61,373 research outputs found
A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks
Approximate solutions of the chemical master equation and the chemical
Fokker-Planck equation are an important tool in the analysis of biomolecular
reaction networks. Previous studies have highlighted a number of problems with
the moment-closure approach used to obtain such approximations, calling it an
ad-hoc method. In this article, we give a new variational derivation of
moment-closure equations which provides us with an intuitive understanding of
their properties and failure modes and allows us to correct some of these
problems. We use mixtures of product-Poisson distributions to obtain a flexible
parametric family which solves the commonly observed problem of divergences at
low system sizes. We also extend the recently introduced entropic matching
approach to arbitrary ansatz distributions and Markov processes, demonstrating
that it is a special case of variational moment closure. This provides us with
a particularly principled approximation method. Finally, we extend the above
approaches to cover the approximation of multi-time joint distributions,
resulting in a viable alternative to process-level approximations which are
often intractable.Comment: Minor changes and clarifications; corrected some typo
Hierarchies of Inefficient Kernelizability
The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam
(STOC 2008) allows us to exclude the existence of polynomial kernels for a
range of problems under reasonable complexity-theoretical assumptions. However,
there are also some issues that are not addressed by this framework, including
the existence of Turing kernels such as the "kernelization" of Leaf Out
Branching(k) into a disjunction over n instances of size poly(k). Observing
that Turing kernels are preserved by polynomial parametric transformations, we
define a kernelization hardness hierarchy, akin to the M- and W-hierarchy of
ordinary parameterized complexity, by the PPT-closure of problems that seem
likely to be fundamentally hard for efficient Turing kernelization. We find
that several previously considered problems are complete for our fundamental
hardness class, including Min Ones d-SAT(k), Binary NDTM Halting(k), Connected
Vertex Cover(k), and Clique(k log n), the clique problem parameterized by k log
n
Pattern formation from consistent dynamical closures of uniaxial nematic liquid crystals
Pattern formation in uniaxial polymeric liquid crystals is studied for
different dynamic closure approximations. Using the principles of mesoscopic
non-equilibrium thermodynamics in a mean-field approach, we derive a
Fokker-Planck equation for the single-particle non-homogeneous distribution
function of particle orientations and the evolution equations for the second
and fourth order orientational tensor parameters. Afterwards, two dynamic
closure approximations are discussed, one of them considering the relaxation of
the fourth order orientational parameter and leading to a novel expression for
the free-energy like function in terms of the scalar order parameter.
Considering the evolution equation of the density of the system and values of
the interaction parameter for which isotropic and nematic phases coexist, our
analysis predicts that patterns and traveling waves can be produced in
lyotropic uniaxial nematics even in the absence of external driving.Comment: 34 pages, 7 figure
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