Fold-change detection and scalar symmetry of sensory input fields

Abstract

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.

Fig. 1.

Dynamics of sensory response to fold change in input. (A) Input signal. Two step changes with identical fold change and different absolute change (note log scale). (B) Output of FCD sensor is identical for the two inputs, including amplitude and adaptation dynamics. (C) Output of a sensor with exact adaptation but no FCD, because peak response and dynamics depend on absolute changes. (D) Output of a sensor with exact adaptation, Weber's law, and no FCD. Weber's law applies, because the peak response depends only on relative change and thus, is equal for both step inputs; however, FCD does not apply, because the temporal adaptation dynamics depend also on absolute input levels.

Fig. 2.

Organisms with movement based on FCD sensors. Schematic of FCD output feeding into the spatial movement of the agent.

Fig. 3.

Bacterial chemotaxis search patterns do not depend on chemoattractant source concentration. (A) Bacterial chemotaxis is comprised of runs and tumbles. When the bacteria sense an increase in attractant (i.e., movement in the right direction), they lower their tumbling frequency and tend to continue in that direction. (B) The experiment by Mesibov et al. (24). Bacteria are allowed to adapt to a background attractant concentration in the plate. After a period of time, a capillary with attractant concentration 3.1 times higher than the background was presented. This formed an attractant gradient, causing the bacteria to migrate to the capillary. The number of bacteria reaching the capillary after 1 h was measured. The experiment was repeated with different background concentrations, keeping the capillary/background concentration ratio constant at 3.1. (C) The number of bacteria that reach the capillary was nearly constant over a three order of magnitude change in background and capillary concentrations adapted from ref. (24) (© Mesibov et al., 1973. J Gen Physiol 62:203–223)]. Plotted on the x axis is the average of the capillary and background concentrations of attractant.

Fig. 4.

Mechanisms for FCD. (A) Incoherent feed-forward loop. (B) Nonlinear integral feedback. (C) Linear integral feedback with log input (satisfies FCD using generalized conditions; Materials and Methods and SI Text).