Oscillations and variability in the p53 system

Understanding the dynamics and variability of protein circuitry requires accurate measurements in living cells as well as theoretical models. To address this, we employed one of the best-studied protein circuits in human cells, the negative feedback loop between the tumor suppressor p53 and the oncogene Mdm2. We measured the dynamics of fluorescently tagged p53 and Mdm2 over several days in individual living cells. We found that isogenic cells in the same environment behaved in highly variable ways following DNA-damaging gamma irradiation: some cells showed undamped oscillations for at least 3 days (more than 10 peaks). The amplitude of the oscillations was much more variable than the period. Sister cells continued to oscillate in a correlated way after cell division, but lost correlation after about 11 h on average. Other cells showed low-frequency fluctuations that did not resemble oscillations. We also analyzed different families of mathematical models of the system, including a novel checkpoint mechanism. The models point to the possible source of the variability in the oscillations: low-frequency noise in protein production rates, rather than noise in other parameters such as degradation rates. This study provides a view of the extensive variability of the behavior of a protein circuit in living human cells, both from cell to cell and in the same cell over time.

1 Oscillations and variability in the p53 system, Geva-Zatorsky et al. 2006 Supplementary figure and movie legends Figure S1. An example of a field of view with several cells (in bright field illumination, gray) overlaid by the fluorescence intensities of p53-CFP and Mdm2-YFP (shown in green and red respectively, their overlap is seen in yellow). Figure S2. Nuclear Mdm2-YFP dynamics of all the measured cells in a single movie exposed to A) 10Gy, and B) no gamma irradiation. Dotted vertical lines indicate division events; solid vertical lines indicate severe blebbing of the cell's membrane. C) Timing of nuclear Mdm2-YFP dynamics in oscillating cells exposed to 10Gy of gamma irradiation (see Fig. 2D). Blue hues indicate low fluorescence levels and yellow-reddish colors indicate high fluorescence levels. Dotted lines show 6 hour intervals. D) Timing of Mdm2-YFP fluctuations in cells not exposed to gamma irradiation. E) Mdm2-YFP dynamics averaged over all cells exposed to 10Gy (solid blue line) and to no gamma irradiation (dotted black line). Top: an example of a cell showing fluctuations with a characteristic frequency of ~10 hours (exposed to 0.3Gy of gamma irradiation), and the power spectrum of the signal (by Fourier transform). Bottom: an example of a cell with multiple oscillations with a period of ~6 hours (exposed to 5Gy), and the power spectrum of the signal (right).   "Oscillations and variability in the p53 system" 2 that this pattern is different from the damped oscillations that result when averaging the dynamics of many cells due to the lack of precise synchrony in the oscillations. Cells which exhibit 2 peaks in the first 16 hours were collected, and the first and second peaks were identified between the consecutive minima. Each peak was normalized to the average peak maximum and shifted in time so that its maximum coincides with the average peak timing.
The average shape for each pulse was calculated by averaging the normalized peaks, and the two peaks were smoothly linked. The second peak was duplicated at equal intervals to get a continuous signal. obtaining a signal with a mean~0, and normalizing its STD to a given value. The multiplicative noise factor that was used was ξ(t)=exp(η(t)). Figure S8. Sensitivity of models to variations in parameters. For each of the oscillating models II-VI, every parameter was varied from 0.1 times to 10 times its best-fit value (Table   1). Dynamics were simulated and peaks were identified. The median value of the p53 (green) and Mdm2 (red) maxima, and the oscillation period (black) were calculated. These values were then divided by the best-fit values, to obtain the relative change. Blue stars mark parameters that significantly changed the amplitude but did not affect the period of the oscillations (by more than ~10%). Lack of data points in certain panels indicate that no oscillations occurred for the respective parameter values.

Supplementary text
Appendix I: Biologically motivated expanded formulations of model VI.
The core models (Fig. 6A), with the 3 dynamic variables, can be readily extended to encompass other important aspects of the p53 system. For example, a recently published model for p53-Mdm2 oscillations Wagner et al., 2005) can be compared to model IV or to model III . Here we present two examples of expanded forms of model VI. In the first example (Fig. S9B), two forms of p53 are explicitly included in the model. Without DNA damage, p53 exists in the inactive form, p53 inactive . The transcription rate of p53 has been shown to be independent of DNA damage (Haupt et al., 1997;Vassilev, 2004). Evidence was presented for an increase in translation rate of p53 following gamma irradiation (Fu and Benchimol, 1997), an effect that can contribute to the oscillatory properties. In order to maintain simplicity in the models we represent production of p53 by a constant rate β p53 . The degradation rate of p53 can depend on numerous factors, including Mdm2. Following DNA damage, Signal, S, is activated (perhaps representing phosphorylated ATM (Bakkenist and Kastan, 2003)). When S accumulates (S>T s ), it modifies inactive p53 (p53 inactive ) converting it into its active form, p53 active (Appella and Anderson, 2001;Unger et al., 1999), at a rate ω. Mdm2 promotes the degradation of p53 (Haupt et al., 1997;Kubbutat et al., 1997). The production rate of Mdm2 is enhanced by p53 (Barak et al., 1993;Wu et al., 1993). Active p53, has a weaker interaction with Mdm2 than inactive p53 and hence a lower degradation rate, α active <α inactive (Shieh et al., 1997). Thus, activation of p53 leads to its accumulation, and as its levels increase, this increases the production rate of Mdm2. Because of the accumulation of active p53, the same results are obtained by the model whether the downstream genes are induced by active p53 alone or by both varieties (p53 active +p53 inactive ). In the transactivation terms of downstream genes we therefore use p53 (without subscript) to denote either of these options. The increased Mdm2 levels in the nucleus lead to faster degradation (or nuclear export) of p53 active . After levels of p53 drop in the nucleus, the production rate of Mdm2 decreases and it is degraded (or exported out of the  For fast modification rates of p53 (large ω, ω>>α active ), with the parameters for the inhibitor equal to the values for Mdm2, the dynamics obtained by this model are equivalent to those obtained by the simplified model VI (Fig. S8A). The inhibitor could be either Mdm2 itself or a different component with similar dynamic properties. Signal thus oscillates, with a shifted phase, such that Signal peaks precede the p53 peaks, and are in an opposite phase to p53downstream genes such as Mdm2 and possibly DNA repair genes. This can create a separation of phases where DNA damage sensing (by Signal) and activity of p53-downstream genes are separated in time. Note that other reported negative feedbacks on p53, such as the feedback through COP1 (Dornan et al., 2004) or Pirh2 (Leng et al., 2003), can be introduced into this model by duplicating the equation for Mdm2 and modifying the interaction parameters to maintain the same dynamic profile for p53 and Mdm2. It would be interesting to examine whether these components of the p53 system do indeed oscillate and whether their dynamics can indeed be modeled in this same fashion.
An alternative formulation places an emphasis on the role of Mdm2 in regulating the p53 response (see reviews in (Bond et al., 2005;Momand et al., 2000)), through varied Mdm2 levels (Stommel and Wahl, 2004), covalent modifications (Maya et al., 2001), or binding of other components such as Arf (Zhang and Xiong, 2001). In this formulation, Mdm2 protein exists in two distinct forms, so that following irradiation and DNA damage the "regular",  Again, for fast modification rates (large ω, ω>>α modified ), the dynamics obtained by this model (Fig. S8C) are equivalent to those obtained by the model above and by the simplified model VI (Fig. S8AB).
Different formulations of the model may be relevant for comparison to different experimental settings and manipulations. One could further expand these to include multiple forms of p53, Mdm2, and other components, and the dynamic properties and response to noise would be preserved, as long as the model maintains its homology to model VI. Such variants of the model can be distinguished by measuring the dynamics of modified components or their interactions, or through the effects of mutations (Bond et al., 2004) or drugs (Vassilev, 2004) on the oscillations. The important distinction between these variants and other models of the p53-Mdm2 feedback (models I-V) is the different connectivity which includes a second, longer negative feedback loop, by inhibition of damage signaling through p53-downstream factors (e.g. inhibitor). Although much evidence has been accumulated for signaling of damage to p53 (Lee and Paull, 2005;Stommel and Wahl, 2004), the feedback from p53 and Mdm2 back to the damage signal is an unknown interaction predicted by this model.