New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology
Geometry-induced protein pattern formation
Edited by Herbert Levine, Rice University, Houston, TX, and approved December 9, 2015 (received for review August 4, 2015)

Significance
Biological cells need the ability to guide intracellular processes to specific spatial locations. This requires biochemical processes to sense and adapt to the geometry of the organism. Previously suggested mechanisms either assume proteins that are able to directly sense membrane curvature or are based on nonlinear diffusion–reaction systems that can generate geometry-adapted patterns. The latter, however, requires fine-tuning of the reaction rates. Here, we show that geometry adaption already follows from generic chemical dynamics. We present a simple reaction module based on generic reactions that establishes geometry-dependent patterns robustly without the need to tune kinetic rates nor any explicit curvature-sensing mechanism.
Abstract
Protein patterns are known to adapt to cell shape and serve as spatial templates that choreograph downstream processes like cell polarity or cell division. However, how can pattern-forming proteins sense and respond to the geometry of a cell, and what mechanistic principles underlie pattern formation? Current models invoke mechanisms based on dynamic instabilities arising from nonlinear interactions between proteins but neglect the influence of the spatial geometry itself. Here, we show that patterns can emerge as a direct result of adaptation to cell geometry, in the absence of dynamical instability. We present a generic reaction module that allows protein densities robustly to adapt to the symmetry of the spatial geometry. The key component is an NTPase protein that cycles between nucleotide-dependent membrane-bound and cytosolic states. For elongated cells, we find that the protein dynamics generically leads to a bipolar pattern, which vanishes as the geometry becomes spherically symmetrical. We show that such a reaction module facilitates universal adaptation to cell geometry by sensing the local ratio of membrane area to cytosolic volume. This sensing mechanism is controlled by the membrane affinities of the different states. We apply the theory to explain AtMinD bipolar patterns in
Protein patterns serve to initiate and guide important cellular processes. A classic example is the early patterning of the Drosophila embryo along its anterior–posterior axis (1). Here, maternal morphogen gradients initiate a complex patterning process that subsequently directs cell differentiation. However, protein patterns play a regulatory role even at the single-cell level. For example, they determine cell polarity and the position of the division plane. In the yeast Saccharomyces cerevisiae, the GTPase Cdc42 regulates cell polarization, which in turn determines the position of a new growth zone or bud site. This pattern-forming process is driven by the interaction between a set of different proteins that cycle between the plasma membrane and the cytoplasm (2, 3). In the rod-shaped bacterium Escherichia coli, Min proteins accumulate at the ends of the cell to inhibit the binding of the division proteins (4, 5). Here, the main player in the pattern-forming process is the ATPase MinD. It attaches to the membrane in its ATP-bound state and recruits MinE and further MinD-ATP from the cytosol (6). Cycling of proteins between membrane and cytosol is mediated by the action of MinE, which stimulates the intrinsic ATPase activity of MinD and thereby initiates its detachment. The ensuing oscillatory pattern directs the division machinery to midcell, enabling proper cell division in two viable daughter cells.
In all of these processes, regulatory proteins establish chemical gradients or patterns that reflect aspects of cell shape. However, how is gradient or pattern formation achieved in the absence of an external template? Many possible mechanisms have been proposed and they are by no means fully classified yet (7, 8). Establishing a pattern involves definition of preferred accumulation points and requires that the symmetry of the homogeneous state is broken. In Bacillus subtilis, there is good evidence suggesting that DivIVA recognizes negative membrane curvature directly by a mechanism that is intrinsic to this cell division protein (7, 9). In contrast, enrichment of MinD at the cell poles in E. coli is an emergent property of the collective dynamics of several proteins. As shown in refs. 10⇓⇓⇓⇓⇓–16, the nonlinear dynamics of the Min system leads to a polar pattern, which oscillates along the long axis and is clearly constrained by cell geometry. A clear disadvantage of such self-organized symmetry breaking through a dynamical instability is that the kinetic parameters must be fine-tuned to allow the establishment of a stable polar pattern.
Here, we show that cell geometry itself can enforce a broken symmetry under generic conditions without any need for fine-tuning. We introduce a class of geometry-sensing protein systems whose only stable state is a spatial pattern that is maintained by energy consumption through an ATPase or GTPase (NTPase). The proposed mechanism is based on a generic property of diffusion: The probability that a protein diffusing through the cytosol will strike (and attach) to the membrane scales with the area of membrane accessible to it. Thus, close to the poles of a rod-shaped cell, most of the trajectories available lead to the membrane. Close to midcell, where the membrane is almost flat, about one-half of the possible paths lead away from the membrane. However, on its own, this mechanism only produces transient patterns on the membrane, as the system approaches a stable, uniform equilibrium in finite time (17). Moreover, patterns only emerge from specific initial conditions. In this paper, we ask, how can this generic property of diffusion be complemented by a minimal set of biomolecular processes to robustly maintain patterns? We show that the NTPase activity of a single protein that cycles between membrane and cytosol is sufficient to achieve this goal. Our analysis shows that an inhomogeneous density profile is established on the membrane in the generic case where the affinities of NTP- and NDP-bound forms differ. Moreover, these membrane-bound patterns are amplified if the proteins are able to bind cooperatively to the membrane (e.g., due to dimerization). This mechanism is highly robust because the stable, uniform equilibrium is simply replaced by a unique, stable patterned state. In particular, the mechanism involves no dynamical instability and requires no fine-tuning of parameters.
Experimental support for the proposed mechanism comes from E. coli mutants in which both EcMinD and EcMinE were replaced by chloroplastic AtMinD (MinD homolog from Arabidopsis thaliana) (18). With this single ATPase (19) the system establishes a bipolar pattern along the long axis, rescuing the
A Generic Reaction Module for Sensing of Cell Geometry
We consider a reaction module comprised of a single type of NTPase that cycles between an NDP-bound (
Minimal reaction module for geometry-induced cell polarity. (A) Illustration of the reaction module: cytosolic
This reaction module serves as a model for the bipolar pattern of AtMinD in E. coli cells (18): AtMinD is an ATPase (19) that has been reported to dimerize (19, 21). This process thus provides evidence for cooperative membrane binding. Unlike EcMinD (20), AtMinD dimerizes even when its Walker-A binding module is inactivated (19), locking the protein in its ADP-bound state. This strongly suggests that also the ADP-bound form of AtMinD exhibits cooperative membrane binding, as we have assumed in the above reaction scheme by introducing a recruitment rate
If not mentioned otherwise, we use the following model parameters, which are set to experimental values acquired for E. coli, if available. The diffusion constants in the cytosol and on the membrane are set to
Results
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 (
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),
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,
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
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
Consider the situation where the attachment rates for
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.
Two-dimensional planar geometry with cytosolic volume (blue) above a membrane at
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
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).
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
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
Discussion
How does protein patterning adapt to cell geometry? Dynamic models for pattern formation often reduce the cytosolic volume to the same dimension as the membrane and focus on the role of nonlinear protein interactions (see, e.g., refs. 11 and 27). At first sight, this appears to make sense, because diffusion coefficients are generically much higher in the cytosol than on the membrane. Indeed, if only attachment and detachment processes are involved, any transient geometry-dependent pattern is rapidly washed out (17).
Here, we have shown that the assumption of a well-mixed cytosolic protein reservoir becomes invalid as soon as cytosolic processes like nucleotide exchange, which alter protein states, become involved. We have introduced a minimal reaction module with a single NTPase that cycles between membrane and cytosol. The fact that cytosolic nucleotide exchange may take place on a diffusive length scale far below cell size has been noted previously (13), and it has been shown that this can be critical for robust, intracellular pattern formation (15). Our analysis reveals that nucleotide exchange leads to an inhomogeneous protein distribution in the cytosol, which is stably maintained and depends strongly on the geometry of the cytosolic space. As a consequence, proteins accumulate on certain membrane regions, depending on the local ratio of membrane area to cytosolic volume. In an elongated cell, this serves as a robust mechanism for proper cell division by facilitating protein accumulation at the poles. The proposed reaction module operates through implicit curvature sensing and does not require that the relevant protein themselves respond to membrane curvature (7, 9) or lipids (28). The degree and the axis of polarization depend on the level of cooperativity in membrane binding, which can be regulated by enzymes.
Our theoretical analysis suggests that evolutionary tuning of this simple reaction module is feasible: because there is no threshold involved, polarity can be improved continuously starting from any parameter configuration. This lack of a threshold can at the same time also be a disadvantage: without a trigger, pattern formation is difficult to induce as response to an upstream event. Another distinctive element of the mechanism is the lack of a characteristic length scale (e.g., as striped Min patterns in E. coli); instead, the pattern scales with the size of the cell. Depending on the functional role, this might be desired or disadvantageous.
The reaction module gives a possible explanation for the bipolar patterns of AtMinD observed in mutant E. coli cells (18). Several experimental tests could be performed to validate the proposed reaction module: one route would be to study spherical E. coli cells. For this geometry, we predict that the polarization of AtMinD should vanish, because the membrane curvature is uniform. This, however, would also be the case if the kinetics of AtMinD binding is directly dependent on membrane curvature, as in the case of DivIVa (7, 9). To rule out this scenario, an in vitro experiment could be conducted, as described in Fig. 3.
In vitro experiments might also serve as a proof of concept for the use of the suggested reaction module in nanoscale self-organization. By enzymatically regulating the kinetic rates of the process, one could induce protein patterns on a membrane, which then serve as templates for the localization of nanoscale structures, e.g., similar to the formation of actin cables close to Cdc42 protein caps in yeast. Localization could either be self-organized or target specific curvatures or be externally controlled by volume exclusion in the cytosolic space. If, in addition, such nanostructures exert forces on the membrane, this self-organization principle could be used to regulate the shape of membranes. Thus, the proposed minimal module might serve as a core network for the design of other geometry-sensing protein networks.
On a more speculative note, geometry-sensing protein networks like the one discussed here would enable a cell to gradually optimize its biological function, because the underlying mechanism does not involve a bifurcation threshold. For example, one could envision a biochemical network containing a protein that is able to trigger hydrolysis-driven detachment. Such a catalytic process could act selectively on the
Finally, due to its generic nature, the proposed mechanism might be involved in many bacterial pattern-forming systems. For instance, the sensitivity to cytosolic reaction volume provides a way to sense large cytosolic structures. This could, for instance, be part of the mechanism that guides PomZ to midcell in Myxococcus xanthus (29). One could also imagine direct feedback mechanisms between force-exerting proteins that regulate cell shape (e.g., FtsZ ring contraction) and proteins that adapt to local cell shape by sensing the local reaction volume, and which guide the downstream accumulation of further force-exerting proteins. In this scenario, cell shape could be controlled (even in a self-organized fashion) by balancing these two processes.
Materials and Methods
The model is mathematically described as a set of reaction–diffusion equations (SI Appendix). All simulations were performed with finite-element methods on a triangular mesh using Comsol Multiphysics 4.3. As initial condition, all proteins were in the NDP state and located in the bulk of the ellipse. In Fig. 3, the particles are initially located on the membrane in the NDP state. For Figs. 1 and 2, the simulation time was 1,000 s, and for Fig. 3, it was 2,000 s. A steady state is reached after ∼100 s.
Acknowledgments
This research was supported by the German Excellence Initiative via the program “NanoSystems Initiative Munich,” and the Deutsche Forschungsgemeinschaft via Project B02 within the SFB 1032 “Nanoagents for Spatio-Temporal Control of Molecular and Cellular Reactions.”
Footnotes
- ↵1To whom correspondence should be addressed. Email: frey{at}lmu.de.
Author contributions: D.T., J.H., and E.F. designed research, performed research, and wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1515191113/-/DCSupplemental.
Freely available online through the PNAS open access option.
References
- ↵
- ↵.
- Wedlich-Söldner R,
- Altschuler S,
- Wu L,
- Li R
- ↵
- ↵
- ↵.
- Raskin DM,
- de Boer PA
- ↵.
- Hu Z,
- Gogol EP,
- Lutkenhaus J
- ↵
- ↵
- ↵.
- Lenarcic R, et al.
- ↵.
- Meinhardt H,
- de Boer PA
- ↵
- ↵
- ↵.
- Huang KC,
- Meir Y,
- Wingreen NS
- ↵.
- Varma A,
- Huang KC,
- Young KD
- ↵
- ↵
- ↵
- ↵
- ↵.
- Aldridge C,
- Møller SG
- ↵.
- Zhou H, et al.
- ↵.
- Fujiwara MT, et al.
- ↵
- ↵
- ↵.
- Shih YL,
- Fu X,
- King GF,
- Le T,
- Rothfield L
- ↵
- ↵.
- Sanamrad A, et al.
- ↵
- ↵
- ↵.
- Treuner-Lange A,
- Søgaard-Andersen L
Citation Manager Formats
Sign up for Article Alerts
Jump to section
You May Also be Interested in
More Articles of This Classification
Physical Sciences
Related Content
- No related articles found.