SUMMARY Many sensory systems, such as vision and hearing, show a response that is proportional to the fold-change in the stimulus relative to the background, a feature related to Weber’s law. Recent experiments suggest such a fold-change detection feature in signaling systems in cells: a response that depends on the fold-change in the input signal, and not on its absolute level (Cohen-Saidon et al., in press; Goentoro and Kirschner, submitted; both in this volume). It is therefore of interest to find molecular mechanisms of gene regulation that can provide such fold-change detection. Here we demonstrate theoretically that fold-change detection can be generated by one of the most common network motifs in transcription networks, the incoherent feedforward loop (I1-FFL), in which an activator regulates both a gene and a repressor of the gene.The fold-change detection feature of the I1-FFL applies to the entire shape of the response, including its amplitude and duration, and is valid for a wide range of biochemical parameters.

Recent studies suggest that certain cellular sensory systems display fold-change detection (FCD): a response whose entire shape, including amplitude and duration, depends only on fold changes in input and not on absolute levels. Thus, a step change in input from, for example, level 1 to 2 gives precisely the same dynamical output as a step from level 2 to 4, because the steps have the same fold change. We ask what the benefit of FCD is and show that FCD is necessary and sufficient for sensory search to be independent of multiplying the input field by a scalar. Thus, the FCD search pattern depends only on the spatial profile of the input and not on its amplitude. Such scalar symmetry occurs in a wide range of sensory inputs, such as source strength multiplying diffusing/convecting chemical fields sensed in chemotaxis, ambient light multiplying the contrast field in vision, and protein concentrations multiplying the output in cellular signaling systems. Furthermore, we show that FCD entails two features found across sensory systems, exact adaptation and Weber's law, but that these two features are not sufficient for FCD. Finally, we present a wide class of mechanisms that have FCD, including certain nonlinear feedback and feed-forward loops. We find that bacterial chemotaxis displays feedback within the present class and hence, is expected to show FCD. This can explain experiments in which chemotaxis searches are insensitive to attractant source levels. This study, thus, suggests a connection between properties of biological sensory systems and scalar symmetry stemming from physical properties of their input fields.adaptation | sensory response | spatial search O rganisms and cells sense their environment using sensory systems. Certain sensory systems have been extensively studied, and their input-output relations are well-characterized, including human senses, such as vision (1, 2), touch, and hearing, and unicellular senses, such as bacterial chemotaxis (3). Many sensory systems have common features. One such feature is exact adaptation in which the output to a change in input to a new constant level gradually returns to a level independent of the input. A second common feature, called Weber's law, states that the maximal response to a change in signal is inversely proportional to the background signal (4): Δy = kΔu/u 0 , where k is a constant, y is the output, and Δu is the signal change over the background u 0 . Weber's law in vision, chemotaxis, and other sensory systems applies over a range of several orders of magnitude of background input levels. Note that this definition stems from current practice that generalizes Weber's original measurements on psychophysical threshold sensitivity (4-7).Recent studies of the input-output properties of certain cellular signaling systems (8, 9) suggest that these systems show a feature called fold-change detection (FCD): a response whose entire shape, including its amplitude and duration, depends only on fold changes in input and not on absolute levels (10) (Fig. 1 A and...

SUMMARY In the canonical Wnt pathway, binding of the Wnt ligand to its transmembrane receptors leads to an inhibition of the degradation of β-catenin; as a result, β-catenin accumulates to a point where it activates target genes. Using mathematical modeling and experiments in mammalian cells, we examined the robustness of the β-catenin response to Wnt stimulation. We found that the final (post-Wnt) level of β-catenin is very sensitive to all perturbations in the Wnt signaling pathway, such that mild genetic or environmental variation would be expected to change the final level of β-catenin, and alter the output of the pathway. By contrast, one unusual parameter was robust: the fold-change in β-catenin (post- Wnt level / pre-Wnt level). Furthermore, in Xenopus embryos, dorsal-anterior development and the corresponding target gene expression are robust to the same perturbations that alter the final level but leave the fold-change intact. These results suggest: First, despite noise and variation, within a range the cell maintains a constant fold-change in β-catenin for a given Wnt stimulation. Second, the transcriptional machinery downstream of the Wnt pathway is constructed to read the robust fold-change and not simply the final level of β-catenin. In analogy to Weber’s law in sensory physiology, some gene transcription networks may be tuned to respond to fold-changes, rather than absolute levels of signals, as a way to reduce the consequences of stochastic, genetic and environmental variation.