Bayesian inference for dynamic transcriptional regulation; the Hes1 system as a case study.

Motivation: In this study we address the problem of estimating the parameters of regulatory networks and provide the first application of Markov chain Monte Carlo (MCMC) methods to experimental data. As a case study we consider a stochastic model of the Hes1 system expressed in terms of stochastic differential equations (SDEs) to which rigorous likelihood methods of inference can be applied. When fitting continuous-time stochastic models to discretely observed time series the lengths of the sampling intervals are important, and much of our study addresses the problem when the data are sparse. Results: We estimate the parameters of an autoregulatory network providing results both for simulated and real experimental data from the Hes1 system. We develop an estimation algorithm using Markov chain Monte Carlo techniques which are flexible enough to allow for the imputation of latent data on a finer time scale and the presence of prior information about parameters which may be informed from other experiments as well as additional measurement error. Availability: Supplementary information is submitted with the paper. Contact

Bayesian inference of biochemical kinetic parameters using the linear noise approximation

Background Fluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data. Results We use the linear noise approximation to model biochemical reactions through a stochastic dynamic model which essentially approximates a diffusion model by an ordinary differential equation model with an appropriately defined noise process. An explicit formula for the likelihood function can be derived allowing for computationally efficient parameter estimation. The proposed algorithm is embedded in a Bayesian framework and inference is performed using Markov chain Monte Carlo. Conclusion The major advantage of the method is that in contrast to the more established diffusion approximation based methods the computationally costly methods of data augmentation are not necessary. Our approach also allows for unobserved variables and measurement error. The application of the method to both simulated and experimental data shows that the proposed methodology provides a useful alternative to diffusion approximation based methods

On the convergence to Walrasian prices in random matching Edgeworthian economies

We show that for a specific class of random matching Edgeworthian economies, the expectation of the limiting equilibrium price coincides with the equilibrium price of the related Walrasian economies. This result extends to the study of economies in the presence of uncertainty within the multi-period Arrow-Debreu model, allowing to understand the dynamics of how beliefs survive and propagate through the market

Bargaining skills in an Edgeworthian economy

We present a model of an Edgeworthian exchange economy where two goods are traded in a market place. For a specific class of random matching Edgeworthian economies, the expectation of the limiting equilibrium price coincides with that of related Walrasian economies. The novelty of our model is that we associate a bargaining skill factor to each participant which brings up a game alike the prisoner’s dilemma into the usual Edgeworth exchange economy. We analyze the effect of the bargaining skill factors in the variations of the individual amount of goods and in the increase of the value of their utilities. Finally, we let the bargaining skills of the participants evolve along the trades and we study their variation

Dynamic analysis of stochastic transcription cycles

In individual mammalian cells the expression of some genes such as prolactin is highly variable over time and has been suggested to occur in stochastic pulses. To investigate the origins of this behavior and to understand its functional relevance, we quantitatively analyzed this variability using new mathematical tools that allowed us to reconstruct dynamic transcription rates of different reporter genes controlled by identical promoters in the same living cell. Quantitative microscopic analysis of two reporter genes, firefly luciferase and destabilized EGFP, was used to analyze the dynamics of prolactin promoter-directed gene expression in living individual clonal and primary pituitary cells over periods of up to 25 h. We quantified the time-dependence and cyclicity of the transcription pulses and estimated the length and variation of active and inactive transcription phases. We showed an average cycle period of approximately 11 h and demonstrated that while the measured time distribution of active phases agreed with commonly accepted models of transcription, the inactive phases were differently distributed and showed strong memory, with a refractory period of transcriptional inactivation close to 3 h. Cycles in transcription occurred at two distinct prolactin-promoter controlled reporter genes in the same individual clonal or primary cells. However, the timing of the cycles was independent and out-of-phase. For the first time, we have analyzed transcription dynamics from two equivalent loci in real-time in single cells. In unstimulated conditions, cells showed independent transcription dynamics at each locus. A key result from these analyses was the evidence for a minimum refractory period in the inactive-phase of transcription. The response to acute signals and the result of manipulation of histone acetylation was consistent with the hypothesis that this refractory period corresponded to a phase of chromatin remodeling which significantly increased the cyclicity. Stochastically timed bursts of transcription in an apparently random subset of cells in a tissue may thus produce an overall coordinated but heterogeneous phenotype capable of acute responses to stimuli