Zuzanna Szymańska

Mathematical modeling of heat shock protein synthesis in response to temperature change.

One of the most important questions in cell biology is how cells cope with rapid changes in their environment. The range of common molecular responses includes a dramatic change in the pattern of gene expression and the elevated synthesis of so-called heat shock (or stress) proteins (HSPs). Induction of HSPs increases cell survival under stress conditions [Morimoto, R.I., 1993. Cells in stress: transcriptional activation of heat shock genes. Science 259, 1409-1410]. In this paper we propose a mathematical model of heat shock protein synthesis induced by an external temperature stimulus. Our model consists of a system of nine nonlinear ordinary differential equations describing the temporal evolution of the key variables involved in the regulation of HSP synthesis. Computational simulations of our model are carried out for different external temperature stimuli. We compare our model predictions with experimental data for three different cases-one corresponding to heat shock, the second corresponding to slow heating conditions and the third corresponding to a short heat shock (lasting about 40 min). We also present our model predictions for heat shocks carried out up to different final temperatures and finally we present a new hypothesis concerning the molecular response to stress that explains some phenomena observed in experiments.

Mathematical modelling of the influence of heat shock proteins on cancer invasion of tissue

Tumour cell invasion is crucial for cancer metastasis, which is the main cause of cancer mortality. An important group of proteins involved in cancer invasion are the Heat Shock Proteins (HSPs). According to experimental data, inhibition of one of these proteins, Hsp90, slows down cancer cells while they are invading tissue, but does not affect the synthesis of matrix metalloproteinases (MMP2 and MMP9), which are very important for cancer metastasis, acting as extracellular matrix (ECM) degrading enzymes. To test different biological hypotheses regarding how precisely Hsp90 influences tumour invasion, in this paper we use a model of solid tumour growth which accounts for the interactions between Hsp90 dynamics and the migration of cancer cells and, alternatively, between Hsp90 dynamics and the synthesis of matrix degrading enzymes (MDEs). The model consists of a system of reaction–diffusion-taxis partial differential equations describing interactions between cancer cells, MDE, and the host tissue (ECM). Using numerical simulations we investigate the effects of the administration of Hsp90 inhibitors on the dynamics of tumour invasion. Alternative mechanisms of reduction of cancer invasiveness result in different simulated patterns of the invading tumour cells. Therefore, predictions of the model suggest experiments which might be performed to develop a deeper understanding of the tumour invasion process.

Modelling the efficacy of hyperthermia treatment.

Multimodal oncological strategies which combine chemotherapy or radiotherapy with hyperthermia, have a potential of improving the efficacy of the non-surgical methods of cancer treatment. Hyperthermia engages the heat-shock response (HSR) mechanism, the main component of which are heat-shock proteins. Cancer cells have already partially activated HSR, thereby hyperthermia may be more toxic to them relative to normal cells. On the other hand, HSR triggers thermotolerance, i.e. hyperthermia-treated cells show an impairment in their susceptibility to a subsequent heat-induced stress. This poses questions about efficacy and optimal strategy for anti-cancer therapy combined with hyperthermia treatment. To address these questions, we adapt our previous HSR model and propose its stochastic extension. We formalize the notion of a HSP-induced thermotolerance. Next, we estimate the intensity and the duration of the thermotolerance. Finally, we quantify the effect of a multimodal therapy based on hyperthermia and a cytotoxic effect of bortezomib, a clinically approved proteasome inhibitor. Consequently, we propose an optimal strategy for combining hyperthermia and proteasome inhibition modalities. In summary, by a mathematical analysis of HSR, we are able to support the common belief that the combination of cancer treatment strategies increases therapy efficacy.

Mathematical modeling of the intracellular protein dynamics: the importance of active transport along microtubules.

In this paper we propose a mathematical model of protein and mRNA transport inside a cell. The spatio-temporal model takes into account the active transport along microtubules in the cytoplasm as well as diffusion and is able to reproduce the oscillatory changes in protein concentration observed in many experimental data. In the model the protein and the mRNA interact with each other that allows us to classify the model as a simple gene regulatory network. The proposed model is generic and may be adapted to specific signaling pathways. On the basis of numerical simulations, we formulate a new hypothesis that the oscillatory dynamics is allowed by the mRNA active transport along microtubules from the nucleus to distant locations.

Large-Scale Parallel Simulations of 3D Cell Colony Dynamics: The Cellular Environment

The authors present a large-scale, hybrid 3D model for simulating dynamics of cell colonies growing in and interacting with a variable environment. For this purpose, they extended an earlier mathematical and computational formulation of a cell colony model to incorporate the cellular environment modeled in a continuous manner. A mathematical description based on partial differential equations is formulated for selected important components of the environment. Such extension is necessary to deal with complex biological processes such as cancer growth, where a number of scales need to be considered (subcellular, cellular, and tissue). The authors show how a continuous model can be efficiently solved on a massively parallel processing system. They also present computational methods that couple discrete and continuous descriptions while maintaining high scalability of the resulting application.

A General Framework for Multiscale Modeling of Tumor–Immune System Interactions

In this paper we review methods that allow the construction of a consistent set of models that may describe the interactions between a tumor and the immune system on microscopic, mesoscopic, and macroscopic scales. The presented structures may be a basis for a description on the sub–cellular, cellular, and macroscopic levels. Important open problems are indicated.

Nonlocal models of biological phenomena: Comment on “On the interplay between mathematical and biology, hallmarks towards a new system biology” by Bellomo, Elaiw, Althiabi and Alghamdi

The Bellomo et al. [3] review provides a general strategy for modelling living systems with particular attention to the description of biological processes at microscopic scale. Descriptions in different scales seem to be deeply justified because biological processes are inherently multi-scale. For instance, if we consider processes such as diseases we find that they are present over many biological scales. First symptoms are almost always observed at the clinical (macroscopic) level, but if we look more closely at the origins of those diseases, it is easy to see that the pathological process often begins with intracellular alterations (microscopic level). Therefore, there is a need for new mathematical tools that are suitable to capture such complexities. The methodology proposed by Bellomo et al. [3] is based on kinetic theory for active particles and multi-scale links between different levels of description. The mathematical structures may be the proper kinetic equations for both closed or open systems. The important step is understanding the essence of multi-scale approaches and applying the asymptotic methods to derive of macro-scale models (cf. [3,4] and also [2,8] and references therein) for which the experimental identification of parameters is usually easier. These macro-scale models may be connected with a microscopic level description, i.e. the level of interacting individual agents (“active particles”) of a living system (cf. [3,4]). Typically, biological processes are so complex that when constructing a mathematical model we need to make many simplifying assumptions. The trick is to propose a simplification which will make the processes easier to understand but that will not counterfeit it. An example of a mathematical technique that can be extremely useful in describing many phenomena in biological systems, and

Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-

Analysis of Immunotherapy Models in the Context of Cancer Dynamics

\upkappa