to a third of the length l of the short cell axis, that maximizes accumulation of PNDP at the cell poles (Fig. 2B, Top). As lλ becomes larger than l, the effect weakens, because the reaction volumes from opposite membrane sites also overlap at midcell. In the limit where lλ is much smaller than the membrane curvature at the poles, the overlap vanishes, and with it the accumulation of PNDP at the poles.

Expressed differently, these heuristic arguments imply that the local ratio of the reaction volume for nucleotide exchange to the available membrane surface is the factor that explains the dependence of the protein distribution on cell geometry. To put this hypothesis to the test, we performed numerical simulations that are in the spirit of a minimal system approach taken by in vitro experiments (23, 25). In our numerical setup, we considered a cytosolic volume adjacent to a flat membrane, as illustrated in Fig. 3. We were interested in how alterations in the volume of cytosol available for protein diffusion and/or nucleotide exchange would affect the density profile on the membrane.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 3.

Two-dimensional planar geometry with cytosolic volume (blue) above a membrane at y=0. The left, right, and top boundaries of the cytosolic regime are reflecting boundaries. Diffusion and nucleotide exchange rates are set to their standard values, and the total number of proteins is set to N=50. Black boxes indicate areas that are not accessible to the proteins and thereby generate excluded reaction volumes; the boundaries of the boxes are assumed to be reflecting. Solid curves show the normalized density of PNDP bound to the membrane: u˜d=ud/udmax. Generally, PNDP accumulates at membrane regions in the vicinity of the cytosolic areas with excluded reaction volumes, with the effect being stronger with larger excluded reaction volumes and closer to the membrane (A, C, and D). In B, the white box indicates that, within its volume, all proteins are allowed to diffuse, but they do not undergo nucleotide exchange. This has a similar but weaker effect to that observed in the other panels: the proteins accumulate at the membrane near the excluded reaction volume. The parameters used in these numerical experiments are summarized in SI Appendix, Table 1.

In accordance with our hypothesis, we find that excluding volume for diffusion in the vicinity of a flat membrane reduces the available reaction volume locally and leads to accumulation of proteins at the membrane (Fig. 3 A, C, and D). The larger the excluded volume, the more proteins accumulated at the membrane. To focus on reaction volume explicitly, we considered a situation in which nucleotide exchange was disabled in a given region of the cytosolic area but proteins could still diffuse in and out of it. Again, we found protein accumulation at the nearby membrane but with reduced amplitude (Fig. 3B). Hence, these numerical studies strongly support our heuristic arguments and lead us to conclude that it is indeed exclusion of the reaction volume for nucleotide exchange that provides for the adaptation of the pattern to the geometry of the setup. Likewise, the membrane patterning in a cell could be effected by the nucleoid if the DNA material acts as a diffusion barrier, although at present this is debated (26). In the SI Appendix, we study how different sizes of effective excluded volume change the membrane pattern. Although bipolarity is still obtained for a broad parameter range, the complex geometry gives rise to a richer spectrum of possible patterns: for large sizes of excluded volume, accumulation at the poles occurs for R<0 (preferential recruitment for PNTP), whereas for R>0 the proteins accumulate at midcell. For intermediate sizes, there are parameter ranges where patterns with several maxima, not necessarily at the poles or midcell, are observed.

Pattern Formation Does Not Require a Dynamical Instability.

The above analysis shows that the difference in local reaction volume for cytosolic nucleotide exchange is the key element of the mechanism underlying geometry sensing. To put this result in perspective with pattern formation mechanisms based on dynamical instabilities, we consolidated the key properties of the spatially extended model in a spatially discretized version amenable to rigorous analytical treatment (Fig. 4A).

• Download figure
• Open in new tab
• Download powerpoint

Fig. 4.

Reduced network model and bifurcation analysis. (A) The full spatiotemporal dynamics in an ellipse is reduced to the nonlinear dynamics of a network of coupled nodes. We take the minimal possible number of nodes reflecting the asymmetry in the ratio of membrane area to bulk volume at the cell poles and midcell. Diffusion in the cytosol is modeled as particle exchange processes between the nodes. The network equations are derived from a discrete time jump process. Because the symmetry of the pattern reflects the symmetry of the ellipse, there is no flux of particles through either midplanes (red dashed lines). Therefore, the network can be further reduced to a single quadrant (black) with the other quadrants (gray) simply mirroring its behavior. (B) The reduced model comprises two membrane nodes at the pole and at midcell, two border nodes connecting membrane and cytosol, as well as three cytosolic nodes. Node 1a captures the increased ratio of membrane area to bulk volume at midcell. As illustrated by the colored nodes, this minimal network model polarizes: the density of PNDP at the pole node is higher than on the membrane node. (C) Density of PNDP at the pole node as a function of the recruitment rate k^dD. The density is given relative to the total number of particles located at the membrane node of the pole. For vanishing attachment rate of PNDP, ω^D+=0, there is a transcritical bifurcation at a critical rate k^dD∗≈8.7 s⁻¹ where a polarized state exchanges stability with an unpolarized state (black lines). In contrast, for the generic case of finite membrane attachment, ω^D+>0, there is only one positive fixed-point solution that is always stable (blue lines).

Diffusion in the cytosol and on the membrane is treated in terms of exchange processes between a network of nodes. A minimal set comprises four nodes on the membrane, two at the poles and two at midcell, and a distribution of nodes in the cytosol, which ensures that the ratio of membrane area to bulk volume at the cell poles is higher than at midcell. Because all observed stationary patterns are symmetrical with respect to both symmetry axes, we can further reduce the network to one quadrant of the ellipse (Fig. 4B). We are now left with a network of one membrane node at a pole and one at midcell, two nodes serving as the interface between membrane and cytosol, and three cytosolic nodes whose distribution reflects the asymmetry in the cytosolic reaction volume between the cell poles and midcell.

We have analyzed the ensuing mathematical model, a system of coupled ordinary differential equations, in the context of dynamic systems theory; for mathematical details and the model parameters used, please refer to the SI Appendix. Confirming our previous reasoning, we found that the reduced network model indeed leads to polarization between cell pole and midcell (Fig. 4B). Moreover, from a bifurcation analysis, we learn that generically the dynamic system does not exhibit a bifurcation: there is only one physically possible solution with positive protein density on the membrane, and this density increases with the recruitment rate (Fig. 4C). Only in the special (nongeneric) case where the attachment rate of PNDP vanishes, ω^D+=0, do we find a transcritical bifurcation. Then, there is a critical recruitment rate k^dD∗ below which the membrane is depleted of PNDP. In other words, generically the system shows an imperfect transcritical bifurcation, which implies robustness of the mechanism linking protein distribution to cell geometry.