A Geometrically-Constrained Mathematical Model of Mammary Gland Ductal Elongation Reveals Novel Cellular Dynamics within the Terminal End Bud
Comparisons between mathematically derived cap cell flux values and values provided by the basal and luminal elongation rates matching condition in the context of the apoptotic correction factor. For each panel green lines represent the experimentally measured rate, black lines represent basal elongation predicted rates and blue lines represent luminal predicted rates. Each bold prediction line is shaded with standard deviation (experimental value) or uncertainty values (model predictions). Model 1 is the “base model”. Model 2 is the “base model + cap cell flux”. Model 3 is the “base model + cap cell flux + apoptotic correction”. Model 4 is the “base model +cap cell flux +apoptotic correction + displacement conversion”. A) Model 1 and 2 outputs are shown. Model 1 predicts two separate rates (no cap cell flux, i.e. X = 0%) that both lie outside the experimental dispersion, whereas Model 2 rates predict coordinated growth (for cap cell flux value X1 = 39%) but a prediction (in red) outside the experimental value error. B) Results from Model 3 which included the required apoptotic correction factor (δ = 0.97) for the luminal layer are shown. Matching condition yields a cap cell flux value of 49% (X2 = 49%) however the predicted rate (in red) still lies outside the experimental dispersion. C) Results from Model 4 are shown after the rates are converted to a displacement rate and δ = 0.97. Matching condition yields a cap cell flux of 50% (X3 = 50%) and the predicted rate (in red) lies within the experimental value. D) A summary of the final model predictions and experimentally measured values are shown. Matching condition values (white dots for minimal value δ = 0.97 and for value δ = 1.55 corresponding to Humphreys et al) and experimentally based model prediction (blue and black dots corresponding to X* = 54%) cluster in a small range of admissible values compatible with the experimentally measured rate.