The authors have declared that no competing interests exist.
Conceived and designed the experiments: PJM PKM REB. Performed the experiments: PJM. Analyzed the data: PJM. Contributed reagents/materials/analysis tools: PJM. Wrote the paper: PJM PKM MVP CMC REB.
The hair follicle system represents a tractable model for the study of stem cell behaviour in regenerative adult epithelial tissue. However, although there are numerous spatial scales of observation (molecular, cellular, follicle and multi follicle), it is not yet clear what mechanisms underpin the follicle growth cycle. In this study we seek to address this problem by describing how the growth dynamics of a large population of follicles can be treated as a classical excitable medium. Defining caricature interactions at the molecular scale and treating a single follicle as a functional unit, a minimal model is proposed in which the follicle growth cycle is an emergent phenomenon. Expressions are derived, in terms of parameters representing molecular regulation, for the time spent in the different functional phases of the cycle, a formalism that allows the model to be directly compared with a previous cellular automaton model and experimental measurements made at the single follicle scale. A multi follicle model is constructed and numerical simulations are used to demonstrate excellent qualitative agreement with a range of experimental observations. Notably, the excitable medium equations exhibit a wider family of solutions than the previous work and we demonstrate how parameter changes representing altered molecular regulation can explain perturbed patterns in Wnt over-expression and BMP down-regulation mouse models. Further experimental scenarios that could be used to test the fundamental premise of the model are suggested. The key conclusion from our work is that positive and negative regulatory interactions between activators and inhibitors can give rise to a range of experimentally observed phenomena at the follicle and multi follicle spatial scales and, as such, could represent a core mechanism underlying hair follicle growth.
Although the molecular interactions that regulate the follicle growth cycle have begun to be uncovered, the fundamental interactions that regulate periodicity remain elusive. In this study we develop a model in which we neglect biophysical effects (and hence morphological changes) by treating each follicle as a functional unit. We then describe caricature interactions at the follicle scale which have the property that a field of coupled follicles can be treated as an excitable medium. We perform a range of simulations that demonstrate qualitative agreement with experimental observations. Furthermore, the modelling results suggest a regulatory mechanism that might represent a key underlying principle in the regulation of hair growth.
Hair is a characteristic feature of mammals and performs a variety of roles, such as thermal insulation, physical protection, camouflage, social interaction and sensory perception
The base of a hair resides in an approximately cylindrically shaped, multicellular mini-organ called a hair follicle that is invaginated in the surface of the skin. Unlike the hair itself, which is composed of dead keratinocytes, hair follicles undergo a process of cyclical regeneration, regulated by an intrinsic clock as well as other extrinsic mechanisms
The follicle growth cycle is traditionally split into three phases: anagen and catagen, when growth and involution occur, respectively, and telogen, a quiescent phase when the follicle is either refractory or awaiting re-entry into anagen
Although it has been established that the hair follicle clock is controlled by interactions local to the hair follicle
The functional phases, P (propagating anagen), A (autonomous anagen), R (refractory telogen) and C (competent telogen) are plotted alongside activator and inhibitor concentrations as a follicle proceeds through a single cycle of the clock. 1,..,5 denote notable transition points in the activator/inhibitor trajectories and are included to aid comparison with the corresponding phase plane diagram presented in
As illustrated in
At the multi follicle scale, it has been observed that follicles can either make the transition from telogen to anagen autonomously or via induction by neighbouring follicles that have themselves just entered anagen
Left: target-like patterns of hair growth arise in the
Whilst the growth cycle of a single hair follicle is dependent on coupled physical (
Mouse model |
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Units | Reference |
Wild-type | 4 | 10 | 28 | 0–60 | d |
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14 | 0 | 12 | 0–15 | d |
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4 | 10 | 6 | 0–5 | d |
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As well as quantifying the excitable dynamics of individual follicles, Plikus et al. have exploited the coupling between anagen and the production of pigmentation
Simulations at the multi follicle scale have previously been modelled using cellular automata
Whilst previous cellular automaton models of hair follicle growth provided a useful framework in which to integrate various experimental data and investigate hypotheses
Before discussing the specifics of the hair follicle system, we provide a brief introduction to the theory of excitable media, a field of study that is used to describe a disparate range of fundamental phenomena in biology, such as nerve signal propagation
The intersection of the solid lines determines the steady-state corresponding to low activator and inhibitor activities. Upon excitation over the threshold (
The central tenet of this study can be described as follows: using an excitable medium framework, a follicle's state is represented by two variables, an activator and an inhibitor of follicle growth. The activator and inhibitor values are correlated with, but not explicitly representative of, the concentrations of known activators and inhibitors of follicle growth, such as members of the Wnt and BMP pathways, respectively. The dynamics of the activators and inhibitors can be described as follows: a follicle has a stable steady-state in which activator and inhibitor activities are low (see
But is there experimental evidence in support of the aforementioned hypothesis? At the multi follicle scale, it is clear that patterns of hair follicle growth share many features observed with patterns arising in excitable media (
We assume that the growth of a single follicle is regulated by the activities of an activator,
(a) We assume a switch-like change in auto-activation rate (see
As the molecular interactions are not yet fully understood in the hair follicle system, in the current study we consider a caricature of excitable dynamics in which the activator dynamics are given by the piece-wise linear function
As the governing equations are linear on the slow time scale, it is straightforward to obtain estimates (see
Notably, the simplified description of activator and inhibitor dynamics allows the derivation of times spent in the propagating and refractory phases of the hair cycle, and thus allows the excitable medium description to be directly related to the follicle-scale measurements used in the PARC model. Furthermore, we highlight that the two-variable model is an abstraction in which the variables
Motivated by the
(a) Activator (solid line) and inhibitor (dashed line) activities plotted against time. (b) Trajectories (solid line) and nullclines in
The bars denote measurements of
Parameter | Description | Unit | Value |
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Natural decay rate | d−1 |
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Inhibition rate | d−1 |
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Activation rate | d−1 |
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Natural decay rate | d−1 |
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Production rate | d−1 |
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Production rate | d−1 |
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Background activator production rate | d−1 |
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Background inhibitor production rate | d−1 |
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Activator threshold | Nondim |
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Activator threshold | Nondim |
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Time scale separation constant | Nondim |
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Noise strength | d−1 |
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Diffusion coefficient |
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Lattice dimension | Nondim |
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The inclusion of stochasticity into the dynamics introduces a noise-dependent characteristic fluctuation for
In order to study populations of interacting follicles on the skin surface, we now consider a two-dimensional field of hair follicles on a regular square
It has previously been suggested that the diffusion of activator and/or inhibitor ligands is a potential mechanism for inter-follicular communication
Using numerical simulations we now demonstrate qualitative agreement between the excitable medium model and a range of experimental observations. Parameters have been chosen such that: (a) the system is in the excitable regime (see
In each of the simulations presented, the governing
In
Activator,
Increasing the coupling strength between neighbouring oscillators, while keeping other parameters fixed, results in propagation of the excited state throughout the follicle field (see
Parameter values as in
We demonstrate border stability by considering initial data in which subpopulations of follicles are in different phases of the cycle. In
Other details as in
In contrast, if the spatial domain is initialised such that the follicles on the left- and right- hand sides are in competent and excited phases, respectively, the excited follicles excite their immediate neighbours and the initial border between competent and excited follicles propagates into the competent domain (see
Other details as in
The simulation results presented to date have shown that the governing excitable medium equations produce phenomena at the individual follicle and follicle population scales that are consistent with the previous PARC model and experimental observations. We now investigate how our model can be applied to perturbed experimental systems, such as the
The implantation of activator- and inhibitor-coated beads into the surface of the skin results in local regions of follicle growth and growth retardation, respectively
The bead is approximated by fixing the activator and inhibitor activities such that
Having developed a model that allows us to relate a molecular description of follicle behaviour with macroscale observations of follicle growth, we now demonstrate the merit of this approach by investigating if changes to the molecular parameters can yield phenotypes observed in perturbed mouse models.
We begin by summarising the experimental observations made by Plikus et al.
It is notable that Wnt7a over-expression does not result in the destruction of the hair follicle cycle, as one might expect if Wnt7a was the only oscillating activator in the system. Rather, it perturbs quantitative features of the hair follicle system but the functional states defined by the P, A, R and C phases remain intact (Plikus et al.
Whilst it is encouraging that an intuitive increase in the activator production rate can recapitulate experimental observations at the single follicle scale in the Wnt7a over-expression mouse, we can use the multi follicle model to predict population scale phenomena and then compare results with the experimental observations. We find that the increased positive feedback rate yields an increased activation front wave speed (see
Activator activities (white - low, black - high) are plotted at
In summary, 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. From these observations we propose that a likely role for Wnt7a expression is to increase positive feedback amongst the different activators of hair follicle growth.
In a similar manner to the previous section, we investigate if the proposed description of follicle excitability can be used to lend insight into observed behaviour in the
As Noggin over-expression does not destroy the hair follicle cycle (anagen and telogen are still well defined), we again note that the activities of activators and inhibitors in the model are not explicitly representative of the expression patterns of individual genes. But can the observations at the follicle and multi follicle scales be interpreted in the proposed excitable medium framework? A striking observation in the
In order to illustrate population-scale behaviour when follicles are in the oscillatory regime, in
Details as in
The motivation for a decrease in the parameter
In this section we suggest a number of further experiments which could help to further determine the excitable properties of the hair follicle system.
A key unvalidated assumption in our model is that there is separation of time scales between activator and inhibitor dynamics. One testable effect of such time scale separation is that spatial gradients of the activator (fast variable) should be much larger than those of the inhibitor (slow variable). In
A snapshot of a slice through the
If the hair follicle system is truly an excitable medium, it should be possible to conduct experiments in which spiral waves, a ubiquitous feature of excitable medium dynamics, are induced. One way in which to do this is to introduce spatial inhomogeneity into the excitable medium and, in our numerical simulations, this can be achieved using the initial data presented in
The right-hand side of the domain is initially in refractory telogen, the left-hand side is in competent telogen and a small region centred at
A defining characteristic of an excitable medium is the phenomenon of thresholding,
The probability that the bead activates neighbouring follicles,
Recent experimental work in the hair follicle system has allowed the gathering of information across a range of spatial scales: at the molecular scale, numerous pathways have been shown to activate and inhibit follicle growth; at the single follicle scale, hair plucking assays have allowed quantification of the time spent in the different phases of the follicle cycle; and at the multi follicle scale, hair clipping assays have allowed the characterisation of population scale behaviours, such as wave propagation. The interdependence between the different scales is only beginning to become understood.
A previous model of mouse hair follicle growth proposed by Plikus et al. related the individual and multi follicle population scales
In this study we propose a stochastic, two-variable, activator-inhibitor model of mouse hair follicle growth dynamics. An important feature of the model is that the functional phases of the hair follicle cycle are emergent, thus allowing us to relate hair plucking measurements at the single follicle scale to underlying molecular regulation. Whilst the two-variable description of molecular events is undoubtedly an abstraction, we believe it is justified in the present case for the following reasons: (a) although the molecular pathways underlying follicle growth are becoming increasingly better understood, the current level of description is qualitative at the molecular scale, making the parameterisation of detailed molecular models difficult; (b) the model is tractable and can thus help to develop insight into how measured effects at different spatial scales are inter-related; and (c) the model can be formulated in a manner allowing comparison with both previous models and experimental observations. We anticipate that increasing quantification at the molecular scale will enable our description of underlying molecular interactions to be fine-tuned in future iterations of the model.
After developing an excitable, stochastic model of a single follicle, we considered a two-dimensional field of diffusively-coupled follicles, as previously suggested by Plikus et al.
We propose that an advantage of the current framework is that it allows one to investigate how changes at the molecular scale might give rise to different patterning phenotypes. In the
A notable feature of our simulations is that competent telogen times must be of the order of
The model presented in this study has a number of limitations. Firstly, we do not have direct estimates of molecular parameters, such as decay rates and cross-activation and -inhibition rates of activators and inhibitors. Secondly, we note that
Before the proposed model is embellished to account for further details of underlying hair follicle biology, there are a number of conceptually simpler experiments that could allow us to further validate the central thesis of this study. Firstly, the separation of time scales in the model allows a clear distinction between excited and competent phases of the cycle. In our model, the activator changes on a much faster time scale than the inhibitor and this is observable by much larger spatial gradients in the activators. Secondly, a ubiquitous feature of excitable media is the presence of spiral waves. If the hair follicle system is an excitable medium, one would expect that particular initiations of oscillators would result in the development of propagating spirals. Finally, excitable media typically exhibit a thresholding property whereby a stimulus of a sufficiently large magnitude is required to excite a given follicle. Hence, one would expect that such a threshold could be identified by examining the behaviour of beads coated with different activator concentrations.
On a concluding note, the regulation of regeneration and renewal is a key characteristic of any homeostatic biological system. In this study we have coupled hair follicle growth to the activity of activator in an excitable medium, a hypothesis that seems particularly attractive given that growth occurs only on a transient time scale. We expect that if further substantiated in the hair follicle system, there may be other instances where an excitable medium framework can be used as a mechanism for regulating regeneration in homeostatic systems.
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