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Two reduction methods to simplify complex ODE mathematical models of biological networks and a case study: The G1/S checkpoint/DNA-damage signal transduction pathways A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy in Computer Sience at Lincoln University by Mutaz Khazaaleh Lincoln University 2018 Abstract of a thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy in Computer Science Two reduction methods to simplify complex ODE mathematical models of biological networks and a case study: The G1/S checkpoint/DNA-damage signal transduction pathways by Mutaz Khazaaleh Model reduction is a hot topic in studies of biological systems. By reducing the complexity of detailed models through finding important elements, developing multi-level models and keeping the soul (biological meaning) of the model, model reduction can help answer many important questions raised about these systems. This thesis addressed several issues related to complex systems, complexity, biological systems complexity and the different reduction methods used to simplify biological models. It gives a brief review of the biological background of the regulation of the cell cycle and proposes two reduction methods to simplify the corresponding complex ODE mathematical models. The first method is based on a hierarchical representation and lumping approach, and the Second method uses a time windows for identifying active and inactive periods of a system and logical models. For the purpose of the current study, biological network model reduction is defined as any method designed to reproduce the original model through a set of smaller models that collectively produce the same behaviour as the original model. It does this, by reducing one or more dimensions of the biological network model’s complexity (i.e., reducing the number of species, number of reactions or model run time). In the first reduction method, based on hierarchical representation and a lumping approach, we used G1/S checkpoint pathway as a case study to present this reduction method. This consisted of two parts; the first part reorganised the biological network as a hierarchy (levels) based on the protein binding relations, and this allowed us to model the biological network with different levels of abstraction. The ii second part applied different levels (level 1, 2, 3) of lumping the species together depending on the level of the hierarchy. We propose and simulate the reduced models for level-1, level-2 and level-3 of lumping for the G1/S checkpoint pathway and evaluate the biological plausibility of the proposed method by comparing the results with the original (ODE) model of Iwamoto et al. (2011). The results of the G1/S checkpoint pathway with or without DNA-damage for reduced models of level-1, level-2 and level-3 of lumping have agreed and are consistent with the original model results and with biological experiments Iwamoto et al. (2011). Therefore, the reduced model (level-1) can be used to evaluate the effects of DNA damage on G1 progression. It is suggested that the proposed method is suitable to reduce complex biological networks. Moreover, the reduced model is more efficient to run and generate solutions than the original ODE model. In second reduction method we used time windows and logical models. Time windows were used to identify regions of slow activity such as gene expression, fast activity such as protein signalling and no activity. In general, most knowledge about regulatory and signalling networks is of a qualitative nature, which allows these networks to be represented by logical models, where the state of a molecule is either 0 (inactive) or 1 (active). These simpler models have many advantages, such as ; they do not require kinetic parameters and are able to capture the essential behaviour of a network; however, they are not able to reproduce detailed time courses for the concentration levels of molecules. Nowadays, however, experiments yield more and more quantitative data, so many quantitative models have been built and most of these models are very complex. An obvious question, therefore, is how to reduce complex quantitative models so they can be used to explain and predict the outcome of these experiments. Here, we present a way of reducing complex quantitative models into logical (Boolean) models, where the use of time windows allows a reduction in time complexity and logical representation allows to do so without kinetic parameters in the model. The method is standardised and can readily be applied to complex quantitative models. Moreover, we discuss and generalise existing theoretical results on the relations between the Boolean and continuous models. As a case study, a continuous ODE model is reduced into a logical model describing the G1/S checkpoint with and without DNA damage. We discuss how this model can explain and predict the G1/S checkpoint behaviour with DNA damage, including oscillations for some molecules and the cell fate. This shows the reduced model is still useful for obtaining biological insights and is easier to run and analyse. This new method greatly helps to simplify complex quantitative models into simpler models and can facilitate the interactions between the modelling and the experiments. Moreover, it helps researchers iii and those who build models to focus on understanding and representing system behaviour rather than on determining the values for the kinetic parameters. While the analysis presented was in terms of biological networks, it should be noted that the specific example used was chosen to explain our two reduction methods. However, the two methods used could be more generally applied to the reduction of ODEs of biological systems and, even more generally, to most complex systems. Relaxing the struggle with the complexity of mathematical models is possible and the proposed reduction methods have the potential to make an impact across many fields of biomedical research. Keywords: Biological systems, biological networks, complexity, model reduction, hierarchy of protein complex formation (abstraction), Ordinary Differential Equations (ODEs), time windows, Boolean model, G1/S checkpoint, DNA damage. iv Acknowledgements First of all, all praise and thanks are due to Allah, the Almighty, having made everything possible by giving me strength and patience to accomplish this work. I must acknowledge and thank the Almighty Allah for blessing and grace. I would like to thank my supervisor, Professor Sandhya Samarasinghe, for the opportunity to undertake this thesis, and for her expert guidance and support in all aspects and stages of research. Without her supervision this research would not have been possible. I would like to thank my co-supervisor, Professor Don Kulasiri, for his expert help and support. To the staff at Lincoln University, thank you. This University offered me services and all the facilities I needed during my PhD thesis. The staff were very friendly and helpful. Special thanks also go to Caitriona Cameron at Teaching and Learning Services. I would like to thank Mom and Dad. Thanks for your love and support. And for teaching me to finish what I start. In addition, I want to direct many thanks to my brothers, sisters and friends, especially Professor Abdel- Rahman Abu-Melhim, for their continuous encouragement and support. Last but not least, great thanks go to my wife, for her love, patience, interest in my work and support. I would like to include my beautiful children (Karam, Jood, Hala, Seraj and Murad) in this acknowledgement. Finally, I dedicate this work to my father, who is a great man. Thank you all! Mutaz Khazaaleh July, 2018 v Table of Contents Abstract ....................................................................................................................................... ii Acknowledgements ...................................................................................................................... v Table of Contents ........................................................................................................................ vi List of Tables ................................................................................................................................ x List of Figures ............................................................................................................................. xii Abbreviations ……......................................................................................................................... xvi Chapter 1 Introduction ............................................................................................................ 1 1.1 General Introduction ................................................................................................................1 1.2 Overview ..................................................................................................................................3