Positional information, positional error, and read-out precision in morphogenesis: a mathematical framework

The concept of positional information is central to our understanding of how cells in a multicellular structure determine their developmental fates. Nevertheless, positional information has neither been defined mathematically nor quantified in a principled way. Here we provide an information-theoretic definition in the context of developmental gene expression patterns and examine which features of expression patterns increase or decrease positional information. We connect positional information with the concept of positional error and develop tools to directly measure information and error from experimental data. We illustrate our framework for the case of gap gene expression patterns in the early Drosophila embryo and show how information that is distributed among only four genes is sufficient to determine developmental fates with single cell resolution. Our approach can be generalized to a variety of different model systems; procedures and examples are discussed in detail

Morphogenesis at criticality

Spatial patterns in the early fruit fly embryo emerge from a network of interactions among transcription factors, the gap genes, driven by maternal inputs. Such networks can exhibit many qualitatively different behaviors, separated by critical surfaces. At criticality, we should observe strong correlations in the fluctuations of different genes around their mean expression levels, a slowing of the dynamics along some but not all directions in the space of possible expression levels, correlations of expression fluctuations over long distances in the embryo, and departures from a Gaussian distribution of these fluctuations. Analysis of recent experiments on the gap gene network shows that all these signatures are observed, and that the different signatures are related in ways predicted by theory. Although there might be other explanations for these individual phenomena, the confluence of evidence suggests that this genetic network is tuned to criticality

The syncytial Drosophila embryo as a mechanically excitable medium

Mitosis in the early syncytial Drosophila embryo is highly correlated in space and time, as manifested in mitotic wavefronts that propagate across the embryo. In this paper we investigate the idea that the embryo can be considered a mechanically-excitable medium, and that mitotic wavefronts can be understood as nonlinear wavefronts that propagate through this medium. We study the wavefronts via both image analysis of confocal microscopy videos and theoretical models. We find that the mitotic waves travel across the embryo at a well-defined speed that decreases with replication cycle. We find two markers of the wavefront in each cycle, corresponding to the onsets of metaphase and anaphase. Each of these onsets is followed by displacements of the nuclei that obey the same wavefront pattern. To understand the mitotic wavefronts theoretically we analyze wavefront propagation in excitable media. We study two classes of models, one with biochemical signaling and one with mechanical signaling. We find that the dependence of wavefront speed on cycle number is most naturally explained by mechanical signaling, and that the entire process suggests a scenario in which biochemical and mechanical signaling are coupled

Dubuis et al. - Data_Fig_2B

Source data for Figure 2B: Invagination of the membrane for 8 embryos (delta x), mean (mean delta) and adjusted mean in fixed tissues (mean delta fixed) as function of time during n.c. 14. NaN indicates that furrow canal could not be detected at that particular time

Data from: Accurate measurements of dynamics and reproducibility in small genetic networks

Quantification of gene expression has become a central tool for understanding genetic networks. In many systems the only viable way to measure protein levels is by immunofluorescence, which is notorious for its limited accuracy. Using the early Drosophila embryo as an example, we show that careful identification and control of experimental error allows for highly accurate gene expression measurements. We generated antibodies in different host species, allowing for simultaneous staining of four Drosophila gap genes in individual embryos. Careful error analysis of hundreds of expression profiles reveals that less than ∼20% of the observed embryo-to-embryo fluctuations stem from experimental error. These measurements make it possible to extract not only very accurate mean gene expression profiles but also their naturally occurring fluctuations of biological origin and corresponding cross-correlations. We use this analysis to extract gap gene profile dynamics with ∼1 min accuracy. The combination of these new measurements and analysis techniques reveals a two-fold increase in profile reproducibility due to a collective network dynamics that relays positional accuracy from the maternal gradients to the pair-rule genes

Dubuis et al. - Source_Data_Files

Complete source data for Dubuis et al. contains two folders: 1. Folder "FC_Calibration" contains the source data used for calibration of the time-dependent furrow canal depth. It displays the chi_2 minimized furrow canal depth (delta_FC) for the 8 live imaged Drosophila embryos as function of time during n.c. 14 (see Materials and Methods). It also shows the mean furrow canal depth averaged over the 8 embryos (mean_delta_FC) and adjusted mean in fixed tissues (mean_delta_FC_fixed) that we used to estimate the age of the 163 fixed embryos. Column 1 contains the time in minutes after the onset of n.c. 14. Columns 2 to 9 contain the chi_2 minimized furrow canal depths (delta_FC_x, x being the embryo number) in for the 8 live imaged embryos. NaN means that we couldn't detect the furrow canal at that particular time. Column 10 contains the mean furrow canal depth averaged over the 8 embryos (mean_delta_FC) measured in micrometers. Column 11 contains the the furrow canal depth, measured in micrometers, which we used to estimate the age of fixed embryos. It was obtained by a 5% shrinkage of the previous column (see Materials and Methods). 2. Folder "Images" contains the original images of 201 Drosophila embryos at blastoderm stage immunostained against the four main gap genes (Kni, Kr, Gt, Hb). For each embryo with provide 5 12-bit images, each corresponding to a different optical channel (see Materials and methods). Channel 0 -- rat @ Kni (488nm); Channel 1 -- bright field (delta_FC); Channel 2 -- guinea pig @ Gt (568nm); Channel 3 -- rabbit @ Kr (594nm); Channel 4 -- mouse @ Hb (647nm). Images were taken with a Leica 20x HC PL APO NA 0.7 oil immersion objective, and with sequential excitation wavelengths of 488, 546, 594 and 633 nm. The bandwidth of the detection filters were set up as shown in Figure 1A to minimize fluorophore cross-talk while still allowing good detection in each optical channel. For each embryo, three high-resolution images (1024x1024 pixels, with 12 bits and at 100 Hz) were taken along the anterior-posterior axis (focused at the midsagittal plane) at 1.7 magnified zoom and averaged together. With these settings, the linear pixel dimension corresponds to 0.44 um

Dubuis et al. - Processed_Profiles

Each file contains the source data for gap gene (Hb, Kr, Gt, Kni) processed dorsal gene expression levels measured using immunofluorescence techniques in 23 Drosophila embryos with FC depth comprised between 10 and 20 microns. We only selected embryos that were imaged close to their midsagittal plane (Orientation '1'). Column 1 contains the embryo number referring to its position on the slide (see Images.zip). Column 2 contains an information about the azimuthal orientation of the embryo on the slide ('1' if the confocal plane is closer to midsagittal plane, '2' if it is closer to the coronal plane). Column 3 contains the furrow canal depth (delta_FC) measured in micrometers. Column 4 contains the corresponding estimated age in minutes. Columns 5 to 1004 contain a 1x1000 vector representing the dorsal time-corrected gene expression level of the gap gene (Hb, Kr, Gt, Kni) in the embryo. The 1000 points are equally spaced along the AP axis. Thus, g345 represents the gene expression level at 34.5%EL. NaN means that we coudn't reliably detect the profile intensity at that position (usually near the edges). Columns 1-4 are identical in the four files

Dubuis et al. - Raw_Profiles

Each file contains the source data for raw gap gene (Hb, Kr, Gt, Kni) dorsal intensity profiles measured using immunofluorescence techniques in 163 Drosophila embryos during n.c. 14. Column 1 contains the embryo number referring to its position on the slide (see Images.zip). Column 2 contains an information about the azimuthal orientation of the embryo on the slide ('1' if the confocal plane is closer to midsagittal plane, '2' if it is closer to the coronal plane). Column 3 contains the furrow canal depth (delta_FC) measured in micrometers. Column 4 contains the corresponding estimated age in minutes. Columns 5 to 1004 contain a 1x1000 vector representing the dorsal intensity profile of the gap gene (Hb, Kr, Gt, Kni) in the embryo. The 1000 points are equally spaced along the AP axis. Thus, Intensity345 represents the intensity at 34.5%EL. NaN means that we couldn't reliably detect the profile intensity at that position (usually near the edges). Columns 1-4 are identical in the four files

Positional information, in bits

Cells in a developing embryo have no direct way of “ measuring ” their physical position. Through a variety of processes, however, the expression levels of multiple genes come to be correlated with position, and these expression levels thus form a code for “ posi- tional information. ” We show how to measure this information, in bits, using the gap genes in the Drosophila embryo as an example. Individual genes carry nearly two bits of information, twice as much as would be expected if the expression patterns consisted only of on/off domains separated by sharp boundaries. Taken to- gether, four gap genes carry enough information to de fi ne a cell ’ s location with an error bar of ∼ 1% along the anterior/posterior axis of the embryo. This precision is nearly enough for each cell to have a unique identity, which is the maximum information the system can use, and is nearly constant along the length of the embryo. We argue that this constancy is a signature of optimality in the transmission of information from primary morphogen inputs to the output of the gap gene network