A two-step patterning process increases the robustness of periodic patterning in the fly eye

Abstract

Complex periodic patterns can self-organize through dynamic interactions between diffusible activators and inhibitors. In the biological context, self-organized patterning is challenged by spatial heterogeneities (‘noise’) inherent to biological systems. How spatial variability impacts the periodic patterning mechanism and how it can be buffered to ensure precise patterning is not well understood. We examine the effect of spatial heterogeneity on the periodic patterning of the fruit fly eye, an organ composed of ∼800 miniature eye units (ommatidia) whose periodic arrangement along a hexagonal lattice self-organizes during early stages of fly development. The patterning follows a two-step process, with an initial formation of evenly spaced clusters of ∼10 cells followed by a subsequent refinement of each cluster into a single selected cell. Using a probabilistic approach, we calculate the rate of patterning errors resulting from spatial heterogeneities in cell size, position and biosynthetic capacity. Notably, error rates were largely independent of the desired cluster size but followed the distributions of signaling speeds. Pre-formation of large clusters therefore greatly increases the reproducibility of the overall periodic arrangement, suggesting that the two-stage patterning process functions to guard the pattern against errors caused by spatial heterogeneities. Our results emphasize the constraints imposed on self-organized patterning mechanisms by the need to buffer stochastic effects.

Author summary

Complex periodic patterns are common in nature and are observed in physical, chemical and biological systems. Understanding how these patterns are generated in a precise manner is a key challenge. Biological patterns are especially intriguing, as they are generated in a noisy environment; cell position and cell size, for example, are subject to stochastic variations, as are the strengths of the chemical signals mediating cell-to-cell communication. The need to generate a precise and robust pattern in this ‘noisy’ environment restricts the space of patterning mechanisms that can function in the biological setting. Mathematical modeling is useful in comparing the sensitivity of different mechanisms to such variations, thereby highlighting key aspects of their design.

We use mathematical modeling to study the periodic patterning of the fruit fly eye. In this system, a highly ordered lattice of differentiated cells is generated in a two-dimensional cell epithelium. The pattern is first observed by the appearance of evenly spaced clusters of ∼10 cells that express specific genes. Each cluster is subsequently refined into a single cell, which initiates the formation and differentiation of a miniature eye unit, the ommatidium. We formulate a mathematical model based on the known molecular properties of the patterning mechanism, and use a probabilistic approach to calculate the errors in cluster formation and refinement resulting from stochastic cell-to-cell variations (‘noise’) in different quantitative parameters. This enables us to define the parameters most influencing noise sensitivity. Notably, we find that this error is roughly independent of the desired cluster size, suggesting that large clusters are beneficial for ensuring the overall reproducibility of the periodic cluster arrangement. For the stage of cluster refinement, we find that rapid communication between cells is critical for reducing error. Our work provides new insights into the constraints imposed on mechanisms generating periodic patterning in a realistic, noisy environment, and in particular, discusses the different considerations in achieving optimal design of the patterning network.

Appendices

Appendix: A: Mathematical model for cluster formation

The formulation of a time-based approach model for understanding the process of cluster formation is described in detail in a separate work (Ref. [15] and [26] in the main text), and is mentioned here briefly for completeness. This model considers three variables: a long-range, non-autonomous activator h, a non-autonomous short-range inhibitor u and a self-inducing activator a. A selected cell is a cell that stably expresses a. The equations that describe the interplay between these variables in one dimension are:

$$\begin{array}{@{}rcl@{}} \tau_{a}\frac{da}{dt}&=&P_{a}\theta \left( a-a_{a} \right)-\lambda_{a}a+G\theta \left( h-h_{1} \right)\left( 1-\theta \left( u-u_{1} \right) \right)\\ \tau_{h}\frac{dh}{dt}&=&P_{h}\theta \left( a-a_{h} \right)-\lambda_{h}a+D_{h}\frac{\partial^{2}h}{\partial x^{2}}\\ \tau_{u}\frac{du}{dt}&=&P_{u}\theta \left( a-a_{u} \right)-\lambda_{u}a+D_{u}\frac{\partial^{2}u}{\partial x^{2}} \end{array} $$

(13)

where τ _a ,τ _h ,and τ _u denote the typical time scales of a, h and u that we used to normalize all other parameters; P _a ,P _h ,and P _u are the respective production constants; λ _a ,λ _h ,and λ _u are the respective degradation rates; D _h ,and D _u are the respective diffusion constants and 𝜃(a) is the Heaviside step function, which is equal to 1 for positive values and 0 otherwise.

Assuming a uniform propagation of h with a speed v, we replace the precise term for h with:

$$\begin{array}{@{}rcl@{}} \tau_{h}\frac{dh}{dt}=P_{h}\theta \left( \textit{v}t-x \right)+D_{h}\frac{\partial^{2}h}{\partial x^{2}}-h \end{array} $$

(14)

which has the solution:

$$h\left( x,t \right)=\left\{\begin{array}{lc} P_{h}-P_{h}\left( \frac{\textit{v}\tau_{h}+c_{1}}{2c_{1}} \right)exp\left[ \left( \frac{-\textit{v} \tau_{h}+c_{1}}{2D_{h}} \right)\bar{x} \right],& \bar{x}<0 \\ P_{h}\left( \frac{-\textit{v}\tau_{h}+c_{1}}{2c_{1}} \right)exp\left[ \left( \frac{-\textit{v}\tau_{h}-c_{1}}{2D_{h}} \right)\bar{x} \right], & \bar{x}>0 \end{array}\right. $$

$$ c_{1}=\sqrt{{(\textit{v}\tau_{h})}^{2}+4D_{h}}. $$

(15)

We can now calculate the time when a cell i at position x is activated (h>h ₁), assuming zero initial level of a:

$$ t_{i}^{ac}=\frac{1}{\textit{v}}\chi+\delta_{0} ; \delta_{0}=\frac{1}{\textit{v}}\frac{2D_{h}}{\textit{v} \tau_{h}+c_{1}}\ln[\frac{h_{1}}{P_{h}}(\frac{2c_{1}}{-\textit{v}\tau_{h}+c_{1}})], $$

(16)

Similarly, the times when each cell becomes selected (a>a _a ) and when the first cell in each cluster reaches the threshold for inhibition production are given by:

$$\begin{array}{@{}rcl@{}} t_{i}^{r}=t_{i}^{ac}+\delta_{1} ; \delta_{1}=\left( \frac{\tau_{a}}{\lambda_{a}} \right)\ln [\frac{G}{G-a_{a}\lambda_{a}}], \end{array} $$

(17)

$$\begin{array}{@{}rcl@{}} t_{i}^{in}={t_{i}^{r}}+\delta_{2} ; \delta_{2}=\left( \frac{\tau_{a}}{\lambda_{a}} \right)\ln [\frac{a_{a}\lambda_{a}-(G+P_{a})}{a_{u}\lambda_{a}-(G+P_{a})}], \end{array} $$

(18)

The number of cells that become refractory to inhibitory signals before the first cell reaches the threshold for inhibition production determines cluster size n. Accordingly,

$$ n\frac{1}{\textit{v}}+{t_{1}^{r}}<t_{1}^{in}. $$

(19)

Substituting (17), (18) and (19), we obtain:

$$ n=\left\lceil \frac{{\textit{v}\tau}_{a}}{\lambda_{a}}\ln {[\frac{a_{a}\lambda_{a}-(G+P_{a})}{a_{u}\lambda_{a}-(G+P_{a})}]} \right\rceil. $$

(20)

Appendix: B: Analytical term for error probability in cluster formation for the case n=2

The error in the general case for n=2, given by (7) can be written analytically as:

$$ \varepsilon =1-\left\{\begin{array}{lll} \delta_{1}+\delta_{2}-\frac{1}{2}\left( \delta_{1}^{2}+{\delta_{2}^{2}} \right)&\tau_{b}>1; \delta_{1}=min\left\{ \delta_{1},1 \right\};\delta_{2}&=min\left\{ \delta_{2},1 \right\}\\ \delta_{1}\cdot \delta_{2}+\frac{1}{2}{\delta_{2}^{2}}+\left( \eta +\gamma + \varphi \right) &\prod\limits_{i=1}^{m} \left( 1+i\left( \delta_{1}+\delta_{2} \right)-t^{in} \right)& \frac{1}{j}<\tau_{b}<\frac{1}{j-1}, \end{array}\right. $$

(21)

where j =2,3,4… and the operators η(t ⁱⁿ),γ(t ⁱⁿ) and φ(t ⁱⁿ) are given by:

$$\eta =\sum\limits_{m=1}^{q} {\{\delta_{1}+\delta_{2}+\min \left[ m-1,j-2 \right]\cdot \left( \delta_{1}+\delta_{2} \right)\}{\int}_{m(\delta_{1}+\delta_{2})}^{\min\{\left( m+1 \right)\left( \delta_{1}+\delta_{2} \right),1+\delta_{1}\}} {dt^{in}}} $$

$$\gamma =\sum\limits_{m=1}^{j-2} {\int}_{m(\delta_{1}+\delta_{2})}^{(m+1)(\delta_{1}+\delta_{2})} dt_{2}^{r}{\int}_{t_{2}^{r}}^{(m+1)(\delta_{1}+\delta_{2})} {dt^{in}} $$

$$ \varphi ={\int}_{(j-1)(\delta_{1}+\delta_{2})}^{1} {{dt}_{2}^{r}\sum\limits_{m=j-1}^{n} {\int}_{\max\{t_{2 ,}^{r}m\left( \delta_{1}+\delta_{2} \right)\}}^{\min\{1+\delta_{1}, \left( m+1 \right)\left( \delta_{1}+\delta_{2} \right)\}}{dt^{in}}} $$

(22)

and where $-1\le \, q=\left \lfloor \frac {1+x}{x+y} \right \rfloor \le j$.

Here, q = j if P(t ⁱⁿ) overlaps with $P(t_{j+2}^{r})$, corresponding with the possible overlap of P(t ⁱⁿ) with $P({t_{4}^{r}})$ in Fig. 3a in the main text.

The error probability ε(τ _b ,δ ₁) above stands for n=2, in which t ⁱⁿ must be chosen prior to any number of overlapping cells chosen from the distribution for ${t}_{3}^{r}$ and the distributions to follow, but succeeds only ${t_{2}^{r}}$. Obtaining ε for a general n (8) requires demanding t ⁱⁿ to succeed a larger number of cells determined by τ _b , the net effect being a higher error probability.

Appendix: C: Analytical term for error probability in cluster refinement

As described in the main text, it is convenient to first define the probability for a cell i to be successfully inhibited by the first cell producing inhibition as:

$$ Q_{i}\left( {t_{I}^{D}} \right)={\int}_{\max\{\bar{{ t}_{i}^{D}} -\tau_{\sigma} ,{t_{I}^{D}}+\tau \}}^{\bar{{ t}_{i}^{D}} +\tau_{\sigma} } {\frac{1}{2\tau_{\sigma} }dt=\quad\left\{{\begin{array}{cl} 1&\qquad \bar{{ t}_{i}^{D}}-\tau_{n}>{t_{I}^{D}}+\tau \\ \frac{\bar{{ t}_{i}^{D}}+\tau_{n}-({t_{I}^{D}}+\tau )}{2\tau_{\sigma} }&\qquad \textit{otherwise} \end{array}}\right. } $$

(23)

The probability for error refinement (after normalizing all terms by$\frac {1}{2\tau _{\sigma } })$, given by (11), can be written analytically as (see Table 1 for integral limits):

$$ \varepsilon =\left\{{\begin{array}{lr} 1-{\int}_{<{t_{1}^{D}}>-\frac{1}{2}}^{\min\{<{t_{1}^{D}}>+\frac{1}{2},T+\theta (\frac{3}{2}-n_{1})\}}dt_{I}^{D}\left( \begin{array}{c} n_{1}\\1 \end{array} \right)Q_{1}^{n_{1}-1}& \tau_{b}>1+\tau \\ 1-\mathrm{\mu -\eta} -\mathrm{\lambda} & \frac{1+\tau}{j}<\tau_{b}<\frac{1+\tau }{j-1} ;j=2,3,4\mathellipsis \end{array}}\right. $$

(24)

Table 1 Limits of the integrals in (24)

$$\mathrm{\mu} =\sum\limits_{m=1}^{j} {\int}_{lim\, 1}^{lim\, 2} {\left( {\begin{array}{c} n_{1}\\ 1 \end{array}} \right){d{t_{I}^{D}}Q}_{1}^{n_{1}-1}\cdot \prod\limits_{k=2}^{m} Q_{k}^{n_{k}}} $$

$$\mathrm{\eta =}\sum\limits_{m=2}^{j-1} \sum\limits_{q=m}^{j} {\int}_{lim 3}^{lim 4} {\left( {\begin{array}{c} n_{m}\\ 1 \end{array}} \right){d{t_{I}^{D}}Q}_{m}^{n_{m}-1}\cdot \prod\limits_{l=1}^{m-1} Q_{l}^{n_{l}} \cdot \prod\limits_{k=m+1}^{q} Q_{k}^{n_{k}}} $$

$$\mathrm{\lambda =}{\int}_{lim 5}^{lim 6} \left( {\begin{array}{c} n_{j}\\ 1 \end{array}} \right) {d{t_{I}^{D}}Q}_{j}^{n_{j}-1}\cdot \prod\limits_{k=1}^{j-1} Q_{k}^{n_{k}} $$

where n ₁…n _j are the number of cells in the columns 1…j. We assumed the inhibition delay to be smaller than the distribution width and the bias (τ< min{τ _b ,1}). $\theta (\frac {3}{2}-n_{1})$ in the upper limit for τ _b >1 + τ is the step function that equals 1 if n ₁=1 (in which case we demand ε=0 regardless of T), or otherwise equals 0.