Logarithmic and power law input-output relations in sensory systems with fold-change detection

PLoS Comput Biol. 2014 Aug 14;10(8):e1003781. doi: 10.1371/journal.pcbi.1003781. eCollection 2014 Aug.


Two central biophysical laws describe sensory responses to input signals. One is a logarithmic relationship between input and output, and the other is a power law relationship. These laws are sometimes called the Weber-Fechner law and the Stevens power law, respectively. The two laws are found in a wide variety of human sensory systems including hearing, vision, taste, and weight perception; they also occur in the responses of cells to stimuli. However the mechanistic origin of these laws is not fully understood. To address this, we consider a class of biological circuits exhibiting a property called fold-change detection (FCD). In these circuits the response dynamics depend only on the relative change in input signal and not its absolute level, a property which applies to many physiological and cellular sensory systems. We show analytically that by changing a single parameter in the FCD circuits, both logarithmic and power-law relationships emerge; these laws are modified versions of the Weber-Fechner and Stevens laws. The parameter that determines which law is found is the steepness (effective Hill coefficient) of the effect of the internal variable on the output. This finding applies to major circuit architectures found in biological systems, including the incoherent feed-forward loop and nonlinear integral feedback loops. Therefore, if one measures the response to different fold changes in input signal and observes a logarithmic or power law, the present theory can be used to rule out certain FCD mechanisms, and to predict their cooperativity parameter. We demonstrate this approach using data from eukaryotic chemotaxis signaling.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Animals
  • Chemotaxis / physiology*
  • Computational Biology
  • Computer Simulation
  • Eukaryota
  • Feedback, Physiological / physiology*
  • Humans
  • Models, Biological*
  • Signal Transduction / physiology*

Grant support

We would like to acknowledge support for this work from the Israel Science Foundations and the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement n° 249919. Uri Alon is the incumbent of the Abisch-Frenkel Professorial Chair. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.