Fisher-Kolmogorov-Petrovsky-Piskunov dynamics mediated by a parent field with a delay

We examine a modification of the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) process in which the diffusing substance requires a parent density field for reproduction. A biological example would be the density of diffusing spores (propagules) and the density of a stationary fungus (parent). The parent produces propagules at a certain rate, and the propagules turn into the parent substance at another rate. We model this evolution by the FKPP process with delay, which reflects a finite time typically required for a new parent to mature before it begins to produce propagules. Although the FKPP process with other types of delays have been considered in the past as a pure mathematical construct, in our paper a delay in the FKPP model arises in a natural science setting. The speed of the resulting density fronts is shown to decrease with increasing delay time and has a nontrivial dependence on the rate of conversion of propagules into the parent substance. Remarkably, the fronts in this model are always slower than Fisher waves of the classical FKPP model. The largest speed is half the classical value, and it is achieved at zero delay and when the two rates are matched.

The Curvature Effect on the Diffusion of Single Brownian Squares on a Cylindrical Surface in the Presence of Depletion Attractions

The diffusion of single micron-sized Brownian square platelets on cylindrical surfaces with different radii of curvature in the presence of depletion attractions was studied experimentally by video microscopy. The translational motion of a square is found to be diffusive along the axial direction of the cylinder but sub-diffusive along the circumferential direction due to the confinement induced by gravity, while its rotational motion displays a sub-diffusive behavior due to the confinement induced by orientation-dependent depletion attractions. Such a confinement effect decreases as the radius of curvature increases and can be tuned both through surface curvatures and/or depletion attractions. Our work provides a new way to control the translational and rotational dynamics of anisotropic particles through curved surfaces in the presence of depletion attractions.

Curvature-assisted self-assembly of Brownian squares on cylindrical surfaces

HYPOTHESIS

We hypothesize that curved surfaces, including cylindrical surfaces, which go beyond prior experiments using flat surfaces, can significantly influence and alter the phase behavior and self-assembly of dense two-dimensional systems of Brownian colloids.

EXPERIMENTS

Here, we report a first experimental study regarding the self-assembly of Brownian square platelets with an edge length L = 2.3 μm on cylindrical surfaces having different curvatures; these platelets are subjected to a depletion attraction in order to form a monolayer above the cylindrical surface, yet have nearly hard interactions within the monolayer. Simulations are also performed to confirm and explain the experimental observations.

FINDINGS

Phase diagrams as a function of curvature are determined experimentally. Interestingly, hexagonal rotator crystal structures are observed for tubes having radii > 10.9L, but a tetratic phase is seen instead for the 10.9L tube at the corresponding platelet area fractions. We show that this transition is caused by the curvature-induced orientation-dependence of the depletion attraction between the squares and the underlying cylindrical surface. Brownian dynamics simulation results confirm the experimental observations and also illustrate helical structures formed by squares packing on cylinders. Our results demonstrate a way towards control over the self-assembly of anisotropic particles through curvature and depletion-attraction-induced orientational confinement.

Chirality provides a direct fitness advantage and facilitates intermixing in cellular aggregates

Chirality in shape and motility can evolve rapidly in microbes and cancer cells. To determine how chirality affects cell fitness, we developed a model of chiral growth in compact aggregates such as microbial colonies and solid tumors. Our model recapitulates previous experimental findings and shows that mutant cells can invade by increasing their chirality or switching their handedness. The invasion results either in a takeover or stable coexistence between the mutant and the ancestor depending on their relative chirality. For large chiralities, the coexistence is accompanied by strong intermixing between the cells, while spatial segregation occurs otherwise. We show that the competition within the aggregate is mediated by bulges in regions where the cells with different chiralities meet. The two-way coupling between aggregate shape and natural selection is described by the chiral Kardar-Parisi-Zhang equation coupled to the Burgers’ equation with multiplicative noise. We solve for the key features of this theory to explain the origin of selection on chirality. Overall, our work suggests that chirality could be an important ecological trait that mediates competition, invasion, and spatial structure in cellular populations.

Is it better to be left- or right-handed? The answer depends on whether the goal is making a handshake or winning a boxing match. The need for coordination favors the handedness of the majority, but being different could also provide an advantage. The same rules could apply to microbial colonies and cancer tumors. Like humans, cells often have handedness (chirality) that reflects the lack of mirror symmetry in their shapes or movement patterns. We find that cells gain a substantial fitness advantage by either increasing the magnitude of their chirality or switching to the opposite handedness. Selection for specific chirality can overcome differences in growth rate and is mediated by the formation of bulges along the colony edge in regions where cells with different chiralities meet.

Fragility of reaction-diffusion models with respect to competing advective processes

We study the coupling of a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation to a separate, advection-only transport process. We find that an infinitesimal coupling can cause a finite change in the speed and shape of the reaction front, indicating the fragility of the FKPP model with respect to such a perturbation. The front dynamics can be mapped to an effective FKPP equation only at sufficiently fast diffusion or large coupling strength. We also discover conditions when the front width diverges and when its speed is insensitive to the coupling. At zero diffusion in our mean-field description, the downwind front speed goes to a finite value as the coupling goes to zero.

Globular Protein Folding In Vitro and In Vivo

In vitro, computational, and theoretical studies of protein folding have converged to paint a rich and complex energy landscape. This landscape is sensitively modulated by environmental conditions and subject to evolutionary pressure on protein function. Of these environments, none is more complex than the cell itself, where proteins function in the cytosol, in membranes, and in different compartments. A wide variety of kinetic and thermodynamics experiments, ranging from single-molecule studies to jump kinetics and from nuclear magnetic resonance to imaging on the microscope, have elucidated how protein energy landscapes facilitate folding and how they are subject to evolutionary constraints and environmental perturbation. Here we review some recent developments in the field and refer the reader to some original work and additional reviews that cover this broad topic in protein science.

Plastic deformation of tubular crystals by dislocation glide

Tubular crystals, two-dimensional lattices wrapped into cylindrical topologies, arise in many contexts, including botany and biofilaments, and in physical systems such as carbon nanotubes. The geometrical principles of botanical phyllotaxis, describing the spiral packings on cylinders commonly found in nature, have found application in all these systems. Several recent studies have examined defects in tubular crystals associated with crystalline packings that must accommodate a fixed tube radius. Here we study the mechanics of tubular crystals with variable tube radius, with dislocations interposed between regions of different phyllotactic packings. Unbinding and separation of dislocation pairs with equal and opposite Burgers vectors allow the growth of one phyllotactic domain at the expense of another. In particular, glide separation of dislocations offers a low-energy mode for plastic deformations of solid tubes in response to external stresses, reconfiguring the lattice step by step. Through theory and simulation, we examine how the tube's radius and helicity affects, and is in turn altered by, the mechanics of dislocation glide. We also discuss how a sufficiently strong bending rigidity can alter or arrest the deformations of tubes with small radii.

In Vivo study of naturally deformed Escherichia coli bacteria

A combination of light-microscopy and image processing has been applied to study naturally deformed Escherichia coli under in vivo condition and at the order of sub-pixel high-resolution accuracy. To classify deflagellated non-dividing E. coli cells to the rod-shape and bent-shape, a geometrical approach has been applied. From the analysis of the geometrical data which were obtained of image processing, we estimated the required effective energy for shaping a rod-shape to a bent-shape with the same size. We evaluated the energy of deformation in the naturally deformed bacteria with minimum cell manipulation, under in vivo condition, and with minimum influence of any external force, torque and pressure. Finally, we have also elaborated on the possible scenario to explain how naturally deformed bacteria are formed from initial to final-stage.

Clustering of vertically constrained passive particles in homogeneous isotropic turbulence

We analyze the dynamics of small particles vertically confined, by means of a linear restoring force, to move within a horizontal fluid slab in a three-dimensional (3D) homogeneous isotropic turbulent velocity field. The model that we introduce and study is possibly the simplest description for the dynamics of small aquatic organisms that, due to swimming, active regulation of their buoyancy, or any other mechanism, maintain themselves in a shallow horizontal layer below the free surface of oceans or lakes. By varying the strength of the restoring force, we are able to control the thickness of the fluid slab in which the particles can move. This allows us to analyze the statistical features of the system over a wide range of conditions going from a fully 3D incompressible flow (corresponding to the case of no confinement) to the extremely confined case corresponding to a two-dimensional slice. The background 3D turbulent velocity field is evolved by means of fully resolved direct numerical simulations. Whenever some level of vertical confinement is present, the particle trajectories deviate from that of fluid tracers and the particles experience an effectively compressible velocity field. Here, we have quantified the compressibility, the preferential concentration of the particles, and the correlation dimension by changing the strength of the restoring force. The main result is that there exists a particular value of the force constant, corresponding to a mean slab depth approximately equal to a few times the Kolmogorov length scale η, that maximizes the clustering of the particles.

Getting into shape: How do rod-like bacteria control their geometry?

Rod-like bacteria maintain their cylindrical shapes with remarkable precision during growth. However, they are also capable to adapt their shapes to external forces and constraints, for example by growing into narrow or curved confinements. Despite being one of the simplest morphologies, we are still far from a full understanding of how shape is robustly regulated, and how bacteria obtain their near-perfect cylindrical shapes with excellent precision. However, recent experimental and theoretical findings suggest that cell-wall geometry and mechanical stress play important roles in regulating cell shape in rod-like bacteria. We review our current understanding of the cell wall architecture and the growth dynamics, and discuss possible candidates for regulatory cues of shape regulation in the absence or presence of external constraints. Finally, we suggest further future experimental and theoretical directions which may help to shed light on this fundamental problem.

Bending forces plastically deform growing bacterial cell walls

Cell walls define a cell's shape in bacteria. The walls are rigid to resist large internal pressures, but remarkably plastic to adapt to a wide range of external forces and geometric constraints. Currently, it is unknown how bacteria maintain their shape. In this paper, we develop experimental and theoretical approaches and show that mechanical stresses regulate bacterial cell wall growth. By applying a precisely controllable hydrodynamic force to growing rod-shaped Escherichia coli and Bacillus subtilis cells, we demonstrate that the cells can exhibit two fundamentally different modes of deformation. The cells behave like elastic rods when subjected to transient forces, but deform plastically when significant cell wall synthesis occurs while the force is applied. The deformed cells always recover their shape. The experimental results are in quantitative agreement with the predictions of the theory of dislocation-mediated growth. In particular, we find that a single dimensionless parameter, which depends on a combination of independently measured physical properties of the cell, can describe the cell's responses under various experimental conditions. These findings provide insight into how living cells robustly maintain their shape under varying physical environments.

Theory of interacting dislocations on cylinders

We study the mechanics and statistical physics of dislocations interacting on cylinders, motivated by the elongation of rod-shaped bacterial cell walls and cylindrical assemblies of colloidal particles subject to external stresses. The interaction energy and forces between dislocations are solved analytically, and analyzed asymptotically. The results of continuum elastic theory agree well with numerical simulations on finite lattices even for relatively small systems. Isolated dislocations on a cylinder act like grain boundaries. With colloidal crystals in mind, we show that saddle points are created by a Peach-Koehler force on the dislocations in the circumferential direction, causing dislocation pairs to unbind. The thermal nucleation rate of dislocation unbinding is calculated, for an arbitrary mobility tensor and external stress, including the case of a twist-induced Peach-Koehler force along the cylinder axis. Surprisingly rich phenomena arise for dislocations on cylinders, despite their vanishing Gaussian curvature.

Dislocation-mediated growth of bacterial cell walls

Recent experiments have illuminated a remarkable growth mechanism of rod-shaped bacteria: proteins associated with cell wall extension move at constant velocity in circles oriented approximately along the cell circumference [Garner EC, et al., (2011) Science 333:222-225], [Domínguez-Escobar J, et al. (2011) Science 333:225-228], [van Teeffelen S, et al. (2011) PNAS 108:15822-15827]. We view these as dislocations in the partially ordered peptidoglycan structure, activated by glycan strand extension machinery, and study theoretically the dynamics of these interacting defects on the surface of a cylinder. Generation and motion of these interacting defects lead to surprising effects arising from the cylindrical geometry, with important implications for growth. We also discuss how long range elastic interactions and turgor pressure affect the dynamics of the fraction of actively moving dislocations in the bacterial cell wall.