Fourier analysis and systems identification of the p53 feedback loop

Abstract

A key circuit in the response of cells to damage is the p53-mdm2 feedback loop. This circuit shows sustained, noisy oscillations in individual human cells following DNA breaks. Here, we apply an engineering approach known as systems identification to quantify the in vivo interactions in the circuit on the basis of accurate measurements of its power spectrum. We obtained oscillation time courses of p53 and Mdm2 protein levels from several hundred cells and analyzed their Fourier spectra. We find characteristic spectra with distinct low-frequency components that are well-described by a third-order linear model with white noise. The model identifies the sign and strength of the known interactions, including a negative feedback loop between p53 and its upstream regulator. It also implies that noise can trigger and maintain the oscillations. The model also captures the power spectra of p53 dynamics without DNA damage. Parameters such as noise amplitudes and protein lifetimes are estimated. This approach employs natural biological noise as a diagnostic that stimulates the system at many frequencies at once. It seems to be a useful way to find the in vivo design of circuits and may be applied to other systems by monitoring their power spectrum in individual cells.U nderstanding the in vivo design of protein circuits is a general biological challenge (1-6). Here, we address this by focusing on one of the best-studied circuits in human cells, the p53-mdm2 feedback loop and its response to DNA damage (7-10). This system responds to DNA double-stranded breaks: breaks are sensed by the kinase ATM, which activates p53 (11). As a result, p53 transcriptionally activates mdm2. Negative feedback is formed because Mdm2 targets p53 for degradation. A second negative feedback loop important for this response is the down-regulation of ATM by wip1, a p53 target gene (12-16).The p53 circuit shows sustained oscillations following gamma irradiation which causes double-stranded DNA breaks (17)(18)(19). In these oscillations, p53 levels periodically rise and fall, with a period of 6-7 h, and Mdm2 oscillates out of phase with p53. The oscillations last for days (19), as discovered at the individual cell level using fluorescently tagged p53 and Mdm2 (18,19). Oscillations with the same period were also seen in whole mice using luciferase reporters of p53 activity (20).Several theoretical models were suggested for these oscillations (9, 17, 21-28). Many of these models rely on a time delay between p53 activation and appearance of Mdm2 proteins, without which the equations show damped (nonsustained) oscillations.To better understand the in vivo design of this circuit, we use here an engineering approach known as systems identification. In engineering, one applies a periodically varying input to a circuit, measures its output at different frequencies, and describes this by linear models of the dynamics (29-31). Such periodic inputs were previously used to analyze biological systems including bacterial chemotaxis (32), yeast osmo-response (2), a...

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