Signal processing by HOG MAP kinase pathway

Signaling pathways relay information about changes in the external environment so that cells can respond appropriately. How
much information a pathway can carry depends on its bandwidth.
We designed a microfluidic device to reliably change the environment of single cells over a range of frequencies. Using this device,
we measured the bandwidth of the Saccharomyces cerevisiae
signaling pathway that responds to high osmolarity. This prototypical pathway, the HOG pathway, is shown to act as a low-pass
filter, integrating the signal when it changes rapidly and following
it faithfully when it changes more slowly. We study the dependence of the pathway’s bandwidth on its architecture. We measure
previously unknown bounds on all of the in vivo reaction rates
acting in this pathway. We find that the two-component Ssk1
branch of this pathway is capable of fast signal integration,
whereas the kinase Ste11 branch is not. Our experimental techniques can be applied to other signaling pathways, allowing the
measurement of their in vivo kinetics and the quantification of
their information capacity.

Limbless undulatory propulsion on land

We analyze the lateral undulatory motion of a natural or artificial
snake or other slender organism that ‘‘swims’’ on land by propagating retrograde flexural waves. The governing equations for the
planar lateral undulation of a thin filament that interacts frictionally with its environment lead to an incomplete system. Closures
accounting for the forces generated by the internal muscles and
the interaction of the filament with its environment lead to a
nonlinear boundary value problem, which we solve using a combination of analytical and numerical methods. We find that the
primary determinant of the shape of the organism is its interaction
with the external environment, whereas the speed of the organism
is determined primarily by the internal muscular forces, consistent
with prior qualitative observations. Our model also allows us to
pose and solve a variety of optimization problems such as those
associated with maximum speed and mechanical efficiency, thus
defining the performance envelope of this mode of locomotion.

The shape and motion of a ruck in a rug

The motion of a ruck in a rug is used as an analogy to explain the role of dislocations in crystalline solids. We take literally one side of this analogy and study the shape and motion of a bump, wrinkle or ruck in a thin sheet in partial contact with a rough substrate in a gravitational field. Using a combination of experiments, scaling analysis and numerical solutions of the governing equations, we quantify the static shape of a ruck on a horizontal plane. When the plane is inclined, the ruck becomes asymmetric and moves by rolling only when the inclination of the plane reaches a critical angle, at a speed determined by a simple power balance. We find that the angle at which rolling starts is larger than the angle at which the ruck stops; i.e., static rolling friction is larger than dynamic rolling friction. We conclude with a generalization of our results to wrinkles in soft adherent extensible films.

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.

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.

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.

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.

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.

Geometric mechanics of periodic pleated origami

Origami structures are mechanical metamaterials with properties that arise almost exclusively from the
geometry of the constituent folds and the constraint of piecewise isometric deformations. Here we
characterize the geometry and planar and nonplanar effective elastic response of a simple periodically
folded Miura-ori structure, which is composed of identical unit cells of mountain and valley folds with
four-coordinated ridges, defined completely by two angles and two lengths. We show that the in-plane and
out-of-plane Poisson’s ratios are equal in magnitude, but opposite in sign, independent of material
properties. Furthermore, we show that effective bending stiffness of the unit cell is singular, allowing us to
characterize the two-dimensional deformation of a plate in terms of a one-dimensional theory. Finally, we
solve the inverse design problem of determining the geometric parameters for the optimal geometric and
mechanical response of these extreme structures