Oscillation patterns in negative feedback loops

Abstract

Organisms are equipped with regulatory systems that display a variety of dynamical behavior ranging from simple stable steady states, to switching and multistability, to oscillations. Earlier work has shown that oscillations in protein concentrations or gene expression levels are related to the presence of at least one negative feedback loop in the regulatory network. Here, we study the dynamics of a very general class of negative feedback loops. Our main result is that, when a single negative feedback loop dominates the dynamical behavior, the sequence of maxima and minima of the concentrations exhibit a pattern that uniquely identifies the interactions of the loop. This allows us to devise an algorithm to (i) test whether observed oscillating time series are consistent with a single underlying negative feedback loop, and if so, (ii) reconstruct the precise structure of the loop, i.e., the activating/repressing nature of each interaction. This method applies even when some variables are missing from the data set, or if the time series shows transients, like damped oscillations. We illustrate the relevance and the limits of validity of our method with three examples: p53-Mdm2 oscillations, circadian gene expression in cyanobacteria, and cyclic binding of cofactors at the estrogen-sensitive pS2 promoter.

Fig. 1.

Oscillations in the p53-Mdm2 system. (a) p53-Mdm2 oscillations as recorded in a fluorescence microscopy experiment (7) and the reconstructed symbolic dynamics. (Inset) The negative feedback loop extracted using the algorithm in the text; the process is shown in b. Here and in subsequent figures, ordinary arrows represent activation, whereas barred arrows represent repression.

Fig. 2.

Examples of negative feedback loops. (a) The simplest case consisting of one activator and one repressor. (b) A three-repressor loop. (c) A general loop with N variables and an odd number of repressors.

Fig. 3.

Qualitative analysis of a two-species negative feedback loop. (a) Schematic diagram of the phase-space of the two-variable negative feedback loop of Fig. 2a. Solid lines show the two nullclines, which intersect and divide the space into four sectors, labeled by the signs of g₁ (Upper) and g₂ (Lower) in that sector. Arrows on the nullclines show the local direction of the vector field, which determines the direction in which the nullcline can be crossed by a trajectory. The dotted line is an example trajectory that follows these rules, while spiralling in toward the stable fixed point. (b) The allowed transitions for moving from one sector to another. The symbolic dynamics of any trajectory has to be consistent with these rules.

Fig. 4.

Examples of the algorithm in action. (a) Circadian rhythms of three kai genes in a Synechocystis cyanobacterial strain (data from ref. 27). (b) Periodic binding of four proteins to the pS2 promoter after addition of estradiol (data from ref. , based on ref. 28). In each case the corresponding symbolic dynamics is also shown, with symbols in the same order as the legend (where maxima/minima of two variables occur very close we have exaggerated the separation between the dotted lines for visual clarity). (Insets) Loop structure deduced from the symbolic dynamics.