Localization of selected candidate genes from the predicted anagen expanding population.

<p>In Situ Hybridization (ISH; RNA) and immunofluorescence (protein) were performed on mouse skin sections taken from telogen (day0) or anagen (Day 16) phases of the hair cycle, determined by Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003914#pcbi.1003914.s008" target="_blank">Figure S8A</a>. DAPI was used as a counterstain for cell nuclei (blue). The expression of candidate genes for ISH is seen as bright foci (red and green) in specific cell types. Note comparisons to technical negative control and positive controls in Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003914#pcbi.1003914.s009" target="_blank">Figure S9</a>. Foxn1 (red) was identified as a candidate matrix derived cell marker and was used here as a positive control for localization to matrix cells <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003914#pcbi.1003914-Mecklenburg1" target="_blank">[50]</a>. (<b>A</b>) ISH: R1 and 2 shows RNA expression for Ovol1 and Smad6 (green), respectively, which were predicted to be expressed in follicle cell populations that expand during the anagen phase. No expression was observed during the telogen phase (Day 0). (<b>B</b>) Protein staining by immunofluorescence: R1 and 2 shows RNA expression for Ovol1 and Smad6 (green), respectively. Again, no expression was observed during the telogen phase (Day 0).</p

Simulation results from mean field coupled oscillator model.

<p>All curves are calculated by solving EQ 2 (for additional details also see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003914#pcbi.1003914.e135" target="_blank">EQ 11</a>). The magnitude of the first order parameter, shown in red, can be easily calculated from the individual order parameters, and . Here, is related to the first order parameter in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003914#pcbi-1003914-g002" target="_blank">Figure 2</a>, also shown in red (note the subscript was dropped for convenience). (<b>A</b>) Simulation results of for configuration one (config 1, solid) and configuration two (confg 2 dashed). Here config 1 relates to cluster one having negative coupling (). Note that the synchronization was stable only in config 1. We also show the incoherent result when configuration two () was set near, the steady-state value. The top plots show the values of for both cluster 1 (green) and 2 (blue) on the unit circle at time = 1, 12 and 40 days. Note that the clusters are out-of-phase. A movie of the individual oscillators corresponding to configuration 1 is available as Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003914#pcbi.1003914.s019" target="_blank">file S2</a>. (<b>B</b>) A simulated bifurcation analysis of the model showing the stable attractors for at different values of (red dots). We note that the simulation results agree with the analytical results of , loss of the incoherent state, and , the upper bound of the wave state. The estimated period of the hair cycle is shown by the dashed line. The values corresponding to the observed hair system are highlighted, note that it is near a critical change in that corresponds to a sharp decrease in the period.</p

Predictions on population dynamics determined by the two population model.

<p>The predicted relative size of the expanding population for both the natural (green) and induced (light gray) hair cycle expression data. The time scale used was set relative to initiation, which was after morphogenesis (postnatal day 23) or after depletion for the natural and induced cycles, respectively.</p

Mouse Hair Cycle Expression Dynamics Modeled as Coupled Mesenchymal and Epithelial Oscillators

<div><p>The hair cycle is a dynamic process where follicles repeatedly move through phases of growth, retraction, and relative quiescence. This process is an example of temporal and spatial biological complexity. Understanding of the hair cycle and its regulation would shed light on many other complex systems relevant to biological and medical research. Currently, a systematic characterization of gene expression and summarization within the context of a mathematical model is not yet available. Given the cyclic nature of the hair cycle, we felt it was important to consider a subset of genes with periodic expression. To this end, we combined several mathematical approaches with high-throughput, whole mouse skin, mRNA expression data to characterize aspects of the dynamics and the possible cell populations corresponding to potentially periodic patterns. In particular two gene clusters, demonstrating properties of out-of-phase synchronized expression, were identified. A mean field, phase coupled oscillator model was shown to quantitatively recapitulate the synchronization observed in the data. Furthermore, we found only one configuration of positive-negative coupling to be dynamically stable, which provided insight on general features of the regulation. Subsequent bifurcation analysis was able to identify and describe alternate states based on perturbation of system parameters. A 2-population mixture model and cell type enrichment was used to associate the two gene clusters to features of background mesenchymal populations and rapidly expanding follicular epithelial cells. Distinct timing and localization of expression was also shown by RNA and protein imaging for representative genes. Taken together, the evidence suggests that synchronization between expanding epithelial and background mesenchymal cells may be maintained, in part, by inhibitory regulation, and potential mediators of this regulation were identified. Furthermore, the model suggests that impairing this negative regulation will drive a bifurcation which may represent transition into a pathological state such as hair miniaturization.</p></div

Periodic gene expression in mouse hair cycle.

<p>An approximation of the hair cycle phase corresponding to the time scale is indicated via a color bar, see legend in lower right. (<b>A</b>) Heat map of actual expression data for probesets identified as periodic. For visualization, the data is normalized relative to the corresponding maximum and minimum values (this normalization was not used in statistical analysis), and sorted first by principal frequency and second by phase. (<b>B</b>) Heat map of expression estimated by the principal periodic component corresponding to A. Sorted and normalized as in A. (<b>C</b>) Histogram of the phases for probeset expression patterns corresponding to the longest period, 31 days, estimated by the principal periodic component.</p

Predictions on expression dynamics determined by the two population model.

<p>(<b>A</b>) Histogram of the coefficient of determination (COD) for model estimated expression, shown for all probesets (dark blue background) and probesets identified as low frequency oscillators (LOF, light pink foreground). (<b>B</b>) The magnitude of the t-statistic used to estimate differential expression between the two estimated populations, shown for all probesets (dark blue background) and probesets identified as low frequency oscillators (LOF, light pink foreground). (<b>C</b>) Normalized expression data, , for the two model populations. The left column shows actual expression data and probesets are ordered by the magnitude of the t-statistic. The right column shows expression estimated by the model ordered as in left column.</p

Localization of selected candidate genes from the predicted static background population markers.

<p>In Situ Hybridization (ISH; RNA) and immunofluorescence (protein) were performed on mouse skin sections taken from telogen (day0) or anagen (Day 16) phases of the hair cycle, determined by Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003914#pcbi.1003914.s008" target="_blank">Figure S8</a> A. DAPI was used as a counterstain for cell nuclei (blue). The expression of candidate genes for ISH is seen as bright foci (red and green) in specific cell types. Note comparisons to technical negative control and positive controls in Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003914#pcbi.1003914.s009" target="_blank">Figure S9</a>. (<b>A</b>) ISH: Fgf7 (red) was used as a positive control marker, which has been reported to be expressed in DP cells <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003914#pcbi.1003914-Rosenquist1" target="_blank">[42]</a>. R1 and 2 shows RNA expression for Stat5a and Fermt2 (green), respectively, which were predicted to be expressed in dermal papilla, or other background population during the telogen and anagen phase. (<b>B</b>) Protein staining by immunofluorescence: Due to technical issues with Fgf7 and Fermt2 antibody selection, morphology was used to determine localization and Vim was chosen as an alternate candidate marker. Vim was predicted to be expressed in dermal papilla or other background population during the telogen and anagen phases. R1 and 2 shows protein expression for Stat5a and Vim (green), respectively.</p