Sources of biological diversity and randomness 

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Multi-level approach for the study of the sea urchin early em-bryogenesis

The sea urchin Paracentrotus lividus is a model organism widely used in developmen-tal biology. Sea urchin’s embryos have many advantages that led to this status of model organism. The eggs are easily accessible, fertilization can be controlled, embryo are trans-parent and develop quickly, enabling their observation with microscopy techniques [69]. The use of the sea urchin for embryology can be traced back to the XIXth century, be-ginning in 1847 with three publications documenting fertilization, « Sur le développement des oursins » (On the development of sea urchins) by Adolphe Dufossé, « Auszug aus einem Berichte des Akameikers v. Baër, aus Triest » (Excerpt from a report by the University Grad-uate von Baër in Trieste), by Karl Ernst von Baër and « Observations sur le mécanisme et les phénomènes qui accompagnent la formation de l’embryon chez l’oursin comestible » (Observations on the mechanism and phenomena accompanying the formation of the embryo of the edible sea urchin), by Alphonse Derbès [31]. These studies were followed by a famous work by Oskar Hertwig on sperm and egg pronuclear fusion in 1876. In 1891, Hans Driesch used the sea urchin to perform experiments on development, showing that a complete embryo could develop from extracted cells refuting preformation and mosaic theories. During the XXth century the sea urchin has been the basis for major discoveries. Tim Hunt and collaborators discovered the role of cyclins in the sea urchin development, which are key regulators of the cell cycle [68]. The sea urchin development has also been used as a basis for the comprehensive study of the gene regulatory network [53]. The most recent development of this approach can be found on the biotapestry website 3.
Figure 2: Diagram representing the first steps of the sea urchin cleavage patterns, from [83] As any multicellular organism, the sea urchin develop from a single fertilized egg. It undergoes a radial holoblastic cleavage, meaning that all the cell divide in a stereotypic way, with division axis either parallel or at right angle with what will become the animal vegetal axis. The first and second cleavages are both meridional and perpendicular to each other, as shown on figure 2. They are followed by an equatorial cleavage perpendicular to them. The three first cleavages divide cells symmetrically, leading to a symmetrical 8-cell blastula. The fourth division round is different of the first three, the animal and the vegetal tier don’t divide similarly. The cells of the animal tier divide meridionally into eight cells with similar volumes called the Mesomeres cells. The cells of the vegetal tier don’t divide symmetrically, the division occur in the equatorial plan, producing four large cells called Macromeres, close to the Mesomeres, and four small cells called Micromeres at the vegetal pole. At this 16-cell stage where the morphological symmetry breakings occur, the cells are well identifiable. The next round of division is equatorial for the Mesomeres, forming two tiers of eight similar cells. The Macromeres divide meridionally, forming a tier of eight cells below the Mesomeres. The Micromeres divide somewhat later forming two sets of cells, four Large Micromeres and four Small Micromeres. The Small Micromeres divide once more, then cease dividing before the larval stage, see figure 3. When the embryo has attained the 32-cells stage, it has begun to form a blastocoel which is a proteinaceous fluid within the embryo, and during the next rounds of division which are less stereotypical, cells organize themselves as a single layered epithelium surrounding the blastocoel.

Multi-level measures and rescaling

The digital reconstruction of a small cohort of developing embryos was analyzed at different levels of observation, from individual cell features to embryo-level dynamics via relevant coarse-grained cell groups. We observed similar patterns of evolution in the mea-sured quantities, such as cell volumes and numbers of neighbors, across all specimens of the cohort. Interindividual variability was captured by a linear temporal rescaling of the cell number charts, and a spatial rescaling of the geometric charts. Spatial symmetries in the embryos defined groups of exchangeable cells that constitute the basis of the proba-bilistic model.

Individual cell features

The digital reconstruction of entire sea urchin specimens allowed us to extract a vari-ety of features at the level of individual cells as presented. For any cell i , we defined the following quantities (Fig. 2.1):
– its life length: xi , corresponding to the time between two consecutive divisions.
– a mitosis time: mi , denoting the moment at which the cell i divides into two daugh- ter cells (and ceases to exist as such).
– its current volume: vi (t ), with the average volume over its life length (using discrete time steps): 1 mi 1 mi −1 i = vi (t )d t ≃ t vi (t ) (2.1) v mi −xi mi xi xi xi −=0.
– its current surface area: si (t ), with the average surface area over its life length: 1 mi −1 si (t ) (2.2).

Intermediate cell groups

Symmetries in the embryo, such as the rotational symmetry around the animal-vegetal (AV) axis, prevent the identification and matching of individual cells from one specimen to another. Unique identification of cells based on their morphological characteristics can-not be done without ambiguity either. To overcome these issues, a generic coarse-grained level of observation was needed. We chose here to define new subsets of L (adding to the list of Section 2.3.1):
– generational cell groups, or simply “cell groups”, the subsets of generation-n cells inside morphogenetic fields k: L n,k = {i ∈ L | ni = n & ki = k}.
Note that L n,k is not the same as L k (t ): the latter is a snapshot of morphogenetic field k at time t and therefore may contain a mix of generations if some cells have divided more frequently than others. Conversely, cells in the former group may have to be taken at dif-ferent times. These sets offer two useful viewpoints on the embryo, one focusing on its global state, the other on cell statistics. In any case, the greater n and t , the more distant these two sets are likely to be. At this intermediate scale, it became possible to identify and map cell groups Len,k across embryos. At the lower level, cells belonging to the same group were expected to display similar individual features. At the sixth generation, each embryo e contained a
total of 32 cells composed of 16 Mes, 8 Mac, 4 LMic, and 4 SMic cells, which can be written: |Le6,1| = 16, |Le6,2| = 8, |Le6,3 | = |Le6,4| = 4 for all e ∈ [1, 5]. Since each cell gives rise to two daughter cells at each cycle and there is no cell death, the number of cells of a given type continued doubling, i.e. |Len,k | = 2n−6|Le6,k | for n ≥ 6 during the period of interest.
The six cell features considered here are the life length xi , mitosis time mi , average cell volume vi , average cell surface area si , and the daughter/mother ratios of average vol-ume ai and surface area bi . The dispersion of these features observed in each group Len,k of the empirical data is represented by distributions, modeled by sequences of random variables. For example, (x1, x2, …, xm ) denotes the sequence of the values that the first fea-ture, life length, takes in the m cells of a given group Len,k . Each xi is the realization of a random variable Xi pertaining to cell i . Cells being indistinguishable within the same group, however, their order in the sequence is irrelevant. This property is referred to as “exchangeability” of the sequence of random variables and is formalized as follows: for any permutation π of the indices [1, m], the joint probability distribution of these vari-ables is the same: P (Xπ(1), Xπ(2), …, Xπ(m)) = P (X1, X2, …, Xm ). (2.5).

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Estimation of cell feature distributions in cell groups

To interpret the empirical feature distributions, capture their evolution and predict the developmental dynamics, it was best to describe them in terms of parametric mod-els. First, however, because of the finite and relatively short duration of the period of observation, some groups Lg were incompletely represented in the dataset. Therefore, a distribution of cell features in a group was considered to be significant only if the num-ber of observed cells constituted at least 95% of the full cardinality |Lg | calculated above. Moreover, certain features, such as the volume vi and surface area si , required the obser-vation of their evolution during whole cell cycles. Incomplete cell cycles were taken into account only if the mean period of observation of individual cell dynamics in a group was at least 20 min. Based on these criteria, we retained 252 distributions of individual cell features across all cell groups Lg (corresponding to a rough average of 4 × 3 significant distribution-generation pairs in each one of the 5 × 4 embryo-cell type fields).
Graphical assessment of these histograms led us to categorize them into two major types: normal distributions (i.e. Gaussian curves) for life lengths xi and mitosis times m i ; and log-normal distributions for average cell volumes vi , average cell surface areas si , daughter/mother volume ratios ai and surface area ratios bi (where a random variable X is said to be log-normally distributed if the random variable Y = log X is normally distributed).

Table of contents :

General Introduction
I Characterizing normality
Introduction
1 Predicting sea urchin’s normal development from a small cohort of digital embryos
1.1 Introduction
1.2 A small cohort of digital sea urchin throughout their cleavage period
1.3 Feature Extraction andMeasuring
1.4 Emergence of embryo-level dynamics from individual cell features
1.5 Spatial modeling
1.6 Discussion
2 Variability in the sea urchin development: A multi-level data driven probabilistic model 
2.1 Introduction
2.2 Image acquisition and digital reconstruction
2.2.1 Image acquisition
2.2.2 Image processing
2.3 Multi-levelmeasures and rescaling
2.3.1 Individual cell features
2.3.2 Intermediate cell groups
2.4 Observation and approximation of multi-level statistics
2.4.1 Estimation of cell feature distributions in cell groups
2.4.2 Cell volume and surface area dynamics
2.4.3 Independence along the lineage
2.5 Multi-level probabilistic model
2.5.1 Prototype
2.6 Biomechanical model description
2.7 Comparison to experimental data
2.7.1 Metrics
2.7.2 Objective functions
2.7.3 Initial State
2.7.4 Validation – Parameter space
3 Perspectives and open problems raised by the probabilistic model of development
3.1 The probabilistic model implies a monoid structure
3.1.1 Monoid Structure
3.1.2 Formalization as a dynamical system
3.1.3 Fluctuation theory and robustness
3.2 Parameters evolution
3.2.1 Waddington’s epigenetic landscape
3.2.2 Kupiec’s ontophylogenesis
Conclusion
II Characterizing diversity 
4 Sources of biological diversity and randomness 
4.1 Sources of variability in biology
4.1.1 Gene mutations
4.1.2 Epigenetic and stochasticity
4.2 Randomness and its formalisms in mathematics and physics
4.2.1 Probability theory
4.2.2 Randomness in algorithmic theories
4.2.3 Randomness in dynamical systems and ergodic theory
4.2.4 Randomness in quantummechanics – Quantum mechanics as a generalized probability theory
4.3 Variability andmodels in biology
4.3.1 Models and simulation as tools for exploring some dynamics of the living
4.3.2 Living organisms are organized objects involving different levels of organization with heterogeneous dynamics
4.4 Conclusion
5 Variable phenotypic expressivity and incomplete penetrance of the zebrafishmutant line squintcz
5.1 Introduction
5.2 Materials and methods
5.3 Results
5.4 Discussion
6 Biological diversity and quantummechanics formalism 
6.1 Introduction
6.2 Variability in biology, emergence of new phenotypes
6.2.1 Squint experiment – an incomplete list of phenotypes
6.3 Correlating several observables
6.3.1 Mendel’s model of inheritance: a formal analog of entanglement
6.4 Discussion
7 Evolution and development: toward an ontogenetic tree 
7.1 Introduction
7.2 Reconstructing the ontogenetic tree of the Danio rerio embryogenesis
7.2.1 The concept of an ontogenetic tree
7.2.2 Formalization of the tree
7.2.3 Observing the phylotypic stage
7.3 Discussion and conclusion
Conclusion
III Quantifying biological shapes 
8 Using persistent homology to quantify tissue shape and organization 
8.1 Introduction
8.2 Global characterization of epithelial tissues
8.2.1 Network of cellular connectivity
8.2.2 Complex networks approach shows some limitations
8.2.3 Persistent homology
8.2.4 Quantitative comparison by computing features on top of barcodes
8.2.5 Classification of tissues
8.2.6 Summary
8.3 Random surfaces with arbitrary degree distribution to model tissue topology
8.3.1 Use of a null model
8.3.2 Topological hypotheses are necessary
8.3.3 Randomly gluing polygons
8.3.4 Topological characteristics of random surfaces
8.3.5 Comparison of the null model and the data for each of the features
8.4 Discussion and Conclusion
9 Tissue shape dynamics: cell proliferation and cell displacements 
9.1 Introduction
9.2 Time evolving networks
9.2.1 Time evolution of static measurements
9.2.2 Looking at spatiotemporal networks
9.3 Using genealogy as a parameter – historical dependency of shape
9.4 Conclusion
Conclusion 
General Conclusion

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