Publications

This work addresses the question of uniqueness of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand -- whose precise form derives directly from the theory of perfect plasticity -- behaves quadratically close to the origin and grows linearly once a specific threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field -- the Cauchy stress in the terminology of perfect plasticity -- which allows us to define characteristic lines, and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study \cite{BF}, we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data.

The decision to stop growing and mature into an adult is a critical point in development that determines adult body size, impacting multiple aspects of an adult’s biology. In many animals, growth cessation is a consequence of hormone release that appears to be tied to the attainment of a particular body size or condition. Nevertheless, the size-sensing mechanism animals use to initiate hormone synthesis is poorly understood. Here, we develop a simple mathematical model of growth cessation in Drosophila melanogaster, which is ostensibly triggered by the attainment of a critical weight (CW) early in the last instar. Attainment of CW is correlated with the synthesis of the steroid hormone ecdysone, which causes a larva to stop growing, pupate, and metamorphose into the adult form. Our model suggests that, contrary to expectation, the size-sensing mechanism that initiates metamorphosis occurs before the larva reaches CW; that is, the critical-weight phenomenon is a downstream consequence of an earlier size-dependent developmental decision, not a decision point itself. Further, this size-sensing mechanism does not require a direct assessment of body size but emerges from the interactions between body size, ecdysone, and nutritional signaling. Because many aspects of our model are evolutionarily conserved among all animals, the model may provide a general framework for understanding how animals commit to maturing from their juvenile to adult form.

During vertebrate organ morphogenesis, large collectives of cells robustly self-organize to form architectural units (bones, villi, follicles) whose form persists into adulthood. Over the past few decades, mechanisms of organ morphogenesis have been developed predominantly through molecular, genetic, and cellular frameworks. More recently, there has been a resurgence of interest in collective cell and tissue mechanics during organ formation. This approach has amplified the need to clarify and unambiguously link events across biological length scales. Doing so may require reassessing canonical models that continue to guide the field. The most recognized model for organ formation centers around morphogens as determinants of gene expression and morphological patterns. The classical view of a morphogen is that morphogen gradients specify differential gene expression in a distinct spatial order. Because morphogen expression colocalizes with emerging feather and hair follicles, the skin has served as a paradigmatic example of such morphogen prepatterning mechanisms.

Molecular electronics break-junction experiments are widely used to investigate fundamental physics and chemistry at the nanoscale. Reproducibility in these experiments relies on measuring conductance on thousands of freshly formed molecular junctions, yielding a broad histogram of conductance events. Experiments typically focus on the most probable conductance, while the information content of the conductance histogram has remained unclear. Here we develop a microscopic theory for the conductance histogram by merging the theory of force-spectroscopy with molecular conductance. The procedure yields analytical equations that accurately fit the conductance histogram of a wide range of molecular junctions and augments the information content that can be extracted from them. Our formulation captures contributions to the conductance dispersion due to conductance changes during the mechanical elongation inherent to the experiments. In turn, the histogram shape is determined by the non-equilibrium stochastic features of junction rupture and formation. The microscopic parameters in the theory capture the junction’s electromechanical properties and can be isolated from separate conductance and rupture force (or junction-lifetime) measurements. The predicted behavior can be used to test the range of validity of the theory, understand the conductance histograms, design molecular junction experiments with enhanced resolution and molecular devices with more reproducible conductance properties.

We study bacterial diffusion in disordered porous media. Interactions with obstacles, at unknown locations, make this problem challenging. We approach it by abstracting the environment to cell states with memoryless transitions. With this, we derive an effective diffusivity that agrees well with simulations in explicit geometries. The diffusivity is non-monotonic, and we solve the optimal run length. We also find a rescaling that causes all of the theory and simulations to collapse. Our results indicate that a small set of microscopic features captures bacterial diffusion in disordered media.

This paper is associated with a poster winner of a 2022 American Physical Society's Division of Fluid Dynamics (DFD) Milton van Dyke Award for work presented at the DFD Gallery of Fluid Motion. The original poster is available online at the Gallery of Fluid Motion.

The propulsion of mammalian spermatozoa relies on the spontaneous periodic oscillation of their flagella. These oscillations are driven internally by the coordinated action of ATP-powered dynein motors that exert sliding forces between microtubule doublets, resulting in bending waves that propagate along the flagellum and enable locomotion. We present an integrated chemomechanical model of a freely swimming spermatozoon that uses a sliding-control model of the axoneme capturing the two-way feedback between motor kinetics and elastic deformations while accounting for detailed fluid mechanics around the moving cell. We develop a robust computational framework that solves a boundary integral equation for the passive sperm head alongside the slender-body equation for the deforming flagellum described as a geometrically nonlinear internally actuated Euler-Bernoulli beam, and captures full hydrodynamic interactions. Nonlinear simulations are shown to produce spontaneous oscillations with realistic beating patterns and trajectories, which we analyze as a function of sperm number and motor activity. Our results indicate that the swimming velocity does not vary monotonically with dynein activity, but instead displays two maxima corresponding to distinct modes of swimming, each characterized by qualitatively different wave forms and trajectories. Our model also provides an estimate for the efficiency of swimming, which peaks at low sperm number.

A growing list of diverse biological systems and their equally diverse functionalities provides realizations of a paradigm of emergent behavior. In each of these biological systems, pervasive ensembles of weak, short-lived, spatially local interactions act autonomously to convey functionalities at larger spatial and temporal scales. In this article, a range of diverse systems and functionalities are presented in a cursory manner with literature citations for further details. Then two systems and their properties are discussed in more detail: yeast chromosome biology and human respiratory mucus

Understanding the transport properties of microorganisms and self-propelled particles in porous media has important implications for human health as well as microbial ecology. In free space, most microswimmers perform diffusive random walks as a result of the interplay of self-propulsion and orientation decorrelation mechanisms such as run-and-tumble dynamics or rotational diffusion. In an unstructured porous medium, collisions with the microstructure result in a decrease in the effective spatial diffusivity of the particles from its free-space value. Here, we analyze this problem for a simple model system consisting of noninteracting point particles performing run-and-tumble dynamics through a two-dimensional disordered medium composed of a random distribution of circular obstacles, in the absence of Brownian diffusion or hydrodynamic interactions. The particles are assumed to collide with the obstacles as hard spheres and subsequently slide on the obstacle surface with no frictional resistance while maintaining their orientation, until they either escape or tumble. We show that the variations in the long-time diffusivity can be described by a universal dimensionless hindrance function f (φ,Pe) of the obstacle area fraction φ and Péclet number Pe, or ratio of the swimmer run length to the obstacle size. We analytically derive an asymptotic expression for the hindrance function valid for dilute media (Peφ 1), and its extension to denser media is obtained using stochastic simulations. As we explain, the model is also easily generalized to describe dispersion in three dimensions.

When a founder cell and its progeny divide with incomplete cytokinesis, a network forms in which each intercellular bridge corresponds to a past mitotic event. Such networks are required for gamete production in many animals, and different species have evolved diverse final network topologies. Although mechanisms regulating network assembly have been identified in particular organisms, we lack a quantitative framework to understand network assembly and inter-species variability. Motivated by cell networks responsible for oocyte production in invertebrates, where the final topology is typically invariant within each species, we devised a mathematical model for generating cell networks, in which each node is an oscillator and, after a full cycle, the node produces a daughter to which it remains connected. These cell cycle oscillations are transient and coupled via diffusion over the edges of the network. By variation of three biologically motivated parameters, our model generates nearly all such networks currently reported across invertebrates. Furthermore, small parameter variations can rationalize cases of intra-species variation. Because cell networks outside of the ovary often form less deterministically, we propose model generalizations to account for sources of stochasticity.