scholarly journals Mixture distributions in a stochastic gene expression model with delayed feedback

2019 ◽  
Author(s):  
Pavol Bokes ◽  
Alessandro Borri ◽  
Pasquale Palumbo ◽  
Abhyudai Singh
Keyword(s):  
Gene Expression ◽  
Differential Equation ◽  
Steady State ◽  
Numerical Solutions ◽  
Stable Steady State ◽  
Active Protein ◽  
Steady State Distribution ◽  
State Distribution ◽  
Asymptotic Results ◽  
Production Delay

AbstractNoise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a quasi-steady-state (QSS) approximation, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable steady state of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable steady states; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.

scholarly journals A stochastic model of gene expression with polymerase recruitment and pause release

2019 ◽  
Author(s):  
Z. Cao ◽  
T. Filatova ◽  
D. A. Oyarzún ◽  
R. Grima
Keyword(s):  
Gene Expression ◽  
Steady State ◽  
Rna Polymerase ◽  
Dosage Compensation ◽  
Gene Dosage ◽  
Relative Sensitivity ◽  
Recruitment Rate ◽  
Time Dependent ◽  
Steady State Distribution ◽  
State Distribution

AbstractTranscriptional bursting is a major source of noise in gene expression. The telegraph model of gene expression, whereby transcription switches between “on” and “off” states, is the dominant model for bursting. Recently it was shown that the telegraph model cannot explain a number of experimental observations from perturbation data. Here we study an alternative model that is consistent with the data and which explicitly describes RNA polymerase recruitment and polymerase pause release, two steps necessary for mRNA production. We derive the exact steady-state distribution of mRNA numbers and an approximate steady-state distribution of protein numbers which are given by generalized hypergeometric functions. The theory is used to calculate the relative sensitivity of the coefficient of variation of mRNA fluctuations for thousands of genes in mouse fibroblasts. This indicates that the size of fluctuations is mostly sensitive to the rate of burst initiation and the mRNA degradation rate. Furthermore we show that (i) the time-dependent distribution of mRNA numbers is accurately approximated by a modified telegraph model with a Michaelis-Menten like dependence of the effective transcription rate on RNA polymerase abundance. (ii) the model predicts that if the polymerase recruitment rate is comparable or less than the pause release rate, then upon gene replication the mean number of RNA per cell remains approximately constant. This gene dosage compensation property has been experimentally observed and cannot be explained by the telegraph model with constant rates.Statement of SignificanceThe random nature of gene expression is well established experimentally. Mathematical modelling provides a means of understanding the factors leading to the observed stochasticity. There is evidence that the classical two-state model of stochastic mRNA dynamics (the telegraph model) cannot describe perturbation experiments and a new model that includes polymerase dynamics has been proposed. In this paper, we present the first detailed study of this model, deriving an exact solution for the mRNA distribution in steady-state conditions, an approximate time-dependent solution and showing the model can explain gene dosage compensation. As well, we use the theory together with transcriptomic data, to deduce which parameters when perturbed lead to a maximal change in the size of mRNA fluctuations.

Kinetic analysis of mechanism of intestinal Na+-dependent sugar transport

1985 ◽  
Vol 248 (5) ◽  
pp. C498-C509 ◽  
Author(s):  
D. Restrepo ◽  
G. A. Kimmich
Keyword(s):  
Experimental Data ◽  
Steady State ◽  
Sugar Transport ◽  
Enzyme Kinetic ◽  
Steady State Distribution ◽  
State Distribution ◽  
Least Squares Fit ◽  
Coupling Stoichiometry ◽  
Rate Limiting ◽  
Kinetics Of

Zero-trans kinetics of Na+-sugar cotransport were investigated. Sugar influx was measured at various sodium and sugar concentrations in K+-loaded cells treated with rotenone and valinomycin. Sugar influx follows Michaelis-Menten kinetics as a function of sugar concentration but not as a function of Na+ concentration. Nine models with 1:1 or 2:1 sodium:sugar stoichiometry were considered. The flux equations for these models were solved assuming steady-state distribution of carrier forms and that translocation across the membrane is rate limiting. Classical enzyme kinetic methods and a least-squares fit of flux equations to the experimental data were used to assess the fit of the different models. Four models can be discarded on this basis. Of the remaining models, we discard two on the basis of the trans sodium dependence and the coupling stoichiometry [G. A. Kimmich and J. Randles, Am. J. Physiol. 247 (Cell Physiol. 16): C74-C82, 1984]. The remaining models are terter ordered mechanisms with sodium debinding first at the trans side. If transfer across the membrane is rate limiting, the binding order can be determined to be sodium:sugar:sodium.

PROCESSOR SHARING G-QUEUES WITH INERT CUSTOMERS AND CATASTROPHES: A MODEL FOR SERVER AGING AND REJUVENATION

2017 ◽  
Vol 31 (4) ◽  
pp. 420-435 ◽  
Author(s):  
J.-M. Fourneau ◽  
Y. Ait El Majhoub
Keyword(s):  
Steady State ◽  
Product Form ◽  
Processor Sharing ◽  
Service Capacity ◽  
Steady State Distribution ◽  
State Distribution ◽  
Signal Arrival ◽  
Open Networks ◽  
Networks Of Queues

We consider open networks of queues with Processor-Sharing discipline and signals. The signals deletes all the customers present in the queues and vanish instantaneously. The customers may be usual customers or inert customers. Inert customers do not receive service but the servers still try to share the service capacity between all the customers (inert or usual). Thus a part of the service capacity is wasted. We prove that such a model has a product-form steady-state distribution when the signal arrival rates are positive.

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