Coupling cell proliferation rates to the duration of recruitment controls final size of the Drosophila wing

Abstract

Organ growth driven by cell proliferation is an exponential process. As a result, even small variations in proliferation rates, when integrated over a relatively long developmental time, will lead to large differences in size. How organs robustly control their final size despite perturbations in cell proliferation rates throughout development is a long-standing question in biology. Using a mathematical model, we show that in the developing wing of the fruit fly, Drosophila melanogaster, variations in proliferation rates of wing-committed cells are inversely proportional to the duration of cell recruitment, a differentiation process in which a population of undifferentiated cells adopt the wing fate by expressing the selector gene, vestigial. A time-course experiment shows that vestigial-expressing cells increase exponentially while recruitment takes place, but slows down when recruitable cells start to vanish, suggesting that undifferentiated cells may be driving proliferation of wing-committed cells. When this observation is incorporated in our model, we show that the duration of cell recruitment robustly determines a final wing size even when cell proliferation rates of wing-committed cells are perturbed. Finally, we show that this control mechanism fails when perturbations in proliferation rates affect both wing-committed and recruitable cells, providing an experimentally testable hypothesis of our model.

1. Introduction

During animal development, both extrinsic and intrinsic cues contribute to the final size of organs [1]. Extrinsic cues such as nutrition, hormones and temperature drive changes in overall body size, allowing organisms to adapt to different environmental conditions and energetic resources, while maintaining functional organs [2]. While size adaptation to genetic fluctuations and changing environments is an important driver of phenotypic evolution, organisms also require a robust control of organ size in order to ensure their proper function. Understanding how body plans are designed to permit variability under certain perturbations while robustly reaching a constant organ size under other perturbations is a fundamental problem in developmental biology [3].

The Drosophila wing disc is a useful model to investigate the mechanisms underlying developmental patterning and organ growth [4–8]. Previous studies suggest that chemical gradients, nutrition, hormones, and mechanical interactions contribute to extrinsic and intrinsic growth, but how all these signals are integrated to result in a wing of a specific size and shape is little understood [9–13]. While extrinsic factors determine the ultimate size of the wing, transplantation experiments of wing discs into the abdomen of an adult fly or experiments in which pupation is halted and the duration of larval development is increased, show that discs are able to stop growing when a particular size is reached [14–16]. Moreover, wing discs stop growing at the pre-determined size even when cell proliferation or cell sizes are perturbed [17], suggesting the presence of intrinsic mechanisms that robustly control the final size of the organ.

Much of the work on intrinsic organ growth has been driven by the study of morphogen gradients that coordinate patterning and cell proliferation during development in a wide variety of contexts [18]. In the Drosophila wing, Decapentaplegic (Dpp) and Wingless (Wg) establish signalling gradients along orthogonal axes of the wing disc [19,–24] and orchestrate tissue patterning and growth [25,–30]. Dpp signalling not only promotes growth but also ensures that cell proliferation is spatially uniform [31–33], and is implicated in models of growth arrest. For instance, the morphogen slope model proposed that if cell proliferation is promoted by spatial differences in Dpp concentrations, as the disc grows and the Dpp gradient becomes increasingly flatter, it eventually arrests growth when the slope drops below a threshold [34]. While experimental data support that cell proliferation is promoted by sharp differences in Dpp expression and proliferation is inhibited when Dpp signalling levels are spatially flat [35], the Dpp gradient seems to scale as the disc grows [33,36–38], arguing against the morphogen slope model. Further experimental and theoretical studies suggested that gradient scaling and uniform proliferation are consistent with a model in which a temporal increase in Dpp signalling levels drives cell proliferation [33,39,40]. Finally, other work has implicated the Dpp morphogen gradient in a mechanical feedback model in which cells that receive the morphogen proliferate until a threshold of mechanical tension is reached [41–43]. On the other hand, the role of the Wg gradient in proliferative growth is more controversial [44,45], perhaps due to the presence of other Wnt ligands in the system [30]. Despite all the literature implicated in morphogen-mediated growth control, it remains unclear how the Drosophila wing disc buffers fluctuations in cell proliferation rates. Because cells in the wing disc divide every 6–10 h over a period of 4 days [31,46], relatively small perturbations in cell proliferation rates (e.g. 10–20% of their normal values) could result in relatively large differences in the final size of the wing [47,48].

In addition to cell proliferation, the Drosophila wing disc grows through cell recruitment, a patterning-driven growth process by which undifferentiated neighbouring cells are incorporated into the wing tissue by the induction of the wing selector gene, vestigial (vg) [49–52]. Cell recruitment occurs during the third larval instar and contributes to about 20% of adult wing size [53]. Moreover, neither of cell proliferation nor recruitment rescues normal wing size in the absence of the other, suggesting that they both contribute additively to growth [53]. However, it is unclear whether cell recruitment and proliferation act as independent mechanisms, or if there is a relationship between their rates in order to achieve a specific developmental goal. Here, we provide support that a crosstalk between cell proliferation and recruitment exists, solely based on prior experimental knowledge about the nature of the recruitment signal and the geometry of the system. Moreover, we propose that this interplay may work as a temporal controller of growth. Particularly, we first used our mathematical model to investigate the dynamics of wing-committed versus recruitable cells and found parameter sets in which the model reproduces experimental observations. With these parameter values, we then show that the population of recruitable cells extinguishes at a time that is inversely proportional to variations in cell proliferation rates. Furthermore, we provide experimental evidence that exponential wing growth slows down as the number of recruitable cells begins to reduce. By updating our model to incorporate this observation, we are able to propose that cell recruitment acts as a temporal controller that allows wings to attain a robust final size despite perturbations in proliferation rates of wing cells, but not when the proliferation of both wing-committed and recruitable cells are simultaneously affected.

2. Results

(a) . Construction of a mathematical model of cell proliferation and recruitment based in the geometry and growth properties of the Drosophila wing primordium

We built a non-spatial dynamical model of the populations of wing-committed, Vg-expressing cells (referred as W cells) versus recruitable, pre-wing cells (referred as R cells), defined as the cells located within the wing pouch that do not express Vg at the moment, but are primed by Dpp and Wg signalling to express it upon reception of the recruitment signal [51] (figure 1a). We start our simulations at the beginning of the third instar (about 80 h after egg laying) where we estimated W₀ = 225 and R₀ = 169 cells (see Methods). The model takes into account cell proliferation in each cell population (at rates α_W and α_R) and cell recruitment (F(W,R); figure 1b), which positively contributes to the number of W cells, but at the same rate reduces the number of R cells. As supported by experimental evidence [52], we assume that cell recruitment is a radial and contact-dependent process. Therefore, F(W,R) is a function that depends on the geometry of the system and the number of cells located at the recruitment front ($\pi \sqrt{W}$; see electronic supplementary material, figure S1, appendix S1). Since cell death is negligible in wild-type wing discs [54], we do not consider apoptosis in our model. Under these assumptions, the dynamics of W(t) and R(t) are given by the following equations (figure 1b; referred as Model 1):

$${\displaystyle \frac{\mathrm{d}W}{\mathrm{d}t}={\alpha }_{W}W\hspace{0.17em}+\hspace{0.17em}\rho \pi \sqrt{W}{\displaystyle \frac{R}{k\pi \sqrt{W}+R}}}$$ 2.1

and

$${\displaystyle \frac{\mathrm{d}R}{\mathrm{d}t}={\alpha }_{R}R-\hspace{0.17em}\rho \pi \sqrt{\mathrm{W}}{\displaystyle \frac{R}{k\pi \sqrt{W}+R},}}$$ 2.2

where ρ (rate units) and k (dimensionless) are constants of proportionality (table 1). Since cell proliferation is nearly homogeneous throughout the disc (except at the DV border), we assume that α_W = α_R, except when proliferation is perturbed in a particular population of cells (see sections below). Parameter values are either extracted from previous experimental studies or explored numerically (table 1).

Figure 1.

Scheme of the Drosophila wing disc and mathematical model of wing and recruitable cells. (a) Cartoon of a Drosophila wing disc, subdivided in three regions (notum, hinge and wing pouch) that correspond to different structures in the adult. The wing blade derives from the wing pouch domain and may be subdivided into wing-committed (W) and pre-wing recruitable (R) cells. W cells are defined by the expression of Vg (red); as the disc grows, the Vg domain expands at the expense of reducing the R domain (green). (b) Dynamic model for the populations of W and R cells. Note that in addition to cell proliferation of W and R cells, the W population grows at the expense of the R population through a recruitment rate function F(W,R); see electronic supplementary material, appendix S1.

Table 1.

Parameters used in the mathematical models of this study.

parameter symbol	meaning	value	justification
αW0, αR0	proliferation rates of W and R cells, respectively	0.0014 min−1	in 72–96 h old, wild-type wing discs, cells divide homogeneously twice every day [46]
k	proportionality constant for half-maximal contribution of R cells to the recruitment rate	1	starting value; we then vary it when we explored parameter space
ρ	maximum percentage of cells from the recruitment boundary that will be recruited per unit of time	0.0039 min−1	starting value; we then vary it when we explored parameter space
μ	half-maximal value of R cells necessary to reduce the growth rate of W in half (equation 2.6)	72 cells	in figure 3d, the slope of the line that connects the third and the fourth point drops to about half with respect to the slope that connects the fourth and the fifth points; thus, we estimate this parameter as the number of R cells in the fourth point in figure 3e, assuming our initial condition of R0 = 169 cells in the first point

(b) . Growth by cell proliferation and cell recruitment of the W population is approximately exponential

We used our mathematical model to explore the dynamics of cell proliferation and cell recruitment during wing growth. Although the system of equations is nonlinear, it is possible to show analytically that W is bounded by an exponential function of time (figure 2a; see electronic supplementary material, appendix S2). Thus, we wonder what would the exponential contribution of cell recruitment be relative to the cell proliferation rate, α_W. In a semi-log plot, the slope of the best-fit line corresponds to an approximate exponential growth rate of the W population (figure 2b)

$$m={\alpha }_W+\epsilon ,$$ 2.3

where ε is approximately the exponential grow rate due to cell recruitment. Thus, the size of the W population is approximately given by (figure 2c)

$$W(t)\approx W_0{\mathrm{e}}^{({\alpha }_{\mathrm{W}}+\epsilon )t}.$$ 2.4

Figure 2.

Dependence of W and R dynamics on system parameters. (a) Numerical solution of W from equations (2.1) to (2.2) using the parameters in table 1 (black curve), compared with an analytical upper bound function (red curve); see electronic supplementary material, appendix 2. (b) Semi-log representation of the numerical solution in a (black curve) compared with the best-fit solution of the linear function P(x) = mx + b. Note that because this is a semi-log plot, m represents the exponential rate of growth that has a contribution of proliferation (α_W) and recruitment (ε). (c) Under the approximation in b, proliferation and recruitment rates additively contribute to exponential wing growth. (d) The dynamics of R can be studied analytically in three different regions of parameter space; see electronic supplementary material, appendix S3. The region in pink corresponds to parameter values in which R is an increasing function of time (cell proliferation of R cells dominates over recruitment). Conversely, the region in green corresponds to parameter values in which R is a decreasing function of time (recruitment dominates over cell proliferation of R cells). The full dynamics of R in the white region cannot be defined analytically, but R is initially increasing. The dotted line represents a subset of parameters defined by k = 1. (e) Numerical solutions for the population of R cells when k and α are fixed, and ρ is varied (colour bar) along the dotted line in d. The dynamic behaviours found in d are reproduced numerically (black curves correspond to parameter values on the black lines in d). Note that within the white region, some solutions are increasing (as in the pink region), but some only increase transiently and then start to decrease (blue curves). The value of ρ reported in table 1 corresponds to the middle value of line segment defined by the dotted line intersected by the two black lines in d.

A numerical estimate of ε (using the parameter values reported in table 1) is 1.0 × 10⁻⁴ min⁻¹, which is about 1/10 of the proliferation rate. This relative growth rate when integrated over the expected duration of the recruitment process during the third larval instar (36 h) is consistent with a previously reported relative contribution of cell recruitment of about 20% [53].

(c) . Analytical and numerical exploration of parameter space reveal the dynamics of R cells

While W is always an increasing function of time in our model, the dynamics of R depends on the parameters of the model. In particular, if cell proliferation of the R population dominates recruitment, we would expect R to be a continuously increasing function of time, a situation that is inconsistent with wild-type development as Vg eventually covers the whole wing pouch at the end of the third larval instar ([53], figure 3). Conversely however, if recruitment dominates cell proliferation in R cells, then the R population will eventually vanish. We explored analytically the specific dependence of this dynamics on parameter values and initial conditions for the particular case in which α = α_W = α_R, a case that appears to be a good approximation in wild-type conditions ([46], see electronic supplementary material, appendix S3). We found three different behaviours for the dynamics of R in parameter space (figure 2d). First, when k > ρ/α, R is an increasing function of time (pink region in figure 2d); in this case, both W and R grow indefinitely, a condition that does not recapitulate the wild-type scenario. Second, when $k<\rho /\alpha -R_0/\pi \sqrt{W_0}$ we found that R is a decreasing function of time (green region in figure 2d). Finally, if $\rho /\alpha >k>\rho /\alpha -R_0/\pi \sqrt{W_0}$, we were not able to demonstrate analytically the full dynamic behaviour of R, but we show that $(\mathrm{d}R/\mathrm{d}t)(t=0)>0$, i.e. R is initially an increasing function of time (white region in figure 2d). We then explore numerically the dynamics in this region of parameter space by fixing k = 1 and α = 0.0014 min⁻¹ and varying ρ (dotted line in figure 2d). We found that for some parameter values R is an increasing function of time for all t, while for others it increases for some time and then it decreases (figure 2e, blue lines).

Figure 3.

Quantification of W and R cells in third instar wing discs. (a–c, left) Vg antibody staining in y,w (considered as wild-type in this study) discs of representative discs of increasing sizes. The yellow dotted line delineates the contour of an ellipse that is chosen as the wing pouch area (see Methods). The areas of the ellipses are shown on the left of each image. (a–c, middle) Binarization of the pixels with or without Vg staining located within the wing pouch area (see Methods). (a–c, right) Noise reduction of the Vg+ isolated pixels (error of the technique) using a particle analysis algorithm (see Methods). (d,e) Quantification of pixels representing W (d) and R (e) cells in discs grouped by wing pouch (ellipse) areas (denoted by a range in μm²), which indirectly correspond to increasing developmental times. The number of discs n used in each group (from left to right) is: 8, 12, 9, 11 and 5). Error bars correspond to the standard error of the mean. Scale bar is shown only in the first image, but it is the same for all images in the first column. Statistical test is an ANOVA one-way test; ns, not statistically significant.

(d) . Time-course analysis of Vg-expressing cells in the wing disc reveals that growth of W slows down while the population of R cells vanishes

In order to compare the dynamics of W and R in our model with experimental data, we approximated the number of cells at different times during the third instar of wild-type discs (figure 3a–c). While W cells are defined by Vg expression, there is no molecular marker of R cells. Therefore, we used the absence of Vg expression in cells within the pouch to quantify R cells (see Methods). We observe that W increases exponentially for much of development, but its growth rate slows down by the end of the third instar (compare the slope of the third to fourth versus the fourth to fifth data points in figure 3d). Conversely, the number of R cells slightly increases or remains constant during the early stages of recruitment before it starts to decrease (figure 3e), a behaviour that is consistent with the results of our model for a specific region of parameter space (white region in figure 2d, blue curves in figure 2e). Note that the decrease in the rate of growth of the W population follows from a dramatic drop in the number of R cells (figure 3d,e), suggesting that the rates of proliferation of W cells and recruitment may be cross-regulated (see Discussion).

(e) . Perturbations in cell proliferation rates of the W population modulate the duration of recruitment

In order to investigate a potential crosstalk between proliferation and recruitment, we examined further the dynamics of our model. We noted that as the population of W cells increases exponentially, the number of recruiter cells, i.e. the W cells located at the recruitment boundary, which for a circular geometry is given by $\pi \sqrt{W}$ (see electronic supplementary material, appendix S1) also increases exponentially, resulting in the acceleration of the recruitment process (figure 4a). As the population of R cells eventually vanishes (figure 3e), the duration of recruitment depends on the proliferation rate of W cells. Indeed, when α_W is perturbed above its wild-type value, α_W0, the duration of recruitment (t_R), defined as the extinction time of the R population (R < 1) is reduced (warm colours in figure 4b); conversely, when α_W is perturbed below wild-type levels, t_R increases (cold colours in figure 4b). From equation (2.4), we ask what is the condition such that the number of W cells at the end of recruitment, W(t_R), is maintained approximately constant to perturbations in α_W0. Wing size at the end of recruitment can be controlled by modulating t_R in a way that is inversely proportional to the perturbation (figure 4c). We then asked if such relationship exists for a set of parameters within those that match the dynamical behaviour in figure 3e (black region in figure 4c). To test this, we performed an optimization procedure that finds the parameter pair (ρ, k) that minimizes the error with the following control function (figure 4c, right; Methods):

$$t_R^{\ast }={\displaystyle \frac{C}{{\alpha }_W+\epsilon },}$$ 2.5

where α_W is the new (perturbed) proliferation rate of W cells, ε is the value obtained by fitting the numerical solution with the perturbed α_W to an exponential as in figure 2b and C = (α_W0 + ε₀)t_R0 is a constant such that the function coincides at the unperturbed value of α_W (figure 4c, right; unperturbed values are denoted with the sub-index 0). Remarkably, for the optimal parameter values, the numerical value of t_R for different perturbations in α_W fits well the function defined by equation (2.5) (figure 4d). This result suggests that cell recruitment is a compensatory mechanism in which the size of W at the end of recruitment is largely invariant to perturbations in cell proliferation rates (figure 4e).

Figure 4.

Cell proliferation rates are inversely proportional to the duration of recruitment. (a) Diagram representing that the recruitment of new cells (yellow) increases as the proliferation rate of W cells increase. Prior W cells are represented in red and R cells represented in green. (b) Numerical examination of R(t) with the parameters reported in table 1, but when α_W is perturbed 25% above and below the wild-type value α_W0. (c, left) Diagram illustrates that the propagation of the recruitment front is accelerated with developmental time. (c, centre) There is a range of parameters in which R cells are extinguished (R ≈ 1) within 24–36 h from the beginning of recruitment (black region). We define this extinction time as t_R. (c, right) Optimization scheme compares how t_R compensates to perturbations in α_W in ideal conditions (red curve) versus the compensation offered by the model (green curve). Assuming the exponential approximation of figure 2b, the t_R that leaves the final size W($t_R^{\ast }$) invariant is such that is inversely proportional to the growth rate α_W + ε (see Methods for details). (d) Plots of the optimal t_R (red curve) and $t_R^{\ast }$ (blue curve; defined in c) versus perturbations in α_W. (e) Plots of W(t_R), versus perturbations in α_W corresponding to optimal t_R (using the numerical solution of equations (2.1)–(2.2); red curve) and $t_R^{\ast }$ (using the equation for W$(t_R^{\ast })$ in c; blue curve).

(f) . Final size invariance is achieved when recruitment temporarily limits cell proliferation rates

So far, our model reveals a relationship between cell proliferation of W cells and the duration of recruitment. Since growth of the W population slows down as R starts to decrease (figure 3d,e), we considered the possibility that the end of recruitment could also be the overall halt of growth. With the aim to explore this possibility, we modified our original model by replacing the parameter α_W in equation (2.1) of Model 1 with a function that assumes that the proliferative growth of W cells slows down to about half its rate when the number of R cells reaches a certain threshold, μ (table 1 and figure 5a)

$${\alpha }_W(R)={\displaystyle \frac{{\alpha }_{W}R}{\mu +R}.}$$ 2.6

Figure 5.

A modified model in which growth is assumed to stop at the end of recruitment explains size invariance with respect to changes in cell proliferation rates. (a) Equations of an alternative model (Model 2) in which the proliferation rate of W cells is no longer a constant, but rather a Hill function of the number of R cells: ${\alpha }_W(R)={\alpha }_{W}R/(\mu +R).$ This function implies that α_W(R) varies continuously (colour bar) of from nearly α_W0 (the wild-type value of this parameter, table 1; darkest colour) when R ≫ μ to zero (white) when R ≪ μ. A plausible mechanistic realization of this model in which R cells secrete a growth factor (signal X, pink arrows) that is received in W cells. When the population of R cells vanishes, no signal X is produced and growth is arrested (STOP sign). (b,c) Temporal dynamics of W (b) and R (c) cells, using the equations in a and the parameter values reported in table 1, when α_W is varied (rainbow colour map) around its wild-type value (α_W0, black curve). (d) Extinction time as defined in figure 4c (which in this case, is also the final time, t_f) obtained from numerically solving the equations in A (using the optimal parameter values from figure 4d,e) as a function of perturbations in α_W. (e) Final size of the W population W_f = W(t_f) as a function of perturbations in α_W (obtained as in d). Perfect control occurs when final size W_f0 is invariant to perturbations in α_W (dotted horizontal line). Wing disc cartoons represent the disc size variations within this range of perturbations in α_W.

This function has the property that α_W(R) ≈ α_W when R ≫ μ, and α_W(R) → 0 when R → 0, i.e. W grows exponentially as in Model 1 when R cells are available but induces growth arrest of W when recruitment is terminated (figure 5a,b). One way in which this assumption could work mechanistically is if R cells produce a secreted signal X that is necessary for growth of W cells (figure 5a; see Discussion). For clarity, we will refer to equations (2.1–2.2) but with α_W replaced by equation (2.6) as Model 2. In Model 1 (§2 (a)), W was always an exponentially increasing function of time (figure 2a,b), whereas in Model 2, W approaches a final value whenever R → 0 (figure 5b). This update in the model permits us to consider the hypothesis that by modulating the speed of recruitment upon perturbations in cell proliferation rates of W cells, wing size could be controlled.

Using the parameters obtained in figure 4, we found that the dynamics of W and R are similar to those observed experimentally (figure 5b,c; compare to figure 3d,e). Moreover, we recovered the property of our original model that perturbations in α_W are inversely compensated by the duration of recruitment (figure 5c,d), but now the time in which recruitment is terminated, t_R, is also the time (t_f) in which W reaches its final size (W_f). Indeed, we note that W_f remains nearly constant over a range of perturbations in α_W (figure 5e), providing support that the relationship between cell proliferation and recruitment works as a growth control mechanism.

(g) . Growth control is lost when cell proliferation rates in both W and R cells are simultaneously perturbed

Experimental testing of our size control hypothesis may be challenging because impairing cell recruitment in R cells and, at the same time, introducing cell proliferation perturbations is not an easy task, even in a widely used genetic model such as the Drosophila wing disc. In order to test our hypothesis, it would be useful to find a perturbation in which the proposed growth control mechanism fails in a way not expected by prior models. In our perturbation simulations, we noted that the compensatory mechanism in which the duration of recruitment is modulated only works when perturbations in cell proliferation affect W cells, but not R cells. Indeed, when perturbations affect both α_W and α_R simultaneously (simply referring to both of the perturbed parameters as α and the unperturbed ones as α₀) in Model 2 (figure 5a), we obtained the opposite behaviour of t_f, compared to the case in which only α_W was affected and α_R was maintained at its unperturbed value of table 1 (compare figure 6a and figure 5d). This is because increasing α_R produce more R cells, so it would take longer to extinguish them by cell recruitment; and the opposite occurs when α_R decreases. As a result of this, W_f is very sensitive to simultaneous perturbations in proliferation rates (figure 6b). This dynamic behaviour occurs independently of the specific choice of parameters (see electronic supplementary material, figure S2). Thus, our model predicts very different behaviours when cell proliferation is perturbed only in W cells than when cell proliferation rates of both W and R cells are affected.

Figure 6.

Size invariance is lost when both α_W and α_R are perturbed simultaneously. (a) Final time t_f (as defined as in figure 5d) versus simultaneous perturbations in α_W and α_R (indistinctly referred as α) around their unperturbed value (α₀) using the parameter values that result from the optimization procedure described in figure 4c. (b) Final size of the W population W_f as a function of perturbations in α₀ using the parameter values obtained in a. Dotted line and wing disc cartoons are as in figure 5e. Note that growth control is completely lost under these simultaneous perturbations. (Online version in colour.)

Importantly, the prediction of this experiment is different to what would be expected by prior growth control models since they do not distinguish between W and R cells, and therefore, affecting both α_W and α_R simultaneously should not be any different than perturbing α_W alone (see Discussion).

3. Discussion

Cell recruitment operates as a patterning-driven growth mechanism in a plethora of systems [55], but whether there is a specific role in developmental growth control is unknown. What advantages would cell recruitment offer as a growth mechanism during development when growth can potentially reach any target size by modulating cell proliferation, cell growth or apoptosis? Recent work in the Drosophila wing shows that cell recruitment has a minor contribution to wing growth compared to cell proliferation, suggesting that perhaps cell recruitment works as a fine-tuning mechanism that provides precision to the final size of the organ [53], but no insights about how such a mechanism could work have been reported. Here, we used an experimentally guided mathematical-modelling approach to show that the rate of cell proliferation of wing-committed cells is coupled to the recruitment of cells into the wing domain. This dynamic interplay between cell proliferation and recruitment has a direct consequence on the duration of the recruitment process. In particular, when cell proliferation of wing-committed cells is faster or slower, the recruitment process terminates sooner or later than in the wild-type condition, respectively (figure 4d). In addition, if termination of recruitment also induces termination of growth, as may be the case in this system (figure 5a; see below), this coupling results in a compensatory mechanism underlying robust size control despite perturbations in cell proliferation rates, a fact that was observed experimentally more than 20 years ago [17], but has not yet been explained mechanistically.

The model relies on some assumptions that deserve further discussion. The assumption that the end of the recruitment process is also the end of overall growth in our updated model is key for our hypothesis that recruitment works as a temporal controller of size in this system. Our experimental observation that growth of the Vg domain significantly slows down when the availability of recruiter cells drops is consistent with this assumption. It could be argued that the drop in the growth rate of W is because there are no more cells to recruit and not because an actual decrease in cell proliferation rates. However, we estimated that recruitment only contributes about 1/10 of the growth rate of W [53], while the observed decrease in the slope in the last two data points in figure 3d is at least two-thirds. Thus, simply excluding cell recruitment in the growth of W cells cannot explain the observed decrease in its growth rate. Further support of this assumption comes from a prior experimental study, which shows that when Vg is broadly overexpressed (i.e. in both W and R cells simultaneously) cell proliferation is dramatically reduced [56]. How could the absence of R cells induce termination of overall growth mechanistically? One explanation is that R cells provide an unidentified growth factor (signal X) that is necessary for proliferation of W cells (figure 5a). In fact, recent work has revealed that termination of cell proliferation relies on a decrease in TORC1 signalling [57], through an unknown signalling pathway. Therefore, an attractive proposal that is consistent with our model is that the signal that sustains cell-autonomous TORC1 expression emanates from recruitable cells and it drops as the number of these cells vanishes with cell recruitment. Finally, our mathematical model assumes a circular geometry, which is essential for increasing the number of recruitable cells and accelerating the recruitment process. Since W grows exponentially, so does the recruitment boundary (see electronic supplementary material, figure S1). How much does the geometry of the system affect the implications of the model? We found that the recruitment function F(W,R) has the same form when the geometry of the system is assumed to be elliptical which resembles more closely the geometry of the wing pouch (see electronic supplementary material, appendix S4).

Our study proposes that cell recruitment not only works as a mechanism to expand the number of Vg-expressing cells, but also, as a time controller of cell proliferation. Thus, our model predicts that upon a proliferation perturbation in the absence of recruitment, the duration of growth will not be tuned by cell proliferation perturbations and final size may be significantly affected. While perturbing proliferation rates and impairing cell recruitment simultaneously could be experimentally challenging, we can estimate analytically the difference in size that would be predicted in a model with versus without recruitment. Under a cell proliferation perturbation ${\alpha }_W$ → Δ${\alpha }_{W0}$, with Δ a number that is slightly greater than 1 (the same argument applies if Δ is slightly less than 1), the adjustment for perfect control would be t_f0/Δ. Therefore, without recruitment-dependent temporal control, the error in final size is approximately given by the additional growth of W that will occur at the unperturbed rate α_W0 + ε for the additional time (1–1/Δ)$t_{\hspace{0.17em}f0}$ is given by

$$W_{\hspace{0.17em}f0}\exp \left(({\alpha }_{W0}+\epsilon )\left(1-{\displaystyle \frac{1}{\mathrm{\Delta }}}\right)t_{\hspace{0.17em}f0}\right),$$

which could be a significant difference if t_f0 is relatively long (i.e. 35–50 h).

What is the range of perturbations in which our size-invariance model works reasonably? Although t_f is a decreasing function of α_W0 for a broad range of perturbations (see electronic supplementary material, figure S3A), a fair control of final size occurs in a limited range (see electronic supplementary material, figure S3B). Particularly, we find that within the range examined in figure 5 (25% above or below the wild-type proliferation rate, α_W0, our model (Model 2) predicts less than 20% difference in final size between these bottom and top perturbations (red and blue curves in figure 5b). Thus, a 50% perturbation in the proliferation rate of W cells results in less than 20% difference in final size (under no control conditions, i.e. exponential growth, the difference would have been exp(0.5α_W0t_f) which for α_W0 = 0.0014 min⁻¹ and t_f ≈ 36 h = 2160 min gives a 411% difference!). In addition, we note that perturbations have an asymmetric effect on final size; in particular, the effect is much larger when α_W0 is reduced than when it is increased (figure 5e). While in the model this is an effect of nonlinearities, biologically, this also has been observed. For instance, in ectotherms, such as Drosophila, reducing the temperature at which animals are reared also reduces overall growth rates, but results in animals with larger organs [58,59], such as we predict in figure 5e. In Drosophila, this effect is asymmetric when rearing temperature is varied above or below the standard condition (25°C); for instance, rearing animals at 16.5°C results in a 33% increase of adult wings, whereas an increase to 29°C only results in slightly smaller wings (about 5%) [60].

Finally, our theoretical work provides guidance into how our hypothesis can be tested experimentally using genetic tools available in Drosophila [61]. Specifically, our mathematical model predicts two totally different outcomes upon perturbations in α_W versus simultaneous perturbations in both α_W and α_R (compare figure 6 with figure 5d,e). These contrasting phenotypes provide a relatively simple way to test our hypothesis versus prior growth control models. For instance, in the morphogen slope model [34,35], the gradient gets flatter with tissue growth and would be predicted to stop when the slope gets sufficiently flat, irrespectively which populations of cells grow faster or slower. A similar argument applies for mechanical feedback models [41–43]. This prediction does not mean, however, that these other models or mechanisms may be simultaneously or independently at work to ensure final size invariance under other genetic and environmental perturbations in this system.

In summary, our work proposes a novel growth control strategy where perturbations in cell proliferation rates of wing-committed cells are coupled to the duration of cell recruitment in the Drosophila wing disc. This strategy could be extended to other developmental or disease-related scenarios. For example, the mechanistic scenario presented in figure 5a could be, in principle, implemented in cell culture using synthetic-biology approaches. Such an implementation may provide insights into the feasibility of this mechanism in other growth control contexts such as in stopping tumour growth [62]. Furthermore, given that cell recruitment is a widespread mechanism in developing organs [55], the mechanism described here provides a previously unidentified role for cell recruitment in the robust determination of size to variations in cell proliferation rates, a long-standing problem in developmental biology.

4. Methods

(a) . Image processing and visualization

All the images were processed and analysed using ImageJ/Fiji software (https://imagej.net/), and the matplotlib (https://matplotlib.org/), pandas (https://pandas.pydata.org/), NumPy (http://www.numpy.org/), SciPy (https://www.scipy.org/) Python packages.

(b) . Quantification of W and R cells

To quantify the number of W and R cells within the wing pouch, we used maximum projection images from a previous publication [53] of mid-to-late third-instar wing discs immunostained with Vg and Wg antibodies, and DAPI (4′,6-diamidino-2-phenylindole; a fluorescent marker of nuclear DNA). Vg staining was used as a marker of W cells, whereas the absence of Vg within a region of recruitment competence (the wing pouch) indirectly marked R cells (unfortunately, we do not have a direct marker of R cells).

(i) . Region of interest

The wing pouch region was determined in the DAPI channel using the ellipse selection tool in ImageJ/Fiji, taking as reference for the minor axis the distance between the epithelial folds that delimit the pouch from the hinge region (since only cells within these limits are competent to become recruited and express Vg). As these folds have a curved shape, we manually adjusted an ellipse to fit these curved lines.

(ii) . Binarization

To remove pixels with unspecific Vg antibody binding, we selected a rectangular region of the wing discs outside our region of interest (ROI), where this protein is not expected to be present and made an average of the intensities within this area to define a threshold value corresponding to background noise in the Vg channel. We then binarized all the pixels within the wing pouch as having (above this threshold) or not having (below this threshold) Vg protein.

(iii) . Particle analysis

Since Vg expression has a nuclear pattern, the absence of Vg does not necessarily mean that a pixel is associated with an R cell. To solve this problem, we masked pixels without Vg (in the binarized image) using the particle analysis tool in ImageJ/Fiji using the following parameters:

• — The size parameter was set to greater than 50 pixels. By manually examining several individual nuclei in the Vg channel, we found that the minimal size of a clearly distinguished nucleus is about 50 pixels;
• — The circularity parameter was allowed to vary between 0 and 1 to admit different nuclear shapes.
• — We selected the include holes option to generate a mask of the total region that corresponds to Vg expressing cells (figure 3a–c, third column).

After this masking procedure, positive pixels within the wing pouch are counted towards W cells and the negative pixels towards R cells.

(iv) . Conversion from pixels to number of cells

We estimated the number of W and R cells within the ROI of every image as follows: (i) Using the binarized image prior to the masking procedure, we counted the number of ‘particles’ in the Vg channel from the particle analysis outlined above; this is the number of W cells. (ii) We then found the number of pixels that would correspond to a single cell in the masked image by dividing the total number of positive pixels by the number of W cells obtained in the previous step. (iii) Finally, the number of R cells is given by the number of negative pixels divided by the number of pixels that corresponds to a single cell in the previous step.

To estimate the initial conditions (W₀ and R₀) that we used in our models, we first manually counted Vg-expressing nuclei in early third-instar wing disc images to obtain W₀. To estimate R₀, we then subtracted the total number of nuclei (marked with DAPI) within the inner ring of Wg expression (since epithelial folds that delimit the pouch from the hinge regions do not appear until the middle of the third instar) to W₀.

(c) . Optimization procedure

For the optimization procedure in figure 4c, we considered parameter pairs (ρ, k) that correspond to a t_R within the range of 24 and 36 h (see electronic supplementary material, figure S4), which is the range of time when we expect recruited cell to be extinguished [53]. For each parameter pair (ρ, k), we perturbed α_W within the range of [0.75α_W0, 1.25α_W0], where α_W0 is the wild-type value reported in table 1. This interval of perturbation is under the normal variability observed in experimental measurements [46]. For each perturbation, we then solved equations (2.1)–(2.2) and computed the duration of recruitment t_R and the recruitment rate ε, as described in figure 2b. We then compared t_R with the time $(t_R^{\ast })$ that would leave the size of W at the end of recruitment, referred as W(t_R), approximately invariant (figure 4c, right) and computed the difference between these values for each perturbed α_W (see electronic supplementary material, figure S5). We defined the optimal parameter set, such that it minimizes this difference (figure 4d,f). We noted that this control works well for the range of perturbations [0.75α_W0, 1.25α_W0], but when this range is expanded, there are significant variations in $t_R-t_R^{\ast }\hspace{0.17em}$ (electronic supplementary material, figure S3). The worst fit case (i.e. the set that maximizes this difference) is shown in electronic supplementary material, figure S6.

Data accessibility

This manuscript includes an electronic supplementary material file. The codes to generate the files and simulations can be obtained from https://github.com/mnahmadb/Interplay-between-cell-proliferation-and-recruitment-controls-the-duration-of-growth-and-final-size- or https://doi.org/10.5061/dryad.8w9ghx3q5 [63].

The data are provided in electronic supplementary material [64].

Authors' contributions

E.D.-T.: conceptualization, formal analysis, investigation, methodology, software, validation, visualization, writing—original draft; L.M.M.-N.: methodology, visualization, writing—review and editing; M.N.: conceptualization, formal analysis, funding acquisition, investigation, project administration, supervision, writing—original draft.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

This work was supported by the Consejo Nacional de Ciencia y Tecnología (Conacyt) of Mexico (grant no. CB-2014-01-236685). E.D.-T. and L.M.M.-N. were recipients of PhD scholarships from Conacyt.