3,476 research outputs found
Quasi-steady state reduction of molecular motor-based models of directed intermittent search
We present a quasiâsteady state reduction of a linear reactionâhyperbolic master equation describing the directed intermittent search for a hidden target by a motorâdriven particle moving on a oneâdimensional filament track. The particle is injected at one end of the track and randomly switches between stationary search phases and mobile, non-search phases that are biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track. Such a scenario is exemplified by the motorâdriven transport of vesicular cargo to synaptic targets located on the axon or dendrites of a neuron. The reduced model is described by a scalar FokkerâPlanck (FP) equation, which has an additional inhomogeneous decay term that takes into account absorption by the target. The FP equation is used to compute the probability of finding the hidden target (hitting probability) and the corresponding conditional mean first passage time (MFPT) in terms of the effective drift velocity V , diffusivity D and target absorption rate λ of the random search. The quasiâsteady state reduction determines V, D and λ in terms of the various biophysical parameters of the underlying motor transport model. We first apply our analysis to a simple 3âstate model and show that our quasiâsteady state reduction yields results that are in excellent agreement with Monte Carlo simulations of the full system under physiologically reasonable conditions. We then consider a more complex multiple motor model of bidirectional transport, in which opposing motors compete in a âtug-of-war,â and use this to explore how ATP concentration might regulate the delivery of cargo to synaptic targets
Random intermittent search and the tug-of-war model of motor-driven transport
We formulate the tug-of-war model of microtubule cargo transport by multiple molecular motors as an intermittent random search for a hidden target. A motor-complex consisting of multiple molecular motors with opposing directional preference is modeled using a discrete Markov process. The motors randomly pull each other off of the microtubule so that the state of the motor-complex is determined by the number of bound motors. The tug-of-war model prescribes the state transition rates and corresponding cargo velocities in terms of experimentally measured physical parameters. We add space to the resulting Chapman-Kolmogorov (CK) equation so that we can consider delivery of the cargo to a hidden target somewhere on the microtubule track. Using a quasi-steady state (QSS) reduction technique we calculate analytical approximations of the mean first passage time (MFPT) to find the target. We show that there exists an optimal adenosine triphosphate (ATP)concentration that minimizes the MFPT for two different cases: (i) the motor complex is composed of equal numbers of kinesin motors bound to two different microtubules (symmetric tug-of-war model), and (ii) the motor complex is composed of different numbers of kinesin and dynein motors bound to a single microtubule(asymmetric tug-of-war model)
Sign rank versus VC dimension
This work studies the maximum possible sign rank of sign
matrices with a given VC dimension . For , this maximum is {three}. For
, this maximum is . For , similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the adjacency
matrix of a regular graph with a second eigenvalue of absolute value
and . We show that the sign rank of the signed
version of this matrix is at least . We use this connection to
prove the existence of a maximum class with VC
dimension and sign rank . This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran
Output-Sensitive Tools for Range Searching in Higher Dimensions
Let be a set of points in . A point is
\emph{-shallow} if it lies in a halfspace which contains at most points
of (including ). We show that if all points of are -shallow, then
can be partitioned into subsets, so that any hyperplane
crosses at most subsets. Given such
a partition, we can apply the standard construction of a spanning tree with
small crossing number within each subset, to obtain a spanning tree for the
point set , with crossing number . This allows us to extend the construction of Har-Peled
and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set
of points in (without the shallowness assumption), a
spanning tree with {\em small relative crossing number}. That is, any
hyperplane which contains points of on one side, crosses
edges of . Using a
similar mechanism, we also obtain a data structure for halfspace range
counting, which uses space (and somewhat higher
preprocessing cost), and answers a query in time , where is the output size
Spectral properties of incommensurate charge-density wave systems
The concept of frustrated phase separation is applied to investigate its
consequences for the electronic structure of the high T_c cuprates. The
resulting incommensurate charge density wave (CDW) scattering is most effective
in creating local gaps in k-space when the scattering vector connects states
with equal energy. Starting from an open Fermi surface we find that the
resulting CDW is oriented along the (10)- and (or) (01)-direction which allows
for a purely one-dimensional or a two-dimensional ``eggbox type'' charge
modulation. In both cases the van Hove singularities are substantially
enhanced, and the spectral weight of Fermi surface states near the M-points,
tends to be suppressed. Remarkably, a leading edge gap arises near these
points, which, in the eggbox case, leaves finite arcs of the Fermi surface
gapless. We discuss our results with repect to possible consequences for
photoemission experiments
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