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The expression of a few key genes determines the body plan of the fruit fly. We show that the spatial expression patterns for several of these genes scale precisely with embryo size. Discrete positional markers such as the peaks in striped patterns or the boundaries of expression domains have positions along the embryo’s major axis proportional to embryo length, accurate to within 1%. Further, the information (in bits) that graded patterns of expression provide about a cell’s position can be decomposed into information about fractional or scaled position and information about absolute position or embryo length; all information available is about scaled position, with $<$2% error. These findings imply that the underlying genetic network’s behavior exhibits scale invariance in a more precise mathematical sense. We argue that models that can explain this scale invariance also have a “zero mode” in the dynamics of gene expression, and this connects to observations on the spatial correlation of fluctuations in expression levels. 

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. Published by the American Physical Society2024 

There is a growing effort in the “physics of behavior” that aims at complete quantitative characterization of animal movements under more complex, naturalistic conditions. One reaction to the resulting explosion of high-dimensional data is the search for low-dimensional structure. Here I try to define more clearly what we mean by the dimensionality of behavior, where observable behavior may consist of either continuous trajectories or sequences of discrete states. This discussion also serves to isolate situations in which the dimensionality of behavior is effectively infinite.