We model DNA as a Kirchhoff rod, making rigorous the
foundations, scales, and assumptions behind the theory.
DNA experiences powerful torsional forces while held in
constrained structures that, on the scales of a few
hundred base pairs, can be viewed as linear. As such,
the structures of DNA can be studied by analysis of a
twisted, intrinsically straight rod under tension. The
novelty of this work lies in several points. First, this
is a fully dynamic approach, including a continuous
injection of twist at one domain boundary where the other
is held fixed from rotation. This is a special case,
most applicable to DNA, but which has not been previously
examined. This work gives careful consideration to the
derivation and novel incorporation of drag forces, and a
rigorous treatment of the relationships between the drag
model and the equations of motion. The model is examined
using linear analysis about the twisted, straight rod
under tension, assymtotics, and numerical analysis of the
driven rod. We then apply the insights obtained through
this work to examine the distribution of a inverted
repeat sequences that are theoretically susceptible to
supercoiling-induced structural transitions to cruciforms
driven by these torsional stresses.
Fitness landscapes are a tool used in theoretical
evolutionary biology to model speciation under various
conditions. A fitness landscape consists of a genotype
space and a function that maps each combination of genes
to a fitness level in the interval $[0, 1]$, which can be
interpreted as the probability of an individual with that
genotype surviving to reproduce. I will present a model
for a fitness landscape that is equivalent to taking a
random site subgraph of a Hamming Torus. This model is
most interesting because it lends itself nicely to an
analysis that uses the theory of multitype branching
processes, and the resulting threshold value for the
emergence of widespread connectivity is interesting and
distinct from the edge subgraph analogue.
Evolutionarily, this small threshold implies that only a
fraction of gene combinations need to be viable for
significant evolution to be possible.
We begin with a description of a mode of signal
transmission among neuron-nodes in a finite
deterministic network. In this mode, an activated node
passes a signal to all its nearest neighbors and
becomes deactivated, unless it simultaneously receives
the signal from one of its nearest neighbors and
reminds activated. By means of a matrix representation,
we show that a connected network equipped with this
mode of signal transmission converges to one of two
states: 1) System-Wide Synchronization (SWS), wherein
all nodes are invariantly activated; and 2) Subgroup
Alternation (SGA), wherein each node falls into an on-
and-off pattern of activation, but not synchronously.
Conditions on wiring configuration required for SWS are
presented.
Fitness Landscapes and Random Site Subgraphs of Hamming Tori
David Sivakoff, J. Gravner
System-wide synchronization in deterministic networks
Michael McAssey, F. Hsieh, E. Ferrer