Distributed Dual Subgradient Algorithms With Iterate-Averaging Feedback for Convex Optimization With Coupled Constraints.

This article considers a general model of distributed convex optimization with possibly local constraints, coupled equality constraints, and coupled inequality constraints, where the coupled equality constraints are affine and the coupled inequality constraints can be nonaffine. To solve this problem, we present two algorithms. The first algorithm is similar to a dual subgradient algorithm that requires a center node in the network. The main advantage of the first algorithm is that it achieves the optimal convergence rate O([1/√k]) . Moreover, it does not require additional treatment for the primal recovery. These merits are achieved by using an iterate-averaging feedback technique on the basis of the dual subgradient method. The second algorithm further removes the requirement of a center node by employing consensus tracking iterates. As a result, the second algorithm is fully distributed at the price of achieving an O([lnk/√k]) convergence rate.

Single-cell stochastic gene expression kinetics with coupled positive-plus-negative feedback.

Here we investigate single-cell stochastic gene expression kinetics in a minimal coupled gene circuit with positive-plus-negative feedback. A triphasic stochastic bifurcation is observed upon increasing the ratio of the positive and negative feedback strengths, which reveals a strong synergistic interaction between positive and negative feedback loops. We discover that coupled positive-plus-negative feedback amplifies gene expression mean but reduces gene expression noise over a wide range of feedback strengths when promoter switching is relatively slow, stabilizing gene expression around a relatively high level. In addition, we study two types of macroscopic limits of the discrete chemical master equation model: the Kurtz limit applies to proteins with large burst frequencies and the Lévy limit applies to proteins with large burst sizes. We derive the analytic steady-state distributions of the protein abundance in a coupled gene circuit for both the discrete model and its two macroscopic limits, generalizing the results obtained by Liu et al. [Chaos 26, 043108 (2016)CHAOEH1054-150010.1063/1.4947202]. We also obtain the analytic time-dependent protein distribution for the classical Friedman-Cai-Xie random bursting model [Friedman, Cai, and Xie, Phys. Rev. Lett. 97, 168302 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.168302]. Our analytic results are further applied to study the structure of gene expression noise in a coupled gene circuit, and a complete decomposition of noise in terms of five different biophysical origins is provided.

Harvesting of interacting stochastic populations.

We analyze the optimal harvesting problem for an ecosystem of species that experience environmental stochasticity. Our work generalizes the current literature significantly by taking into account non-linear interactions between species, state-dependent prices, and species seeding. The key generalization is making it possible to not only harvest, but also 'seed' individuals into the ecosystem. This is motivated by how fisheries and certain endangered species are controlled. The harvesting problem becomes finding the optimal harvesting-seeding strategy that maximizes the expected total income from the harvest minus the lost income from the species seeding. Our analysis shows that new phenomena emerge due to the possibility of species seeding. It is well-known that multidimensional harvesting problems are very hard to tackle. We are able to make progress, by characterizing the value function as a viscosity solution of the associated Hamilton-Jacobi-Bellman equations. Moreover, we provide a verification theorem, which tells us that if a function has certain properties, then it will be the value function. This allows us to show heuristically, as was shown by Lungu and Øksendal (Bernoulli 7(3):527-539, 2001), that it is almost surely never optimal to harvest or seed from more than one population at a time. It is usually impossible to find closed-form solutions for the optimal harvesting-seeding strategy. In order to by-pass this obstacle we approximate the continuous-time systems by Markov chains. We show that the optimal harvesting-seeding strategies of the Markov chain approximations converge to the correct optimal harvesting strategy. This is used to provide numerical approximations to the optimal harvesting-seeding strategies and is a first step towards a full understanding of the intricacies of how one should harvest and seed interacting species. In particular, we look at three examples: one species modeled by a Verhulst-Pearl diffusion, two competing species and a two-species predator-prey system.

Stochastic population growth in spatially heterogeneous environments: the density-dependent case.

This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout n patches whose population abundances are modeled as the solutions of a system of nonlinear stochastic differential equations living on [Formula: see text]. We prove that r, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter r can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if [Formula: see text], the population abundances converge polynomially fast to a unique invariant probability measure on [Formula: see text], while when [Formula: see text], the population abundances of the patches converge almost surely to 0 exponentially fast. This generalizes and extends the results of Evans et al. (J Math Biol 66(3):423-476, 2013) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. We also show that the stochastic growth rate depends continuously on the coefficients. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. We show how one can adapt the nondegenerate results to the degenerate setting. As an example we fully analyze the two-patch case, [Formula: see text], and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al. which show that sometimes coupling by dispersal can make the system persistent.

Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations.

The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J. Appl. Math. 56, 1766-1793 (1996); ibid. 56, 1794-1819 (1996)] to average out the fast-reacting variables. This reduction method leads to a limit averaging system, which is an approximation of the slow reactions. Because in the stochastic chemical kinetics, the CLE is seen as the approximation of the SSA, the limit averaging system can be treated as the approximation of the slow reactions. As an application, we examine the reduction of computation complexity for the gene regulatory networks with two-time scales driven by intrinsic noise. For linear and nonlinear protein production functions, the simulations show that the sample average (expectation) of the limit averaging system is close to that of the slow-reaction process based on the SSA. It demonstrates that the limit averaging system is an efficient approximation of the slow-reaction process in the sense of the weak convergence.

Muscarinic antagonist control of myopia: a molecular search for the M1 receptor in chick.

PURPOSE: Pirenzepine, an M1 selective muscarinic antagonist, is effective in slowing the progression of myopia in both humans and experimental animals, including chick. As an M1 selective antagonist, pirenzepine is considered to mediate its effect through M1 receptors. However, there is currently no report of the M1 receptor in chicken. Therefore, if the mechanism of action of pirenzepine is similar across species, either the drug mediates its effect through a non-M1 mechanism, or M1 muscarinic receptors are present in chicken. The aim of the present study was to determine whether a genetic template for the M1 receptor was expressed, or even present, in chick. METHODS: Polymerase Chain Reaction (PCR), and Southern and northern blotting analyses were used to search for M1 mRNA in chick ocular and brain tissues. PCR and Southern analyses were then used for searching the chick M1 gene and promoter. Appropriate rat positive controls were included throughout the study. RESULTS: Direct mRNA detection by northern analysis showed no evidence of M1 mRNA expression in the chick ocular and brain tissues studied. Identical results were obtained from PCR amplification and were further confirmed by Southern analysis. Similarly, no M1 gene or promoter sequences were detected by PCR or Southern analyses. Our methods were validated in every case by a positive finding in equivalent rat tissue and by detection of M2 and M4 mRNA expression in chick retina. CONCLUSIONS: Findings in this study suggest that the chick does not possess an M1 receptor. This finding is of primary interest to vision researchers in that it suggests pirenzepine is unlikely to mediate its inhibitory effect on the progression of myopia through an M1 receptor in chick. Alternative mechanisms of action are discussed.

Lack tolerance of morphine induced spasm of Guinea pig sphincter of Oddi

To determine whether guinea pigs chronically exposed to morphine would develop tolerance to the morphine-induced contraction of the sphincterof Oddi (SO), adult male guinea pigs were implanted with morphine pellets (100 mg morphine). The effect of increasing IV doses of morphine on the SO was assessed by determining the duration of which saline perfusate stopped flowing into the duodenum of morphine-treated guinea pigs (MTGP) vs monimplanted animals (non-MTGP). Isolated bovine and guinea pigs SO were also challenged with morphine. Int the in vivo experiments the spasmogenic response of the SO from MTGP to morphine was greater than of SO from non-MTGP. However, morphine had no effect on isolated SO. These results indicate that chronic morphine exposure does not results in tolerance of the SO to the spasmogenic effects of morphine. On the contrary, chronic morphine even sensitized the SO to morphine, in addition, the in vitro data indicated that morphine does not act directly on the smooth muscle of SO to cause spasmogenic effect