Quantifying the genetic origins of body plan scaling

Many of the higher organisms have remarkably consistent body proportions where, within a given species, the sizes of the different body parts, such as the head or the legs, scale with the size of the organism. To probe the origins of body plan scaling, one should study early stages of development where this property is established. Drosophila melanogaster is arguably the most suited organism for such a study since decades of research have enabled measurements of the relevant developmental markers with unprecedented precision. Specifically, we now have access to the expression profiles of all developmental genes that jointly orchestrate the body plan formation of the fruit fly through a complex network of interactions. The richness and quality of measurement data in the fly embryo call for a comparable level of quantitative rigor in studying the question of spatial scaling—a challenge that Nikolić et al. embark on in their recent PNAS publication (1).

The body plan segmentation of the developing fly embryo is coordinated by a set of pair-rule genes, the expression of which is controlled by gap genes. The latter, in turn, are regulated by maternally deposited morphogens (Fig. 1). The authors first investigated the scaling behavior of the bottom two layers of this cascade, formed by the gap and pair-rule genes, respectively. They started by revisiting what is arguably the most straightforward way to probe the scaling of the expression profiles, which is to measure specific markers of these profiles, such as their peak positions or the positions at which they reach their half maximum values. Spatial scaling means that the absolute positions $x_p$ of these positional markers scale with the embryo length $L$, such that their scaled positions $f_p=x_p/L$ are the same across embryos of different length (Fig. 1). Consistent with earlier work (2–5), they saw for both the gap genes and the pair-rules a near-perfect scaling of the absolute positions of these positional markers with the length of the embryo, with the error in the scaled positions $f_p$ being only $\approx$1%. This error is low and already hints at scaling. In fact, the error is so low that they could rule out an alternative mechanism that lacks scaling. Specifically, if instead of scaling with $L$, each positional marker $x_p$ were perfectly pinned at an absolute distance from the anterior or the posterior pole irrespective of the embryo size, then the variability in the fractional position $x_p/L$ stemming purely from the variations in embryo length would significantly exceed the measured value ($\approx$1%) for a wide range of $x_p$. Together, these two results already provide convincing evidence for spatial scaling.

Fig. 1.

The expression profiles of gap and pair-rule genes scale with the length of the embryo $L$ despite the fact that their maternal inputs do not exhibit such scaling. When plotted against the real coordinate $x$ along the anterior–posterior axis, the material gradient (illustrated for Bicoid in red) shows no significant variations between embryos of different lengths, while the gap gene profile (illustrated for Giant in blue) does vary with length. In contrast, when plotting the profiles against the scaled coordinate $x_s=x/L$, the gap gene levels in embryos with different lengths closely resemble each other, while the profiles of the material input are different.

The authors, however, take the question to another level of quantitative rigor. While the positional markers do represent a select set of key locations in the morphogen profiles, this set is limited and may miss important features in profiles that do not have sharp peaks of localized expression. A more complete statement about scale invariance should therefore concern the entire spatial distribution of expression levels along the length of the embryo.

The richness and quality of measurement data in the fly embryo call for a comparable level of quantitative rigor in studying the question of spatial scaling—a challenge that Nikolić et al. embark on in their recent PNAS publication.

Nikolić et al. employ information theory as a principled and interpretable framework for addressing this challenge (1). Perfect scaling means that the gene expression profiles $\mathbf{g}$ encode information about the scaled position $x/L$ rather than separately about the absolute position $x$ and embryo length $L$. Conversely, it implies that the scaled position $x/L$ determines $\mathbf{g}$, and that $\mathbf{g}$ cannot be specified more sharply by providing $x$ and $L$ separately. Expressed mathematically, this condition of perfect scaling can be written as $I(\mathbf{g};\{x,L\})=I(\mathbf{g};x/L)$, where $I(a;\{b,c\})$ stands for the mutual information between $a$ and the pair $\{b,c\}$. Any significant deviation from this condition would indicate imperfect scaling.

Computing the mutual information is generally a formidable task when it concerns multidimensional variables (in this case, the array of expression levels for different genes along the embryo). However, as evidenced by the high-precision measurements, the gap gene expression levels have, to a very good approximation, a jointly Gaussian distribution at fixed positions along the embryo—a convenient simplification that allowed the authors to obtain accurate estimates of the information metrics. More concretely, the authors estimated $I(\mathbf{g};\{x,L\})$, $I(\mathbf{g};x/L)$, and, as a measure of deviation from perfect scaling, their difference $\mathrm{\Delta }I$. While $I(\mathbf{g};x/L)$ is $\approx 4.2$ bits, enough for the cells to decode their position by $\approx$1% accuracy along the embryo, $\mathrm{\Delta }I$ is only $\approx$0.04 $\pm$ 0.04 bits, which is less than 2% of $I(\mathbf{g};x/L)$. These observations are stunning, for two reasons: By quantifying the mutual information with an accuracy of a few percent of a bit, they have dramatically raised the bar for computing this quantity from biological data. Second, the deviation from perfect scaling is vanishingly small, so we can conclude that the profiles are truly scale-invariant.

These results show that the scaling of the body plan already exists at the level of the gap-gene network. But does it also originate here? Or does it originate higher up, at the top of the information cascade?

One might think that the property of scale invariance originates at the top layer of the information cascade represented by the maternal inputs. However, this does not appear to be the case. By studying the spatial profile of Bicoid—one of the main maternal morphogens, the authors find that, in contrast to gap gene profiles, it contains more information about the absolute position $x$ compared to the scaled position $x/L$, with the difference between the two being statistically significant. A simpler test also shows that Bicoid concentration profiles extracted from embryos of different lengths nicely overlap when plotted against the absolute position (see also ref. 2). The picture that thus emerges from these observations is that the maternal inputs to the gap-gene network do not scale with embryo length; instead, scale invariance emerges from the interactions between the gap-genes, which is then passed on, via the pair-rule genes, to the body plan.

The next question is how scaling emerges from gap gene interactions. The authors do not provide a mechanistic explanation. However, they do show that scale invariance has an important implication for the dynamics of the gap-gene network. In particular, they show that scaling implies the existence of a “zero mode.” In general, the dynamics of a chemical system near steady state can be decomposed into that of modes, which are spatial variations of concentrations that regress independently back to steady state, akin to normal modes in mechanical systems. Typically, these modes experience a linear restoring force close to steady state, such that their amplitudes decay exponentially in time. The unique characteristic of a zero mode is that it has no linear restoring force; even very close to steady state, the mode decays nonlinearly. The fluctuations along such a zero mode are not only large in magnitude compared to other modes but also decay algebraically rather than exponentially in time; moreover, they generate long-range spatial correlations between gene expression levels. Intriguingly, there is experimental evidence that the gap gene system exhibits such dynamics (5–7).

While Nikolić et al. demonstrate unambiguously that the early developmental gene-expression patterns of the fruit fly exhibit scale invariance, their study also raises new questions. First of all, what does the presence of a zero mode mean for the stability of the system? Not only are their fluctuations large, their slow algebraic decay also hampers the implementation of a widely used mechanism for noise reduction, namely time averaging. Other questions concern the role of the maternal inputs. How can the gap-gene network exhibit scaling while its inputs do not (8–11)? Moreover, what is the role of these maternal inputs? The gap-gene system consists of interlocked motifs of mutual repression (12, 13), which means that several gene-expression patterns could be (meta)stable. Do the maternal inputs merely serve to break this symmetry, by pinning the concentration profiles at the boundaries? Or do they also play a key role in stabilizing the profiles deep inside the embryo, by containing the movement of the expression boundaries? In this context, it is interesting to note that a recent computational study provides evidence for the idea that the gap gene system forms stable patterns even without maternal inputs, suggesting that their principal role is indeed to break the symmetry (13).

The manuscript of Nićolic et al. has shown how very precise quantitative measurements, combined with rigorous theory, allow us to draw firm conclusions and at the same time raise new and interesting questions. We hope that this approach will be used more widely in the field.