Mathematical modelling in systems biology : cell cycle regulation during leaf development in Arabidopsis

In this thesis, we studied a mathematical modelling approach of systems biology in plants. We have concentrated on two different issues related to the cell cycle and cell division (especially in the plant Arabidopsis). The first issue is that of the epidermal cell population in the Arabidopsis leaf and the second issue deals with gene networks which play an important role during the cell cycle. The chapters are grouped into four parts. In Part I, we described the cell cycle as the series of events that takes place in a cell leading to its division and duplication. We also stated the general objective of the study. We addressed the various aspects of the problems and the key factors that are assumed to influence or cause the problems. We provided a comprehensive mathematical framework in Part II to be used in the other chapters for the modelling, simulation and analysing purposes. Here we introduced and studied Michaelis-Menten kinetics (a model of enzyme kinetics) and the quasi-steady state assumption to reduce the complexity of the model. We also introduced two basic mathematical models for the growth of cell size in plants. In Part III, we considered two case studies related to the cell cycle and cell division in Arabidopsis. The first case study is the temporal control of epidermal cell divisions in the Arabidopsis leaf. The growth of plant organs is the result of two processes acting on the cellular level, namely cell division and cell expansion. The precise nature of the interaction between these two processes is still largely unknown as it is experimentally challenging to disentangle them. The lower epidermal tissue layer of the Arabidopsis leaf is composed of two cell types, puzzle shaped pavement cells and guard cells, which build the stomata. We determined the cell number and the individual cell areas separately for both cell types during development. To dissect the rules whereby different cell types divide and expand, the experimental data were fit into a computational model that describes all possible changes a cell can undergo from a given day to the next day. The model allows to calculate the probabilities for a precursor cell to become a guard or pavement cell, the maximum size at which it can divide into two pavement cells or two guard cells, the cell cycle duration and two different growth rates for two kinds of cells (pavement and guard cells) in one population. The second case study deals with the fact that atypical E2F activity restrains APC/CCCS52A2 function obligatory for endocycle onset. We have demonstrated that the atypical E2F transcription factor E2Fe/DEL1 controls the onset of the endocycle through a direct transcriptional control of APC/C activity. Because E2Fe/DEL1 represses the CCS52A2 promoter, we hypothesize that its level must drop below a critical threshold to allow sufficient accumulation of CCS52A2 during late S and G2 phase for cells to proceed from division to endoreduplication. We built a mathematical model to analyse the above hypothesis. The model is based on an ODE model used for the binding of ligands to proteins with the help of Hill functions. This mathematical model helps to understand mechanistically how decreasing E2Fe/DEL1 levels can account for the division-to-endoreduplication transition. Finally, in Part IV, we discussed some future work to extend the above research. We suggested several ideas to design new experiments and increase the value of the models. The term ”we” is used throughout the text to underline the fact that every single result of the author’s work as represented in the thesis was only possible because of the provision of experiments, equipment, materials and scientific input from others

Towards an optimal sampling strategy for assessing genetic variation within and among white clover (Trifolium repens L.) cultivars using AFLP

Cost reduction in plant breeding and conservation programs depends largely on correctly defining the minimal sample size required for the trustworthy assessment of intra- and inter-cultivar genetic variation. White clover, an important pasture legume, was chosen for studying this aspect. In clonal plants, such as the aforementioned, an appropriate sampling scheme eliminates the redundant analysis of identical genotypes. The aim was to define an optimal sampling strategy, i.e., the minimum sample size and appropriate sampling scheme for white clover cultivars, by using AFLP data (283 loci) from three popular types. A grid-based sampling scheme, with an interplant distance of at least 40 cm, was sufficient to avoid any excess in replicates. Simulations revealed that the number of samples substantially influenced genetic diversity parameters. When using less than 15 per cultivar, the expected heterozygosity (He) and Shannon diversity index (I) were greatly underestimated, whereas with 20, more than 95% of total intra-cultivar genetic variation was covered. Based on AMOVA, a 20-cultivar sample was apparently sufficient to accurately quantify individual genetic structuring. The recommended sampling strategy facilitates the efficient characterization of diversity in white clover, for both conservation and exploitation

Using MatCont in a two-parameter bifurcation study of models for cell cycle controls

A recent application field of bifurcation theory is in modelling the cell cycle. We refer in particular to the work of J.J. Tyson and B. Novak where the fundamental idea is that the cell cycle is an alternation between two stable steady states of a system of kinetic equations. We study and extend the basic model of Tyson and Novak using the Matlab numerical bifurcation software MatCont, in a two-parameter setting and highlight several new features. We show that the limit point curves in the two-variable model behave in an ungeneric way under variation of the natural parameters and that the hysteresis loop in the model is not the usual loop caused by the existence of a codimension-2 cusp point. We continue orbits homoclinic-to-saddle-node (HSN) in the three-variable model and find that these orbits die in a non-central orbit homoclinic-to-saddle-node under a natural parameter variation. As an extension we introduce a model in which cell division appears as a continuous-in-time limit cycle. We perform a continuation of this limit cycle under a natural parameter variation and show that it loses stability in a limit point of cycles bifurcation. Alternatively, we study the cell cycle as a boundary value problem as proposed by Tyson and Novak. This leads to an interpretation of the cell as a slow-fast system and we derive several conclusions on the relation between the growth rate of the cells and the cell size at division, and on the controllability of the process

Model-Based Analysis of Arabidopsis Leaf Epidermal Cells Reveals Distinct Division and Expansion Patterns for Pavement and Guard Cells1[W][OA]

To efficiently capture sunlight for photosynthesis, leaves typically develop into a flat and thin structure. This development is driven by cell division and expansion, but the individual contribution of these processes is currently unknown, mainly because of the experimental difficulties to disentangle them in a developing organ, due to their tight interconnection. To circumvent this problem, we built a mathematic model that describes the possible division patterns and expansion rates for individual epidermal cells. This model was used to fit experimental data on cell numbers and sizes obtained over time intervals of 1 d throughout the development of the first leaf pair of Arabidopsis (Arabidopsis thaliana). The parameters were obtained by a derivative-free optimization method that minimizes the differences between the predicted and experimentally observed cell size distributions. The model allowed us to calculate probabilities for a cell to divide into guard or pavement cells, the maximum size at which it can divide, and its average cell division and expansion rates at each point during the leaf developmental process. Surprisingly, average cell cycle duration remained constant throughout leaf development, whereas no evidence for a maximum cell size threshold for cell division of pavement cells was found. Furthermore, the model predicted that neighboring cells of different sizes within the epidermis expand at distinctly different relative rates, which could be verified by direct observations. We conclude that cell division seems to occur independently from the status of cell expansion, whereas the cell cycle might act as a timer rather than as a size-regulated machinery