A Geometrically-Constrained Mathematical Model of Mammary Gland Ductal Elongation Reveals Novel Cellular Dynamics within the Terminal End Bud - PubMed (original) (raw)
A Geometrically-Constrained Mathematical Model of Mammary Gland Ductal Elongation Reveals Novel Cellular Dynamics within the Terminal End Bud
Ingrid Paine et al. PLoS Comput Biol. 2016.
Abstract
Mathematics is often used to model biological systems. In mammary gland development, mathematical modeling has been limited to acinar and branching morphogenesis and breast cancer, without reference to normal duct formation. We present a model of ductal elongation that exploits the geometrically-constrained shape of the terminal end bud (TEB), the growing tip of the duct, and incorporates morphometrics, region-specific proliferation and apoptosis rates. Iterative model refinement and behavior analysis, compared with biological data, indicated that the traditional metric of nipple to the ductal front distance, or percent fat pad filled to evaluate ductal elongation rate can be misleading, as it disregards branching events that can reduce its magnitude. Further, model driven investigations of the fates of specific TEB cell types confirmed migration of cap cells into the body cell layer, but showed their subsequent preferential elimination by apoptosis, thus minimizing their contribution to the luminal lineage and the mature duct.
Conflict of interest statement
I have read the journal's policy and the authors of this manuscript have the following competing interests: MTL is a Limited Partner in StemMed Ltd. and a Manager in StemMed Holdings L.L.C.
Figures
Fig 1. Morphology characterization of the terminal end bud.
Glands from 5 and 6 week old mice were embedded, sectioned and stained for epithelial markers and measurements were taken of both cells and the TEB structure. A) Regional measurements of the TEB are represented in a scaled schematic (n values: Regions 1, 2, 5, and 6 = 15 TEBs, Region 3 = 23 TEBs, Region 4 = 8 TEBs, Region 7 layer diameter = 24 TEBs, lumen = 15 TEBs, length = 29 TEBs, Region 8 = 8 ducts). B) Mean cellular dimensions of the cap and myoepithelial cells, Regions 1–4 (whiskers denote range, n values: Region 1 length = 15 TEBs/ 45 cells, Region 1 width = 10 TEBs/ 126 cells. Region 2 length = 15 TEBs/ 56 cells, Region 2 width = 10 TEBs/ 115 cells. Region 3 length = 23 TEBs/ 273 cells, Region 3 width = 9 TEBs/ 36 cells. Region 4 length = 8 TEBs/ 197 cells, Region 4 width = 8 TEBs/ 197 cells). C) Mean cellular dimensions of body and luminal cells, Regions 5–8 (whiskers denote range, n values: Region 5 width/length = 5 TEBs/ 221 cells. Region 6 width/length = 5 TEBs/ 316 cells. Region 7 length/width = 4 TEBs/ 204 cells. Region 8 length/width = 6 ducts/ 301 cells). See also Table A in S1 Text and S2 Fig.
Fig 2. Determination of proliferation rate and cell cycle dynamics.
Proliferation rates were analyzed by performing a dual labeling experiment with thymidine analogs EdU and BrdU. Mice were given a pulse of EdU at time “0”, then pulsed with BrdU every 2 hours for 24 hours. Mice were harvested 2 hours after pulsing with BrdU. A) Representative images of TEBs from each time point stained for EdU and BrdU incorporation. B) Quantification of EdU and Brdu single and double-positive populations throughout the time course indicate an S phase duration of 6 hours and total cell cycle time of 16 hours (n = 3 mice, 12 glands, 10 TEBs). See also S3 Fig.
Fig 3. Analysis of apoptosis by Cleaved Caspase III (CC3) staining.
Apoptotic levels were determined by cleaved caspase III staining. A) Representative images of TEBs stained for CC3 and EdU incorporation. B) Quantification of CC3 positive cells indicates the majority of apoptosis occurs in Regions 5, 6, and 7 (n = 50 TEBs). See also S4 Fig.
Fig 4. Cap cells migrate into body cell layer and undergo apoptosis.
Proliferation status and apoptotic status of SMA+ cells were investigated in both the outer regions (Region 1 and 2) and the inner regions (Regions 5 and 6) of the TEB. A) Representative image of a TEB triple stained for SMA, BrdU, and CC3. B) Mean number of SMA+ cells present in each region, (whiskers denote range) n = 34 TEBs. C) SMA+ cells also positive for BrdU or pHH3 in each region presented as mean ±SEM, *p < .05, ****p < .0001, n = 34 TEBs. D) Quantification of SMA+ cells positive for CC3 in each region presented as mean, ****p<0.0001 (whiskers denote range) n = 34 TEBs.
Fig 5. Experimental determination of TEB displacement rate.
Quantification of the distance from the nipple to the ductal boundary in 5 (n = 18), 6 (n = 18), 7 (n = 18), and 8 (n = 8) week FVB mice. A) Representative inguinal glands from 5, 6, 7 and 8 week old FVB mice are pictured with a dashed line demarcating the ductal front. B) Quantification of the outgrowths at each time point are fitted with a best fit line (one-way ANOVA p<0.0001, R2 = .8274). On average during puberty, the duct grows at a rate of 0.54 mm per day.
Fig 6. Assessment of TEB path tortuosity.
The angle of deflection caused by bifurcation, the frequency of bifurcation, and the disparity between displacement and path tracing measurements were measured. A) Representative image of a TEB bifurcation event with the original path of the TEB noted as a solid line, the new path of each TEB as a result of the bifurcation are noted as dotted lines and the angles measured noted. B) Quantification of the angle of deflection presented as mean (whiskers denote range)(n = 62). C) Representative image of TEB’s growth path with 2 bifurcation events noted and the length of duct between noted with a solid line. D) Quantification of the total length of duct between bifurcation events is presented as mean (whiskers denote range)(n = 23). E) Total length of duct measurements were compared to corresponding displacement measurements. Displacement measurements consistently underestimated the total length by 6.1% (±0.9) (p = .0001, paired ratios t-Test, n = 23).
Fig 7. Model iteration summary and fit versus data-informed results.
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 _X_1 = 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% (_X_2 = 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% (_X_3 = 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.
References
- Thompson DAW. On growth and form Cambridge Eng.: University press; 1917. xv, 793 p. p.
- Turing AM. The chemical basis of morphogenesis. 1953. Bulletin of mathematical biology. 1990;52(1–2):153–97; discussion 19–52. Epub 1990/01/01. - PubMed
- Takaki R. Can morphogenesis be understood in terms of physical rules? Journal of biosciences. 2005;30(1):87–92. Epub 2005/04/13. - PubMed
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