152 research outputs found
Adiabatic reduction of models of stochastic gene expression with bursting
This paper considers adiabatic reduction in both discrete and continuous
models of stochastic gene expression. In gene expression models, the concept of
bursting is a production of several molecules simultaneously and is generally
represented as a compound Poisson process of random size. In a general
two-dimensional birth and death discrete model, we prove that under specific
assumptions and scaling (that are characteristics of the mRNA-protein system)
an adiabatic reduction leads to a one-dimensional discrete-state space model
with bursting production. The burst term appears through the reduction of the
first variable. In a two-dimensional continuous model, we also prove that an
adiabatic reduction can be performed in a stochastic slow/fast system. In this
gene expression model, the production of mRNA (the fast variable) is assumed to
be bursty and the production of protein (the slow variable) is linear as a
function of mRNA. When the dynamics of mRNA is assumed to be faster than the
protein dynamics (due to a mRNA degradation rate larger than for the protein)
we prove that, with the appropriate scaling, the bursting phenomena can be
transmitted to the slow variable. We show that the reduced equation is either a
stochastic differential equation with a jump Markov process or a deterministic
ordinary differential equation depending on the scaling that is appropriate.
These results are significant because adiabatic reduction techniques seem to
have not been applied to a stochastic differential system containing a jump
Markov process. Last but not least, for our particular system, the adiabatic
reduction allows us to understand what are the necessary conditions for the
bursting production-like of protein to occur.Comment: 24 page
Boundary value for a nonlinear transport equation emerging from a stochastic coagulation-fragmentation type model
We investigate the connection between two classical models of phase
transition phenomena, the (discrete size) stochastic Becker-D\"oring, a
continous time Markov chain model, and the (continuous size) deterministic
Lifshitz-Slyozov model, a nonlinear transport partial differential equation.
For general coefficients and initial data, we introduce a scaling parameter and
prove that the empirical measure associated to the stochastic Becker-D\"oring
system converges in law to the weak solution of the Lifshitz-Slyozov equation
when the parameter goes to 0. Contrary to previous studies, we use a weak
topology that includes the boundary of the state space (\ie\ the size )
allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov
model in the case of incoming characteristics. The condition reads where is the volume distribution
function, solution of the Lifshitz-Slyozov equation, and the
aggregation and fragmentation rates, the concentration of free particles
and a nucleation constant emerging from the microscopic model. It is
the main novelty of this work and it answers to a question that has been
conjectured or suggested by both mathematicians and physicists. We emphasize
that this boundary value depends on a particular scaling (as opposed to a
modeling choice) and is the result of a separation of time scale and an
averaging of fast (fluctuating) variables.Comment: 42 pages, 3 figures, video on supplementary materials at
http://yvinec.perso.math.cnrs.fr/video.htm
The Becker-Döring process: law of large numbers and non-equilibrium potential
In this note, we prove alaw of large numbersfor an infinite chemical reactionnetwork for phase transition problems called the stochastic Becker-Döring process.Under a general condition on the rate constants we show the convergence in lawand pathwise convergence of the process towards the deterministic Becker-Döringequations. Moreover, we prove that the non-equilibrium potential, associated to thestationary distribution of the stochastic Becker-Döring process, approaches the rela-tive entropy of the deterministic limit model. Thus, the phase transition phenomenathat occurs in the infinite dimensional deterministic modelis also present in the finitestochastic model.In this note, we prove alaw of large numbersfor an infinite chemical reactionnetwork for phase transition problems called the stochastic Becker-Döring process.Under a general condition on the rate constants we show the convergence in lawand pathwise convergence of the process towards the deterministic Becker-Döringequations. Moreover, we prove that the non-equilibrium potential, associated to thestationary distribution of the stochastic Becker-Döring process, approaches the rela-tive entropy of the deterministic limit model. Thus, the phase transition phenomenathat occurs in the infinite dimensional deterministic modelis also present in the finitestochastic model
Adiabatic reduction of a model of stochastic gene expression with jump Markov process
This paper considers adiabatic reduction in a model of stochastic gene
expression with bursting transcription considered as a jump Markov process. In
this model, the process of gene expression with auto-regulation is described by
fast/slow dynamics. The production of mRNA is assumed to follow a compound
Poisson process occurring at a rate depending on protein levels (the phenomena
called bursting in molecular biology) and the production of protein is a linear
function of mRNA numbers. When the dynamics of mRNA is assumed to be a fast
process (due to faster mRNA degradation than that of protein) we prove that,
with appropriate scalings in the burst rate, jump size or translational rate,
the bursting phenomena can be transmitted to the slow variable. We show that,
depending on the scaling, the reduced equation is either a stochastic
differential equation with a jump Poisson process or a deterministic ordinary
differential equation. These results are significant because adiabatic
reduction techniques seem to have not been rigorously justified for a
stochastic differential system containing a jump Markov process. We expect that
the results can be generalized to adiabatic methods in more general stochastic
hybrid systems.Comment: 17 page
On the bursting of gene products
In this article we demonstrate that the so-called bursting production of
molecular species during gene expression may be an artifact caused by low time
resolution in experimental data collection and not an actual burst in
production. We reach this conclusion through an analysis of a two-stage and
binary model for gene expression, and demonstrate that in the limit when mRNA
degradation is much faster than protein degradation they are equivalent. The
negative binomial distribution is shown to be a limiting case of the binary
model for fast "on to off" state transitions and high values of the ratio
between protein synthesis and degradation rates. The gene products population
increases by unity but multiple times in a time interval orders of magnitude
smaller than protein half-life or the precision of the experimental apparatus
employed in its detection. This rare-and-fast one-by-one protein synthesis has
been interpreted as bursting.Comment: 13 page
Human Luteinizing Hormone and Chorionic Gonadotropin Display Biased Agonism at the LH and LH/CG Receptors.
Human luteinizing hormone (LH) and chorionic gonadotropin (hCG) have been considered biologically equivalent because of their structural similarities and their binding to the same receptor; the LH/CGR. However, accumulating evidence suggest that LH/CGR differentially responds to the two hormones triggering differential intracellular signaling and steroidogenesis. The mechanistic basis of such differential responses remains mostly unknown. Here, we compared the abilities of recombinant rhLH and rhCG to elicit cAMP, β-arrestin 2 activation, and steroidogenesis in HEK293 cells and mouse Leydig tumor cells (mLTC-1). For this, BRET and FRET technologies were used allowing quantitative analyses of hormone activities in real-time and in living cells. Our data indicate that rhLH and rhCG differentially promote cell responses mediated by LH/CGR revealing interesting divergences in their potencies, efficacies and kinetics: rhCG was more potent than rhLH in both HEK293 and mLTC-1 cells. Interestingly, partial effects of rhLH were found on β-arrestin recruitment and on progesterone production compared to rhCG. Such a link was further supported by knockdown experiments. These pharmacological differences demonstrate that rhLH and rhCG act as natural biased agonists. The discovery of novel mechanisms associated with gonadotropin-specific action may ultimately help improve and personalize assisted reproduction technologies
Molecular Distributions in Gene Regulatory Dynamics
We show how one may analytically compute the stationary density of the
distribution of molecular constituents in populations of cells in the presence
of noise arising from either bursting transcription or translation, or noise in
degradation rates arising from low numbers of molecules. We have compared our
results with an analysis of the same model systems (either inducible or
repressible operons) in the absence of any stochastic effects, and shown the
correspondence between behaviour in the deterministic system and the stochastic
analogs. We have identified key dimensionless parameters that control the
appearance of one or two steady states in the deterministic case, or unimodal
and bimodal densities in the stochastic systems, and detailed the analytic
requirements for the occurrence of different behaviours. This approach
provides, in some situations, an alternative to computationally intensive
stochastic simulations. Our results indicate that, within the context of the
simple models we have examined, bursting and degradation noise cannot be
distinguished analytically when present alone.Comment: 14 pages, 12 figures. Conferences: "2010 Annual Meeting of The
Society of Mathematical Biology", Rio de Janeiro (Brazil), 24-29/07/2010.
"First International workshop on Differential and Integral Equations with
Applications in Biology and Medicine", Aegean University, Karlovassi, Samos
island (Greece), 6-10/09/201
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