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# First-order patterning transitions on a sphere as a route to cell morphology

Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved March 15, 2016 (received for review January 8, 2016)

## Significance

Pollen grains, insect eggshells, and mite carapaces of different species exhibit an amazing variety of surface patterning, despite having similar developmental characteristics and material properties. This pattern formation is robust enough to warrant its use in taxonomic classification. Focusing on pollen, we propose a theory of transitions to spatially modulated phases on spheres to explain both the variability and robustness of the patterns. We find that the sphere geometry allows for a wider variety of patterns compared with planar surfaces. A species may robustly “choose” among the possibilities by locally nucleating a patch of the pattern. We expect our theory to describe a wide variety of pattern-forming processes on spherical geometries.

## Abstract

We propose a general theory for surface patterning in many different biological systems, including mite and insect cuticles, pollen grains, fungal spores, and insect eggs. The patterns of interest are often intricate and diverse, yet an individual pattern is robustly reproducible by a single species and a similar set of developmental stages produces a variety of patterns. We argue that the pattern diversity and reproducibility may be explained by interpreting the pattern development as a first-order phase transition to a spatially modulated phase. Brazovskii showed that for such transitions on a flat, infinite sheet, the patterns are uniform striped or hexagonal. Biological objects, however, have finite extent and offer different topologies, such as the spherical surfaces of pollen grains. We consider Brazovskii transitions on spheres and show that the patterns have a richer phenomenology than simple stripes or hexagons. We calculate the free energy difference between the unpatterned state and the many possible patterned phases, taking into account fluctuations and the system’s finite size. The proliferation of variety on a sphere may be understood as a consequence of topology, which forces defects into perfectly ordered phases. The defects are then accommodated in different ways. We also argue that the first-order character of the transition is responsible for the reproducibility and robustness of the pattern formation.

Surface patterning in many animal and plant species, including insect eggshells, pollen grains, fungal spores, and mite carapaces, is extremely diverse. Stripes, spikes, pores, ridges, and other decorations (1, 2), illustrated for pollen in Fig. 1*A*, all present very different geometries. Paradoxically, the distinct morphologies may develop via the same sequence of developmental stages (3⇓⇓–6), although the patterns are distinctive enough to be used for taxonomic classification over eons. In this paper, we propose a general model of the formation of these patterns and speculate that the origin of some of these counterintuitive features relies upon fluctuation effects leading to global pattern nucleation.

We focus on a class of biological surface patterns observed in many disparate taxa (fungi, arachnids, insects, angiosperms) consisting of spikes, hexagons, and stripes of cross-linked polysaccharide material tiled on a spherical cell. The surface pattern formation of these biological systems typically involves many cell components, including the cytoskeleton, plasma membrane, and cell wall (callose wall in pollen, cuticle in arthropod cuticles and fungal spores) (7⇓–9). Without some physical coupling, coordination among these many parts would require complex biological signaling across large regions of the organism. Hence, the patterns seem more plausibly to develop via a simple physical process. We are already familiar with complex, self-organized patterning via relatively simple processes in the natural world: convection cells at a Rayleigh–Bénard instability (10), the patterning of pigments in animals (11), and hexagonal patterning of dried mud or the basalt columns of the Giant’s Causeway (12).

Whereas patterning on flat, planar substrates is expected to yield striped or hexagonal patterns (13), we demonstrate that the analogous transition on a sphere has a much richer phenomenology. The spherical geometry introduces topological defects, yielding a varied set of pattern possibilities. Also, because the transition we describe here has a first-order character, it is possible to produce a particular pattern by templating a small patch, which would then induce pattern growth over the entire surface via nucleation dynamics (10). The patterning inside the nucleation region itself could be controlled by local surface chemistry of the plasma membrane, allowing for pattern reproducibility within a species.

Although our theory may be applicable in any of the biological cases stated above, for simplicity we consider the biochemical details of pollen below, as shown in Fig. 1*A*, and refer to the general case of such decorated cells as pollen. One of the earliest indications of patterning in pollen begins with plasma membrane undulations (8), as shown in Fig. 1*B*. This distortion of the local membrane curvature is also implicated in other iterations of this pattern-forming process, such as insect and arachnid cuticle development (9, 14). Here, we present a model for pattern formation via a phase transition at the plasma membrane. We show that the characteristic size of the membrane undulations, *λ*, is a function of physical parameters of the membrane. Hence, the membrane tension and elasticity, lipid and protein density, or osmolarity of the surrounding fluid could all vary among species and contribute to diversity in the final, observed cuticle and cell wall patterns.

Mechanical buckling is another microscopic mechanism that may plausibly cause surface patterns in the biological systems. However, we believe our model of pattern formation may be especially applicable to systems like pollen, because the transition to patterning may occur locally, without the homogeneous long-range forces in existing models of elastic buckling (15). Another characteristic suggestive of a phase transition is that all these systems have a cross-linked polymeric layer secreted on the surface of the cell membrane.

We derive from the microscopic model a more general, coarse-grained description, which turns out to be the spherical analog of the Brazovskii model (13). Such models describe a wide variety of systems (16), including block copolymer assembly (17), crystallizing Bose–Einstein condensates in optical cavities (18), and cholesteric liquid crystals (19). Such systems on a sphere might also be excellent experimental test beds for our theory. Although there have been recent numerical investigations of such models on a sphere via numerical methods (20), our analysis goes beyond this theory by incorporating fluctuations and providing a broader understanding of such transitions through analytical methods. The fluctuations lead to first-order behavior, suggesting a nucleation and growth scenario (10, 21).

## A Microscopic Model

As a microscopic model, consider a concentration field **1** is the appropriate spherical measure *R* is the radius of the sphere. We expand **1** favors modes with

The membrane itself fluctuates away from its spherical shape, so that the radius varies with *θ* and *ϕ*, *u* may also then be expanded in spherical harmonics with modes **2**. Although there are many possible models for spherical lipid membranes, outlined in ref. 23, for example, the specific form does not matter for our purposes, because the result will be general. All models will typically have a bending term with a bending rigidity *κ* and a surface tension *σ*. Generically, the field *u* by introducing a spontaneous curvature: It is reasonable that the inhomogeneity introduced by a local excess of *μ* will depend on the microscopic details of how the spontaneous curvature is induced by the inhomogeneity.

Our total, microscopic free energy is **1**. Note that for

Crucially, *λ* of the pattern, because *λ* using typical parameters for lipid membranes gives the right order of magnitude for pollen pattern features (

The preceding discussion shows that the effective free energy for the field modes *K* and *τ* are new coupling constants that depend on the microscopic parameters in Eqs. **1** and **3** (22). The interaction terms **1**. The key feature of the effective free energy in Eq. **5** is the gradient term (the term depending explicitly on

Before continuing, we note that the precise microscopic model for pollen is not known, and there are many possibilities (7, 25). However, our final result in Eq. **5** is not contingent on the particular details of our phase separation model, and we expect that the coarse-grained features of many microscopic models will obey Eq. **5**, but with different dependencies of the coupling constants *K*, *τ*, and *A*. We now use the final result in Eq. **5** to demonstrate that robustness and variability are general features of this mechanism of pattern formation. In the following, we set

As in the flat case (13), fluctuations will induce phase transitions to ordered states. In preparation, we expect ordered states of the form*m* that indicate the direction of the ordered state in the *m*s. An ordered state consisting of a single spherical harmonic mode contribution (*m*) is the analog of the striped phase *R* will introduce a new length scale into the problem and finite size effects at small *R*. In the following we construct finite-size crossover scaling functions that capture both the large and small *R* behavior at a fixed pattern wavelength.

## Fluctuation-Induced First-Order Transition

Consider the transition to an ordered state in our general free energy in Eq. **5**. The interaction terms **1**), we expect a second-order transition. This *τ* changes signs. When *τ*. In this situation, the patterned and unpatterned state minima never coexist and the pattern would have to develop homogeneously over the entire sphere surface, with no nucleation process. However, we shall see that fluctuations modify this picture and instead induce a first-order transition.

To facilitate computations, it is convenient to define a “bare” propagator or two-point correlation function*SI Text*. We always work in the limit that the coupling coefficients

The HF approximation of the renormalized propagator is written as a self-consistency condition on *τ* in the disordered state:**11** is the discrete analog of an integration of the propagator over all modes (i.e., a one-loop correction). The prime on the sum indicates a regularization procedure where the divergence associated with large *SI Text*). Note that the function in Eq. **11** captures both a large radius regime,

Eq. **11** admits only positive solutions for *τ*. Hence, fluctuations prevent the temperature-like term from changing sign. If a cubic term were present, then a first-order transition is possible if

Turning to the four-point vertex function **7** where the denominator is smallest near

The three *m*-dependent Gaunt coefficient terms in Eq. **12** are three different angular momentum “channels” that contribute to the vertex. A single momentum channel contributes whenever **12** vanish. Then, the renormalized vertex **9** [because **12** that if two or more channels contribute and if **6** and are the spherical analogs of the cosine standing waves of the flat-space Brazovskii analysis.

We now examine the most divergent piece of the fluctuation correction **13**. Setting **6** is expected to change sign due to fluctuations, consistent with the Brazovskii result.

We have now shown that our model generically exhibits a first-order transition to a patterned phase. In the absence of a cubic term in the terms

## SI Text

*SI Text* provides the details of the derivation of the loop corrections, the equation of state, and the change in the thermodynamic potential, *Useful Identities and Relations*, *The Disordered State and Loop Corrections*, and *The Ordered State and**.* In the latter two sections, Feynman diagram techniques are used to facilitate the computations. Our goal is to derive all of the equations in the main text in more detail.

### Useful Identities and Relations.

Recall that the Gaunt coefficients in the main text were defined as

The Gaunt coefficients that appear in Eq. **S1** satisfy triangle relations given by**S1** has a special form and implies the following selection rule:

Like the

To expand the cubic and quartic terms in our Hamiltonian **1** in the main text), it is necessary to compute the integral of a product of three and four spherical harmonics *θ* (colatitude) and *ϕ* (longitude). To make our notation more compact, we introduce a vector of indexes **S12** is arbitrary. Hence, we may rearrange the **S14** any way we like. This will be an important symmetry of these Gaunt coefficients that we will use when calculating the loop corrections in the next section.

### The Disordered State and Loop Corrections.

We now calculate the two-point correlation function or propagator **5** in the main text. Expanding the quartic term calculated as shown in Eq. **S14**, we find

We now define the Feynman rules to construct our diagrams. The first major component comes from the quadratic piece of the Hamiltonian, from which we derive the free propagator, denoted by a line:[S18]To simplify formulas that appear throughout the rest of this text, we make the definition

Let’s begin with corrections to the inverse propagator. Using the geometric series for the propagator (30), it is possible to write the fully renormalized inverse propagator diagramatically as[S21]where the fully renormalized propagator is denoted by a double line, and the second term on the right-hand side is the sum of all of the two-point amputated one-particle irreducible (1PI) graphs. These are the graphs that cannot be cut into two subgraphs by removing a single propagator link. There are many of these graphs that one would have to calculate. However, we simplify the calculation by looking at just the one-loop correction. If we include the cubic term, there are two different kinds of loop corrections:[S22]In Brazovskii’s analysis (13), he argues that the first loop correction may be neglected relative to the second in Eq. **S22** because the loop integration in the first diagram contributes only over a narrow set of directions. This is more difficult to see in our spherical harmonic expansion, but we may neglect this diagram in our analysis, as well. We return to this point later (Eq. **S37**).

We can actually include an even larger set of diagrams if we replace the propagator in the loop with the renormalized propagator *g* to yield a self-consistent equation,[S23]where we have neglected the first loop diagram in Eq. **S22** that we expect to be small. The renormalized propagator *g* in this approximation has a new temperature-like parameter *τ*), where the subscript reminds us that we are in the disordered state. Hence, when calculating the loop in Eq. **S23**, we have to replace the *τ* in the original propagator with **S7** to sum on *m*, we find[S25]where we used the special value of the Gaunt coefficient in Eq. **S6** and identified

Now we must grapple with the sum **S25**. There is a logarithmic divergence that occurs for large **S25** is convergent and, with some assistance from a computer algebra system (Mathematica v10.1; Wolfram Research), we compute**S27** is large and we may make use of an asymptotic series for **S28**: **S29** into Eq. **S25** and evaluating the latter equation at **11**) via Eq. **S23**. Alternatively, Eq. **S23** may be written as an equation for the fluctuation-renormalized propagator **S29**.

The vertex function **S8** and identified our loop summation

The most divergent contribution to the sums over **S33** comes from **S33** contains no divergences, so we set **S33**. The sum **S34** does not need regularization and reads

Finally, let us return briefly to our neglected loop correction to the propagator. Now that we have calculated **S36**. Therefore, at our momenta of interest **S25** because **S25** increases linearly with

### The Ordered State and Δ Φ .

Recall that in the ordered state, we have to expand around a new potential minimum, so that our Hamiltonian has a different form and a different set of Feynman rules. First, instead of the fields *ψ* away from the ordered state **S15**. However, there are new cross terms coming from powers of the expanded modes *ψ* describe fluctuations away from the potential minimum. So, we have*ψ*, as these do not contribute to any correlation functions of the *ψ* fields. These new terms introduce three new kinds of vertices, with three or two legs that we may contract. We denote these vertices as[S39][S40][S41]where the circles on the legs indicate the insertion of an ordered field mode *ψ*, these two new vertices must be included in the Feynman rules already defined in the previous section.

The vertex in Eq. **S39** is the next-lowest order contribution to the three-point function **S40** contributes a new term to the propagator equation. Before calculating any loop corrections, let us study the scaling properties of these two new vertices for small **S60** below). Hence, the circles in the new vertices will bring in scaling factors of *h*,*ψ*, because *τ* in the ordered state, **S30**. In this diagonal approximation, Eq. **S42** reduces to

Our equation of state, Eq. **S43**, depends only on the two-point function **S23**) with nontrivial *m* dependence. First, there are two new diagrams without any loops,[S45]and[S46]where the external legs have indexes **15**). As usual, this contribution will be important for the special modes with **S46** is the most important difference between the propagators in the ordered and disordered states. The contribution from Eq. **S46** scales like

The cubic term, Eq. **S45**, also contributes. However, note that by the property of the Gaunt coefficients, it contributes only for a single, special nonzero ordered state mode **S46** will have contributions from all ordered state modes. So, we neglect this cubic term contribution for now and then check that this is a reasonable approximation within our isotropic approximation (Eq. **S63**). The same argument applies for the last loop correction in Eq. **S44**, which is also generated by the cubic term. We expect it to be negligible relative to the other loop contributions. For now, we focus on the contribution in Eq. **S46**.

For ordered states with a single mode, **S46** vanishes except when

The first loop correction in Eq. **S44** is the same Hartree–Fock contribution we found for the disordered state in Eq. **S25**. So, there is nothing new here except for a replacement of **S32**):[S50]

We now explicitly show that this loop correction is negligible compared with Eq. **S49** when **S50**. As discussed in the main text and above, we neglect the off-diagonal contributions to the two-point function, so we may set the external leg indexes **S32** and **S34**,[S51]where we have indicated the appropriate *m* indexes on the external legs of the diagram. This contribution includes a special function *m* dependence to the diagram:**S51**, it is easy to see that this new loop correction scales in the same way as the loop correction in Eq. **S49**. So, we will have to analyze the function **S51** contributes significantly only for special values of *m*s that contributes to the ordered state

We can check explicitly that Eq. **S51** does not contribute significantly. The ratio of the two loop contributions for an arbitrary state **S60**) is given by[S53]where we have assumed *A* for a single nonzero ordered state mode **S50** are aligned with the reciprocal lattice vectors of the patterned phase. To check that *B* the ratio of scattering functions **S51** relative to Eq. **S49** even when

Thus, we have (partially) justified our neglect of the loop correction in Eq. **S50** when computing the propagator in the ordered state. This is also consistent with the Brazovskii analysis. So, going back to our equation for the propagator, we find[S54]where we have used Eq. **S49** for the loop correction. It is clear that Eq. **S54** reduces to Eq. **15** in the main text. Finally, we may evaluate the inverse propagator at *τ*, denoted by **S47**. So, Eq. **S54** reduces to**S43**:

Now everything is in place to solve for the field modes **S57** into Eq. **S43** and find an equation for **S59** we assumed the ordered state modes

It is clear from Eq. **S59** that **S59**). Dividing Eq. **S59** by **S60** depend on the function **S56**. This could be done numerically, but we are interested in an analytically tractable approximation. Hence, to make progress, we look for an isotropic approximation to Eq. **S56** and replace *m*-independent approximation to the coefficient **S56**. The simplest solution is to average **S48**) contributes the most, as can be verified numerically. Then, because **S56** is a reasonable approximation. After regularizing the propagator sum as in the disordered state calculation (Eq. **S29**), we find an *m*-independent solution for **S45**) vanishes in this approximation because it contributes the following to the renormalized parameter **S44** vanishes in this isotropic approximation.

We now calculate the change in potential energy per unit area *h*, so that the ordered state modes *a* increase from 0 to *t* changes from **18** in the main text and Eqs. **S59** and **S60**, we find*a* to *t* in the leftover integral and found the quartic term contribution

We now need the Jacobian factor *a* and *t* are connected via Eq. **S62**, generalized to the varying ordered state modes **17** in the main text. We now differentiate both sides of this equation with respect to *t* and rearrange the terms to find our Jacobian **S64** produces the final result,

In the planar limit, the free energy change in Eq. **S69** does not reduce to the Brazovskii result in an obvious way because it depends on the directions *τ*. Eq. **S69** may now be used in conjunction with the solutions for the ordered states

## Patterned States

We now consider an ordered state *h* to *h* to generate the free energy, *h* satisfies

Through a change of variables in the functional integral for the partition function, we expand *ψ* around *h* to *h* (13). Because the unstable modes have **6**: *h* to tilt the potential so that, for *h* large enough, the ordered state becomes the minimum, and then return *h* to 0. During this process, the amplitude *a* changes from

A field *h* in the direction of the state *SI Text*, Eq. **S59**. To simplify calculations and facilitate analytic solutions, we consider the states that satisfy this condition by pairwise cancellation of two of the modes, e.g., via

Calculating the fluctuation-corrected free energy requires the fluctuation-corrected two-point function *τ* is sufficiently small (13), which we assume in the following.

Substituting Eq. **15** into Eq. **14** and making an isotropic approximation to the propagator *τ* dependence in Eq. **14**, leaving the following equation of state,*t* that satisfies the equation**17** reduces to Eq. **11**. In the ordered state, we find a different temperature-like parameter

Now we compute the change in free energy **11**. We parameterize *a* that will increase from *h* is turned off, **16** with *m*. A convenient choice for the final amplitude is **16**. In the absence of a cubic term (**8**) may break the degeneracy, but many patterns are likely nearly degenerate on a sphere. When

We may construct ordered states with arbitrary numbers of nonzero *τ* correspond to stable patterns. Integrating up the free energy changes, we find**14** into Eq. **18** yields a complex expression for *SI Text*, Eq. **S69**)—finding the values of

Roughly speaking, when

Because multiple modes contribute to the ordered state for large *R* and fixed pattern wavelength *h* via a biochemically controlled process to force the pattern into a particular metastable ordered state. The pollen may then “quench” this pattern, forcing it to spread over the surface via a nucleation process.

In conclusion, we have developed a phenomenological theory of pattern formation on a sphere. This theory provides a plausible explanation of the physical origins of micron-scale surface textures found on cell walls and cuticles of distantly related taxa such as plants, mites, fungi, and insects. We showed how this mechanism may originate in plasma membrane undulations coupled to the phase separation of polysaccharide materials, which later coordinate the deposition of a tough exterior wall. Our theory predicts that the pollen grain surface is quenched below a first-order transition point during development, and we have argued that a patterned phase can spread after the quench via a nucleation process. A given species may specify one of these many patterned modes via a nucleation site defined by one or more of several possible cell-biological mechanisms. For example, a localized site could be designated by the local surface chemistry of the plasma membrane relative to one pole of the cell or by crowding at the cell surface of nascent pollen caused by ordered packing in the developing anther. We showed that the first-order character of the transition will be maintained even when the free energy has no cubic term. We also argued that the theory without a cubic term may be particularly relevant because the plasma membrane composition in vivo may be tuned to a critical point (29).

Whereas the first-order character of this transition may explain the reproducibility of a pattern in one species, the theory may also provide an answer to why there is so much pattern variability among different species. First, a wide variety of patterns is possible by modifying the nucleation pattern, which, once formed, allows the rest of the pattern to propagate rapidly and robustly across the surface. Second, pattern formation on a sphere is intrinsically varied because, in contrast to the planar case, the ordered states on the sphere must accommodate defects, providing a larger space of possible patterns. By contrast, butterfly wing scale development may be an example of patterning on a flat substrate via this mechanism; the distal surface of the wing scale forms exclusively striped patterns and the plasma membrane has also been implicated in the initial pattern templating (14).

There is much room for future work: A detailed phase diagram might be constructed using numerical techniques described in ref. 20 and incorporating fluctuation corrections. It would also be helpful to study the dynamics to understand how a nucleation region might be specified, leading to a particular global pattern. There has already been progress on this in the planar case (10), providing a starting point for the spherical case.

## Acknowledgments

We thank S. A. Brazovskii, S. Gopalakrishnan, and D. Audus for encouraging comments and valuable discussions. This work was supported, in part, by the National Science Foundation through Grant DMR-1262047 (to R.D.K.), a Packard Foundation Fellowship (to A.M.S.), and a Kaufman Foundation New Research Initiative award. R.D.K. was partially supported by a Simons Investigator grant from the Simons Foundation.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: kamien{at}physics.upenn.edu.

Author contributions: M.O.L., E.M.H., A.R., A.M.S., and R.D.K. designed research; M.O.L., E.M.H., A.R., and R.D.K. performed research; M.O.L. and E.M.H. contributed new reagents/analytic tools; A.R. analyzed data; and M.O.L., E.M.H., A.R., A.M.S., and R.D.K. 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.1600296113/-/DCSupplemental.

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