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Streszczenie:
The existence and properties of intracellular waves of increased free cytoplasmic calcium concentration (calcium waves) are strongly affected by the binding and unbinding of calcium ions to a multitude of different buffers in the cell. These buffers can be mobile or immobile and, in general, have multiple binding sites that are not independent. Previous theoretical studies have focused on the case when each buffer molecule binds a single calcium ion. In this study, we analyze how calcium waves are affected by calcium buffers with two non-independent binding sites, and show that the interactions between the calcium binding sites can result in the emergence of new behaviors. In particular, for certain combinations of kinetic parameters, the profiles of buffer molecules with one calcium ion bound can be non-monotone.

Słowa kluczowe:
Reaction-diffusion systems, Buffered calcium systems

Streszczenie:
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.

Słowa kluczowe:
Propagation, sphere, bistable kinetics

Streszczenie:
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.

Słowa kluczowe:
discontinuous reaction term, stationary fronts, sphere

Streszczenie:
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.

Słowa kluczowe:
stationary wave, sphere, bistable kinetics

Streszczenie:
Accumulating data indicate that cancer stem cells contribute to tumor chemoresistance and their persistence alters clinical outcome. Our previous study has shown that ovarian cancer may be initiated by ovarian cancer initiating cells (OCIC) characterized by surface antigen CD44 and c-KIT (CD117). It has been experimentally demonstrated that a microRNA, namely miR-193a, targets c-KIT mRNA for degradation and could play a crucial role in ovarian cancer development. How miR-193a is regulated is poorly understood and the emerging picture is complex. To unravel this complexity, we propose a mathematical model to explore how estrogen-mediated up-regulation of another target of miR-193a, namely E2F6, can attenuate the function of miR-193a in two ways, one through a competition of E2F6 and c-KIT transcripts for miR-193a, and second by binding of E2F6 protein, in association with a polycomb complex, to the promoter of miR-193a to down-regulate its transcription. Our model predicts that this bimodal control increases the expression of c-KIT and that the second mode of epigenetic regulation is required to generate a switching behavior in c-KIT and E2F6 expressions. Additional analysis of the TCGA ovarian cancer dataset demonstrates that ovarian cancer patients with low expression of EZH2, a polycomb-group family protein, show positive correlation between E2F6 and c-KIT. We conjecture that a simultaneous EZH2 inhibition and anti-estrogen therapy can constitute an effective combined therapeutic strategy against ovarian cancer.

Słowa kluczowe:
Ovarian cancer, Messenger RNA, Epigenetics, DNA methylation, Estrogens, Gene expression, Cancer treatment, Carcinogenesis