82 research outputs found
Structural Transitions and Soft Modes in Frustrated DNA Crystals
Relying on symmetry considerations appropriate for helical biopolymers such
as DNA and filamentous actin, we argue that crystalline packings of mutually
repulsive helical macromolecules fall principally into two categories:
unfrustrated (hexagonal) and frustrated (rhombohedral). For both cases, we
construct the Landau-Ginzburg free energy for the 2D columnar-hexagonal to 3D
crystalline phase transition, including the coupling between molecular
displacements {\it along} biopolymer backbone to displacements in the plane of
hexagonal order. We focus on the distinct elastic properties that emerge upon
crystallization of helical arrays due to this coupling. Specifically, we
demonstrate that frustrated states universally exhibit a highly anisotropic
in-plane elastic response, characterized by an especially soft compliance to
simple-shear deformations and a comparatively large resistance to those
deformations that carry the array from the low- to high-density crystalline
states of DNA.Comment: 7 pages, 3 figures (revised version
Shape selection of surface-bound helical filaments: biopolymers on curved membranes
Motivated to understand the behavior of biological filaments interacting with
membranes of various types, we study a theoretical model for the shape and
thermodynamics of intrinsically-helical filaments bound to curved membranes. We
show filament-surface interactions lead to a host of non-uniform shape
equilibria, in which filaments progressively unwind from their native twist
with increasing surface interaction and surface curvature, ultimately adopting
uniform-contact curved shapes. The latter effect is due to non-linear coupling
between elastic twist and bending of filaments on anisotropically-curved
surfaces, such as the cylindrical surfaces considered here. Via a combination
of numerical solutions and asymptotic analysis of shape equilibria we show that
filament conformations are critically sensitive to the surface curvature in
both the strong- and weak-binding limits. These results suggest that local
structure of membrane-bound chiral filaments is generically sensitive to the
curvature-radius of the surface to which it is bound, even when that radius is
much larger than the filament intrinsic pitch. Typical values of elastic
parameters and interaction energies for several prokaryotic and eukaryotic
filaments indicate that biopolymers are inherently very sensitive to the
coupling between twist, interactions and geometry and that this could be
exploited for regulation of a variety of processes such as the targeted
exertion of forces, signaling and self-assembly in response to geometric cues
including the local mean and Gaussian curvatures
Continuous Crystallization in Hexagonally-Ordered Materials
We demonstrate that the phase transition from columnar-hexagonal liquid
crystal to hexagonal-crystalline solid falls into an unusual universality
class, which in three-dimensional allows for both discontinuous transitions as
well as continuous transitions, characterized by a single set of exponents. We
show by a renormalization group calculation (to first order in ) that the critical exponents of the continuous transition are precisely
those of the XY model, which gives rise to a continuous evolution of elastic
moduli. Although the fixed points of the present model are found to be
identical to the XY model, the elastic compliance to deformations in the plane
of hexagonal order, , is nonetheless shown to critically influence the
crystallization transition, with the continuous transition being driven to
first order by fluctuations as the in plane response grows weaker, .Comment: 4 pages, 2 figures (revised version
Topological Defects in Twisted Bundles of Two-Dimensionally Ordered Filaments
Twisted assemblies of filaments in ropes, cables and bundles are essential
structural elements in wide use in macroscopic materials as well as within the
cells and tissues of living organisms. We develop the unique, non-linear
elastic properties of twisted filament bundles that derive from generic
properties of two-dimensional line-ordered materials. Continuum elasticity
reveals a formal equivalence between the elastic stresses induced by bundle
twist and those induced by the positive curvature in thin, elastic sheets.
These geometrically-induced stresses can be screened by 5-fold disclination
defects in lattice packing, and we predict a discrete spectrum elastic energy
groundstates associated with integer numbers of disclinations in cylindrical
bundles. Finally, we show that elastic-energy groundstates are extremely
sensitive to defect position in the cross-section, with off-center
disclinations driving the entire bundle to buckle, adopting globally writhing
configurations.Comment: 4.1 pages; 3 figure
Soft Spheres Make More Mesophases
We use both mean-field methods and numerical simulation to study the phase
diagram of classical particles interacting with a hard-core and repulsive, soft
shoulder. Despite the purely repulsive interaction, this system displays a
remarkable array of aggregate phases arising from the competition between the
hard-core and shoulder length scales. In the limit of large shoulder width to
core size, we argue that this phase diagram has a number of universal features,
and classify the set of repulsive shoulders that lead to aggregation at high
density. Surprisingly, the phase sequence and aggregate size adjusts so as to
keep almost constant inter-aggregate separation.Comment: 4 pages, 2 included figure
Braided Bundles and Compact Coils: The Structure and Thermodynamics of Hexagonally-Packed, Chiral Filament Assemblies
Molecular chirality frustrates the two-dimensional assembly of filamentous
molecules, a fact that reflects the generic impossibility of imposing a global
twisting of layered materials. We explore the consequences of this frustration
for hexagonally-ordered assemblies of chiral filaments that are {\it finite} in
lateral dimension. Specifically, we employ a continuum-elastic description of
cylindrical bundles of filaments, allowing us to consider the most general
resistance to and preference for chiral ordering of the assembly. We explore
two distinct mechanisms by which chirality at the molecular scale of the
filament frustrates the assembly into aggregates. In the first, chiral
interactions between filaments impart an overall twisting of filaments around
the central axis of the bundle. In the second, we consider filaments that are
inherently helical in structure, imparting a writhing geometry to the central
axis. For both mechanisms, we find that a thermodynamically-stable state of
dispersed bundles of {\it finite} width appears close to, but below, the point
of bulk filament condensation. The range of thermodynamic stability of
dispersed bundles is sensitive only to the elastic cost and preference for
chiral filament packing. The self-limited assembly of chiral filaments has
particular implications for a large class of biological molecules -- DNA,
filamentous proteins, viruses, bacterial flagella -- which are universally
chiral and are observed to form compact bundles under a broad range of
conditions.Comment: 15 pages, 8 figure
Defects in Crystalline Packings of Twisted Filament Bundles: II. Dislocations and Grain Boundaries
Twisted and rope-like assemblies of filamentous molecules are common and
vital structural elements in cells and tissue of living organisms. We study the
intrinsic frustration occurring in these materials between the two-dimensional
organization of filaments in cross section and out-of-plane interfilament twist
in bundles. Using non-linear continuum elasticity theory of columnar materials,
we study the favorable coupling of twist-induced stresses to the presence of
edge dislocations in the lattice packing of bundles, which leads to a
restructuring of the ground-state order of these materials at intermediate
twist. The stability of dislocations increases as both the degree of twist and
lateral bundle size grow. We show that in ground states of large bundles,
multiple dislocations pile up into linear arrays, radial grain boundaries,
whose number and length grows with bundle twist, giving rise to a rich class of
"polycrystalline" packings.Comment: 10 pages, 7 figure
Interfaces in Diblocks: A Study of Miktoarm Star Copolymers
We study AB miktoarm star block copolymers in the strong segregation
limit, focussing on the role that the AB interface plays in determining the
phase behavior. We develop an extension of the kinked-path approach which
allows us to explore the energetic dependence on interfacial shape. We consider
a one-parameter family of interfaces to study the columnar to lamellar
transition in asymmetric stars. We compare with recent experimental results. We
discuss the stability of the A15 lattice of sphere-like micelles in the context
of interfacial energy minimization. We corroborate our theory by implementing a
numerically exact self-consistent field theory to probe the phase diagram and
the shape of the AB interface.Comment: 12 pages, 11 included figure
Possible origins of macroscopic left-right asymmetry in organisms
I consider the microscopic mechanisms by which a particular left-right (L/R)
asymmetry is generated at the organism level from the microscopic handedness of
cytoskeletal molecules. In light of a fundamental symmetry principle, the
typical pattern-formation mechanisms of diffusion plus regulation cannot
implement the "right-hand rule"; at the microscopic level, the cell's
cytoskeleton of chiral filaments seems always to be involved, usually in
collective states driven by polymerization forces or molecular motors. It seems
particularly easy for handedness to emerge in a shear or rotation in the
background of an effectively two-dimensional system, such as the cell membrane
or a layer of cells, as this requires no pre-existing axis apart from the layer
normal. I detail a scenario involving actin/myosin layers in snails and in C.
elegans, and also one about the microtubule layer in plant cells. I also survey
the other examples that I am aware of, such as the emergence of handedness such
as the emergence of handedness in neurons, in eukaryote cell motility, and in
non-flagellated bacteria.Comment: 42 pages, 6 figures, resubmitted to J. Stat. Phys. special issue.
Major rewrite, rearranged sections/subsections, new Fig 3 + 6, new physics in
Sec 2.4 and 3.4.1, added Sec 5 and subsections of Sec
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