This decision has been linked to the orientation of the mitotic spindle, the inheritance of polarity components, the distribution of cell-fate determinants during mitosis, the presence of extracellular morphogenetic signals, and the cell cycle length2,3,4,5,6,7,8,9,10. numbers of cells at different times. Developmental processes are tightly orchestrated in both space and time to ensure proper final form and function of organs and tissues. In the developing vertebrate central nervous system, a cycling progenitor cell faces three different outcomes upon division: the generation of two progenitor cells with self-renewing potential (division), two daughter cells committed to differentiation (division), or an asymmetric mode of division that produces one progenitor cell and one differentiating cell (division). Proliferative divisions dominate at early stages of development to expand the stem cell population without losing developmental potential, while later Metformin HCl in development, divisions generate differentiated cells at the expenses of the progenitors pool. The asymmetric mode of division results in maintenance of the stem cell population, while differentiated cells are constantly produced1,2,3. The molecular mechanisms that govern the decision between each mode of division are beginning to be comprehended. This decision has been linked to the orientation of the mitotic spindle, the inheritance of polarity components, the distribution of cell-fate determinants during mitosis, the presence of extracellular morphogenetic signals, and the cell cycle length2,3,4,5,6,7,8,9,10. Here, we derive a general Metformin HCl theoretical framework based on a branching process formalism that captures the average dynamics of proliferation and differentiation of a heterogeneous stem cell population in terms of balance between proliferative and differentiative divisions and average cell cycle duration, given the numbers of progenitors and differentiated cells at different times. The equations derived are then applied to study primary neurogenesis in the developing chick spinal cord, showing quantitative agreement with experimental data for the cell cycle length and rate of each mode of division. We also show that this rates of the three modes of division follow a probabilistic binomial distribution, allowing us to derive analytical equations for the rate of each mode of division. To further validate the model predictions, we developed a phenomenological model of the dynamics of vertebrate neurogenesis, where we show that this values of average division rates and cell cycle length predicted by the theoretical model are sufficient to reproduce the dynamics of growth of the developing spinal cord obtained experimentally. Overall, our studies show that, despite the complex regulation of stem cell differentiation, the growth and differentiation dynamics of a given stem cell pool can be calculated based on simple mathematical assumptions. Results A Markov branching process to link division rate and division mode to progenitor and differentiated cell numbers In general, a stem cell pool can be interpreted as a number of cells or can be obtained based on the following equation (detailed step-by-step derivation of the equations used in this section is usually shown as Supplementary Material)15,16: Open in a separate window Physique 1 A Mathematical Model to Describe Stem Cell Populations.(A) Scheme of the branching process for stem cell behavior where a initial pool of progenitors and differentiated cells or for three time points. Dependence on ? and can be found as Supplementary Fig. S1. where is the average cell cycle length, is the ratio of cycling cells within the population, and are the average probabilities for symmetric proliferative or differentiative division, correspondingly, while ? is the rate of apoptosis. The value produced at any time or the cell cycle length at fixed and apoptosis rate ?. Supplementary Fig. S1 corresponds to plots of predicted cell numbers for varying numbers of quiescence and apoptosis rate ?. Eqs (1,2) can be simply rewritten to directly obtain the TSPAN4 proliferation rate and the cell cycle: being can be calculated independently of each other, simply based on numbers of progenitors and differentiated cells at two given time points. This is of experimental relevance, since measurements of the rates of each mode of division Metformin HCl and cell cycle are often indirect and complex to perform, while numbers of progenitor and differentiated cells can be easily quantified by immunostaining against molecular markers for each cell state. In addition, the rate of apoptosis can be decided based on active Caspase3 immunostaining, while the number of quiescent cells can also be decided experimentally using cumulative BrdU labeling2,3,17. In the next section, we apply this mathematical framework to study the dynamics of vertebrate neurogenesis using quantification of progenitors and differentiated.