Basolateral Junction Proteins Regulate Competition for the Follicle Stem Cell Niche in the Drosophila Ovary

Epithelial stem cells are routinely lost or damaged during adult life and must therefore be replaced to maintain homeostasis. Recent studies indicate that stem cell replacement occurs through neutral competition in many types of epithelial tissues, but little is known about the factors that determine competitive outcome. The epithelial follicle stem cells (FSCs) in the Drosophila ovary are regularly lost and replaced during normal homeostasis, and we show that FSC replacement conforms to a model of neutral competition. In addition, we found that FSCs mutant for the basolateral junction genes, lethal giant larvae (lgl) or discs large (dlg), undergo a biased competition for niche occupancy characterized by increased invasion of neighboring FSCs and reduced loss. Interestingly, FSCs mutant for a third basolateral junction gene, scribble (scrib), do not exhibit biased competition, suggesting that Lgl and Dlg regulate niche competition through a Scrib-independent process. Lastly, we found that FSCs have a unique cell polarity characterized by broadly distributed adherens junctions and the lack of a mature apical domain. Collectively, these observations indicate that Lgl and Dlg promote the differentiation of FSC progeny to a state in which they are less prone to invade the neighboring niche. In addition, we demonstrate that the neutral drift model can be adapted to quantify non-neutral behavior of mutant clones.

. Estimators for the parameters R and b can be evaluated as those maximizing L (R,b).
We evaluated the maximum Likelihood numerically using the MATLAB Optimization Toolbox subroutine fminunc (Tables S1-S3). To perform the maximization procedure, two additional implicit parameters must also be maximized, namely the (unknown) initial labeling frequencies ! ! | , (k=1,2). Taking t 0 =7 days, we find that these implicit parameters are very close to the empirical frequencies at 7 days post-labeling. From Eq.
[ †] one also obtains the standard error and the 95% confidence intervals for R, b (Tables  S2,S3). The standard error is evaluated by inverting the Hessian matrix of L (R,b). We obtained the Hessian matrix numerically in MATLAB (subroutine fminunc). To calculate the 95% confidence intervals for each of the parameters, we projected the Likelihood function for each parameter, defining The 95% confidence intervals for parameter X∈{R,b} satisfy = ! !" !" !! with q=0.025 for the lower bound and q=0.975 for the upper bound. Figure S5 shows an example of the Likelihood function, the projected Likelihood functions, and fits with Maximum Likelihood parameters.

Null hypothesis testing
For each data set, we tested the significance level of the null hypothesis b=0, i.e. that competition is neutral. The p-values (Table S3) were calculated using the Likelihood Ratio Test (Wasserman, 2004). For this, we calculated the maximum Likelihood from Eq. [ †], as before, but now assuming zero bias. The p-value was then evaluated as, where Prob ! ! > is the chi-squared distribution with q degrees of freedom, and q=1 reflecting the number of degrees of freedom constrained in the null hypothesis relative to the unconstrained model.

A generalized model for competitive bias in a multi-stem cell niche
This section is intended for the more mathematically oriented reader, to extend the modeling framework used in the main part of the paper to a niche containing more than two stem cells. The key findings are summarized as follows: (1) we note that the same long-time clonal behavior emerges independently of the number of stem cells in the niche, and (2) the apparent bias b and replacement rate R fitted to an experiment through Eq. (*) of the Methods section are effective parameters (not true values of R,b), which depend on the number of stem cells and the geometry of the stem cell niche. (3) If more than two stem cells are present, the apparent average replacement rate R will depend on the competitive bias, b, so that introduction of any bias (positive or negative) may appear to accelerate the rate of stem cell replacement. Expressions for R,b, are derived below for two stereotypical niche anatomies: one allowing nearest neighbor stem cell competition, and the other allowing all stem cells to compete with each other.
To fit the clonal dynamics arising from competition between stem cells in a confined niche of N S stem cells, we study the probability distribution for the number of labeled stem cells per clone using a Master Equation for the fraction of clones ! ( ) containing n stem cells at time t post-labeling, The transition probability ! is determined by the particular geometry of the niche: if the differentiation of a stem cell can be compensated by the division of any other stem cell in the niche, we have a logistic term ! = ( ! − )/ ! , which describes the Moran Process (Moran, 1957); by contrast, if each stem cell competes with just two neighbors, as seen in the mouse intestinal crypt, we have (in a ring-like geometry) ! = {1 for n=1,…,N S -1; 0 for n=0}, (Klein et al., 2010;Lopez-Garcia et al., 2011;Snippert et al., 2011).
In solving Eq. (1) for different niche geometries, we are particularly interested in the long-time behavior of the clone size distribution, as it is found to follow a simple exponential relaxation of the type seen in the data. In particular, it will later become apparent that the fraction of niches that are either mosaic ( mos. -containing labeled and unlabeled cells), fully-labeled ( all ), or fully-unlabeled ( none ), asymptote to: This dynamics has the same form as Eq. (*) (main text), and holds for any number of stem cells. Thus, when fitting clone fate data, the fractions mos.
(!"".!) , all (eff.0) appear to correspond to initial fraction of mosaic and fully labeled crypts in an experiment -but such a direct interpretation of the data is incorrect. Instead, mos.
(!"".!) , all (eff.0) depend on the history of clonal expansion and extinction at early times, and thus vary with the physiological parameters R, b, and N S . Likewise, in general the effective parameters eff. , and eff both differ from the replacement rate R and bias b.
The goal below is to relate the effective parameters in Eq.
(2) to the underlying parameters R, b, and N S , and so to allow R, b to be inferred from relatively simple measurements. We begin with a case of nearest-neighbor stem cell competition, and then consider a case of competition between all stem cells in a niche.

3a. Stem cell competition with replacement of nearest-neighbors
This case has been recently explored in studies focusing on the mouse spermatogenesis and intestinal crypt turnover (Klein et al., 2010;Lopez-Garcia et al., 2011;Klein and Simons, 2011;Vermeulen et al., 2013;Snippert et al., 2014). We derive the previously published solution here for completeness, and then apply it to focus on the late-time behavior of the clonal dynamics.

General solution:
For this case, Eq. (1) becomes To solve Eq. (3), we first identify the modal solutions !,! = with pre-factors ! set by the initial condition. Introducing an initial condition ! 0 = !,! ! corresponding to exactly ! stem cells contiguously labeled at t=0, the clone size distribution is found to be, recovers Eq. (*) given in the Methods section.

Late time dynamics:
At late times the lowest mode (k=1) in Eq. (4) dominates, so the clonal dynamics converge onto the same form as Eq. (2) for any value of ! . From Eq. (4) we find the effective initial labeling fractions [see Eq.
Similarly, the parameters eff. , eff apparent from the data do not give the true average replacement rate and bias. Instead, from Eq. (2) the effective quantities are related to the true parameters through the relations, Note that eff. , eff. are both independent of the initial distribution of labeled stem cells. One also sees a clear role for the niche size in establishing the outcome of competition, since a small bias b is amplified with increasing N S .

3b. Stem cell competition with random replacement
The case where stem cells may be replaced by any other stem cells, irrespective of their position, corresponds to the Moran Process that has been widely studied in population genetics. Here, Eq. (1) hosts a logistic term ! = ( ! − )/ ! .
Despite the long history of this model (Moran, 1958), no studies of the Moran process have been carried out specifically tracking clonal dynamics as relevant to the stem cell system. We are therefore not aware of a closed form solution that relates the experimentally-accessible parameters of Eq. (2) (1) leads to Heun's equation, for which no closed form solution is currently known. We provide below an approximation for the effective quantity eff. , and a heuristic expression for eff , although a full treatment of the problem is beyond the scope of this paper.
A well-known approximation to the Moran Process can be extracted from the seminal studies of Motoo Kimura in population genetics (Kimura, 1964), who solved Eq. (1) exactly in the limit of a large population ( ! ≫ 1) and small bias ( !~1 ). In this limit, Kimura showed that the eigenvalues R k of Eq.
(1) are related to those of the oblate spheroid wave equation through the relationship, where the parameter s as originally defined by Kimura is simply related to our bias parameter as s=2b (see Appendix I of Houchmandzadeh and Vallade, 2010); the quantity λ k,m (x) is the eigenvalue of the oblate spheroid wave equation. Although there is no closed form expression for !,! , the asymptotic behaviors are known, The case ! 1 is reviewed in (Kimura, 1964), and the case ! 1 is derived from the asymptotic expansion of λ k,m (x) in (Do-Nhat, 2001). Finally, the effective bias obtained by fitting Eq. (2)