On the relation between fluctuation and response in biological systems

doi:10.1073/pnas.2334996100

On the relation between fluctuation and response in biological systems

^*Department of Pure and Applied Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan; ^†Department of Biotechnology, Graduate School of Engineering, and ^§Department of Bioinformatic Engineering, Graduate School of Information Science and Technology, Osaka University, 2-1, Yamadaoka, Suita, Osaka 565-0871, Japan; and ^‡Intelligent Cooperation and Control Project, Precursory Research for Embryonic Science and Technology, Japan Science and Technology Corporation, 2-1, Yamadaoka, Suita, Osaka 565-0871, Japan

Edited by John Ross, Stanford University, Stanford, CA, and approved September 17, 2003 (received for review August 18, 2003)

Abstract

A general relationship between fluctuation and response in a biological system is presented. The fluctuation is given by the variance of some quantity, whereas the response is given as the average change of that quantity for a given parameter change. We propose a relationship where the two are proportional, in a similar way to the fluctuation–dissipation theorem in physics. By studying an evolution experiment where fluorescence of protein in bacteria increases, we confirm our relation by observing a positive correlation between the speed of fluorescence evolution and the phenotypic fluctuation of the fluorescence over clone bacteria. The generality of the relationship as well as its relevance to evolution is discussed.

Footnotes

↵ ¶ To whom correspondence should be addressed. E-mail: kaneko{at}complex.c.u-tokyo.ac.jp.
This paper was submitted directly (Track II) to the PNAS office.
↵ ∥ The derivation could be a little simplified, if we assume the Gaussian distribution completely and take the Δa → 0 limit in the beginning, as is often adopted in standard statistical physics. We adopted the present derivation to see under what conditions the linearity approximation holds and also to conveniently discuss the decrease of variance through evolution observed in the experiment (to be shown later). Further elaboration on our framework and the linearity condition should be pursued in the future.