bibkey: baratoseifert2015skewness
authors: Andre C. Barato, Udo Seifert
year: 2015
title: "Skewness and Kurtosis in Statistical Kinetics"
doi: 10.1103/PhysRevLett.115.188103
url: https://arxiv.org/abs/1510.03279v1
claim: "For unicyclic enzymatic schemes with an irreversible product release, the randomness parameter, skewness and kurtosis of the cycle completion time are bounded above by 1, 2 and 9; for a four-state two-cycle scheme the paper finds numerically that a constraint S <= 2 keeps the randomness parameter at most 1, and leaves open whether this ordering holds for arbitrary networks that do not consist of a single cycle."
strata_touched:
  - D5/S3/StatisticalMechanics/WaitingTimeRandomnessSkewnessRefutation
license: citation-only
triage: anchor

Barato–Seifert, skewness and kurtosis in statistical kinetics

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DOI: 10.1103/PhysRevLett.115.188103

Source: https://arxiv.org/abs/1510.03279v1

Phys. Rev. Lett. 115, 188103 (2015).

A. C. Barato and U. Seifert, arXiv:1510.03279v1 [cond-mat.stat-mech] (12 October 2015, the only version); Phys. Rev. Lett. 115, 188103 (2015). Quotations are from the v1 TeX source skewkurtstatkinarxiv.tex, with macros for $\mathcal{R},\mathcal{S},\mathcal{K}$ written out.

The cycle completion time and its moments (l. 132–153):

The probability that a cycle is completed at time $t$ is denoted by $\psi (t)$. Its moment of order $n$ is defined as $t_n\equiv {\int }_0^{\infty }\psi (t)t^n$.

$\mathcal{R}\equiv {\sigma }^2/t_1^2$, where $\sigma \equiv \sqrt{t_2-t_1^2}$. […] the skewness $\mathcal{S}\equiv (t_3-3t_1{\sigma }^2-t_1^3)/{\sigma }^3$, and the kurtosis $\mathcal{K}\equiv (t_4-4t_3{t}_1+6{\sigma }^2{t}_1^2+3t_1^4)/{\sigma }^4$.

A cycle is completed by the irreversible product-release transition (eq. fullreaction: “The last transition is taken to be irreversible”).

The ordering and the open question (l. 341–358):

The maxima of randomness parameter $\mathcal{R}$, skewness $\mathcal{S}$ and kurtosis $\mathcal{K}$ for a unicyclic network are $\mathcal{R}=1$, $\mathcal{S}=2$, and $\mathcal{K}=9$ […] This numerical investigation consists of, for example, maximizing the skewness with the constraint that $\mathcal{R}\le 1$, which leads to a skewness exceeding $2$. The opposite however is not possible, if the constraint $\mathcal{S}\le 2$ is satisfied, the randomness parameter never goes above $1$. […] It remains as an open question whether this property is true for arbitrary networks that do not consist of a single cycle.

Encoding and boundary

A network is a finite rate matrix on Fin m with nonnegative off-diagonal rates and a marked product-release transition $p\to r$ with positive rate, from which $p$ is reachable from every state. The cycle completion time is the first time the marked transition fires after a release into $r$. Its moments are the first-passage moments $m_n$ with $m_0=1$ and $(-G)m_n=n{m}_{n-1}$, where $G$ is the generator with the marked transition sent to a cemetery; $t_n=m_n(r)$. This is the standard phase-type characterization of the moments of $\psi$, used in place of the source’s generating-function route; $\mathcal{R}$ and $\mathcal{S}$ are the source’s formulas in these moments.

The repository refutes the ordering for arbitrary networks with an eight-state network of two routes sharing the product edge (a fast route through one intermediate and a slow five-stage route), for which $\mathcal{R}=\frac{2257}{1089}>1$ while $0<\mathcal{S}<2$. The unicyclic bounds, the kurtosis part of the ordering and reversible perturbations are outside this conclusion.