Sławomir Białecki, M.Sc., Eng.

Department of Biosystems and Soft Matter (ZBiMM)
Division of Modelling in Biology and Medicine (PMBM)
position: specialist
telephone: (+48) 22 826 12 81 ext.: 448, 450
room: 321
e-mail: sbialeck

Recent publications
1.Kaźmierczak B., Tsai J.C., Białecki S., The propagation phenomenon of solutions of a parabolic problem on the sphere, MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES, ISSN: 0218-2025, DOI: 10.1142/S0218202518500483, Vol.28, No.10, pp.2001-2067, 2018
Abstract:

In this paper, we study propagation phenomena on the sphere using the bistable reaction–diffusion formulation. This study is motivated by the propagation of waves of calcium concentrations observed on the surface of oocytes, and the propagation of waves of kinase concentrations on the B-cell membrane in the immune system. To this end, we first study the existence and uniqueness of mild solutions for a parabolic initial-boundary value problem on the sphere with discontinuous bistable nonlinearities. Due to the discontinuous nature of reaction kinetics, the standard theories cannot be applied to the underlying equation to obtain the existence of solutions. To overcome this difficulty, we give uniform estimates on the Legendre coefficients of the composition function of the reaction kinetics function and the solution, and a priori estimates on the solution, and then, through the iteration scheme, we can deduce the existence and related properties of solutions. In particular, we prove that the constructed solutions are of C2,1 class everywhere away from the discontinuity point of the reaction term. Next, we apply this existence result to study the propagation phenomenon on the sphere. Specifically, we use stationary solutions and their variants to construct a pair of time-dependent super/sub-solutions with different moving speeds. When applied to the case of sufficiently small diffusivity, this allows us to infer that if the initial concentration of the species is above the inhomogeneous steady state, then the species will exhibit the propagating behavior.

Keywords:

Propagation, sphere, bistable kinetics

Affiliations:
Kaźmierczak B.-IPPT PAN
Tsai J.C.-National Chung Cheng University (TW)
Białecki S.-IPPT PAN
2.Białecki S., Kaźmierczak B., Lipniacki T., Polarization of concave domains by traveling wave pinning, PLOS ONE, ISSN: 1932-6203, DOI: 10.1371/journal.pone.0190372, Vol.12, No.12, pp.e0190372-1-10, 2017
Abstract:

Pattern formation is one of the most fundamental yet puzzling phenomena in physics and biology. We propose that traveling front pinning into concave portions of the boundary of 3-dimensional domains can serve as a generic gradient-maintaining mechanism. Such a mechanism of domain polarization arises even for scalar bistable reaction-diffusion equations, and, depending on geometry, a number of stationary fronts may be formed leading to complex spatial patterns. The main advantage of the pinning mechanism, with respect to the Turing bifurcation, is that it allows for maintaining gradients in the specific regions of the domain. By linking the instant domain shape with the spatial pattern, the mechanism can be responsible for cellular polarization and differentiation.

Affiliations:
Białecki S.-IPPT PAN
Kaźmierczak B.-IPPT PAN
Lipniacki T.-IPPT PAN
3.Białecki S., Kaźmierczak B., Nowicka D., Tsai J.-C., Regularity of solutions to a reaction–diffusion equation on the sphere: the Legendre series approach, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.4390, Vol.40, No.14, pp.5349-5369, 2017
Abstract:

In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two-dimensional sphere close to its poles. This equation is the counterpart of the well-studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C1 smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods.

Keywords:

discontinuous reaction term, stationary fronts, sphere

Affiliations:
Białecki S.-IPPT PAN
Kaźmierczak B.-IPPT PAN
Nowicka D.-IPPT PAN
Tsai J.-C.-National Chung Cheng University (TW)
4.Białecki S., Kaźmierczak B., Tsai J-C., Stationary waves on the sphere, SIAM JOURNAL ON APPLIED MATHEMATICS, ISSN: 0036-1399, DOI: 10.1137/140999384, Vol.75, No.4, pp.1761-1788, 2015
Abstract:

In this paper, we investigate stationary waves on the sphere using the bistable reaction-diffusion system. The motivation of this study arises from the study of activation waves of B cells in immune systems. We analytically establish (i) the existence and uniqueness of stationary waves; (ii) the limiting wave profile for small diffusivity of diffusing species; and (iii) the stability of the constructed stationary waves. The stability result may suggest the critical role of stationary waves in the determination of initial data for initiating propagating waves on the sphere, which is consistent with the numerical results for the B-cell activation model.

Keywords:

stationary wave, sphere, bistable kinetics

Affiliations:
Białecki S.-IPPT PAN
Kaźmierczak B.-IPPT PAN
Tsai J-C.-National Chung Cheng University (TW)

Conference papers
1.Białecki S., Kaźmierczak B., CALCIUM OSCILLATIONS IN A SPATIALLY EXTENDED THREE COMPARTMENT CELL MODEL, KKZMBM2014, Krajowa Konferencja Zastosowań Matematyki w Biologii i Medycynie, 2014-09-23/09-27, Łochów (PL), pp.15-20, 2014
Abstract:

We derive a spatially extended three compartment cell model for evolution of calcium ions concetrations.
To obtain specific form of the fluxes between the compartments, we compare it with the model proposed by
Marhl et al. (2000). We examine numerically the period and shape of oscillations as a function of diffusion
coefficients. We demonstrate a decay of the oscillations at the critical value of diffusion of free calcium ions.

Affiliations:
Białecki S.-IPPT PAN
Kaźmierczak B.-IPPT PAN