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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - TEMPOMATH (Temporal delays in mathematical models of cell biology processes)

Teaser

Summary

The primary aim of the action was the construction and analysis of new predictive and verifiable mathematical models that can uncover the effects of time delays upon various cell biology processes. By applying cutting edge functional differential equation techniques to interrogate a number of cell biology processes of huge current interest, including collective cell movement, cancer invasion, and stem cell maturation. The ultimate goal of this research was to develop new mathematical ways to understand and control cell biology processes. We developed mathematical models, tools and theories that can be applied to a wide variety of biological and medical problems, whenever temporal delays are relevant.

Work performed

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

Final results

â€¢ For individual based models of cell migration and proliferation, non-Markovian effects are often ignored, despite their clear biological relevance. The SotA to take into account non-Markovianity was to introduce multiple artificial stages, for example into the cell cycle, and allow linear transition along this chain of stages. We constructed a new family of agent-based models (ABMs) of go-or-grow type, that is flexible enough to incorporate any time distribution for cell biology processes, thus avoiding the use of a large number of non-biological variables.
â€¢ We designed new non-Markovian stochastic simulation algorithms for these ABMs.
â€¢ We derived the corresponding mean-field models, which are expressed by a system of delay differential equations (or integro-differential equations), and developed mathematical techniques for their analysis.
â€¢ We performed a complete global analysis of a novel delay logistic equation with discrete and distributed delays, constructing a heteroclinic orbit and describing the global. We found long lasting oscillatory transients.
â€¢ We discovered a class of blow-up solutions for delay logistic equations with instantaneous positive feedback, and found that, contrary to the classical delay logistic equation, here local stability does not imply global stability.
â€¢ We proposed new chaos control methods based on monotone semiflows and a domain decomposition method. It was applied to the celebrated Mackey-Glass equation of blood cell production, that can support chaotic trajectories. A new chaos control method was proposed based on state dependent delays.
â€¢ Our framework allows the more accurate modelling of various cell biology processes, such as cancer invasion. A better understanding of cell population models with `high proliferation-low migration\' and `low proliferation-high migration\' phenotypes, that are important in the progression of aggressive cancers, may help us improving cancer treatments in the future.