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Convex Hulls of L\'evy Processes
Let , , be a L\'evy process in starting at the
origin. We study the closed convex hull of . In
particular, we provide conditions for the integrability of the intrinsic
volumes of the random set and find explicit expressions for their means
in the case of symmetric -stable L\'evy processes. If the process is
symmetric and each its one-dimensional projection is non-atomic, we establish
that the origin a.s. belongs to the interior of for all . Limit
theorems for the convex hull of L\'evy processes with normal and stable limits
are also obtained.Comment: 11 page
Model of ionic currents through microtubule nanopores and the lumen
It has been suggested that microtubules and other cytoskeletal filaments may
act as electrical transmission lines. An electrical circuit model of the
microtubule is constructed incorporating features of its cylindrical structure
with nanopores in its walls. This model is used to study how ionic conductance
along the lumen is affected by flux through the nanopores when an external
potential is applied across its two ends. Based on the results of Brownian
dynamics simulations, the nanopores were found to have asymmetric inner and
outer conductances, manifested as nonlinear IV curves. Our simulations indicate
that a combination of this asymmetry and an internal voltage source arising
from the motion of the C-terminal tails causes a net current to be pumped
across the microtubule wall and propagate down the microtubule through the
lumen. This effect is demonstrated to enhance and add directly to the
longitudinal current through the lumen resulting from an external voltage
source, and could be significant in amplifying low-intensity endogenous
currents within the cellular environment or as a nano-bioelectronic device.Comment: 43 pages, 6 figures, revised versio
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