Increasing stability with complexity in a system composed of unstable subsystems

AbstractWe examine stability of Hoffman's symmetric model of the immune system ẋ = Si − xi∑j=1n Kji xj; xi > 0; i= 1,2, …, n; (1) where Si > 0, Kij = Kji ⩾ 0. This paper gives necessary and sufficient conditions on {Si} and {Kij} for Eq. (1) to have a unique, stable, steady-state solution. Determining existence of a steady-state solution requires a theorem delimiting the range R of a function F: D ⊆ Rn → R ⊆ Rn, where D is a (possibly proper) subset of Rn. This theorem may be new.If off-diagonal elements {Kij: i ≠ j} are non-zero with probability C and 0 < Smin ⩽ Si ⩽ ϱSmin, ϱ a fixed integer, we let P(n, C) be the probability that Eq. (1) does not have a stable, steady-state solution. Let T(n) = (ϱ + 1)2ϱln nn (2) As n → ∞, CT(n) → r > 1 implies P(n, C) → 0. If we set {Kii = 0; i = 1, 2,…, n}, this result shows that accumulating more unstable subsystems increases the probability of stability of this system

Finite-size corrections to Poisson approximations in general renewal-success processes

AbstractConsider a renewal process, and let K⩾0 denote the random duration of a typical renewal cycle. Assume that on any renewal cycle, a rare event called “success” can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence can be relatively slow, because each success corresponds to a time interval, not a point. If K is an arithmetic variable, a “finite-size correction” (FSC) is known to speed Poisson convergence by providing a second, subdominant term in the appropriate asymptotic expansion. This paper generalizes the FSC from arithmetic K to general K. Genomics applications require this generalization, because they have already heuristically applied the FSC to p-values involving absolutely continuous distributions. The FSC also sharpens certain results in queuing theory, insurance risk, traffic flow, and reliability theory

Effect of Non Gaussian Noises on the Stochastic Resonance-Like Phenomenon in Gated Traps

We exploit a simple one-dimensional trapping model introduced before, prompted by the problem of ion current across a biological membrane. The voltage-sensitive channels are open or closed depending on the value taken by an external potential that has two contributions: a deterministic periodic and a stochastic one. Here we assume that the noise source is colored and non Gaussian, with a $q$-dependent probability distribution (where $q$ is a parameter indicating the departure from Gaussianity). We analyze the behavior of the oscillation amplitude as a function of both $q$ and the noise correlation time. The main result is that in addition to the resonant-like maximum as a function of the noise intensity, there is a new resonant maximum as a function of the parameter $q$.Comment: Communication to LAWNP01, Proceedings to be published in Physica D, RevTex, 8 pgs, 5 figure

Trapping Dynamics with Gated Traps: Stochastic Resonance-Like Phenomenon

We present a simple one-dimensional trapping model prompted by the problem of ion current across biological membranes. The trap is modeled mimicking the ionic channel membrane behaviour. Such voltage-sensitive channels are open or closed depending on the value taken by a potential. Here we have assumed that the external potential has two contributions: a determinist periodic and a stochastic one. Our model shows a resonant-like maximum when we plot the amplitude of the oscillations in the absorption current vs. noise intensity. The model was solved both numerically and using an analytic approximation and was found to be in good accord with numerical simulations.Comment: RevTex, 5 pgs, 3 figure

Lifetime of a target in the presence of N independent walkers

We study the survival probability of an immobile target in presence of N independent diffusing walkers. We address the problem of the Mean Target Lifetime and its dependence on the number and initial distribution of the walkers when the trapping is perfect or imperfect. We consider the diffusion on lattices and in the continuous space and we address the bulk limit corresponding to a density of diffusing particles and only one isolated trap. Also, we use intermittent motion for optimization of search strategies.Comment: 18 pages, 5 figures. Accepted for publication in Physica A

NEXT-Peak: A Normal-Exponential Two-Peak Model for Peak-Calling in ChIP-seq Data

Background: Chromatin immunoprecipitation followed by high-throughput sequencing (ChIP-seq) can locate transcription factor binding sites on genomic scale. Although many models and programs are available to call peaks, none has dominated its competition in comparison studies. Results: We propose a rigorous statistical model, the normal-exponential two-peak (NEXT-peak) model, which parallels the physical processes generating the empirical data, and which can naturally incorporate mappability information. The model therefore estimates total strength of binding (even if some binding locations do not map uniquely into a reference genome, effectively censoring them); it also assigns an error to an estimated binding location. The comparison study with existing programs on real ChIP-seq datasets (STAT1, NRSF, and ZNF143) demonstrates that the NEXT-peak model performs well both in calling peaks and locating them. The model also provides a goodness-of-fit test, to screen out spurious peaks and to infer multiple binding events in a region. Conclusions: The NEXT-peak program calls peaks on any test dataset about as accurately as any other, but provides unusual accuracy in the estimated location of the peaks it calls. NEXT-peak is based on rigorous statistics, so its model also provides a principled foundation for a more elaborate statistical analysis of ChIP-seq data

Interparticle interaction and structure of deposits for competitive model in (2+1)- dimensions

A competitive (2+1)-dimensional model of deposit formation, based on the combination of random sequential absorption deposition (RSAD), ballistic deposition (BD) and random deposition (RD) models, is proposed. This model was named as RSAD$_{1-s}$(RD$_f$BD$_{1-f}$)$_s$. It allows to consider different cases of interparticle interactions from complete repulsion between near-neighbors in the RSAD model ($s=0$) to sticking interactions in the BD model ($s=1, f=0$) or absence of interactions in the RD model ($s=1$, $f=0$). The ideal checkerboard ordered structure was observed for the pure RSAD model ($s=0$) in the limit of $h \to \infty$. Defects in the ordered structure were observed at small $h$. The density of deposit $p$ versus system size $L$ dependencies were investigated and the scaling parameters and values of $p_\infty=p(L=\infty)$ were determined. Dependencies of $p$ versus parameters of the competitive model $s$ and $f$ were studied. We observed the anomalous behaviour of the eposit density $p_\infty$ with change of the inter-particle repulsion, which goes through minimum on change of the parameter $s$. For pure RSAD model, the concentration of defects decreases with $h$ increase in accordance with the critical law $\rho\propto h^{-\chi_{RSAD}}$, where $\chi_{RSAD} \approx 0.119 \pm 0.04$.Comment: 10 pages,4 figures, Latex, uses iopart.cl

Stochastic Aggregation of Point Particles Revisited

In the present letter we employ the method of the dynamical renormalisation group to compute the average mass distribution of aggregating point particles in 2 dimensions in the regime when the effects of local mass distribution fluctuations are essentialComment: 9 pages, submitted to Phys. Letters