Optimal vein density in artificial and real leaves

The long evolution of vascular plants has resulted in a tremendous
variety of natural networks responsible for the evaporatively
driven transport of water. Nevertheless, little is known about the
physical principles that constrain vascular architecture. Inspired by
plant leaves, we used microfluidic devices consisting of simple
parallel channel networks in a polymeric material layer, permeable
to water, to study the mechanisms of and the limits to evaporationdriven flow. We show that the flow rate through our biomimetic
leaves increases linearly with channel density (1/d) until the distance between channels (d) is comparable with the thickness of the
polymer layer (), above which the flow rate saturates. A comparison with the plant vascular networks shows that the same optimization criterion can be used to describe the placement of veins
in leaves. These scaling relations for evaporatively driven flow
through simple networks reveal basic design principles for the
engineering of evaporation–permeation-driven devices, and highlight the role of physical constraints on the biological design of

Flip-flop-induced relaxation of bending energy: implications for membrane remodeling

Cellular and organellar membranes are dynamic materials that underlie many aspects of cell biology. Biological
membranes have long been thought of as elastic materials with respect to bending deformations. A wealth of theory and experimentation on pure phospholipid membranes provides abundant support for this idea. However, biological membranes are not
composed solely of phospholipids—they also incorporate a variety of amphiphilic molecules that undergo rapid transbilayer
flip-flop. Here we describe several experimental systems that demonstrate deformation-induced molecular flip-flop. First we
use a fluorescence assay to track osmotically controlled membrane deformation in single component fatty acid vesicles, and
show that the relaxation of the induced bending stress is mediated by fatty acid flip-flop. We then look at two-component phospholipid/cholesterol composite vesicles. We use NMR to show that the steady-state rate of interleaflet diffusion of cholesterol is
fast relative to biological membrane remodeling. We then use a Fo¨rster resonance energy transfer assay to detect the transbilayer movement of cholesterol upon deformation. We suggest that our results can be interpreted by modifying the area difference
elasticity model to account for the time-dependent relaxation of bending energy. Our findings suggest that rapid interleaflet diffusion of cholesterol may play a role in membrane remodeling in vivo. We suggest that the molecular characteristics of sterols make
them evolutionarily preferred mediators of stress relaxation, and that the universal presence of sterols in the membranes of
eukaryotes, even at low concentrations, reflects the importance of membrane remodeling in eukaryotic cells.

How kelp produce blade shapes suited to different flow regimes: A new wrinkle

Synopsis Many species of macroalgae have flat, strap-like blades in habitats exposed to rapidly flowing water, but have
wide, ruffled ‘‘undulate’’ blades at protected sites. We used the giant bull kelp, Nereocystis luetkeana, to investigate how
these ecomorphological differences are produced. The undulate blades of N. luetkeana from sites with low flow remain
spread out and flutter erratically in moving water, thereby not only enhancing interception of light, but also increasing
drag. In contrast, strap-like blades of kelp from habitats with rapid flow collapse into streamlined bundles and flutter at
low amplitude in flowing water, thus reducing both drag and interception of light. Transplant experiments in the field
revealed that shape of the blade in N. luetkeana is a plastic trait. Laboratory experiments in which growing blades from
different sites were subjected to tensile forces that mimicked the hydrodynamic drag experienced by blades in different
flow regimes showed that change in shape is induced by mechanical stress. During growth experiments in the field and
laboratory, we mapped the spatial distribution of growth in both undulate and strap-like blades to determine how these
different morphologies were produced. The highest growth rates occur near the proximal ends of N. luetkeana blades of
both morphologies, but the rates of transverse growth of narrow, strap-like blades are lower than those of wide, undulate
blades. If rates of longitudinal growth at the edges of a blade exceed the rate of longitudinal growth along the midline of
the blade, ruffles along the edges of the blade are produced by elastic buckling. In contrast, flat blades are produced when
rates of longitudinal growth are similar across the width of a blade. Because ruffles are the result of elastic buckling,
a compliant undulate N. luetkeana blade can easily be pushed into different configurations (e.g., the wavelengths of the
ruffles along the edges of the blade can change, and the whole blade can twist into left- and right-handed helicoidal
shapes), which may enhance movements of the blade in flowing water that reduce self-shading and increase mass
exchange along blade surfaces

Botanical ratchets

Ratcheting surfaces are a common motif in nature and appear in plant awns and grasses. They are known
to proffer selective advantages for seed dispersion and burial. In two simple model experiments, we show
that these anisotropically toothed surfaces naturally serve as motion rectifiers and generically move in a
unidirectional manner, when subjected to temporally and spatially symmetric excitations of various
origins. Using a combination of theory and experiment, we show that a linear relationship between awn
length and ratchet efficiency holds under biologically relevant conditions. Grass awns can thus efficiently
transform non-equilibrium environmental stresses from such sources as humidity variations into useful
work and directed motion using their length as a fluctuation amplifier, yielding a selective advantage to
these organelles in many plant species.

Self-organization of a mesoscale bristle into ordered hierarchical helical assemblies

Mesoscale hierarchical helical structures with diverse functions are abundant in nature. Here
we show how spontaneous helicity can be induced in a synthetic polymeric nanobristle
assembling in an evaporating liquid. We use a simple theoretical model to characterize the
geometry, stiffness, and surface properties of the pillars that favor the adhesive self-organization
of bundles with pillars wound around each other. The process can be controlled to yield highly
ordered helical clusters with a unique structural hierarchy that arises from the sequential assembly
of self-similar coiled building blocks over multiple length scales. We demonstrate their function
in the context of self-assembly into previously unseen structures with uniform, periodic patterns
and controlled handedness and as an efficient particle-trapping and adhesive system.

Unfolding the sulcus

Sulci are localized furrows on the surface of soft materials that form by a compression-induced
instability. We unfold this instability by breaking its natural scale and translation invariance, and compute
a limiting bifurcation diagram for sulcfication showing that it is a scale-free, subcritical nonlinear
instability. In contrast with classical nucleation, sulcification is continuous, occurs in purely elastic
continua and is structurally stable in the limit of vanishing surface energy. During loading, a sulcus
nucleates at a point with an upper critical strain and an essential singularity in the linearized spectrum. On
unloading, it quasistatically shrinks to a point with a lower critical strain, explained by breaking of scale
symmetry. At intermediate strains the system is linearly stable but nonlinearly unstable with no energy
barrier. Simple experiments confirm the existence of these two critical strains.

On the growth and form of the gut

The developing vertebrate gut tube forms a reproducible looped pattern as it grows into the body cavity. Here we use
developmental experiments to eliminate alternative models and show that gut looping morphogenesis is driven by the
homogeneous and isotropic forces that arise from the relative growth between the gut tube and the anchoring dorsal
mesenteric sheet, tissues that grow at different rates. A simple physical mimic, using a differentially strained composite
of a pliable rubber tube and a soft latex sheet is consistent with this mechanism and produces similar patterns. We devise
a mathematical theory and a computational model for the number, size and shape of intestinal loops based solely on the
measurable geometry, elasticity and relative growth of the tissues. The predictions of our theory are quantitatively
consistent with observations of intestinal loops at different stages of development in the chick embryo. Our model also
accounts for the qualitative and quantitative variation in the distinct gut looping patterns seen in a variety of species
including quail, finch and mouse, illuminating how the simple macroscopic mechanics of differential growth drives the
morphology of the developing gut.

Geometric mechanics of curved crease origami

Folding a sheet of paper along a curve can lead to structures seen in decorative art and utilitarian
packing boxes. Here we present a theory for the simplest such structure: an annular circular strip that is
folded along a central circular curve to form a three-dimensional buckled structure driven by geometrical
frustration. We quantify this shape in terms of the radius of the circle, the dihedral angle of the fold, and
the mechanical properties of the sheet of paper and the fold itself. When the sheet is isometrically
deformed everywhere except along the fold itself, stiff folds result in creases with constant curvature and
oscillatory torsion. However, relatively softer folds inherit the broken symmetry of the buckled shape with
oscillatory curvature and torsion. Our asymptotic analysis of the isometrically deformed state is
corroborated by numerical simulations that allow us to generalize our analysis to study structures with
multiple curved creases.

Elastohydrodynamics of wet bristles, carpets and brushes

Surfaces covered by bristles, hairs, polymers and other filamentous structures arise in
a variety of natural settings in science such as the active lining of many biological
organs, e.g. lungs, reproductive tracts, etc., and have increasingly begun to be used in
technological applications. We derive an effective field theory for the elastohydrodynamics
of ordered brushes and disordered carpets that are made of a large number of elastic
filaments grafted on to a substrate and interspersed in a fluid. Our formulation for the
elastohydrodynamic response of these materials leads naturally to a set of constitutive
equations coupling bed deformation to fluid flow, accounts for the anisotropic properties
of the medium, and generalizes the theory of poroelasticity to these systems. We use the
effective medium equations to study three canonical problems—the normal settling of
a rigid sphere onto a carpet, the squeeze flow in a carpet and the tangential shearing
motion of a rigid sphere over the carpet, all problems of relevance in mechanosensation
in biology with implications for biomimetic devices.

The branch with the furthest reach

How should a given amount of material be moulded into a cantilevered beam clamped
at one end, so that it will have the furthest horizontal reach? Here, we formulate and solve this
variational problem for the optimal variation of the cross-section area of a heavy cantilevered beam
with a given volume V , Young’s modulus E, and density ρ, subject to gravity g. We find that
the cross-sectional area should vary according a universal profile that is independent of material
parameters, with both the length and maximum reach-out distance of the branch that scale as
(EV /ρg)
, with a universal self-similar shape at the tip with the area of cross-section a ∼ s
, s
being the distance from the tip, consistent with earlier observations of tree branches, but with a
different local interpretation than given before. A simple experimental realization of our optimal
beam shows that our result compares favorably with that of our observations. Our results for the
optimal design of slender structures with the longest reach are valid for cross-sections of arbitrary
shape that can be solid or hollow and thus relevant for a range of natural and engineered systems.