Extinction of metastable stochastic populations

We investigate extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state n=0 is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n=0. In scenario B there is an intermediate repelling point n=n_1 between the attracting point n=0 and another attracting point n=n_2 in the vicinity of which the metastable population resides. The crux of the theory is WKB method which assumes that the typical population size in the metastable state is large. Starting from the master equation, we calculate the quasi-stationary probability distribution of the population sizes and the (exponentially long) mean time to extinction for each of the two scenarios. When necessary, the WKB approximation is complemented (i) by a recursive solution of the quasi-stationary master equation at small n and (ii) by the van Kampen system-size expansion, valid near the fixed points of the deterministic rate equation. The theory yields both entropic barriers to extinction and pre-exponential factors, and holds for a general set of multi-step processes when detailed balance is broken. The results simplify considerably for single-step processes and near the characteristic bifurcations of scenarios A and B.Comment: 19 pages, 7 figure

First passage time distribution for a random walker on a random forcing energy landscape

We present an analytical approximation scheme for the first passage time distribution on a finite interval of a random walker on a random forcing energy landscape. The approximation scheme captures the behavior of the distribution over all timescales in the problem. The results are carefully checked against numerical simulations.Comment: 16 page

Computational Modeling of Fluid Flow and Intra-Ocular Pressure following Glaucoma Surgery

Background Glaucoma surgery is the most effective means for lowering intraocular pressure by providing a new route for fluid to exit the eye. This new pathway is through the sclera of the eye into sub-conjunctival tissue, where a fluid filled bleb typically forms under the conjunctiva. The long-term success of the procedure relies on the capacity of the sub-conjunctival tissue to absorb the excess fluid presented to it, without generating excessive scar tissue during tissue remodeling that will shut-down fluid flow. The role of inflammatory factors that promote scarring are well researched yet little is known regarding the impact of physical forces on the healing response. Methodology To help elucidate the interplay of physical factors controlling the distribution and absorption of aqueous humor in sub-conjunctival tissue, and tissue remodeling, we have developed a computational model of fluid production in the eye and removal via the trabecular/uveoscleral pathways and the surgical pathway. This surgical pathway is then linked to a porous media computational model of a fluid bleb positioned within the sub-conjunctival tissue. The computational analysis is centered on typical functioning bleb geometry found in a human eye following glaucoma surgery. A parametric study is conducted of changes in fluid absorption by the sub-conjunctival blood vessels, changes in hydraulic conductivity due to scarring, and changes in bleb size and shape, and eye outflow facility. Conclusions This study is motivated by the fact that some blebs are known to have ‘successful’ characteristics that are generally described by clinicians as being low, diffuse and large without the formation of a distinct sub-conjunctival encapsulation. The model predictions are shown to accord with clinical observations in a number of key ways, specifically the variation of intra-ocular pressure with bleb size and shape and the correspondence between sites of predicted maximum interstitial fluid pressure and key features observed in blebs, which gives validity to the model described here. This model should contribute to a more complete explanation of the physical processes behind successful bleb characteristics and provide a new basis for clinically grading blebs

On population extinction risk in the aftermath of a catastrophic event

We investigate how a catastrophic event (modeled as a temporary fall of the reproduction rate) increases the extinction probability of an isolated self-regulated stochastic population. Using a variant of the Verhulst logistic model as an example, we combine the probability generating function technique with an eikonal approximation to evaluate the exponentially large increase in the extinction probability caused by the catastrophe. This quantity is given by the eikonal action computed over "the optimal path" (instanton) of an effective classical Hamiltonian system with a time-dependent Hamiltonian. For a general catastrophe the eikonal equations can be solved numerically. For simple models of catastrophic events analytic solutions can be obtained. One such solution becomes quite simple close to the bifurcation point of the Verhulst model. The eikonal results for the increase in the extinction probability caused by a catastrophe agree well with numerical solutions of the master equation.Comment: 11 pages, 11 figure

Transport in molecular states language: Generalized quantum master equation approach

A simple scheme capable of treating transport in molecular junctions in the language of many-body states is presented. An ansatz in Liouville space similar to generalized Kadanoff-Baym approximation is introduced in order to reduce exact equation-of-motion for Hubbard operator to quantum master equation (QME)-like expression. A dressing with effective Liouville space propagation similar to standard diagrammatic dressing approach is proposed. The scheme is compared to standard QME approach, and its applicability to transport calculations is discussed within numerical examples.Comment: 10 pages, 3 figure

Controlling Al–M Interactions in Group 1 Metal Aluminyls (M = Li, Na, and K). Facile Conversion of Dimers to Monomeric and Separated Ion Pairs

The aluminyl compounds [M{Al(NONDipp)}]2 (NONDipp = [O(SiMe2NDipp)2]2–, Dipp = 2,6-iPr2C6H3), which exist as contacted dimeric pairs in both the solution and solid states, have been converted to monomeric ion pairs and separated ion pairs for each of the group 1 metals, M = Li, Na, and K. The monomeric ion pairs contain discrete, highly polarized Al–M bonds between the aluminum and the group 1 metal and have been isolated with monodentate (THF, M = Li and Na) or bidentate (TMEDA, M = Li, Na, and K) ligands at M. The separated ion pairs comprise group 1 cations that are encapsulated by polydentate ligands, rendering the aluminyl anion, [Al(NONDipp)]− “naked”. For M = Li, this structure type was isolated as the [Li(TMEDA)2]+ salt directly from a solution of the corresponding contacted dimeric pair in neat TMEDA, while the polydentate [2.2.2]cryptand ligand was used to generate the separated ion pairs for the heavier group 1 metals M = Na and K. This work shows that starting from the corresponding contacted dimeric pairs, the extent of the Al–M interaction in these aluminyl systems can be readily controlled with appropriate chelating reagents

Evolution of a Complex Locus: Exon Gain, Loss and Divergence at the Gr39a Locus in Drosophila

Background. Gene families typically evolve by gene duplication followed by the adoption of new or altered gene functions. A different way to evolve new but related functions is alternative splicing of existing exons of a complex gene. The chemosensory gene families of animals are characterised by numerous loci of related function. Alternative splicing has only rarely been reported in chemosensory loci, for example in 5 out of around 120 loci in Drosophila melanogaster. The gustatory receptor gene Gr39a has four large exons that are alternatively spliced with three small conserved exons. Recently the genome sequences of eleven additional species of Drosophila have become available allowing us to examine variation in the structure of the Gr39a locus across a wide phylogenetic range of fly species. Methodology/Principal Findings. We describe a fifth exon and show that the locus has a complex evolutionary history with several duplications, pseudogenisations and losses of exons. PAML analyses suggested that the whole gene has a history of purifying selection, although this was less strong in exons which underwent duplication. Conclusions/Significance. Estimates of functional divergence between exons were similar in magnitude to functional divergence between duplicated genes, suggesting that exon divergence is broadly equivalent to gene duplication.Publisher PDFPeer reviewe

Few-Qubit lasing in circuit QED

Motivated by recent experiments, which demonstrated lasing and cooling of the electromagnetic modes in a resonator coupled to a superconducting qubit, we describe the specific mechanisms creating the population inversion, and we study the spectral properties of these systems in the lasing state. Different levels of the theoretical description, i.e., the semi-classical and the semi-quantum approximation, as well as an analysis based on the full Liouville equation are compared. We extend the usual quantum optics description to account for strong qubit-resonator coupling and include the effects of low-frequency noise. Beyond the lasing transition we find for a single- or few-qubit system the phase diffusion strength to grow with the coupling strength, which in turn deteriorates the lasing state.Comment: Prepared for the proceedings of the Nobel Symposium 2009, Qubits for future quantum computers, May 2009 in Goeteborg, Sweden. Published versio

Large fluctuations in stochastic population dynamics: momentum space calculations

Momentum-space representation renders an interesting perspective to theory of large fluctuations in populations undergoing Markovian stochastic gain-loss processes. This representation is obtained when the master equation for the probability distribution of the population size is transformed into an evolution equation for the probability generating function. Spectral decomposition then brings about an eigenvalue problem for a non-Hermitian linear differential operator. The ground-state eigenmode encodes the stationary distribution of the population size. For long-lived metastable populations which exhibit extinction or escape to another metastable state, the quasi-stationary distribution and the mean time to extinction or escape are encoded by the eigenmode and eigenvalue of the lowest excited state. If the average population size in the stationary or quasi-stationary state is large, the corresponding eigenvalue problem can be solved via WKB approximation amended by other asymptotic methods. We illustrate these ideas in several model examples.Comment: 20 pages, 9 figures, to appear in JSTA