For the sake of computational efficiency and for theoretical purposes, in mathematical modelling, the DiracDelta distributions are often utilized as a replacement for cells or vesicles, since the... Show moreFor the sake of computational efficiency and for theoretical purposes, in mathematical modelling, the DiracDelta distributions are often utilized as a replacement for cells or vesicles, since the size of cells or vesicles is much smaller than the size of the surrounding tissues. Here, we consider the scenario that the cell or the vesicle releases the diffusive compounds to the immediate environment, which is modelled by the diffusion equation. Typically, one separates the intracellular and extracellular environment and uses homogeneous Neumann boundary condition for the cell boundary (so-called spatial exclusion model), while the point source model neglects the intracellular environment. We show that extra conditions are needed such that the solutions to the two models are consistent. We prove a necessary and sufficient condition for the consistency. Suggested by the numerical results, we conclude that an initial condition in the form of Gaussian kernel in the point source model compensates for a time-delay discrepancy between the solutions to the two models in the numerical solutions. Various approaches determining optimal amplitude and variance of the Gaussian kernel have been discussed. Show less
Epithelial-mesenchymal transition (EMT) and immunoevasion through upregulation of programmed death-ligand 1 (PD-L1) are important drivers of cancer progression. While EMT has been proposed to... Show moreEpithelial-mesenchymal transition (EMT) and immunoevasion through upregulation of programmed death-ligand 1 (PD-L1) are important drivers of cancer progression. While EMT has been proposed to facilitate PD-L1-mediated immunosuppression, molecular mechanisms of their interaction remain obscure. Here, we provide insight into these mechanisms by proposing a mathematical model that describes the crosstalk between EMT and interferon gamma (IFN gamma)-induced PD-L1 expression. Our model shows that via interaction with microRNA-200 (miR-200), the multi-stability of the EMT regulatory circuit is mirrored in PD-L1 levels, which are further amplified by IFN gamma stimulation. This IFN gamma-mediated effect is most prominent for cells in a fully mesenchymal state and less strong for those in an epithelial or partially mesenchymal state. In addition, bidirectional crosstalk between miR-200 and PD-L1 implies that IFN gamma stimulation allows cells to undergo EMT for lower amounts of inducing signal, and the presence of IFN gamma accelerates EMT and decelerates mesenchymal-epithelial transition (MET). Overall, our model agrees with published findings and provides insight into possible mechanisms behind EMT-mediated immune evasion, and primary, adaptive, or acquired resistance to immunotherapy. Our model can be used as a starting point to explore additional crosstalk mechanisms, as an improved understanding of these mechanisms is indispensable for developing better diagnostic and therapeutic options for cancer patients. Show less
We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right... Show moreWe investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity λ. The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say ν∗λνλ∗. The aim of this paper is to prove that the map λ↦ν∗λλ↦νλ∗ is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression. Show less
An algorithm is proposed for multi-objective optimisation of Lipschitz objective functions that each satisfy a Lipschitz condition of which a Lipschitz constant is a priori known. The number of... Show moreAn algorithm is proposed for multi-objective optimisation of Lipschitz objective functions that each satisfy a Lipschitz condition of which a Lipschitz constant is a priori known. The number of function evaluations is reduced by determining a good next point of evaluation using an Expected Hypervolume Improvement (EHVI) approach. It is closely related to Shubert’s Algorithm for single objective optimisation on one-dimensional decision space, but sampling sequences can be slightly different. Show less
Gulgowksi, J.; Hille, S.C.; Szarek, T.; Ziemlanska, M.A. 2019
