Developmental graphs comparison strategy for analysis of pattern formation and phylogeny.

In most taxa of plant and animal kingdoms the initial steps of embryogenesis and the final morphology of an organism are strongly determined. However, these two phenomena do not correlate from phylogenetic point of view, namely, different unrelated taxa can have the same type of early embryogenesis, while there can be different types of cleavage inside one taxon. Here we discuss an approach enabling giving an insight into the understanding of this phenomenon. First, we propose a strategy for constructing developmental graphs (trees) that provide mathematical formalization of a process of embryogenesis. Second, we suggested an algorithm of trees comparison, developed specifically for this type of labeled graphs, which allows calculating a distance between two developmental trees, and thus clustering them into groups. Next we performed the analysis of correspondence between the obtained clusters and the inception of morphological characters in given clustered groups of organisms, which allows describing several particular cases of interrelation between developmental trends and formation of morphological structures. Here we present some illustrations of the suggested methodology on the analysis of plant angiosperm species belonging to different taxa of various ranks.

Determinism and variability of the morphogenesis pathways.

Along with a strict determinism of early embryogenesis in most living organisms, some of them exhibit variability of cell fates and developmental pathways. Here we discuss the phenomena of determinism and variability of developmental pathways, defining its dependence upon cell potency, cell sensitivity to the external signals and cell signaling. We propose a set of conjectures on the phenomenon of variability of developmental pathways, and denote a difference between a normal (local) variability, leading to an invariant final structure (e.g., embryo shape), and fundamental one, which is a switching between different developmental pathways, leading to different possible structures. For illustrating our conjectures, we analyzed early developmental stages of plant embryos with different levels of variability of morphogenesis pathways, and provide a set of computational experiments by Morphogenesis Software.

Model of Morphogenesis.

We build a theoretical model of morphogenesis. This model describes cell fate in the developing organism using the notion of epigenetic code of each cell. Namely, given the epigenetic spectra of a cell and its neighboring cells, we can determine the corresponding cell event it will perform. This means that the properties of a group of cells (comprising an embryo or its part) at any time point are also known, and thus, the evolution of an embryo can be described. By this strategy, it is possible to establish the tissue, organ, or embryo shapes at any time, starting from a zygote. As an essential part of the model, the formalization of the notion of cell potency is introduced, and the related properties are discussed.

Morphogenesis software based on epigenetic code concept.

The process of morphogenesis is an evolution of shape of an organism together with the differentiation of its parts. This process encompasses numerous biological processes ranging from embryogenesis to regeneration following crisis such as amputation or transplantation. A fundamental theoretical question is where exactly do these instructions for (re-)construction reside and how are they implemented? We have recently proposed a set of concepts, aiming to respond to these questions and to provide an appropriate mathematical formalization of the geometry of morphogenesis [1]. First, we consider a possibility that the evolution of shape is determined by epigenetic information, responsible for realization of different types of cell events. Second, we suggest a set of rules for converting this epigenetic information into instructive signals for cell event for each cell, as well as for transforming it after each cell event. Next we give notions of cell state, determined by its epigenetic array, and cell event, which is a change of cell state, and formalize development as a graph (tree) of cell states connected by 5 types of cell events, corresponding to the processes of cell division, cell growth, cell death, cell movement and cell differentiation. Here we present a Morphogenesis software capable to simulate an evolution of a 3D embryo starting from zygote, following a set of rules, based on our theoretical assumptions, and thus to provide a proof-of-concept of the hypothesis of epigenetic code regulation. The software creates a developing embryo and a corresponding graph of cell events according to the zygotic epigenetic spectrum and chosen parameters of the developmental rules. Variation of rules influencing the resulting shape of an embryo may help elucidating the principal laws underlying pattern formation.

Mathematical modeling reveals the factors involved in the phenomena of cancer stem cells stabilization.

Cancer Stem Cells (CSC), a subset of cancer cells resembling normal stem cells with self-renewal and asymmetric division capabilities, are present at various but low proportions in many tumors and are thought to be responsible for tumor relapses following conventional cancer therapies. In vitro, most intriguingly, isolated CSCs rapidly regenerate the original population of stem and non-stem cells (non-CSCs) as shown by various investigators. This phenomenon still remains to be explained. We propose a mathematical model of cancer cell population dynamics, based on the main parameters of cell population growth, including the proliferation rates, the rates of cell death and the frequency of symmetric and asymmetric cell divisions both in CSCs and non-CSCs sub-populations, and taking into account the stabilization phenomenon. The analysis of the model allows determination of time-varying corridors of probabilities for different cell fates, given the particular dynamics of cancer cells populations; and determination of a cell-cell communication factors influencing these time-varying probabilities of cell behavior (division, transition) scenarios. Though the results of the model have to be experimentally confirmed, we can anticipate the development of several fundamental and practical applications based on the theoretical results of the model.

Theory of Morphogenesis.

A model of morphogenesis is proposed based on seven explicit postulates. The mathematical import and biological significance of the postulates are explored and discussed.