On the relation between fluctuation and response in biological systems

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.

Fig. 1.

Histogram of the logarithm of fluorescence intensity for each generation, in the experiment described in the text. The number above the peak of each distribution indicates its generation number. We evaluated the fluorescence intensity of each E. coli in each generation by dividing its measured fluorescence intensity value (FL) by its forward scatter (FS) value measured with the cytometry, because the FS value roughly indicates the size of E. coli, whereas the FL value is usually proportional to the FS value.

Fig. 2.

The average (black squares) and variance (red circles) of fluorescence over clone cells selected at each generation, plotted vs. the generation number. The average values and variances are computed from the Gaussian-like distributions as their peak position and half-width in Fig. 1. The synonymous mutation rate for each generation, which was determined by DNA sequencing, is also plotted as blue triangles.

Fig. 3.

σ² (variance of fluorescence intensity) multiplied by Δa (synonymous mutation rate) is plotted vs. the change of average fluorescence intensity as red circles by using the data from Fig. 2. The red line is a linear fit to the data, which turn out to pass through the origin. For reference, the synonymous mutation rate Δa vs. the change of average fluorescence intensity value is also plotted as black squares. The correlation coefficient for the linear fit is 0.79, whereas that for the synonymous mutation rate (black squares) is 0.21.