How far is neuroepithelial cell proliferation in the developing central nervous system a deterministic process? Or, to put it in a more precise way, how accurately can it be described by a deterministic mathematical model? To provide tracks to answer this question, a deterministic system of transport and diffusion partial differential equations, both physiologically and spatially structured, is introduced as a model to describe the spatially organized process of cell proliferation during the development of the central nervous system. As an initial step towards dealing with the three-dimensional case, a unidimensional version of the model is presented. Numerical analysis and numerical tests are performed. In this work we also achieve a first experimental validation of the proposed model, by using cell proliferation data recorded from histological sections obtained during the development of the optic tectum in the chick embryo.
The present work aims at modeling the spatial organization of the neuroepithelial (NE) cell proliferation in a developing cortical structure along an early and brief developmental period by using sets of quantitative data empirically obtained from a standardized experimental model: the developing chick optic tectum (OT). The mathematical model is based on a deterministic approach that uses the formalism of partial differential equations.
1.1 Biological background: Developmental neurobiology 1.1.1 Relevance of a spatial and temporal organization in a developing system
The appropriate number of cells of each terminally differentiated cell type and also the spatial patterns they exhibit within the different tissues and organs composing pluricellular organisms are governed by interactive self-regulating behaviors that the developing cells exhibit during the embryonic development [1]. The increase in supracellular complexity generated during development requires the temporally and spatially organized operation of specific developmental cell behaviors (DCBs). These DCBs are usually reciprocally regulated and operate simultaneously and interactively [2]. Every developing cell population can be considered as both emitter and receiver of developmental regulatory signals having, on the one hand, informative and, on the other hand, structural roles. Thus, a central hypothesis in Developmental Biology proposes that the space-time organized operation of developmental cell behaviors depends on the cooperative establishment of spatially organized cell signaling networks mediated by diffusing informative molecules [3,4,5,6]. Molecular diffusion results in asymmetric distribution of developmentally active informative signals. This asymmetry plays a fundamental role in establishing temporal and/or spatial organization of specific DCBs that result in the whole developmental process in the organized patterns of cells, tissues and organs, that living organisms eventually exhibit in their final, terminally differentiated, state.
Not all developing cell populations possess similarly relevant informative roles. There exist specific transient cell populations, the so-called organizers, that primarily play informative roles influencing or regulating the developmental behavior of the other cells. By means of installing asymmetric distributions of developmentally active signals the organizers serve as instruments of an informative reference system in the establishment of spatially organized processes of cell determination and differentiation [7,8]. Amongst the most complex biological structures, the multilayered concentric neuronal organization of the central nervous system (CNS), i.e., brain cortex, cerebellum cortex etc., occupy a privileged position. The development of such a structural and functional complexity, the so-called corticogenesis, requires the organized operation of several DCBs. Amongst these DCBs: (a) the cell proliferation (CP), counteracted by apoptosis or programmed cell death, is involved in the generation of the appropriate number of neurons for each cortical area and each cortical layer; (b) the directed cell migration, a process mediated by specific interfacial interactions between cell surface and extracellular matrix components, controls the correct position of each specific neuronal type along the CNS spatial axes; (c) cell determination or commitment, a process mediated by irreversible genetic information reprogramming, allows different cell populations to select one out of a set of multiple developmental pathways; (d) cell differentiation, a process mediated by selective gene activation and selective protein synthesis, warrants the expression of different specific neuronal phenotypes; finally, (e) the development of the neural processes (neuritogenesis), dendrites and axons and the establishment of specific synaptic contacts (synaptogenesis) conclude this complex self-organizing and interactively regulated process of corticogenesis.
All these DCBs are properly organized in time and space. During the early stages, several organizers strategically positioned along the cephalic-caudal and dorsal-ventral axes of the developing CNS establish the primitive pattern of each region and subregion, as well as the identity of the neuromeres. Later, cell proliferation, migration and differentiation controlled by reciprocal interactions allow the whole system to expand and differentiate into several structurally and functionally integrated systems of interconnected neuronal circuits. Abundant evidence indicate that all these processes are spatially and temporally organized [9,10,11,12,13,14].
During the early development, the primitive CNS primordium, the so-called neural tube (NT), is almost exclusively composed of proliferative neuroepithelial (NE) cells. During this early proliferative phase the NE cells behave as a population of self-renewing stem cells that divide symmetrically, i.e. from each dividing NE cell
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