2,367 research outputs found
Comparing open and closed molecular self-assembly
We study theoretically in the present work the self-assembly of molecules in
an open system, which is fed by monomers and depleted in partial or complete
clusters. Such a scenario is likely to occur for example in the context of
viral self-assembly. We provide a general formula for the mean-field size
distribution which is valid both at equilibrium in a closed system, and in the
stationary state in an open system. This allows us to explore in a simple way
out-of-equilibrium features for self-assembly and compare them to equilibrium
properties. In particular, we identify a region of parameter space for which
the out-of-equilibrium size distribution in the presence of external fluxes is
equal to the equilibrium size distribution in the absence of external fluxes,
up to a constant renormalization factor. The range of validity of this result
and its consequences are discussed.Comment: PACS 81.16.Fg - Supramolecular and biochemical assembly PACS 82.39.-k
- Chemical kinetics in biological systems PACS 05.65.+b - Self-organized
system
The distribution of word matches between Markovian sequences with periodic boundary conditions
Word match counts have traditionally been proposed as an alignment-free measure of similarity for biological sequences. The D2 statistic, which simply counts the number of exact word matches between two sequences, is a useful test bed for developing rigorous mathematical results, which can then be extended to more biologically useful measures. The distributional properties of the D2 statistic under the null hypothesis of identically and independently distributed letters have been studied extensively, but no comprehensive study of the D2 distribution for biologically more realistic higher-order Markovian sequences exists. Here we derive exact formulas for the mean and variance of the D2 statistic for Markovian sequences of any order, and demonstrate through Monte Carlo simulations that the entire distribution is accurately characterized by a PĂłlya-Aeppli distribution for sequence lengths of biological interest. The approach is novel in that Markovian dependency is defined for sequences with periodic boundary conditions, and this enables exact analytic formulas for the mean and variance to be derived. We also carry out a preliminary comparison between the approximate D2 distribution computed with the theoretical mean and variance under a Markovian hypothesis and an empirical D2 distribution from the human genome
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