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

Optimal decoding of information from a genetic network

Gene expression levels carry information about signals that have functional significance for the organism. Using the gap gene network in the fruit fly embryo as an example, we show how this information can be decoded, building a dictionary that translates expression levels into a map of implied positions. The optimal decoder makes use of graded variations in absolute expression level, resulting in positional estimates that are precise to ~1% of the embryo's length. We test this optimal decoder by analyzing gap gene expression in embryos lacking some of the primary maternal inputs to the network. The resulting maps are distorted, and these distortions predict, with no free parameters, the positions of expression stripes for the pair-rule genes in the mutant embryos

Fly wing vein patterns have spatial reproducibility of a single cell

Developmental processes in multicellular organisms occur in fluctuating environments and are prone to noise, yet they produce complex patterns with astonishing reproducibility. We measure the left-right and inter-individual precision of bilaterally symmetric fly wings across the natural range of genetic and environmental conditions and find that wing vein patterns are specified with identical spatial precision and are reproducible to within a single-cell width. The early fly embryo operates at a similar degree of reproducibility, suggesting that the overall spatial precision of morphogenesis in Drosophila performs at the single-cell level. Could development be operating at the physical limit of what a biological system can achieve

Maternal Origins of Developmental Reproducibility

Cell fate decisions during multicellular development are precisely coordinated, leading to highly reproducible macroscopic structural outcomes [1-3]. The origins of this reproducibility are found at the molecular level during the earliest stages of development when patterns of morphogen molecules emerge reproducibly [4, 5]. However, although the initial conditions for these early stages are determined by the female during oogenesis, it is unknown whether reproducibility is perpetuated from oogenesis or reacquired by the zygote. To address this issue in the early Drosophila embryo, we sought to count individual maternally deposited bicoid mRNA molecules and compare variability between embryos with previously observed fluctuations in the Bicoid protein gradient [6, 7]. Here, we develop independent methods to quantify total amounts of mRNA in individual embryos and show that mRNA counts are highly reproducible between embryos to within similar to 9%, matching the reproducibility of the protein gradient. Reproducibility emerges from perfectly linear feedforward processes: changing the genetic dosage in the female leads to proportional changes in both mRNA and protein numbers in the embryo. Our results indicate that the reproducibility of the morphological structures of embryos originates during oogenesis, which is when the expression of maternally provided patterning factors is precisely controlled

Optimal decoding of cellular identities in a genetic network

In developing organisms, spatially prescribed cell identities are thought to be determined by the expression levels of multiple genes. Quantitative tests of this idea, however, require a theoretical framework capable of exposing the rules and precision of cell specification over developmental time. We use the gap gene network in the early fly embryo as an example to show how expression levels of the four gap genes can be jointly decoded into an optimal specification of position with 1% accuracy. The decoder correctly predicts, with no free parameters, the dynamics of pair-rule expression patterns at different developmental time points and in various mutant backgrounds. Precise cellular identities are thus available at the earliest stages of development, contrasting the prevailing view of positional information being slowly refined across successive layers of the patterning network. Our results suggest that developmental enhancers closely approximate a mathematically optimal decoding strategy

Finding the Last Bits of Positional Information

In a developing embryo, information about the position of cells is encoded in the concentrations of morphogen molecules. In the fruit fly, the local concentrations of just a handful of proteins encoded by the gap genes are sufficient to specify position with a precision comparable to the spacing between cells along the anterior-posterior axis. This matches the precision of downstream events such as the striped patterns of expression in the pair-rule genes, but is not quite sufficient to define unique identities for individual cells. We demonstrate theoretically that this information gap can be bridged if positional errors are spatially correlated, with correlation lengths approximately 20 % of the embryo length. We then show experimentally that these correlations are present, with the required strength, in the fluctuating positions of the pair-rule stripes, and this can be traced back to the gap genes. Taking account of these correlations, the available information matches the information needed for unique cellular specification, within error bars of approximately 2 % . These observation support a precisionist view of information flow through the underlying genetic networks, in which accurate signals are available from the start and preserved as they are transformed into the final spatial patterns

Functional and ultrastructural analysis of reafferent mechanosensation in larval zebrafish

All animals need to differentiate between exafferent stimuli, which are caused by the environment, and reafferent stimuli, which are caused by their own movement. In the case of mechanosensation in aquatic animals, the exafferent inputs are water vibrations in the animal's proximity, which need to be distinguishable from the reafferent inputs arising from fluid drag due to locomotion. Both of these inputs are detected by the lateral line, a collection of mechanosensory organs distributed along the surface of the body. In this study, we characterize in detail how hair cells-the receptor cells of the lateral line-in zebrafish larvae discriminate between such reafferent and exafferent signals. Using dye labeling of the lateral line nerve, we visualize two parallel descending inputs that can influence lateral line sensitivity. We combine functional imaging with ultra-structural EM circuit reconstruction to show that cholinergic signals originating from the hindbrain transmit efference copies (copies of the motor command that cancel out self-generated reafferent stimulation during locomotion) and that dopaminergic signals from the hypothalamus may have a role in threshold modulation, both in response to locomotion and salient stimuli. We further gain direct mechanistic insight into the core components of this circuit by loss-of-function perturbations using targeted ablations and gene knockouts. We propose that this simple circuit is the core implementation of mechanosensory reafferent suppression in these young animals and that it might form the first instantiation of state-dependent modulation found at later stages in development