Cell proliferation and motility are key processes that govern cancer invasion or wound healing. The go-or-grow hypothesis postulates that proliferation and migration spatiotemporally exclude each other. This has been acknowledged, for example, for glioblastoma. In general, two phenotypes that can be of particular importance to progression of aggressive cancers are `high proliferation-low migration' and `low proliferation-high migration', and the mechanisms governing this switching are of great interest in current medical research. A simplification of this phenomenon is assuming that motile cells stop for a period of time to complete cell division, upon which they switch back into the migratory phenotype. A number of biological hypotheses can be formulated to describe what happens when the target site for the daughter cell becomes occupied during the period of cell division. We constructed a family of on-lattice individual based models (IBMs), and derived the mean field approximations, which were expressed by systems of nonlinear delay differential equations or integro-differential equations.
When cell division is aborted when the target site is occupied, the derived equation is a novel delay logistic equation including terms with discrete and distributed delays. The global dynamics was completely described, and it was proven that all feasible solutions converge to the positive equilibrium. The main tools of our proof rely on persistence theory, comparison principles and an L2-perturbation technique. Using local invariant manifolds, a unique heteroclinic orbit was constructed that connects the unstable zero and the stable positive equilibrium, and we showed that these three complete orbits constitute the global attractor. Despite global attractivity, the dynamics is not trivial as we can observe long-lasting transient oscillatory patterns of various shapes.
The Gillespie-algorithm is routinely applied to model biological systems by means of stochastic simulations of IBMs, but this tool was developed for Markovian models. We needed to modify it and design new algorithms. We performed a large number of simulations for a variety of our IBMs, creating a huge synthetic dataset. We systematically explored the behaviours of the IBMs under different biological hypotheses, and investigated whether mean field approximation provides an acceptable prediction of the cell population dynamics. We also investigated numerically the wave-speeds of invasion fronts. A crucial point of cancer research is to find ways to reduce the invasion speed. What we found is that the dependence of the speed of the traveling waves in non-Markovian models with volume exclusion is deviating from what we expect from classical theory. It may have important implications for cancer treatment, since it appears that, counter-intuitively, supressing cell proliferation may be detrimental to the patient under some special circumstances.
The Mackey-Glass equation describes a physiological control systems, where there is a delay in the production of blood cells from the stem cells of the bone marrow. It is a simple looking time delay system with very complicated behavior. We developed a novel approach for chaos control: we proved that with well-chosen control parameters, all solutions of the system can be forced into a domain where the feedback is monotone, and by the theory of monotone semiflows we can guarantee that the system is not chaotic any more. We showed that this domain decomposition method is applicable with the most common control terms, and we proposed a completely new chaos control scheme based on state dependent delays.
Dissemination
The results have been presented at several plenary, keynote and minisymposium talks at international conferences (ICDDEA Timisora, BIOMAT Moscow, ECMI Budapest, ECMTB Lisbon, Complex Networks Cambridge), and published in high profile journals such as Proc Royal Soc A, Complexity, Comm Pure Appl Anal, Chaos. Our results about controlling physiological chaos has been reported in media outlets such as National Geographic.
Exploitation
Based on the results of this Action, the researcher won the most prestigious research grant in Hungary, called Élvonal (Frontier), to build a research team and kickstart this line of research in Hungary. It is expected that long term collaborations will be maintained between the U of Szeged (institution of the researcher) and the U of Oxford (host institution of this Action), as well as the U of St. Andrews (host institution of the Secondment).
