Instytut Podstawowych Problemów Techniki
Polskiej Akademii Nauk

Partnerzy

A. Bobrowski

Lublin University of Technology (PL)

Ostatnie publikacje
1.  Bobrowski A., Lipniacki T., Robin-type boundary conditions in transition from reaction-diffusion equations in 3D domains to equations in 2D domains, Journal of Differential Equations, ISSN: 0022-0396, DOI: 10.1016/j.jde.2019.08.022, Vol.268, No.1, pp.239-271, 2019

Streszczenie:
We consider a singular limit of diffusion equations in 3D domains of thickness converging to zero. In the 2D limit the resulting reaction-diffusion equation has a source term resulting from the Robin-type boundary conditions imposed on boundaries of the original 3D domain. The proposed approach can be applied to constructing approximate solutions of diffusion problems in thin planar, cylindrical, or spherical layers between two membranes. As an example we refer to the problem of activation of B lymphocytes, which typically have large nuclei and a thin cytoplasmic layer which can be considered as a spherical shell. For this example, assuming additionally axial symmetry we provide a rigorous convergence theorem in the language of semigroups of operators.

Słowa kluczowe:
semigroups of operators, degenerate convergence, singular perturbation, boundary conditions, thin layers, signaling pathways, phosphorylation–dephosphorylation cycle

Afiliacje autorów:
Bobrowski A. - Lublin University of Technology (PL)
Lipniacki T. - IPPT PAN
140p.
2.  Bobrowski A., Kaźmierczak B., Kunze M., An averaging principle for fast diffusions in domains separated by semi-permeable membranes, MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES, ISSN: 0218-2025, DOI: 10.1142/S0218202517500130, Vol.27, No.4, pp.663-706, 2017

Streszczenie:
We prove an averaging principle which asserts convergence of diffusion processes on domains separated by semi-permeable membranes, when diffusion coefficients tend to infinity while the flux through the membranes remains constant. In the limit, points in each domain are lumped into a single state of a limit Markov chain. The limit chain's intensities are proportional to the membranes' permeability and inversely proportional to the domains' sizes. Analytically, the limit is an example of a singular perturbation in which boundary and transmission conditions play a crucial role. This averaging principle is strongly motivated by recent signaling pathways models of mathematical biology, which are discussed toward the end of the paper.

Słowa kluczowe:
Convergence of sectorial forms and of semigroups of operators, diffusion processes, boundary and transmission conditions, Freidlin–Wentzell averaging principle, singular perturbations, signaling pathways, kinase activity, intracellular calcium dynamics, neurotransmitters

Afiliacje autorów:
Bobrowski A. - Lublin University of Technology (PL)
Kaźmierczak B. - IPPT PAN
Kunze M. - University of Konstanz (DE)
45p.
3.  Bobrowski A., Lipniacki T., Piechór K., Rudnicki R., Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, ISSN: 0022-247X, DOI: 10.1016/j.jmaa.2006.11.043, Vol.333, No.2, pp.753-769, 2007

Streszczenie:
The paper is devoted to a stochastic process introduced in the recent paper by Lipniacki et al. [T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A.R. Brasier, M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol. 238 (2006) 348–367] in modelling gene expression in eukaryotes. Starting from the full generator of the process we show that its distributions satisfy a (Fokker–Planck-type) system of partial differential equations. Then, we construct a c0 Markov semigroup in L1 space corresponding to this system. The main result of the paper is asymptotic stability of the involved semigroup in the set of densities.

Słowa kluczowe:
Piece-wise deterministic process, Stochastic gene expression, Semigroups of operators, Feller semigroups, Dual semigroups, Markov semigroups, Asymptotic stability

Afiliacje autorów:
Bobrowski A. - Lublin University of Technology (PL)
Lipniacki T. - IPPT PAN
Piechór K. - IPPT PAN
Rudnicki R. - Institute of Mathematics, Polish Academy of Sciences (PL)

Kategoria A Plus

IPPT PAN

logo ippt            ul. Pawińskiego 5B, 02-106 Warszawa
  +48 22 826 12 81 (centrala)
  +48 22 826 98 15
 

Znajdź nas

mapka
© Instytut Podstawowych Problemów Techniki Polskiej Akademii Nauk 2024