A multi-scale model for hair follicles reveals heterogeneous domains driving rapid spatiotemporal hair growth patterning

The control principles behind robust cyclic regeneration of hair follicles (HFs) remain unclear. Using multi-scale modeling, we show that coupling inhibitors and activators with physical growth of HFs is sufficient to drive periodicity and excitability of hair regeneration. Model simulations and experimental data reveal that mouse skin behaves as a heterogeneous regenerative field, composed of anatomical domains where HFs have distinct cycling dynamics. Interactions between fast-cycling chin and ventral HFs and slow-cycling dorsal HFs produce bilaterally symmetric patterns. Ear skin behaves as a hyper-refractory domain with HFs in extended rest phase. Such hyper-refractivity relates to high levels of BMP ligands and WNT antagonists, in part expressed by ear-specific cartilage and muscle. Hair growth stops at the boundaries with hyper-refractory ears and anatomically discontinuous eyelids, generating wave-breaking effects. We posit that similar mechanisms for coupled regeneration with dominant activator, hyper-refractory, and wave-breaker regions can operate in other actively renewing organs. DOI: http://dx.doi.org/10.7554/eLife.22772.001

modeling framework for the HF that captures: (i) activator/inhibitor signaling dynamics in a single HF, (ii) cyclic growth of a single HF, and (iii) coupling between multiple HFs through diffusive signals. Using this model, we reveal that skin as a whole behaves as a heterogeneous regenerative field, where: (a) dominant hair cycle waves start in the ventrum, (b) propagate dorsally in a bilateral pattern, (c) stop at the boundary with hyper-refractory ear skin, and (d) break at non-propagating anatomical landmarks, such as eyelids and ears. We also show that WNT and BMP serve as a universal activator/inhibitor signaling pair, whose varying activities underlie distinct hair regeneration dynamics in all anatomical locations studied. These results provide new understanding of how the entire skin of the animal manages all of its hair regeneration.

Results
A multi-scale model recapitulates a single growing HF, as well as HF-to-HF communication First, in modeling the geometry of a single HF, we considered four key expression sites for activator/ inhibitor ligands, antagonists, and receptors along the HF axis: bulge, HG, matrix, and DP ( Figure 1A). During the cycle, the bulge (assigned as Region I) remains relatively static, whereas the DP moves up and down along the HF axis. Also dynamic are the HG and matrix. The former only exists during telogen, while the latter only exists during anagen. The HG grows down to make matrix during anagen onset, whereas during catagen, the matrix collapses, and a new HG reforms. Simplistically, cyclic HGfimatrixfiHG dynamics are coordinated with the DP; thus, in the model we identify them jointly as Region II. Next, we considered that both regions produce signaling factors. Although a biological simplification, we assumed that Region I does so at a rather constant rate, while Region II shows distinct temporal dynamics (Appendix 2-table 4). We also assumed that Region II is essential for sending hair cycle-promoting signal(s), while Region I is the primary signal target. In short, we hypothesized that the essential temporal molecular dynamics in the HF operate as follows: Region II generates a signaling ligand (L) gradient; Region I detects it and transmits it into ligand-bound receptors (LR) that then, through a series of intermediate signaling steps not captured in the model directly (such as activities of the downstream signaling pathways and involvement of additional cell eLife digest Skin includes hundreds of thousands of hair follicles that cycle through different stages of activity. Each follicle grows hair, sometimes (in the case of long hairs like human head hair and horse tail hairs) for several years, before losing it. The follicle then goes through a resting stage before starting to grow another hair. To achieve high hair density, the follicles need to coordinate their hair-making activities. If they all worked independently from one another, bald patches would inevitably form that would compromise how effectively the skin works.
Groups of cells can communicate using a variety of chemical signals. It was not known whether cells in hair follicles from different regions of the skin rely on the same signals to communicate, and whether follicles in neighboring regions are able to 'understand' one another.
Through a combination of mathematical modeling and experimental results from mice, Wang, Oh et al. now show that hair follicles across the body use a common signaling system. This system consists of a pair of signals: 'activators' that stimulate hair growth, and 'inhibitors' that prevent it. The balance between these two signals affects the pattern of hair growth. For example, higher levels of activators allow fur to grow thickly on the belly of the mouse, likely to protect against heat loss and injuries from the ground. By contract, higher levels of inhibitors make the hairs on the ear sparse, which may prevent them from interfering with hearing.
There is little evidence that hair follicles on the scalp communicate in adult humans. Learning to activate and control communication between these follicles could provide a way to treat male pattern baldness and similar conditions. Understanding how hair follicles communicate may also help researchers to develop ways of regenerating other fast-renewing organs, such as the gut and bone marrow. Figure 1. Model recapitulates hair cycling and its associated activator and inhibitor signaling dynamics. (A) Schematic depiction of HF growth dynamics during telogen and anagen. Telogen and anagen HFs are shown on the left and in the center, respectively. In both hair cycle phases, Region I (purple) represents bulge and Region II (orange) represents DP with HG during telogen phase, and DP with matrix during anagen phase. On the right, schematic drawing of diffusive activator (Act. L. in green) and inhibitor (Inh. L. in red) interactions with their corresponding receptors (Act. R and Inh. R, Figure 1 continued on next page populations), regulates cyclic HF growth ( Figure 1A). The molecular signaling events, either activating or inhibitory, can be summarized as: ½L ¼ Diffusion þ Production þ Reaction of L and R (1) q qt ½LR ¼ Reaction of L and R þ Degradation þ Extra Source (2) where L, R and LR stand for ligands, receptors, and ligand-bound receptors, respectively. In the dynamics of LR (Equation 2), the 'Extra Source' describes stochastic signaling effects due to noise, and potential signaling contributions from Region I (Appendix 2-Governing equations for activators and inhibitors). As Equations 1 and 2 show, ligand-receptor interactions in the model take place only for the same signaling pathway, and no direct pathway cross-talk is set to occur. This, again, is a biological simplification. Recently, evidence for pathway interactions have emerged (Kandyba et al., 2013), and its effect is explored in Appendix 2-Possible interactions between the activator and inhibitor pathways do not qualitatively alter the HF dynamics. Our model integrates key signaling features of the hair growth cycle: strong activator signals enhance HF growth, while strong inhibitor signals prevent it. We modeled HF growth through the spatial average of LR concentration differences between the levels of activator and inhibitor in Region I (Equation 7 in Appendix 2-Modeling HF phases by concentration difference). We assumed the hair cycle has two critical 'checkpoints': (i) the event in late competent telogen, when production of activator starts to increase (Chen et al., 2014;Greco et al., 2009;Oshimori and Fuchs, 2012;Plikus et al., 2008b), and (ii) the event of anagen termination, when the HF starts to involute. Thus, our model recognizes two phases determined by these checkpoints:~anagen, starting from the moment of activator amplification until anagen termination, and~telogen, lasting until the next activator amplification event. In the context of the conventional hair growth cycle,~anagen incorporates the late portion of competent telogen and the entire anagen, while~telogen includes catagen, refractory telogen and the remainder of competent telogen (Plikus et al., 2011;Plikus and Chuong, 2014;Plikus et al., 2008b) (Appendix 2-Modeling HF phases by concentration difference; Appendix 2-figure 2).
Model simulations produce several emergent behaviors. The cycle becomes autonomous -that is, it displays stable periodicity and excitability emerges naturally without a built-in 'clock' ( Figure 1B). Cycling is maintained within a range of parameter values, allowing testing for various intrinsic and extrinsic signaling scenarios ( Figure 1C). Associated with these dynamics are periodic changes in the system's geometry -the signaling source in Region II moves cyclically. Simulations indicate that the moving HF geometry in the model is critical, greatly contributing to the regulation of the cycle. In a single HF model, activator/inhibitor diffusion occurs only along the HF axis. When a HF population is modeled, hair-to-hair communication emerges naturally as ligand diffusion from neighbors supplements intrinsic HF ligand levels. As such, hair cycle pace depends on interactive signaling between neighboring HFs -a feature that we explore below. Figure 1 continued not depicted) that form ligand-bond-receptors (Act. LR and Inh. LR) and their coupling with physical growth of the HF (blue) is shown. (B) Typical noisefree dynamics of the activator (green) and inhibitor (red) and cyclic HF growth (blue) are shown. X-axis is time in simulated days. Y-axis for activator and inhibitor shows simulated signaling levels, and for HF growth -simulated length of the HF. Grey area demarcates one modeled hair growth cycle. (C) The duration of~anagen and~telogen phases as the function of inhibitor signaling strengths. X-axis shows modeled inhibitor levels with '0' being an arbitrary baseline levels. Y-axis shows time in simulated days. Upon stronger inhibitory signaling (high Inh. L level)~anagen shortens (yellow) and telogen lengthens (purple). The entire cycle (blue) becomes longer either with stronger or weaker inhibitory signaling. When inhibitory signaling becomes either very strong or very weak, the excitability of the system breaks down and HFs equilibrate in one state (grey regions). Also see Appendix 2-tables 1, 2 and 4. (D-E'') A total of 236 putative activator genes (green) and 122 putative inhibitor genes (red) available from a whole skin microarray dataset were identified to recapitulate temporal dynamics of the simulated activator (D) and inhibitor (E), respectively. Multiple WNT pathway members are in the putative activator gene set (D', D''), while BMP pathway members are among the putative inhibitor genes (E', E''). See gene list in Dataset 1. For all genes log-transformed, zero-mean expression profile values were calculated using colorimetric ratio-scale algorithm as reported in (Lin et al., 2009). DOI: 10.7554/eLife.22772.003 HF cycling emerges from the growth-mediated coupling of activator and inhibitor Our model predicts that HF cycling occurs only within a certain range of signal strengths, that is the excitable regime ( Figure 1C, white region). Within this regime, activator and inhibitor are predicted to inversely modulate duration of both~telogen and~anagen phases. At certain, either too high or too low signal strengths, the excitability is predicted to break down and the HF is expected to enter a non-cycling state of equilibrium ( Figure 1C, grey regions). For example, when inhibitor levels are very high, the HF is predicted to equilibrate in an extended telogen (Appendix 2-figure 5A), while extended anagen is predicted for the opposite signaling condition (Appendix 2- figure 5B). Next, we used bioinformatic and experimental approaches to validate the model's key prediction that the same activator or inhibitor pathway can inversely modulate telogen and anagen phase duration. Considering the established roles for BMP and WNT as respective inhibitor and activator pathways regulating telogen duration in the dorsal skin, we explored if they can also regulate anagen duration in the same skin region in a model-predicted fashion. First, we found that model-predicted temporal dynamics for inhibitor and activator during~anagen ( Figure 1D and 1E) match the actual anagen expression dynamics for multiple BMP and WNT pathway members established on a highly temporally resolved whole-tissue dorsal skin microarray dataset (Lin et al., 2009) Appendix 1-Identifying model predicted hair cycle activators and inhibitors). We also show that perturbing BMP (for details see Appendix 1-Validating model-predicted roles for BMP signaling in hair cycle control) and WNT in transgenic mice (for details see Appendix 1-Validating model-predicted roles for WNT signaling in hair cycle control) alters dorsal anagen phase duration and leads to hair length defects in a way that is consistent with the model's predictions. Overall, this data shows that our model generates biologically meaningful outcomes and that its predictive power is robust.

Model reveals skin is a heterogeneous regenerative field
Next, we set out to explore novel aspects of hair regeneration at the population level. For this purpose, we modeled a linear array of HFs (i.e. two-dimensional organization; Appendix 2- figure 4A) and a grid of HFs (i.e. three-dimensional organization; Appendix 2- figure 4B). In both cases, the diffusion of activators and inhibitors accompanying each HF during growth naturally led to HF coupling (Appendix 1-Validating model-predicted roles for BMP signaling in hair cycle control) and emergence of several known features of collective hair growth behavior, including spontaneous anagen initiation and anagen wave spreading (Appendix 2-figures 11, 12). We then focused on the phenomenon of bilaterally symmetric hair growth that is prominent in young mice (Plikus et al., 2008b) yet remains unexplained. Conventionally, first anagen in the dorsal skin of newborn mice is considered synchronous. On the other hand, adult mice display fully asynchronous and asymmetric dorsal hair growth patterns (Chen et al., 2014;Plikus and Chuong, 2008a;Plikus et al., 2009). This, however, is preceded by prominent bilateral symmetry, which often persists into the fourth hair cycle (Plikus and Chuong, 2008a). We now show that in the three-dimensional model where all HFs are assumed to be identical, full asynchrony evolves within just one cycle, and bilateral symmetry cannot be achieved (Appendix 2-Dorsal and ventral HF patterns; Appendix 2-figures 11, 12; Appendix 2-video 1). Therefore, we hypothesized that first anagen is inherently asynchronous as a result of spatially patterned HF development. Indeed, spatial distribution of early anagen HFs in the dorsal skin of newborn mice (Figure 2A-D) reveals head-to-tail and subtle lateral-to-medial asynchronies. We modeled the impact of these asynchronies on hair growth pattern evolution. Simulations reproduced head-to-tail asynchrony (Appendix 2-Dorsal and ventral HF patterns; Appendix 2-figures 14, 15; Appendix 2-video 3); however, it persisted for at least 10 cycles, which is far more than the 3-4 cycles observable in mice. Moreover, prominent bilateral symmetry failed to form.
We note that the above and previous simulations (Murray et al., 2012;Plikus et al., 2011) were performed on homogenous HF populations, where all HFs are assumed to be identical. We then considered that novel patterns might develop upon interaction of two or more HF populations, whose activator/inhibitor signaling levels are inherently different. In principle, dorsal skin HFs can interact with HFs from other body regions, such as ventral skin, where hair cycle dynamics are potentially distinct. Because all skin is continuous and forms an approximation of a cylinder, we modeled it as an unrolled sheet, where two Ventral sub-domains flank a rectangular Dorsal domain ( Figure 2E). For initial modeling conditions (Appendix 2-Dorsal and ventral HF patterns), we considered that: (i) the first cycle on the dorsal skin has built-in head-to-tail asynchrony, and that (ii) ventral HFs develop with a 3-to 4-day delay relative to dorsal HFs (Appendix 1-figures 7-9). Because ventral HFs are known to produce distinctly shorter hairs (Candille et al., 2004), in the model we assumed that they have faster cycle dynamics compared to dorsal HFs (Appendix 2-Changes in the total amount of activator and inhibitor receptors results in different sensitivity of~anagen and~telogen lengths to signaling changes, Appendix 2-Dorsal and ventral HF patterns; Appendix 2-figures 8,9). Indeed, in this configuration, our model readily reproduced patterns with aspects of bilateral symmetry already in the second cycle as the result of dominant waves spreading from the Ventral to the Dorsal domain ( Figure 2F, t68-78.5). Importantly, after the second cycle, the effect of the initial built-in head-to-tail asynchrony started to disappear. Instead, the interaction between Ventral and Dorsal HFs continued to produce prominent bilateral symmetry in the third (Figures 2F, and later cycles (Appendix 2-video 4). Taken together, the model predicts that rapid hair growth pattern evolution requires interaction of two or more skin domains with distinct hair cycle parameters.

Ventral-dorsal interactions produce bilaterally symmetric hair growth patterns
Next, we imaged Flash mice, whose luciferase reporter produces skin-specific WNT activity signal and allows to sensitively and non-invasively determine the location and percentage of anagen HFs across the entire body (Hodgson et al., 2014) (Figure 3A-C). Luminescence levels were measured both dorsally and ventrally and mice were followed up until day P119, encompassing up to five hair growth cycles. Combined analysis from multiple mice reveals prominent phase advancement in ventral over dorsal anagen, specifically during the second, third, and fourth hair cycles ( Figure 3B, blue area). Additionally, the spatial luminescence signal mapping reveals distinct ventral-to-dorsal anagen propagation with features of bilateral symmetry during second ( Figure 3C; Appendix 1-figure 6) and third cycles ( Figure 3C'), supporting the patterning mechanism predicted by the model ( Figure 2F). We also mapped body-wide hair growth patterns on the basis of anagen HF pigmentation between days P0-P55 (Figure 3D-G; Appendix 1-figures 7-12). This analysis confirms ventral over dorsal anagen phase advancement starting from the second cycle and provides the following additional insights: (i) Ventrally, anagen phase is the shortest in the 'chin domain', ending around P10. It is longer in the 'ventral domain' proper, ending in the genital area around P14 and in the chest area around P17 ( Figure 3D).
(iii) First ventral telogen is shorter than dorsal telogen. Second anagen initiates in the chin and ventral domains already between P21-24 and then spreads toward ventral-dorsal boundaries in form of two bilaterally symmetric waves ( Figure 3E; Appendix 1-figure 12). Second anagen also ends faster in the ventral skin, maintaining ventral-dorsal asynchrony and bilateral symmetry ( Figure 3F).
(iv) Third anagen initiates the fastest in the chin domain, as early as P42 ( Figures 3G and 4H). When transplanted onto the back of pigmented SCID mice, chin skin grafts (n = 8) showed faster cycling compared to dorsal skin grafts (n = 8). While first post-transplantation anagen started with used for this analysis is provided at the bottom on the panel. (D) Analysis of the whole mount dorsal skin samples from P0 220bpMsx2-hsplacZ mice, where lacZ reporter activates in anagen HFs starting from phase IIIb, confirms head-to-tail asynchrony. (E, F) Modeling rapid hair growth pattern evolution in the context of two heterogeneous domains. (E) Schematic depiction of the modeling conditions with central Dorsal domain flanked by two lateral Ventral sub-domains with coupling between Dorsal and Ventral HFs. (F) Compared to Dorsal domain HFs, Ventral domain HFs were assigned with higher levels of total available activator and inhibitor receptors, allowing shorter~anagen and~telogen duration. Furthermore, hair cycle asynchrony was introduced into Dorsal domain to model the initial head-to-tail asynchrony. In simulations, interactions between HFs across domain boundaries result in bilateral symmetry during the second cycle (simulated time t68-78.5). Also, initial asynchrony breaks down in the cycle 3 (t130), and partial bilateral symmetry maintains into the late cycles (see Appendix 2-video 4). Scale bars: A, A', D -5 mm. Images on A, A' and D are composites. DOI: 10.7554/eLife.22772.004 Flash mouse between days P5-P48. Bioluminescent signal is color-coded according to the colorimetric scale on the right. (B) Combined temporal dynamics (from 6 Flash mice) of the bioluminescent signal-based anagen measurements over four hair cycles (days P5-P119). Dorsal skin dynamics are in brown and ventral skin dynamics are in blue. Prominent temporal advancement of ventral over dorsal anagen initiation can be seen during second, third and fourth cycles (light blue areas). This advancement is accompanied by dominant ventral-to-dorsal anagen wave spreading. (C, C') Mapping of Flash-based anagen reveals ventral-to-dorsal hair growth wave propagation and bilateral pattern symmetry. New anagen areas for each time point are color-coded. Second anagen initiation is shown on panel C, and third anagen initiation on panel C'. Also see Appendix 1- figure 6. (D-G) Hair growth distribution patterns on P17 (D), P21 (E), P39 (F) and P55 (G). Three mice were analyzed at each time point. Inverted whole mount skin samples from representative mice are shown. Schematic pattern maps are provided with color-coded anagen (green), catagen (yellow) and telogen (red) regions. Also see Appendix 1-figures 7-12. (H, I) HF cycling dynamics in chin skin grafts remain faster compared to dorsal skin grafts. After transplantation, first anagen initiated similarly in both chin and dorsal skin grafts, however, second anagen initiated significantly faster in chin grafts. Representative chin and Figure 3 continued on next page similar timing in both chin and dorsal grafts, consecutive anagen started significantly faster in chin grafts ( Figure 3H and I; Appendix 1- figure 13). Furthermore, in many instances, grafts induced anagen in the surrounding dorsal host skin. Taken together, these data support that dominant ventral-to-dorsal hair wave spreading drives rapid hair growth pattern evolution and bilateral symmetry. Underlying this behavior are faster hair growth cycle dynamics in chin and ventral HFs, a property that is partially maintained upon skin grafting.
Next, we asked if faster hair cycle dynamics in chin and ventral domains correlate with distinct molecular dynamics in putative activators and inhibitors. We performed RNA-seq profiling of whole skin from chin, ventral and dorsal domains at six hair cycle time points: first (aka competent) telogen, early anagen, mid-anagen, late anagen, catagen and early second (aka refractory) telogen. Analysis revealed non-overlapping transcriptomic trajectories of the hair cycle between the three domains ( Figure 4A-B'') and domain-specific expression patterns for multiple putative activator and inhibitor genes at all hair cycle time points (Appendix 1-figures 14-19; Dataset 2). We then asked if refractory properties of early telogen differ between the domains. Differential gene expression analysis ( Figure 4C-D) revealed enrichment in chin and ventral domains for gene ontologies related to macrophage function and lipid storage, and enrichment in chin domain for muscle-related genes ( Figure 4E). Consistently, chin skin shows more contractile cells around HFs, and chin and ventral skin have thicker dermal adipose tissue and substantially more CD11b + ;F4/80 + macrophages as compared to dorsal skin (Appendix 1-figures 20, 21). Furthermore, dorsal early telogen skin shows gene expression changes consistent with higher refractivity -it is enriched for several BMP ligands, and depleted for BMP antagonists and WNT ligands ( Figure 4F). Consistently, in Axin2-lacZ WNT reporter mice, many more HFs with WNT-active DPs are seen in chin and ventral as compared to dorsal skin at P36 ( Figure 4G; Appendix 1- figure 22A). WNT activity increases in dorsal skin to the levels of ventral skin only by P42 ( Figure 4H; Appendix 1- figure 22B). Furthermore, in P42 BRE-gal BMP reporter mice, many more HFs with BMP-active bulges are seen in dorsal as compared to chin and ventral skin ( Figure 4I; Appendix 1- figure 22C). In Krt14-Wnt7a mice, spontaneous anagen initiation sites in dorsal skin overrun ventral-to-dorsal wave dominance ( Figure 4J; Appendix 1- figure 23). In contrast, in Krt14-Bmp4 mice, ventral-dorsal hair growth waves stall and asymmetric anagen patches form instead ( Figure 4K). Together, this data confirms that lower refractivity and the underlying differences in BMP and WNT activities form the bases for ventral-dorsal hair growth dominance.

Ear pinna behaves as a hyper-refractory skin domain
Our model also predicts conditions when hair cycling stops and HFs equilibrate in an extended telogen, such as due to high levels of inhibitors (Appendix 2-Hyper-refractory domain; Appendix 2-figure 5A, 16). We profiled mouse skin for the existence of such behavior and found ears to match such prediction. In the ear skin, HF morphogenesis begins between days P2-P4, and HFs remain in anagen until about P15 ( Figure 5A). After first anagen, and for at least three months, they remain in an extended telogen, while at the same time dorsal HFs have already reached their third cycle ( Figure 5B and C). Seldom, solitary anagen HFs can be found, but no coordinated hair growth waves, characteristic to other skin regions, are observed ( Figure 5B, day P95). Moreover, anagen waves spreading from cranial skin could not propagate into ear skin ( Figure 7E). These observations are consistent with the possibility that ear skin is hyper-refractory. Next, we examined ear HFs' responses to several potent anagen inducers: cyclosporin A (Maurer et al., 1997;Paus et al., 1989), smoothened agonist (SAG) (Paladini et al., 2005) and hair plucking (Chen et al., 2015). We show that while dorsal telogen HFs readily respond to cyclosporin A (Appendix 1-figure 27B), ear HFs remain quiescent even 3 weeks after treatment ( Figure 5F). Anagen can be induced in response to SAG; however, this response occurs late, after 3 weeks, and remains restricted to the medial side of the ear ( Figure 5E). This is contrasted by rapid SAG-induced anagen in dorsal skin (Appendix 1-  Plucking induces anagen along the medial side of the ear; however, there is no anagen wave spreading into the unplucked region, a feature common in dorsal skin (Chen et al., 2015) ( Figure 5D; Appendix 1-figures 25A-B, 26). Furthermore, whole ear plucking experiments reveal very sparse anagen activation along the lateral side (Appendix 1- figure 25C). These data demonstrate that physiologically, adult ear HFs equilibrate in a hyper-refractory telogen state, yet in principle remain capable of regenerative cycling in response to selective external stimuli.
To understand how ear HF hyper-refractivity relates to activator and inhibitor signaling levels, we compared on RNA-seq refractory telogen dorsal skin with telogen ear skin and, additionally, cartilage/muscle complex, a structure unique to ears. We show that, transcriptionally, these three tissue types are distinct ( Figure 6A), containing large number of differentially expressed genes ( Figure 6B; Dataset 3) enriched for distinct gene ontologies ( Figure 6C). Analysis of the signaling pathways implicated in the hair cycle control revealed a number of differentially expressed WNT and BMP pathway ligands and antagonists ( Figure 6D). Compared to dorsal skin, ear skin is enriched for transcripts for several WNT antagonists, including Dkkl1, Dkk2 and Sfrp2, as well as collagen Col17a1, implicated in HF stem cell maintenance (Matsumura et al., 2016). Cartilage/muscle complex is prominently enriched for Bmp5, and multiple WNT antagonists, including Frzb, Sfrp2, Sfrp5 and Wif1. Additionally, it showed upregulated expression of other known hair cycle inhibitors Fgf18 (Kimura-Ueki et al., 2012;Leishman et al., 2013) and Ctgf (Liu and Leask, 2013).
We validated WNT and BMP changes from RNA-seq by studying relevant pathway reporters and measuring changes in ear hair cycling in mutant mouse models. Using Axin2-lacZ reporter mice, we show isolated sites of WNT activity in ear skin dermis, and a lack of activity in telogen HFs as well as in the cartilage and muscle ( Figure 6G and G'). Using BRE-gal reporter mice, we show high levels of BMP activity in telogen ear HFs (in the bulge), as well as in the cartilage and muscle ( Figure 6F and F'). Overexpression of the BMP antagonist Noggin in Krt14-Noggin mice partially rescued the hyper-refractory state -substantially more spontaneous anagen HFs can be found in Krt14-Noggin ears as compared to wild-type control ( Figure 6H and H'; Appendix 1-figure 28B, D). Wnt7a overexpression in Krt14-Wnt7a mice also reactivated anagen in ear skin, albeit to a lesser extent compared to Noggin overexpression ( Figure 6H''; Appendix 1-figure 28C, D). Together, these results support that hyper-refractivity of ear HFs depends on higher levels of BMP ligands and WNT antagonists, in part produced by the cartilage/muscle complex ( Figure 6E).

Hair growth waves distort around hyper-refractory and hairless skin regions
Lastly, our model predicts that hair growth waves can form distorted patterns around non-propagating skin regions, such as hyper-refractory hair-bearing skin or hairless skin ( Figure 7A and B; Appendix 2-Hyper-refractory domain and Wave breaker; Appendix 2-figure 17; Appendix 2-video 5). We considered that pattern distortion could occur in the cranial skin at the boundaries with hyperrefractory ears and eyelids -naturally occurring physical breaks in the skin. Indeed, we observe that hair growth waves prominently break around the eyelids and ears -anagen waves propagate faster through the hair-bearing skin around the eyelids and ears, and then distort into the spaces in front of these anatomical structures ( Figure 7C and E). Similar patterns are also observed for the ventralto-dorsal hair growth wave around the limbs ( Figure 7D). We conclude that distortions of hair growth waves around anatomical structures with temporary or permanent non-propagating properties contribute to rapid body-wide hair growth pattern evolution.

Discussion
Growth-regulated parallel signaling makes the HF an excitable medium Previous mathematical models have recapitulated cycling of a single HF (Al-Nuaimi et al., 2012;Halloy et al., 2000) or in HF populations in two dimensions (Murray et al., 2012;Plikus et al., 2011). Here, we developed a multiscale model where coupling of activator and inhibitor signals with the movements of a HF in a three-dimensional space simulates cyclic growth and communication between neighboring HFs. In a single HF regime, our model faithfully predicts the effects that changes in WNT and BMP signaling can exert on the length of the anagen phase of the hair cycle. Similar to the FHN generic excitable media model (Murray et al., 2012), our model also recapitulates several known population-level features of the HF system such as spontaneous hair growth initiation and hair wave spreading. Importantly, however, only our model allows incorporation of differential HF growth in space, a feature required for simulating heterogeneous skin properties such as interactions between skin domains with different hair cycle frequencies or the hair wave distortion effect. Thus, while the multiscale nature and non-linearity make our model more difficult to derive analytical results, its heterogeneous domain feature allows studying complex skin-wide hair growth dynamics (see Appendix 2-Comparison with FitzHugh-Nagumo (FHN) model).

HF morphogenesis across mouse skin is spatially asynchronous
Hair growth in newborn mice is commonly thought to occur simultaneously across the entire skin. In fact, we show that the first cycle is already distinctly patterned: at birth, anagen HFs in dorsal skin have head-to-tail and lateral-to-medial asynchronies, while first anagen entry by ventral HFs is delayed by approximately 3 days and proceeds as a concentric lateral-to-midline wave. Similarly delayed by 6 days are ear HFs. First anagen naturally follows the process of HF morphogenesis, which is known to be temporarily asynchronous, and to occur, at least in the dorsal skin, in three successive waves (reviewed in Clavel et al., 2012). Pattern-wise, development of HFs relies on reactiondiffusion (Sick et al., 2006) and on space-filling expansion-induction mechanisms (Cheng et al., 2014). Importantly, models for both mechanisms assume spatially synchronous HF morphogenesis. Our findings of spatial asynchrony of the first anagen indicate spatial asynchrony of HF morphogenesis. Future studies will be required to understand the modeling and signaling aspects of such phenomenon.

Hair cycle patterns evolve from the interaction of heterogeneous skin domains
Our data reveal prominent regional differences in hair cycle dynamics and show that interaction between HFs across domain boundaries drives rapid evolution of complex hair growth patterns. Specifically, we show that during early postnatal cycles, chin and ventral domains become the dominant sources of skin-wide anagen waves. Such dominant behavior of chin and ventral domains is accompanied by distinct activity dynamics for WNT and BMP, putative hair cycle activators and inhibitors, respectively. Transgenic mouse studies further confirm the functional importance of differential WNT and BMP activities in setting distinct hair growth pace across discrete anatomical skin regions. Admittedly, an in-depth follow-up study will be necessary to identify and verify the major site-specific cellular sources for WNT and BMP ligands and antagonists.
We also show that ear skin behaves as a hyper-refractory domain, where telogen HFs are resistant to anagen-inducing stimuli and cannot participate in hair growth wave propagation. We reveal that such hyper-refractivity relates to high levels of BMP ligands and WNT antagonists, in part produced by the cartilage/muscle complex, a structure unique to the ear skin. Thus, novel behaviors can be produced by the cooption of signals from new tissue modules, rather than by the modification of preexisting ones. This finding parallels the modulatory effects of non-HF cell types on the dorsal skin hair cycle, including adipose progenitors (Festa et al., 2011;Rivera-Gonzalez et al., 2016), mature adipocytes (Plikus et al., 2008b), and resident macrophages (Castellana et al., 2014;Chen et al., 2015). Finally, we show that anatomically defined structures that cannot propagate hair growth waves, namely ears and eyelids, can generate a 'wave-breaker' effect. Similar distortion effects are likely to occur around other anatomical structures, such as the tail and genitals, and around skin defects, such as scars, and can jointly contribute to rapid hair growth pattern evolution.
Taken together, our study reveals that the skin as a whole functions as a complex regenerative landscape with regions of fast, slow, and very slow hair renewal (Appendix 1-figure 29). We show that this behavior produces a fur coat with variable hair density, which likely serves an adaptive role, such as in thermoregulation. Mechanistically, we show that the WNT/BMP activator/inhibitor signaling pair modulates hair regeneration in all skin regions studied. This suggests that the WNT/BMP 'molecular language' for hair growth is general, rather than a special case for a specific body site. Its generality allows for hair-to-hair communications to arise across anatomic domain boundaries, which, in turn, enables novel hair growth dynamics not obvious from prior work -fast cycling skin regions (such as chin skin) function as a kind of hair growth pacemaker. Furthermore, our findings on ear hair cycle expand the repertoire of tissues with signaling macro-environment function to include any closely-positioned anatomic structures with signaling properties, such as cartilage.
We posit that some of the newly found hair regeneration features can have analogs in other organs. For instance, dominant anatomically defined pacemakers are common in the electrically coupled muscle-based tissues, including heart and stomach, where they generate directional contractile rhythmicity. Other actively regenerating organs, such as the intestines and bone marrow, can likely contain anatomic regions of faster and slower regeneration and, conceivably, they can be coupled to work in coordination. Knowledge learned from the skin system in the current study can guide the search for regenerative landscapes in these and other organs. Because coordination principles observed in the skin may be universal, the likelihood of them operating in other organs is substantial despite prominent anatomical differences.

Computational modeling
The modeling framework is based on a hybrid approach, with individual HFs modeled as an expanding or contracting one-dimensional line and with the diffusive molecules described in reaction-diffusion equations (Appendix 2, Equations 1-4). The latter are solved using a finite difference scheme with the standard central difference approximation on the diffusion (see Appendix 2-1-dimensional (1D) HF model to Numerical methods in Appendix 2).  (Hodgson et al., 2014) were used. For Dkk1 induction, P30 Krt5-rtTA;tetO-Dkk1 mice were placed on 2 mg/ml Doxycycline-containing water ab libitum, and skin was collected at P44 for histology and at P50 for hair length measurements.

Skin grafting
5 Â 5 mm skin grafts from chin and dorsal domains of P21 C57BL/6J male mice were transplanted onto the dorsum of gender-matched pigmented P50 SCID recipients. At the time of grafting, donor skin was in first telogen and recipient skin was in second telogen.

Hair plucking
In dorsal skin, club hairs were plucked from 5 Â 5 mm areas. In the ear pinna, plucking was done on the caudal skin. For quantitative plucking, approximately 500 club hairs we plucked along the medial ear side.

Topical drug treatment
Cyclosporin A: for the dorsal skin, 100 ul of Cyclosporin A solution (1, 5, and 10 mg/ml) was applied topically once a day for 7 days. For the ear pinna, caudal skin was treated with 100 ul of 10 mg/ml of Cyclosporin A once a day for 7 days. Smoothened agonist (SAG): for the dorsal skin, 120 uM of SAG in DMSO/acetone was applied topically once a day for 4 days as described (Paladini et al., 2005). For the ear pinna, caudal skin was treated with 25 ul of SAG solution once a day for 4 days.

Hair length measurements and club hair counting
Guard, awl, auchene and zigzag club hair types were photographed, traced and calibrated using Adobe Illustrator software. See Appendix 1-table 1. Club hair density was evaluated on wholemount telogen skin samples that were pre-treated with 1 mg/mL Collagenase/Dispase and counterstained with hematoxylin.

Whole mount in vivo bioluminescence imaging
Whole body imaging of Flash mice was performed as previously described (Hodgson et al., 2014). Briefly, mice were injected with 150 mg/kg of firefly D-luciferin substrate and imaged with the Xenogen IVIS Spectrum system.

FACS and analysis
Second telogen skin from C57BL/6J male mice was treated with Dispase to separate epidermis from dermis. Epidermis was digested with Accutase and dermis with Collagenase. Epidermal and dermal cell suspensions were combined and stained with anti-CD11b (eBioscience) and anti-F4/80 antibodies (eBioscience). Due to small tissue size, chin skin cells from three mice were combined for each experiment. FACS data were analyzed using FlowJo.

RNA-sequencing and analyses
Total RNA was isolated using the RNeasy Mini Kit (Qiagen). RNA samples with RIN >8.0 were considered for cDNA library preparation. Full-length cDNA library amplification and tagmentation was performed as previously described (Picelli et al., 2014). Libraries were multiplexed and sequenced as paired-end on an Illumina Next-Seq500 platform. Paired-end reads were aligned to the mouse genome (mm10/gencode.vM8) and quantified using the RNA-seq by Expectation-Maximization algorithm (RSEM) with standard parameters (version 1.2.25) (Li and Dewey, 2011). Samples were batcheffect corrected. EdgeR (version 3.14.0) was employed to identify differentially expressed genes (DEGs) across samples of interest. FPKM values were taken as inputs for PCA analysis and DEG analyses. Data is available at GEO: GSE85039.

activators and inhibitors
To find out which signaling pathway activities fit the temporal dynamics of activator and inhibitor predicted by the model (Figure 1D and E), we examined a highly temporally resolved whole-skin microarray dataset, which includes nine consecutive time points: five for anagen, three for catagen, and one for telogen (Lin et al., 2009). We identified two sets of 236 and 122 genes whose temporal dynamics recapitulate those of the simulated activator ( Figure 1D' and D'') and inhibitor signals ( Figure 1E' and E''; Dataset S1), respectively. Focusing on major signaling pathways, we show that multiple members of WNT and BMP are represented in each gene set. The putative activator set includes WNT ligands (Wnt2, Wnt9a, Wnt7b), soluble WNT activator r-Spondin1, multiple WNT-specific Tcf transcription factors, and soluble BMP antagonists, Follistatin and Sostdc1. The putative inhibitor set includes multiple BMP ligands (Bmp2, Bmp4, Bmp6, Bmp7), BMP receptors (Bmpr1a, Bmpr1b and Bmpr2), BMP-specific transcription factors (Smad2, Smad5 and Smad7), and soluble (Wif1, Sfrp1), transmembrane (Kremen1), and intracellular WNT antagonists (Nkd2, Prickle1).

Validating model-predicted roles for BMP signaling in hair cycle control
For BMP signaling, we examined Krt14-Bmp4 (Guha et al., 2004) and Krt14-Noggin mice (Plikus et al., 2004(Plikus et al., , 2005 overexpressing BMP ligand and soluble antagonist, respectively. Consistent with the notion of BMP acting as anagen inhibitor, Krt14-Bmp4 mice show shortened pelage, prominently on their ears, tail, and paws, and partial baldness on the trunk (Appendix 1- figure 1A, 2). Fully grown dorsal hairs in Krt14-Bmp4 are significantly shorter compared to control across all hair types by 35-47% (p<0.05) (Appendix 1- figure  1B). This is accompanied by the shortened anagen duration as established by histology on day P15 (Appendix 1- figure 3B). In contrast, Krt14-Noggin mice show a general increase in hairiness, prominent on their ears, tail, and paws (Appendix 1- figure 1A, 2), and significantly longer than WT dorsal hairs: 12% longer for guard (p<0.01) and 7% longer for auchene type (p<0.01) (Appendix 1- figure 1B). In parallel, we observe longer anagen phase duration as revealed by histology on day P19 (Appendix 1- figure 3A). Hair length changes in Krt14-Noggin mice were not statistically significant for the zigzag and awl types.
Appendix 1-figure 1. BMP changes affect hair length. Grossly, Krt14-Noggin mice display longer than normal pelage (A). In contrast, Krt14-Bmp4 mice display generalized pelage shortening. Also see Appendix 1- figure 2. (B) Compared to control, hairs are longer in Krt14-Noggin and shorter in Krt14-Bmp4 mutants. Krt14-Noggin hairs lengthen by up to~12%, while Krt14-Bmp4 hairs shorten by 35-46%. Arrows mark hair ends. Scale bars: B -1 mm. Images on B are composites. DOI: 10.7554/eLife.22772.013 Appendix 1-figure 2. Changes in BMP signaling result in morphological pelage length defects. Compared to control, Krt14-Noggin mice display generalized increase in pelage length, prominently on the tail and paws. Note fused paws, short digits, lack of claws and polydactyly, phenotypes that were reported previously. Krt14-Bmp4 mice display visibly short pelage, patches of prominent thinning on the trunk, as well as short hairs on the tail and paws. Also note other paw defects, longer digits and long, curved claws. Appendix 1-figure 6. Spatial luminescence signal mapping across hair cycles. Spatial maps of the luminescence signal in individual Flash mouse across three consecutive hair cycles are shown. Luminescence signal is represented as a 'heatmap' ranging from low (blue) to strong (red). For each time point, both ventral and dorsal views of the animal are provided. Images are grouped into two types of sets: (i) 'propagation waves', which show telogen-to-anagen wave spreading and (ii) 'regression waves', which show anagen-to-catagen-to-telogen wave spreading. During each hair cycle, both propagation and regression waves demonstrate notable temporal phase advancement in the ventral as compared to dorsal skin. Also see main Appendix 1-figure 8. Distribution of the hair cycle stages in P2 mouse skin. On day P2, initial head-to-tail hair cycle asynchrony in the dorsal skin becomes less prominent, while dorsal-toventral hair cycle asynchrony is maintained (inserts 2 and 4). DOI: 10.7554/eLife.22772.020 Appendix 1-figure 10. Distribution of hair cycle stages in P15 mouse skin. By day P15, HFs in the chin domain and most of the cranial domain undergo catagen-to-telogen transition, whereas HFs in the dorsal and ventral domains are in anagen. At this time, mouse-to-mouse pattern variability starts to become prominent, with more or less (as compared to the example pattern shown here) of the chin and cranial domains having transitioned to telogen. DOI: 10.7554/eLife.22772.022 ventral and most of the dorsal skin is in second anagen. One prominent exception is the cranial domain, situated in the space between eyes and ears, which at this time is still in the first telogen (insert 1). In some cases (as in the skin sample shown here), small regions of skin in the most posterior portion of the dorsal domain are also in the first telogen (insert 6). Appendix 1-figure 29. Landscape model of hair growth pattern formation. Mouse skin is represented as a landscape with peaks (ventral and chin skin), valleys (dorsal and cranial skin) and obstacles (eye openings, ear pinnae and limbs). In analogy to water streaming downward, hair growth waves (green) preferentially spread from ventral to dorsal skin, producing bilateral symmetry. In analogy to water waves diffracting around physical obstacles, hair growth waves (red) diffract around hyper-refractory ear pinnae, limb skin, and eye openings. DOI: 10.7554/eLife.22772.041 Appendix 1-table 1. Numbers of club hairs used for analysis.

Hair type Control
Krt14-Noggin Modeling: A multi-scale model for hair follicles reveals heterogeneous domains driving rapid spatiotemporal hair growth patterning 1-dimensional (1D) HF model

Geometry of a hair follicle
The 1D HF model is on a computation domain along the z-axis: z 2 ½Z; 0 (Appendix 2- figure  1). HF growth, characterized by elongation of the bottom part of the HF, ranges in a region ½h min ; h max & ½Z; 0. Region I is the bulge region, which does not move associatively with HF growth, and is located at ½h I ; h I þ d I . Region II includes HG and DP during telogen, or matrix and DP during anagen. Region II is the bottom region of the HF, hence it constantly moves in association with HF growth. Computationally, Region II is located at ½hðtÞ; hðtÞ þ d II , where hðtÞ marks the bottom tip of the HP. We have the following relation: h min ¼ h I À d II .

Equations for activators and inhibitors
The dynamical system consists of the diffusion of activator (Act L ) and inhibitor ligands (Inh L ), the reactions between ligands and receptors of the same species, where Act R /Inh R denotes the free activator/inhibitor receptors; and Act LR /Inh LR , which denotes the activator/inhibitor ligand-bound receptors. We assume that the total amount of free and bound receptors for either activator or inhibitor are preserved at any z-level, i.e., ½Act R ðz; tÞ þ ½Act LR ðz; tÞ R A tot ; ½Inh R ðz; tÞ þ ½Inh LR ðz; tÞ R I tot The dynamical system includes the following equations: 1. Two stochastic PDEs on the whole computational domain ½Z; 0, depicting the production (in region II) and the diffusion of the ligands, and their binding to corresponding receptors (in region I): where d I and d II indicates region I and II: Functions I A ðhÞ and I I ðhÞ give the ligand production rates, which will be discussed in Appendix 2-Modeling the production of ligands.
2. Two stochastic ODEs in region I, evaluated at each z 2 ½h I ; h I þ d I ½h max ; 0, depicting the binding reaction between ligands and corresponding receptors, and the degradation of (ligand-bound) receptors: where b A and b I represent the constant contributions due to the reactions between the ligands produced from and receptors located at region I, or any extra sources that may be added to the system; s A ; s I are noise terms. In our simulations, unless specified, we use multiplicative noise, that is where l A ; l I are the noise strength, and dt is the time step.
3. Boundary conditions for PDEs: The parameter values related to the above equations are given in Appendix 2-

Modeling HF phases by concentration difference
Let D be the difference of the average of ½Act LR and ½Inh LR in region I: Modeled HF growth relies on DðtÞ, subject to the following rules: during telogen, HF rests at its minimum length, meanwhile D progressively increases; once D exceeds a certain threshold D þ at very late telogen, the activator senses it and amplifies its own production (i.e. positive feedback). The latter results in a quick increase in D, which in turn induces anagen initiation and HF growth toward its maximal length. At that point in time, inhibitor starts to accumulate while activator degrades, leading to a decrease of D. Once D deceases back to D À ¼ 0 (i.e., activator and inhibitor levels balance out), HF starts to involute toward its original minimal length. In other words, HF completes its cycle and returns to telogen (Appendix 2- figure 2). There are two critical time points associated with these dynamics: t þ , the moment when D reaches D þ and activator production amplifies; and t À , the moment of anagen termination, when D decreases back to D À . On these bases, we divided modeled hair growth cycle into two phases: the period from a t þ to its following t À and referred to as ~anagen, which includes very late competent telogen (C) and the full of anagen -both propagating (P) and autonomous anagen (A); and the period from a t À to the next t þ is referred to as~telogen, which includes catagen (Cat), refractory (R) telogen and most of competent telogen (Appendix 2-figure 2).

Modeling the production of ligands
The production rates of Act L and Inh L are modeled in relative to the follicle growth hðtÞ based on experimental observation presented in Appendix 2-table 4, where we qualitatively evaluate the activity strengths of activator/inhibitor ligands (L), antagonists (A), receptors (R), and ligand-bound receptors (LR), in region I and region II during different phases within a full HF cycle. We simplify our model by eliminating the antagonist (A in Appendix 2-table 4), and estimate the net ligand production rate using the following formula: where the values of L, A and LR are from Appendix 2-table 4, depending on the phase of hair growth. Appendix 2- figure 3A shows the temporal pattern estimated from the above equation, produced by data from Appendix 2-table 4.
The production rates of activator (I A ) and inhibitor (I I ) are given by the following equations: For illustration of modeling of h, please see Appendix 2-figure 1 and 2.

2-dimensional (2D) and 3-dimensional (3D) HF models
The dynamics of 2D and 3D HF models are the same as the 1D HF model; however, in the 1D model we are considering a single HF in a computation domain of ½Z; 0, whereas in the 2D and 3D models we consider an array of HFs in a computation domain of ½0; X Â ½Z; 0 (Appendix 2- figure 4A) and of ½0; X Â ½0; Y Â ½Z; 0 (Appendix 2- figure 4B), respectively.
The diffusion terms in Appendix 2-Equations (1, 2) will switch to: The boundary conditions at z ¼ 0 and z ¼ Z are the same as Appendix 2-Equations (5,6).
In 2D model, we apply no leak boundary conditions on x ¼ 0; X: In 3D model, we apply periodic or no leak boundary conditions on x ¼ 0; X, and no leak boundary conditions on y ¼ 0; Y, depending on the detailed model. For example, when we apply periodic and no leak boundary conditions on x ¼ 0; X and y ¼ 0; Y, respectively, we have: ½Act L ðx ¼ 0; y; z; tÞ ¼ ½Act L ðx ¼ X; y; z; tÞ ½Inh L ðx ¼ 0; y; z; tÞ ¼ ½Inh L ðx ¼ X; y; z; tÞ Parameter values used in 2D and 3D simulations are given in Appendix 2-table 1 -3, 5.

Numerical methods
In 1D and 2D model, we use implicit finite difference to discretize the diffusion term and forward Euler methods to discretize the production and reaction terms. For example, the discretized equation of Appendix 2-Equation (1) is in the form of where u m n stands for ½Act L m n , and f represents the production and reaction terms in Appendix 2-Equation (1). It is unconditionally stable, and the truncation error is OðDtÞ þ OðDx 2 Þ in 1D, or d~OðDtÞ þ OðDx 2 Þ þ OðDy 2 Þ in 2D.
In 3D model, implicit finite difference method is not practical, since it requires solution of a system of N x Â N y Â N z equations. Instead we use explicit finite difference method. The truncation error is d~OðDtÞ þ OðDx 2 Þ þ OðDy 2 Þ þ OðDz 2 Þ. To make this method stable, we require that For the discretization parameter values we used in simulations (Appendix 2-table 3), we have n~0:13, which satisfies the stability requirement.

Effects of noise on HF dynamics
As we discussed in the main text, cycling occurs only within a range of signal strengths ( Figure 1C, white region), and when activator/inhibitor signals are either too strong or too weak ( Figure 1C, grey regions), HF reaches a stable equilibrium state and fails to cycle. When inhibitor is very strong or activator is very weak, the signal difference threshold required for spontaneous excitation cannot be reached (Appendix 2-figure 5A), and HF equilibrates in a prolonged telogen-like state. On the other hand, when inhibitor is very low, it fails to catch up with the activator, and HF equilibrates in a prolonged anagen-like state instead (Appendix 2- figure 5B).
Within the excitable region on which the HF can cycle, we used noise-free model to show the effects of signal levels on~anagen and~telogen lengths: higher inhibitor level results in shorter~anagen and longer~telogen ( Figure 1C, yellow and magenta lines).
Qualitatively similar results were also obtained in the stochastic regime with multiplicative noise in both activator and inhibitor signaling (Appendix 2- figure 6A).
We investigate how noise in signaling might affect the HF growth by adding multiplicative noise with different strengths in both activator and inhibitor signaling. Simulations show that strong signaling noise shortens the average length of both~anagen and~telogen, and at the same time, increases their variability (Appendix 2- figure 6B). This is interesting, although not surprising, because both~anagen and~telogen checkpoints are largely determined by the differential level between the activator and inhibitor signals. Stronger noise is likely to increase the probability of the signal reaching the critical differential level, advancing HFs to the next phase -an irreversible process.
Next we repeat the above simulations with additive noise instead. With additive noise added to both activator and inhibitor LR equations,~anagen is shortened while~telogen is extended, the same as Appendix 2-figure 6A Appendix 2- figure 6B. However, the change of Std of additive noise is not as much as when it is from multiplicative noise (Appendix 2- figure 6C,D). On the other hand, for medium inhibitor level but different noise strength, stronger additive noise also leads to shorter~anagen and~telogen, but the change is not as large as it is for multiplicative noise -mostly due to the reason that we choose same noise strength l A and l I for both multiplicative and additive noise to make a comparison, in which case multiplicative noise would cause bigger effects on the activator and inhibitor levels.
Finally we would like to point out that we use multiplicative noise in all 2D and 3D simulations. However, if additive noise is adopted instead, the results will not change essentially. For example, we repeat the simulation of interaction between dorsal and ventral HF waves with additive noise having strength l A ¼ l I ¼ 0:05. Similar to that with multiplicative noise (Figure 2F), interaction between dorsal and ventral HF waves last for two cycles, then breaks into more stochastic waves with bilateral symmetry (Appendix 2- figure 7).
Changes in the total amount of activator and inhibitor receptors results in different sensitivity of~anagen and~telogen lengths to signaling changes While higher inhibitor levels or lower activator levels always result in shorter~anagen and telogen, we find that by altering certain background parameter values, HF may achieve different sensitivity to such signal changes. One such pair of parameters we find is the total amount of activator and inhibitor receptors (R A tot and R I tot ).
We investigate~anagen and~telogen lengths with either both low amounts of activator and inhibitor receptors (R A tot ¼ R I tot ¼ 6), or both high amounts (R A tot ¼ R I tot ¼ 60). The result is shown in Appendix 2-figure 8. First, under the same signaling levels, high R A tot and R I tot always result in both short~anagen and~telogen. This difference between receptor levels in ventral and dorsal domains was used in the 3D modeling of HF wave interactions across dorsal and ventral domains (see Appendix 2-Dorsal and ventral HF patterns); as shown by experimental observations, ventral HFs have both shorter anagen and telogen than dorsal HFs. Next, these two modeling HF types also show different sensitivity to inhibitor signaling changes. When the receptor levels are both low, both~anagen and~telogen phases react more dramatically to changes in signaling (Appendix 2- figure 8A,B, solid bars). For instance, after the same net increase in the inhibitor ligand value, simulated~anagen phase decreased by~16 days and~telogen phase increased by~23.8 days under low R A tot and R I tot regime. However, under high R A tot and R I tot regime,~anagen phase decreased by~6.4 days and~telogen phase increased by~14.8 days. Appendix 2-figure 8C,D exemplifies changes in activator/inhibitor LR profiles and accompanying changes in HF growth dynamics under low R A tot and R I tot regime, while Appendix 2- figure 8E,F under high R A tot and R I tot regime. Based on this, we speculate that higher R A tot and R I tot abundance confers HFs with better ability to read out signal changes, and increases the robustness of~anagen and telogen durations.
We also explore all four possible combinations of low vs. high activator and inhibitor profiles. From the results in Appendix 2- figure 9 we see that each combination gives different~anagen and~telogen lengths, and that each combination results in different sensitivity to inhibitor level changes.

Possible interactions between the activator and inhibitor pathways do not qualitatively alter the HF dynamics
In our model the temporal patterns of activator and/or inhibitor ligand activities are dependent on HF growth, which is based on experimental observation. This is a phenomenological description, part of which may come out from any interaction between the pathways.
Recently, Kandyba et al. (2013) discovered the inhibition from BMP (inhibitor) to WNT (activator); other than this, so far there is no solid support that other interactions between activator and inhibitor exist. We incorporate this inhibitory interaction into our model in two ways: 1. the inhibitor LR inhibits the ligand-receptor binding of activator (Appendix 2- figure  10A): or 2. the inhibitor ligands inhibits the production of activator ligands (Appendix 2- figure  10B): However, in either way, please be aware that the original model which gives the phenomenological description already involves possibly existing interactions, and adding such an inhibitory feedback only enhances the effect.
In either Appendix 2-Equation (14) or (15), we run simulations with m ¼ 2; b I ¼ À4 for a wide range of a, where small/large a corresponds to weak/strong inhibition of the inhibitor (BMP) to the activator (WNT), with a ¼ 0 indicating that the inhibition does not exist, and the dynamics return to Appendix 2- Equation (1-4). Simulation results show that in either mode of inhibition,~anagen length will decrease and~telogen length will increase. (Appendix 2- figure 10C,D). Moreover, as the inhibition becomes stronger,~anagen length quickly reaches a minimum (Appendix 2- figure 10C,D, green lines); however, telogen length increases gradually (Appendix 2- figure 10C,D, red lines). Observing the dynamical patterns of HF growth and activator and inhibitor LR, we find that when the inhibition is extremely strong, the inhibition of the inhibitor to the activator push the activator LR almost flat (Appendix 2- figure 10E, green line), while the inhibitor LR still shows the hill-valley pattern (Appendix 2- figure 10E, red line).
When the activator is inhibited, a single HF is still able to present three dynamic regimes upon changes of signal levels: an excitable one and two equilibrium ones, the same as in Figure 1C. Appendix 2- figure 10F gives the profile of the excitable regime, from which we see that the dynamics react to inhibitor levels in a same way as before, with the only difference that the excitable regime requires a weaker inhibitor level now: b I 2 ½À13; À4 with the inhibitory feedback in L-R binding with a strength a ¼ 10, compared to b I 2 ½À9; 0:5 without any feedback ( Figure 1C).

Dorsal and ventral HF patterns
It is observed on mouse dorsal skin that during the first 2-to-3 cycles, synchronized anagen waves propagate from head to tail, followed by more spontaneous, asynchronous waves from the fourth cycle onwards, while still showing some bilateral symmetry. To study the cause of such patterns, we propose four scenarios. The first three of them consider a uniform HF domain simulating the dorsal domain only, and they are simulated by both the 2D and 3D models. The fourth scenario involves interactions between different HF domains: that is simulating the interactions between dorsal and ventral domains; hence, it is only simulated in the 3D model. Below we list certain modeling details of the 3D simulations, with the 2D simulations designed in similar ways: show up in later waves. However, the initial head-to-tail wave pattern persists for at least 10 consecutive cycles despite constant disruption from spontaneous initiations, which violates biological observation where the steady head-to-tail pattern breaks down as early as the third wave. Moreover, 3D simulation shows no bilateral symmetry.
4. Finally we consider the scenario where dorsal HFs interact with ventral HFs, where dorsal HFs near the head have built-in temporal phase advancement than other dorsal HFs. Moreover, the first anagen entrance of ventral HFs occurs later than in dorsal HFs, and ventral HFs have shorter anagen and telogen phases. We consider a 100 Â 100 domain, where the 60 Â 100 area in the middle is to simulate the dorsal domain, with its HFs having less available receptors (R A tot ¼ R I tot ¼ 15) resulting in longer anagen and telogen; the two side-domains, each with a size of 20 Â 100 simulate the ventral domain, which flank the dorsal domain and are connected via periodic boundary condition, and HFs in them have more available receptors (R A tot ¼ R I tot ¼ 50). In the beginning, for all HFs on the top row in the dorsal domain (20 x 80; y ¼ 100), we give a one time increase in the activator production (I A ! I A þ 1000), and we block the growth of ventral HFs by setting all HF length h ¼ h min in these areas during t 14. Multiplicative noise terms s A and s I (l A ¼ l I ¼ 0:02) are added to the activator and inhibitor LR Appendix 2-Equations (3,4). In this case, we see wave patterns similar to those in our biological observations, shown in Figure 2F and Appendix 2-video 4 and discussed in the main text.

Hyper-refractory domain
In studying the ear pinna skin behaving as hyper-refractory HF domain, we model two phenomena: 1) the high level of inhibitor in ear skin results in hair cycling termination, and HFs equilibrate in an extended telogen; 2) anagen waves spreading from dorsal skin could not propagate into the ear skin due to its hyper-refractory state.
We model the first phenomenon on a 100 Â 100 domain, with parameter values given in Appendix 2-table 1 (3D dorsal column), 1 and 5, except Noise is able to randomly initiate the first HF growth cycle, but the high inhibitor levels prevent HF wave propagating, and no other HF cycles appear, HFs stay in an extended telogen (Appendix 2- figure 16).
For the second phenomenon, that is no HF wave propagating from dorsal skin into the ear skin, we model on a 100 Â 100 domain. Parameter values are given in Appendix 2-table 1 3D Dorsal column, with l A ¼ l I ¼ 0:02, and the region ½30; 70 Â ½30; 60 and with elevated inhibitor level (a 0 I ¼ 2 in this region) to simulate the ear skin, while the other part the dorsal domain. We stimulate a top-to-bottom HF wave by giving a one-time increase (at t ¼ 0:1) in the activator ligand production in the top row (I A ! I A þ 1000 at y ¼ 100). We see steady HF wave propagate through the dorsal region for several cycles along with stochastic initiations; however, no HF wave is able to propagate into the ear skin (Appendix 2- figure 17).

Wave breaker
Hairless skin domains and physical breaks in the skin can break homogenous wave spreading. To model this wave-breaker phenomenon, we consider a 100 Â 100 domain to simulate the Cranial domain, with parameter values given in Appendix 2-table 1 (the 3D dorsal column), 3 and 5, with l A ¼ l I ¼ 0:02. The ½30; 72 Â ½56; 60 region is 'cut-off' (Figure 7A, black region), that is no HF growth permitted and no molecules are allowed to diffuse into this region. Additionally, we apply no-slip boundary conditions to the four boundaries of this strip. We stimulate the top row of HF growth at the beginning of the simulation by giving it a short-time increase in activator production (for 0 t 0:1, set I A ! I A þ 400 for y ¼ 100), which creates a steady HF wave propagating from top to bottom. The 'eyelid' region breaks the wave propagation, creating the distortion-like effect which is particularly clear during the first few cycles.
In comparison, we also model another wave-breaker scenario: while still no HF growth is allowed in the wave-breaker region, molecules are allowed to diffuse into this region. This might be a similar scenario to a skin where a part of it has the HF growth impaired. We block the HF growth in this region by setting hðtÞ h min at all times. Other parameter values are the same as in the previously described scenario. Simulation results show that such a growth-impaired region also creates a distortion-like effect which is quite similar to the 'eyelid' simulation results (Appendix 2-figure 18).

Sensitivity test of parameters
In this part, we test the sensitivity of several crucial parameters in our model.

Threshold D þ
At single HF level, D þ has great effect on~telogen length, but little effect on~anagen length. For instance, whereas increasing D þ greatly increases~telogen length, it slightly increases anagen length (Appendix 2- figure 19). For different values of D þ , increasing inhibitor level will always result in shorter~anagen and longer~telogen, although to different extents (Appendix 2-figure 20).

Threshold D À
On the other hand, at the single HF level, increasing D À greatly decreases~anagen length while slightly decreasing~telogen length (Appendix 2- figure 21). For different values of D À , increasing inhibitor level will always result in shorter~anagen and longer~telogen, although to different extents (Appendix 2- figure 22).
Growth parameters H þ ; H À ; m þ ; m À Parameters H þ ; m þ determine the growth rate of a HF during phase P, while H À ; m À determine its degeneration rate during catagen. The HF dynamics are not sensitive to these parameters. With fixed activator and inhibitor levels, changing either of these four parameters have little effect on~anagen/~telogen length (Appendix 2-figure 23). Increasing inhibitor level will always result in shorter~anagen and longer~telogen, and the pattern does not change much under different values of these parameters (Appendix 2- figure 24).

Maximum HF growth length h max
At the single HF level, increasing h max will decrease~telogen length, with a relatively slight effect on~anagen length (Appendix 2- figure 25). For different values of h max , increasing inhibitor level will always result in shorter~anagen and longer~telogen, although to different extents (Appendix 2-figure 26).
At HF wave level, we test the effect of h max in noise-free 3D model on a 60 Â 100 domain, with parameter values given in Appendix 2-table 1 (3D dorsal column), 3 and 5, except l A ¼ l I ¼ 0, and D þ ¼ 2:6 or 2:7 as shown above. At t ¼ 0:1, we give a one time increase in the activator production (I A ! I A þ 3000) for all HFs on the top row (y ¼ 100) to generate steady head-to-tail HF waves. We tracked the time when the second HF wave reaches y ¼ 60 and y ¼ 40 (denoted as T 2 60 and T 2 40 ); when it leaves y ¼ 40 (denoted as T 2 40 ); and when the third HF wave reaches y ¼ 40 (denoted as T 3 40 ). We measured the wave speed, wave interval and wave length in the following way: for different values of h max . Simulation results show that longer HF (i.e., larger jh max j) results in a longer wave interval and a wider wave, but the wave speed is hardly affected (Appendix 2-figure 27).
Total available receptors R A tot and R I tot As we discussed in section Section 5, the total amount of activator and inhibitor receptors results in different~anagen and~telogen lengths, and we use this property to model the difference between dorsal and ventral HFs. In Figure 2F, the dorsal-ventral interaction is simulated with R A tot ¼ R  Figure 2F where R A tot ¼ R I tot ¼ 50 for the same domain. However, during later cycles, with large ventral R A tot and R I tot , there is clearly a delay between ventral and dorsal HF wave development (Appendix 2-figure 28, 29, 30).

Comparison with FitzHugh-Nagumo (FHN) model
The FitzHugh-Nagumo model is a 2D model that describes an excitable medium, and it has also been used in the study of HF behavior (Murray et al., 2012). Below we provide a detailed comparison between the FHN model and our model. While there are many similarities between the two models in predicting experimental observations, we do find there are several important differences between them at both single and population HF levels. The codes and parameters of the FHN model are based on the original ones from Murray et al. (2012).

Single HF
Main differences between our model and the FHN model at the single HF level: 1. FHN model cannot predict reasonable anagen and telogen times as observed in experiment. Plikus et al. (2008b) reported 14 and 28 days for anagen and refractory telogen for WT mice, respectively, while competent anagen may vary in the range of 0-60 days. It is reported in Murray et al. (2012) that such a time scale cannot be reached: '. . .in order for the proposed model to yield stochastic excitations at a rate in agreement with observations we find that Tc (competent telogen time) in the model must be of the order of 10 5 days. . .' 'A notable feature of our simulations is that competent telogen times must be of the order of~10 5 days such that the frequency of stochastic excitations across a population of follicles is comparable to the population scale patterns measured by Plikus et al. However, at the single follicle scale Plikus et al. have measured competent telogen times in the range of 0-60 days. In fact, when we used these much shorter competent telogen times the simulations are dominated by stochastic excitations in a manner inconsistent with population scale measurements from wild-type mice (data not shown).' However, we would like to point out that Chen et al. (2014) use the FHN model and reach biologically observed time scales, as shown in Chen et al. (2014), with a specific choice of parameters, the authors reach T PA (propagating anagen)~15 days, T R (refractory telogen) 30 days and T C (competent telogen)~10 days. However, we fail to reproduce their results without further information about parameter values. Moreover, to reach such a state it needs to sacrifice the sensitivity of telogen length -in particular, that of competent telogen length -with respect to the change of activator level.
2. The dynamics of the FHN model are either periodic or excitable. In the periodic regime there is no competent telogen phase, while in the excitable regime sustained excitability relies on noise or exogenous activating factors, which makes the competent telogen too sensitive to certain core parameters. Our model, however, in its periodic dynamics provides competent telogen with variable length, and noise and/or external factors affect the length of different phases instead of determining the system moves on to the next cycle. In the FHN model, when the nullcline _ w ¼ 0 intersects with the other nullcline _ v ¼ 0 in its first piece, that is v ss < v 0 , which is the scenario shown in Appendix 2-figure 31A, the system becomes an excitable medium. This is the scenario discussed in Murray et al. (2012), and to end the competent telogen, the noise has to be strong enough to pass over the threshold (Appendix 2- figure 31B). For such an excitable scenario, the mean time spent in competent telogen is estimated by Murray et al. (2012). When the noise is not strong enough, the system will stay in competent telogen for a long time. In contrast, when the inhibitor nullcline _ w ¼ 0 intersects the activator nullcline _ v ¼ 0 in its second piece (Appendix 2-figure 31C), that is v 0 <v ss <v 1 , the system will deterministically undergo cycles, since there is no more threshold to pass over that relies on the help of noise. Appendix 2- figure 31D shows the deterministic trajectory (black line) which clearly goes in cycles, unlike the stochastic trajectory in Appendix 2-figure 31B which has to wait in competent telogen for a long time waiting for excitation.
To study a HF system with all four phases (PARC) using the FHN model, it has to be in the excitable regime and depend on noise and/or external factors to get the system excited. In contrast, most simulations of our model are done in the periodic regime, which also provide the competent regime that is missing from a periodic FHN model. In addition, the length of the competent regime in the periodic regime of our model can vary a lot, allowing noise greatly to adjust the length of different phases ( Figure 1C).
3. Subjected to changes in the activator/inhibitor production rate (I=J), in FHN model refractory telogen reacts differently to experimental observations, while competent telogen acts too sensitively. Plikus (2012) showed that anagen, refractory and competent telogen lengths change associated to activator and/or inhibitor level changes. Such an activator/ inhibitor level change can be simulated via changes of activator/inhibitor production rate I=J, or the positive feedback in the activator pathway a 1 -the latter will be discussed later.
Regarding change of I, we did not find published data. For change of J, as is pointed out in When the positive feedback in the activator pathway a 1 is not too low, refractory telogen length (dashed line, T R ) barely changes with activator/inhibitor production rate change I=J (Appendix 2- figure 32B,C,E,F), which is stated in Murray et al. (2012) and against experimental observations. For small a 1 , refractory telogen length increases as activator production rate increases (Appendix 2- figure 32A) and decreases as inhibitor production rate increases (Appendix 2- figure 32D), also against biological observations. Competent telogen length (dash-dot line, T C ) reacts dramatically to activator/inhibitor production rate changes: it decreases from the order of 10 4À5 down to 0 quickly as I increases (Appendix 2- figure 32A,B,C) or as J decreases (Appendix 2- figure 32D,E,F). The change rate of T C is affected by the value of a 1 : with bigger a 1 (Appendix 2- figure  32C,F) the change rate is slightly milder comparing to extremely abrupt change when a 1 is small (Appendix 2- figure 31A,D). In fact, indicating by Appendix 2-figure 32, only within a small window of (I; J) could T C stay in a reasonable scale. Although the reaction trends coincide with experimental observations -T C decreases as I increases or J deceases; however, it might be too sensitive to changes of I and/or J, comparing to a reasonable change of T PA . If we restrict to the small window of (I; J) where T C behaves reasonable, then T PA reacts too insensitive to changes in I and/or J.
4. The FHN model is unable to reach an extended anagen scenario. Our model gives two equilibrium regimes besides the excitable regime ( Figure 1C): one relates to extended telogen achieved with high inhibitor or low activator level (Appendix 2- figure 5A), which can also be achieved in FHN model; however, the other equilibrium regime relates to extended anagen achieved with low inhibitor or high activator level (Appendix 2- figure  5B), this scenario may explain phenomena such as human scalp hair anagen, which can last for years, or the long hair observed in angora rabbits. The FHN model is unable to simulate such an extended anagen scenario. 5. In order to reach both short anagen and telogen in the FHN model, it needs many parameter values changed, while in our model we can easily achieve this state by changing one or two parameters. Thus, our model is a convenient tool to model the interactions between dorsal and ventral domains. In vivo experiments support fast hair cycle dynamics in the ventral domain, more specifically, ventral HFs are short in both anagen and telogen lengths. While in both the FHN and our model we discover parameters that result in short anagen and long telogen (or vice versa), in our model we also have parameters that can be modified to alter anagen and telogen length, R A tot and R I tot (Appendix 2-figure 8, 9), or h max (Appendix 2-figure 25). In contrast, in the FHN model we do not know if by adjusting one or two parameters we could obtain such a fast cycling HF type. This directly affects the simulation of the interaction between ventral and dorsal domains.
Below are some similarities between the FHN model and our model at a single HF level: 1. Subjected to changes in the activator/inhibitor production rate (I=J), anagen length reacts similar to what is predicted by our model. Despite the noncoincidence of refractory and competent telogen to experimental observations and our model predictions, in FHN model, anagen length increases as activator production rate increases (Appendix 2-figure 32A-C) and decreases as inhibitor production rate increases (Appendix 2-figure 32D-F), which qualitatively coincides with our model predictions. Moreover, the change rate of anagen length to I or J is not greatly affected by the value of a 1 .
2. Anagen/telogen length change subjected to the positive feedback in the activator pathway a 1 coincides with experimental observations. In the FHN model, if the anagen/ telogen length change is adjusted via a 1 instead of the production rates I=J, the simulation results coincide with experimental observations. However, since in our model there is no direct cross-talk between the activator and inhibitor pathways, there is no way to directly compare the two models regarding such a positive feedback effect.
As summarized in Murray et al. (2012), 'increased positive feedback in the activator dynamics results in the observed phenomena of faster activation wavefronts, shorter refractory and competent telogen times, unchanged anagen time, increased spontaneous initiation rates and the emergence of target patterns at the population scale.' We reproduce the simulation results of phase length to positive feedback (a 1 ) relation under different values of I and J, shown in Appendix 2-figure 33.
As a 1 increases, anagen length (solid line, T PA ) increases slightly while refractory telogen length (dashed line, T R ) clearly decreases; both coincide with experimental observations. Depending on the base values of I and J, competent telogen length shows totally different reactions to changes in a 1 . When J is not high compared to I (Appendix 2-figure 33A-C), the competent telogen length mostly stays at 0, that is, the system dynamics are periodic, absent of competent telogen. On the other hand, when J is relatively high to I, that is, the inhibitor level is high in the system (Appendix 2- figure 33D), competent telogen increases dramatically as a 1 decreases. That is, in a system where inhibitor level (J) is high, as the positive feedback in the activator pathway (a 1 ) gets weaker, the mean time spent for the system waiting to get excited increases extremely fast. However, there are certain values of I and J that allow the system -in particular, competent telogen reacts in a similar way to what is observed experimentally or predicted in our model (Appendix 2-figure 33E,F). In these scenarios, competent telogen length decreases at a rate comparable to that of refractory telogen as a 1 increases.
In Appendix 2-figure 34, we present the nullclines and trajectories in v À w phase corresponding to values of I, J and a 1 in Appendix 2-figure 33.

HF wave
Main differences between our model and the FHN model at the HF wave level: 1. In the context of the heterogeneous two-domain model (i.e. dorsal and ventral domains), the dorsal head-to-tail asynchrony resulted from dorsal-ventral interactions breaks down within approximately 2-3 cycles in our model, which is consistent with the biological observations. Similar breakdowns take many more cycles to achieve in the FHN model. Experimental observations show ventral-dorsal interactions leading to dorsal head-to-tail asynchrony in the first 2-3 cycles, which quickly degrade into spontaneous initiations yet possessing bi-lateral symmetry.
Our model simulated similar patterns ( Figure 2F) -the dorsal head-to-tail asynchrony resulted from dorsal-ventral interactions lasts for approximately 2-3 cycles, after which it turns into spreading waves with centers located on the ventral-dorsal borders showing bilateral , together with spontaneous initiations. Using the FHN model, we see the similar results; however, the dorsal-ventral interaction always lasts for more cycles (Appendix 2figure 35, dorsal-ventral interactions persists for five cycles). We tried different parameter values but the dorsal-ventral interaction always lasts for more than three cycles, as is observed experimentally or predicted by our model.
A possible reason behind this might be that while our model allows us to modulatẽ anagen and~telogen lengths by adjusting R A tot and R I tot to approximate ventral HF behavior, that is shorter~anagen and~telogen, the FHN model has no easy way to modulate them. Therefore, we have to set HFs in ventral domain to have shorter telogen in sacrifice of slightly longer anagen compared to dorsal HFs.
2. In the wave-breaker simulations, we are able to investigate two scenarios using our model: a physical region cut-off from the HF wave domain (like eyes, ears), or a region permeable to signal molecules with disabled HF growth (by wound, for example). The FHN model is unable to distinguish these two scenarios.
3. Experiments also imply ventral HFs are shorter than dorsal HFs, combined with our single HF analysis that within a region, short HF have shorter anagen and telogen (Appendix 2figure 25), in the future extension of our study, we will investigate the interaction between dorsal and ventral domains characterized by different HF length. The FHN model does not permit for this. 4. In a homogeneous domain with HFs near the head having temporal phase advancement built-in, both models reproduce the head-to-tail hair cycle asynchrony during early cycles, which will degenerate into random initiations with no bilateral symmetry (Appendix 2- figure 14, 15, Appendix 2-video 3, Appendix 2- figure 36). However, the early head-totail steady pattern fades quickly in the FHN model while in ours it degenerates gradually. Simulations show that in our model simulations, even after 10 cycles, the wave may still resemble the early head-to-tail pattern. This is because we are sitting in a periodic state with non-zero competent telogen length, and noise is still able to initiate waves allowing the system show excitable properties, and the memory of an early wave pattern can persist through later waves.
On the other hand, in the FHN model, the early head-to-tail pattern may be lost as early as the 2nd cycle. Appendix 2- figure 36 shows the time course of a typical simulation by the FHN model, in this simulation, the head-to-tail pattern persists for the first three cycles, while stochastic initiation takes over starting in the fourth cycle. The reason for such a quick loss of the early memory in the FHN model is because for the early head-to-tail pattern to survive, the system should possess a periodic property; however, the system is in an excitable regime where stochastic effects dominate, hence the stochastic initiations quickly wash away the head-to-tail pattern.
Below are some similarities between the FHN model and our model at HF wave level: 1. In a homogeneous domain, if all HFs are synchronized in telogen, they have equal probability of anagen initiation, in both models we see stochastic initiations dominate and there is no bilateral symmetry (Appendix 2-figure 11, 12, Murray et al., 2012).
2. In a homogeneous domain, when HFs near the head have higher built-in levels of activator signaling, both models show highly successive waves, that is target-like pattern (Appendix 2-figure 13, Figure S4 in Murray et al., 2012). showing some bilateral symmetries. The full time course is shown in Appendix 2-video 1. See Appendix 2-tables 1 -3 and 5 for parameter values, except l A ¼ l I ¼ 0:035. DOI: 10.7554/eLife.22772.054

Supplementary figures
Appendix 2-figure 13. 2D (A) and 3D (B) simulations of dorsal HF waves, when HFs near the head have higher build-in levels of activator signaling. Here we see highly successive steady waves characterized as the 'target-like wave' pattern, also against experimental observations. The full time course of the 3D simulation is shown in Appendix 2-video 2. See Appendix 2-tables 1 -3 and 5 for parameter values, except l A ¼ l I ¼ 0:02 and D þ ¼ 1:8 in the 3D simulation 13B. DOI: 10.7554/eLife.22772.055