Session W02: Patterns I: Pattern Formation in Biological Systems

What is the correct physical picture of reflectin protein self-assembly?

Squids and octopuses occupy every optical niche of the ocean, from mud flats to the midwater abyss.  In every one of these optical scenarios, each species' highly sophisticated camouflage is driven by a layer of living cells in the skin containing ordered, sub-wavelength arrays of a high-refractive-index protein called reflectin. In some shallow contexts, this layer matches the albedo of a given species' background, while in open ocean contexts, bioluminescent light when scattered through this layer matches the radiance of the surrounding water.  In all cases, these feats of photonic engineering are achieved via self-assembly of "reflectin" protein. Although almost two decades of research have elapsed since reflectins' discovery, we still do not have a clear enough physical picture of reflectin protein-protein interactions to know why some mixtures of these proteins make mirrors in vivo, while other mixtures of similar proteins make light guides.  This talk explores and weighs the evidence for several novel hypotheses of reflectin assembly mechanisms, including the free energy of the proteins' association with lipid bilayers and the bilayers' corresponding physical phases; the evidence for true patchy-colloid physics in reflectins; the possibility of a surface-induced phase transition; and the possible role of flexible metal coordination.  None of these possibilities are mutually exclusive.  While we do not yet have a complete picture of this system, it is clear that squid are leveraging assembly constructs at the cutting edge of current physics knowledge that are worth serious consideration.   

Sensing the shape of a cell: reaction-diffusion and energy minimization

How can a cell sense its own shape? How can cells pattern their proteins over a rough complex surface? Some dividing cells sense their shape, becoming polarized along their long axis. We study polarity arising from Rho GTPase proteins cycling between active membrane-bound forms and inactive cytosolic forms – a wave-pinning reaction-diffusion process. We show that wave pinning senses the cell's long axis. Simulating wave-pinning on a curved surface, high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, simply minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. However, on rough surfaces, shape sensing is disrupted, and high-activity domains localize to locations other than the global peaks and valleys of the surface. Simultaneously, the perimeter minimization rule fails. We study how domains evolve on rough surfaces, falling into local minima in a corrugated energy landscape. The effectiveness of the shape sensing can be controlled by altering the protein's diffusion coefficient. Our results help understand the factors that allow cells to sense their own shape – and the limits that membrane roughness can place on this process.   

Robust pattern formation in reaction-diffusion systems in the presence of heterogeneity

The emergence of spatial patterns in biological systems plays a key role in many important functions. Reaction-diffusion systems have been used to explain the emergence of biological patterns such as stripes, hexagons, fronts, and spirals through interacting chemical species. These systems must be robust to microscopic spatial fluctuations in reactant properties in order to form stable patterns. Our aim is to build such robustness by modifying local interaction rules such that small changes in reactant properties do not affect the scale of the formed pattern. Further, we aim for larger changes in reactant properties to result in discrete jumps in the pattern scale, corresponding to the formation of a discrete new module of the underlying system. The Gierer-Meinhardt (GM) model is a canonical example of a two-species reaction-diffusion system that produces periodic spatial patterns. We augment the GM model with additional species not central to pattern formation but which stabilize the formed patterns and provide the desired robustness properties. To demonstrate these results in our augmented model, we show that adding a continuous gradient in the intrinsic activator and inhibitor length scales leads to discrete steps in the emergent scale of the formed pattern.   

Ballistic deposition with memory:a new universality class of surface growth with a new dynamical scaling

Motivated by recent experimental studies in microbiology, we suggest a modification of the classic ballistic deposition model of surface growth, where memory of a deposition at a site induces more depositions at that site or its neighbors. By studying the statistics of surfaces in this model, we obtain three independent critical exponents: the growth exponent $\beta =5/4$, the roughening exponent $\alpha = 2$, and the new (size) exponent $\gamma = 3/4$. The model requires a modification to the Family-Vicsek scaling, resulting in the dynamical exponent $z = \frac{\alpha-\gamma}{\beta} = 1$. This modified scaling collapses the surface width vs time curves for various lattice sizes. This is a previously unobserved universality class of surface growth where the KPZ universality is an unstable fixed point of the dynamics.     

Mechanochemical coupling enables propagating patterns without diffusion

Morphogenesis and morphological events during embryo development involve coordinated cell and tissue shape changes. They are driven by mechanical forces, which in turn are triggered by complex chemical signaling. The chemical signals form gradients that provide positional information in the tissue, and in principle are also dependent on mechanical state of cells. Using mathematical modeling, we demonstrate how chemical signaling coupled with long-range buckling deformations can induce dynamic spatial patterns in an elastic shell even without a diffusive or advective transport of the chemicals. This mechanism is a complementary proposal to the Turing patterns, but it does not require interactions between multiple chemical species with very different diffusivity. We demonstrate that depending on key parameters, for example the threshold in activation of cell's mechanical response or the shell thickness, the mechanochemical feedback gives rise to qualitatively different deformation patterns such as ridges and spots that are robust to changes in shell geometry. Our findings open new routes to designing an excitable medium with tunable pattern formation that has potential applications in tissue engineering and fabrication of functional surfaces.   

Towards a Quantitative Understanding of Spontaneous Mitotic Waves

Various models have been developed to capture the mechanism of the mitotic clock. Taken together, they describe how the cyclinB-Cdk1 complex directs the cell through a series of steps which define one mitotic cycle. This regulation facilitates the longevity of multicellular organisms by enforcing a regular, clock-like timing of mitotic events. When a collection of these oscillators couple, they synchronize. In particular, in various systems—e.g. Drosophila and Xenopus—early embryogenesis is marked by a series of synchronous cell divisions across the embryo, exceding the scope of a diffusive signal. Instead, we explore a known spatial coordination mechanism: waves. Using Xenopus extracts and a Cdk1-FRET sensor, we exploit a novel approach utilizing metaphase-arrested extracts to produce one-dimensional directed mitotic waves. We probe the possible differences between waves in systems with or without reconstituted nuclei. We find the speed of the former to be significantly slower, suggesting an active role for nuclei in wave propogation. Moreover, we quantify the wave speed  over time, showing a clear relationship between speed and the time Cdk1 spends in the active state. In total, we display a unique method for directly visualzing and characterizing biochemical mitotic waves.   

Honeycomb formation under geometric frustration

The wax–made comb of the honeybee is a masterpiece of animal architecture. As honeybees build their nests in pre-existing tree cavities, they grew accustomed to dealing with the presence of geometric constrains, resulting in non-regular hexagons and topological defects. In this work, we study how bees collectively adapt their environment to regulate and heal the honeycomb structure. Specifically, we identify the irregularities in honeycomb structure in the presence of various types of geometric frustrations. We 3D-print our experimental frames with imprinted foundations. The resulting constructed comb show clear evidence of reoccurring, self-organized patterns built by the bees in response to specific geometric frustrations in the starter frames. Our computational model can recreate and predict these self-organized patterns which can effectively solve various geometrical miss-match problems while optimizing the cost for comb building. Understanding how bees adapt the topology and geometry of the lattice to conform to different constrains will lead to a set of rules to systematically generate biologically inspired designs in the fields of swarm robotics, collective construction, and lightweight cellular structures.   

Frustrated frustules: geometric frustration in diatom frustules

Diatoms are single-celled organisms with a remarkable cell wall made of silica called the frustule. The frustule is porous, with multi-scale nano- and micro-pores. While diatoms have fascinated scientists for centuries, it is still unknown how the cell wall structures are created. In our work we investigate the intricate arrangement of the microstructure pattern of the diatom C.granii and related species. Here, we used confocal fluorescence microscopy to obtain high-resolution 3-D images of fluorescently-labeled C.granii diatom, and precisely located the positions of their structural features. We find that the micro-pores are arranged in a radially aligned hexagonal lattice. Tiling a circular structure with a radially aligned hexagonal lattice requires insertion of lattice defects. We find that the micro-pore structure has geometrically necessary defects, reminiscent of a circular crochet pattern, where the number of defects increases linexarly with size.   

Biological pattern formation in spatio-temporally fluctuating environments

The study of biological pattern-forming systems has revealed generic features that promote robustness with respect to variations between cells or populations in morphogen and protein production rates. By contrast, less is understood about how such systems respond to spatio-temporal fluctuations in morphogen concentrations within individual cells or populations over timescales faster than or similar to growth. Such fluctuations can be caused by growth, motility or changes in the external environment. Here, we formulate a general theory of biological pattern formation in spatio-temporally fluctuating environments. Using our framework, we show how pattern-forming systems can be reduced to a low-dimensional representation based on their time-dependent response to fluctuations, complementing recent work on low-dimensional representations in fixed conditions.   

Directing protein-based patterns with advective bulk flow

We theoretically predict and experimentally show that the propagation direction of Min-protein patterns in vitro can be controlled by a hydrodynamic flow of the bulk solution, and that the response to flow depends on the concentration ratio of MinE to MinD. For low E:D ratios, the membrane-bound wave patterns propagate downstream relative to the bulk flow while they propagate upstream for large E:D ratios. For intermediate E:D ratios, we find a multistability of both propagation directions relative to the flow. Theoretical analysis of the mathematical model reveals the mechanism underlying upstream propagation of Min patterns and links it to the fast conformational switching of MinE in the bulk. Accordingly, a MinE mutant without that switch exhibits only downstream propagation in experiments. Our work demonstrates how hydrodynamic flow can be used to control protein-based pattern formation and to gain insight into the underlying pattern forming mechanisms. From a broader perspective, the bulk flow is a perturbation that breaks the mirror symmetry of the system and therefore is a nonequilibrium analog of an external magnetic field applied to a ferromagnet.   

Organizing principles shared between intracellular protein patterns and intracellular condensates

Intracellular protein pattern formation organizes many cell functions like cell division, cell polarity and bud-site selection. Key questions concern the nature of patterns formed, how they are oriented within the cell, and how the number and size of pattern domains are controlled. The main organizing principles are captured by (nearly) mass-conserving two-component reaction–diffusion systems. Although these reaction–diffusion dynamics operate far from equilibrium, it has much in common with the phenomenology of phase separation. In particular, their long-term dynamics can be understood as a coarsening process and its arrest. Since coarsening is a characteristic behavior of phase-separating systems, the questions of size control and number selection arise analogously for intracellular condensates formed by liquid-liquid phase separation. We show that similar concepts underlie both cellular self-organization processes and that these parallels allow us to use the same basic approaches to understand liquid-liquid phase separation and protein pattern formation.   

Reconstructing spatiotemporal protein patterns in heterogeneous systems

Many cellular processes, such as cell division and cell motility, are spatially and temporally organized by protein patterns. Such self-organized patterns can be mathematically described by reaction-diffusion equations, which has greatly advanced our understanding of how spontaneous spatial patterns emerge from homogeneity. Biological systems, however, are intrinsically heterogeneous, such that patterns may involve multiple spatial and temporal scales, thus complicating their theoretical analysis. We will discuss multiscale patterns of the Min protein system, a paradigmatic model for pattern formation, in a spatially heterogeneous setup. Building up on a recently developed theoretical framework for mass-conserving reaction-diffusion systems [1], we show that the intricate dynamics is well described by the spatiotemporal evolution of the total densities, which we identify as the relevant degrees of freedom at large length and time scales. Moreover, we show that the spatiotemporal pattern-forming dynamics at small scales can be reconstructed and even predicted from the spatial distribution of the total densities in the system.

[1] Phase-space geometry of mass-conserving reaction-diffusion dynamics, F. Brauns, J. Halatek, and E. Frey., Phys. Rev. X 10, 041036 (2020)