Scaling and front dynamics in Ising quantum chains

We study the relaxation dynamics of a quantum Ising chain initially prepared in a product of canonical states corresponding each to an equilibrium state of part of the chain at a given temperature. We focus our attention on the transverse magnetization for which a general expression is given. Explicite results are given for the completely factorized initial state, corresponding to a situation where all the spins are thermalized independently, and for the two-temperatures initial state, where part of the chain called the system is thermalized at a temperature $T_s$ and the remaining part is at a temperature $T_b$.Comment: 7 pages, submitted to EPJ

Analytical results for a stochastic model of gene expression with arbitrary partitioning of proteins

In biophysics, the search for analytical solutions of stochastic models of cellular processes is often a challenging task. In recent work on models of gene expression, it was shown that a mapping based on partitioning of Poisson arrivals (PPA-mapping) can lead to exact solutions for previously unsolved problems. While the approach can be used in general when the model involves Poisson processes corresponding to creation or degradation, current applications of the method and new results derived using it have been limited to date. In this paper, we present the exact solution of a variation of the two-stage model of gene expression (with time dependent transition rates) describing the arbitrary partitioning of proteins. The methodology proposed makes full use of the the PPA-mapping by transforming the original problem into a new process describing the evolution of three biological switches. Based on a succession of transformations, the method leads to a hierarchy of reduced models. We give an integral expression of the time dependent generating function as well as explicit results for the mean, variance, and correlation function. Finally, we discuss how results for time dependent parameters can be extended to the three-stage model and used to make inferences about models with parameter fluctuations induced by hidden stochastic variables.Comment: 15 pages, 6 figure

Regulation by small RNAs via coupled degradation: mean-field and variational approaches

Regulatory genes called small RNAs (sRNAs) are known to play critical roles in cellular responses to changing environments. For several sRNAs, regulation is effected by coupled stoichiometric degradation with messenger RNAs (mRNAs). The nonlinearity inherent in this regulatory scheme indicates that exact analytical solutions for the corresponding stochastic models are intractable. Here, we present a variational approach to analyze a well-studied stochastic model for regulation by sRNAs via coupled degradation. The proposed approach is efficient and provides accurate estimates of mean mRNA levels as well as higher order terms. Results from the variational ansatz are in excellent agreement with data from stochastic simulations for a wide range of parameters, including regions of parameter space where mean-field approaches break down. The proposed approach can be applied to quantitatively model stochastic gene expression in complex regulatory networks.Comment: 4 pages, 3 figure

Relaxation in the XX quantum chain

We present the results obtained on the magnetisation relaxation properties of an XX quantum chain in a transverse magnetic field. We first consider an initial thermal kink-like state where half of the chain is initially thermalized at a very high temperature $T_b$ while the remaining half, called the system, is put at a lower temperature $T_s$. From this initial state, we derive analytically the Green function associated to the dynamical behaviour of the transverse magnetisation. Depending on the strength of the magnetic field and on the temperature of the system, different regimes are obtained for the magnetic relaxation. In particular, with an initial droplet-like state, that is a cold subsystem of finite size in contact at both ends with an infinite temperature environnement, we derive analytically the behaviour of the time-dependent system magnetisation

Exact protein distributions for stochastic models of gene expression using partitioning of Poisson processes

Stochasticity in gene expression gives rise to fluctuations in protein levels across a population of genetically identical cells. Such fluctuations can lead to phenotypic variation in clonal populations, hence there is considerable interest in quantifying noise in gene expression using stochastic models. However, obtaining exact analytical results for protein distributions has been an intractable task for all but the simplest models. Here, we invoke the partitioning property of Poisson processes to develop a mapping that significantly simplifies the analysis of stochastic models of gene expression. The mapping leads to exact protein distributions using results for mRNA distributions in models with promoter-based regulation. Using this approach, we derive exact analytical results for steady-state and time-dependent distributions for the basic 2-stage model of gene expression. Furthermore, we show how the mapping leads to exact protein distributions for extensions of the basic model that include the effects of post-transcriptional and post-translational regulation. The approach developed in this work is widely applicable and can contribute to a quantitative understanding of stochasticity in gene expression and its regulation.Comment: 10 pages, 5 figure

"The greatest Poet that has [n]ever existed" -- A Narrative Networks Analysis of the Poems of Ossian

Surprising as it may seem, applications of statistical methods to physics were inspired by the social sciences, which in turn are linked to the humanities. So perhaps it is not as unlikely as it might first appear for a group of statistical physicists and humanists to come together to investigate one of the subjects of Thomas Jefferson's poetic interests from a scientific point of view. And that is the nature of this article: a collaborative interdisciplinary analysis of the works of a figure Jefferson described as a ''rude bard of the North'' and ''the greatest Poet that has ever existed.'' In 2012, a subset of this team embraced an increase in interdisciplinary methods to apply the new science of complex networks to longstanding questions in comparative mythology. Investigations of network structures embedded in epic narratives allowed universal properties to be identified and ancient texts to be compared to each other. The approach inspired new challenges in mathematics, physics and even processes in industry, thereby illustrating how collaborations of this nature can be mutually beneficial and can capture the attention of a public, often ill-served by academic communication and dissemination. This article derives from these works, and from our consistent objective to help bridge the perceived gap between the natural sciences and the humanities. First we discuss the history of relationships between the two. Then we discuss the origins of the poems of Ossian and Jefferson's interests. We follow with our statistical approach in the next section. In the final section, we explore ideas for future research on these themes and discuss the potential of collaborative pursuits of human curiosity to overcome the two cultures dichotomy and embrace a scientific- and humanities-literate information age.Comment: Contriution to book chapte

Work fluctuations in quantum spin chains

We study the work fluctuations of two types of finite quantum spin chains under the application of a time-dependent magnetic field in the context of the fluctuation relation and Jarzynski equality. The two types of quantum chains correspond to the integrable Ising quantum chain and the nonintegrable XX quantum chain in a longitudinal magnetic field. For several magnetic field protocols, the quantum Crooks and Jarzynski relations are numerically tested and fulfilled. As a more interesting situation, we consider the forcing regime where a periodic magnetic field is applied. In the Ising case we give an exact solution in terms of double-confluent Heun functions. We show that the fluctuations of the work performed by the external periodic drift are maximum at a frequency proportional to the amplitude of the field. In the nonintegrable case, we show that depending on the field frequency a sharp transition is observed between a Poisson-limit work distribution at high frequencies toward a normal work distribution at low frequencies.Comment: 10 pages, 13 figure

Exact Distributions for Stochastic Gene Expression Models with Bursting and Feedback

Stochasticity in gene expression can give rise to fluctuations in protein levels and lead to phenotypic variation across a population of genetically identical cells. Recent experiments indicate that bursting and feedback mechanisms play important roles in controlling noise in gene expression and phenotypic variation. A quantitative understanding of the impact of these factors requires analysis of the corresponding stochastic models. However, for stochastic models of gene expression with feedback and bursting, exact analytical results for protein distributions have not been obtained so far. Here, we analyze a model of gene expression with bursting and feedback regulation and obtain exact results for the corresponding protein steady-state distribution. The results obtained provide new insights into the role of bursting and feedback in noise regulation and optimization. Furthermore, for a specific choice of parameters, the system studied maps on to a two-state biochemical switch driven by a bursty input noise source. The analytical results derived thus provide quantitative insights into diverse cellular processes involving noise in gene expression and biochemical switching.Comment: Accepted in Phys. Rev. Let