Appendix to "Approximating perpetuities"

An algorithm for perfect simulation from the unique solution of the distributional fixed point equation $Y=_d UY + U(1-U)$ is constructed, where $Y$ and $U$ are independent and $U$ is uniformly distributed on $[0,1]$. This distribution comes up as a limit distribution in the probabilistic analysis of the Quickselect algorithm. Our simulation algorithm is based on coupling from the past with a multigamma coupler. It has four lines of code

A Dominated Coupling From The Past algorithm for the stochastic simulation of networks of biochemical reactions

<p>Abstract</p> <p>Background</p> <p>In recent years, stochastic descriptions of biochemical reactions based on the Master Equation (ME) have become widespread. These are especially relevant for models involving gene regulation. Gillespie’s Stochastic Simulation Algorithm (SSA) is the most widely used method for the numerical evaluation of these models. The SSA produces exact samples from the distribution of the ME for finite times. However, if the stationary distribution is of interest, the SSA provides no information about convergence or how long the algorithm needs to be run to sample from the stationary distribution with given accuracy. </p> <p>Results</p> <p>We present a proof and numerical characterization of a Perfect Sampling algorithm for the ME of networks of biochemical reactions prevalent in gene regulation and enzymatic catalysis. Our algorithm combines the SSA with Dominated Coupling From The Past (DCFTP) techniques to provide guaranteed sampling from the stationary distribution. The resulting DCFTP-SSA is applicable to networks of reactions with uni-molecular stoichiometries and sub-linear, (anti-) monotone propensity functions. We showcase its applicability studying steady-state properties of stochastic regulatory networks of relevance in synthetic and systems biology.</p> <p>Conclusion</p> <p>The DCFTP-SSA provides an extension to Gillespie’s SSA with guaranteed sampling from the stationary solution of the ME for a broad class of stochastic biochemical networks.</p

Randomized stopping times and provably secure pseudorandom permutation generators

Conventionally, key-scheduling algorithm (KSA) of a cryptographic scheme runs for predefined number of steps. We suggest a different approach by utilization of randomized stopping rules to generate permutations which are indistinguishable from uniform ones. We explain that if the stopping time of such a shuffle is a Strong Stationary Time and bits of the secret key are not reused then these algorithms are immune against timing attacks. We also revisit the well known paper of Mironov~\cite{Mironov2002} which analyses a card shuffle which models KSA of RC4. Mironov states that expected time till reaching uniform distribution is $2n H_n - n$ while we prove that $n H_n+ n$ steps are enough (by finding a new strong stationary time for the shuffle). Nevertheless, both cases require $O(n \log^2 n)$ bits of randomness while one can replace the shuffle used in RC4 (and in Spritz) with a better shuffle which is optimal and needs only $O(n \log n)$ bits

Major Cellular and Physiological Impacts of Ocean Acidification on a Reef Building Coral

As atmospheric levels of CO2 increase, reef-building corals are under greater stress from both increased sea surface temperatures and declining sea water pH. To date, most studies have focused on either coral bleaching due to warming oceans or declining calcification due to decreasing oceanic carbonate ion concentrations. Here, through the use of physiology measurements and cDNA microarrays, we show that changes in pH and ocean chemistry consistent with two scenarios put forward by the Intergovernmental Panel on Climate Change (IPCC) drive major changes in gene expression, respiration, photosynthesis and symbiosis of the coral, Acropora millepora, before affects on biomineralisation are apparent at the phenotype level. Under high CO2 conditions corals at the phenotype level lost over half their Symbiodinium populations, and had a decrease in both photosynthesis and respiration. Changes in gene expression were consistent with metabolic suppression, an increase in oxidative stress, apoptosis and symbiont loss. Other expression patterns demonstrate upregulation of membrane transporters, as well as the regulation of genes involved in membrane cytoskeletal interactions and cytoskeletal remodeling. These widespread changes in gene expression emphasize the need to expand future studies of ocean acidification to include a wider spectrum of cellular processes, many of which may occur before impacts on calcification

Modeling CICR in rat ventricular myocytes: voltage clamp studies

<p>Abstract</p> <p>Background</p> <p>The past thirty-five years have seen an intense search for the molecular mechanisms underlying calcium-induced calcium-release (CICR) in cardiac myocytes, with voltage clamp (VC) studies being the leading tool employed. Several VC protocols including lowering of extracellular calcium to affect <it>Ca</it>²⁺loading of the sarcoplasmic reticulum (SR), and administration of blockers caffeine and thapsigargin have been utilized to probe the phenomena surrounding SR <it>Ca</it>²⁺release. Here, we develop a deterministic mathematical model of a rat ventricular myocyte under VC conditions, to better understand mechanisms underlying the response of an isolated cell to calcium perturbation. Motivation for the study was to pinpoint key control variables influencing CICR and examine the role of CICR in the context of a physiological control system regulating cytosolic <it>Ca</it>²⁺concentration ([<it>Ca</it>²⁺]<it>_myo</it>).</p> <p>Methods</p> <p>The cell model consists of an electrical-equivalent model for the cell membrane and a fluid-compartment model describing the flux of ionic species between the extracellular and several intracellular compartments (cell cytosol, SR and the dyadic coupling unit (DCU), in which resides the mechanistic basis of CICR). The DCU is described as a controller-actuator mechanism, internally stabilized by negative feedback control of the unit's two diametrically-opposed <it>Ca</it>²⁺channels (trigger-channel and release-channel). It releases <it>Ca</it>²⁺flux into the cyto-plasm and is in turn enclosed within a negative feedback loop involving the SERCA pump, regulating[<it>Ca</it>²⁺]<it>_myo</it>.</p> <p>Results</p> <p>Our model reproduces measured VC data published by several laboratories, and generates graded <it>Ca</it>²⁺release at high <it>Ca</it>²⁺gain in a homeostatically-controlled environment where [<it>Ca</it>²⁺]<it>_myo</it>is precisely regulated. We elucidate the importance of the DCU elements in this process, particularly the role of the ryanodine receptor in controlling SR <it>Ca</it>²⁺release, its activation by trigger <it>Ca</it>²⁺, and its refractory characteristics mediated by the luminal SR <it>Ca</it>²⁺sensor. Proper functioning of the DCU, sodium-calcium exchangers and SERCA pump are important in achieving negative feedback control and hence <it>Ca</it>²⁺homeostasis.</p> <p>Conclusions</p> <p>We examine the role of the above <it>Ca</it>²⁺regulating mechanisms in handling various types of induced disturbances in <it>Ca</it>²⁺levels by quantifying cellular <it>Ca</it>²⁺balance. Our model provides biophysically-based explanations of phenomena associated with CICR generating useful and testable hypotheses.</p

Mod-ϕ Convergence, II: Estimates on the Speed of Convergence

In this paper, we give estimates for the speed of convergence towards a limiting stable law in the recently introduced setting of mod-ϕ convergence. Namely, we define a notion of zone of control, closely related to mod-ϕ convergence, and we prove estimates of Berry–Esseen type under this hypothesis. Applications include: - the winding number of a planar Brownian motion; - classical approximations of stable laws by compound Poisson laws; - examples stemming from determinantal point processes (characteristic polynomials of random matrices and zeroes of random analytic functions); - sums of variables with an underlying dependency graph (for which we recover a result of Rinott, obtained by Stein’s method); - the magnetization in the d-dimensional Ising model; - and functionals of Markov chains