Piecewise Deterministic Processes in Biological Models

Rudnicki, Ryszard; Tyran‐Kamińska, Marta

doi:10.1007/978-3-319-61295-9

Cited by 54 publications

(80 citation statements)

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“…In particular, generators of substochastic semigroups are resolvent positive and the Hille-Yosida theorem implies the following result (see e.g. [34,Theorem 4.4]): A linear operator (G, D(G)) is the generator of a substochastic semigroup on L 1 if and only if D(G) is dense in L 1 , the operator G is resolvent positive, and…”

Section: Preliminariesmentioning

confidence: 99%

“…Although stability and ergodicity of PDMPs are developed in great generality in [15], the general problem of existence of absolutely continuous invariant measures has not been treated at all except for specific examples, see [30] for a recent account of different models where the existence is known. If we know already that the process induces a substochastic semigroup then we can use the methods presented in [32,34] to get existence of invariant densities. To complete our general approach we also study in Section 2.4 relationships between invariant densities of the continuous time process and of the process observed at jump times; our results correspond to the results from [14,17], but we do not assume that the process is non-explosive and we look for absolutely continuous invariant measures.…”

Section: Introductionmentioning

confidence: 99%

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Densities for piecewise deterministic Markov processes with boundary

Gwiżdż

Tyran‐Kamińska

2019

Journal of Mathematical Analysis and Applications

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We study the existence of densities for distributions of piecewise deterministic Markov processes. We also obtain relationships between invariant densities of the continuous time process and that of the process observed at jump times. In our approach we use functional-analytic methods and the theory of linear operator semigroups. By imposing general conditions on the characteristics of a given Markov process, we show the existence of a substochastic semigroup describing the evolution of densities for the process and we identify its generator. Our main tool is a new perturbation theorem for substochastic semigroups, where we perturb both the action of the generator and of its domain, allowing to treat general transport-type equations with non-local boundary conditions. A couple of particular examples illustrate our general results.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Densities for piecewise deterministic Markov processes with boundary

Gwiżdż

Tyran‐Kamińska

2019

Journal of Mathematical Analysis and Applications

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“…where y is the pre-burst level and x > y is any admissible post-burst level. Next we discuss how post-protein level x can be sampled from the distribution (29). One option would be to use the inversion sampling method directly on (29).…”

Section: 1mentioning

confidence: 99%

“…Next we discuss how post-protein level x can be sampled from the distribution (29). One option would be to use the inversion sampling method directly on (29). This would involve equating the right-hand side of (29) to a randomly drawn variate from the unit interval and solving in terms of the post-protein level x.…”

Section: 1mentioning

confidence: 99%

“…to the binary variable indicating whether the gene is On and Off, there is another time-dependent numerical variable indicating the amount of protein in the cell. In many models the protein amount is considered to be a discrete variable [18][19][20][21], but here we use a hybrid discrete-continuous formalism [22][23][24][25][26][27][28][29] and treat protein level as a continuous variable; we refer to it also as concentration. The protein concentration increases whenever the gene is On and decreases when the gene is Off.…”

Section: Introductionmentioning

confidence: 99%

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Maintaining Gene Expression Levels by Positive Feedback in Burst Size in the Presence of Infinitesimal Delay

Bokes

2018

Preprint

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Synthesis of individual molecules in the expression of genes often occurs in bursts of multiple copies. Gene regulatory feedback can affect the frequency with which these bursts occur or their size. Whereas frequency regulation has traditionally received more attention, we focus specifically on the regulation of burst size. It turns out that there are (at least) two alternative formulations of feedback in burst size. In the first, newly produced molecules immediately partake in feedback, even within the same burst. In the second, there is no within-burst regulation due to what we call infinitesimal delay. We describe both alternatives using a minimalistic Markovian drift-jump framework combining discrete and continuous dynamics. We derive detailed analytic results and efficient simulation algorithms for positive noncooperative autoregulation (whether infinitesimally delayed or not). We show that at steady state both alternatives lead to a gamma distribution of protein level. The steady-state distribution becomes available only after a transcritical bifurcation point is passed. Interestingly, the onset of the bifurcation is postponed by the inclusion of infinitesimal delay.

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