Figure 1.
(A) When a cell is exposed to radiation, the p53 levels oscillate in a sustained (undamped) way. (B) A schematic illustration showing that as the extent of DNA damage increases (increased radiation dose) the average number of oscillations (the number of peaks in the figure) also increases but the average duration (period) of each oscillation remains nearly constant. (C) The p53-MDM2 feedback loop. p53 activates MDM2 while MDM2 suppresses p53. NUMB activates p53 by suppressing MDM2. (D) Three types of fluctuation: intrinsic noise or fast aperiodic fluctuations (top), extrinsic noise or slow aperiodic fluctuations (middle), and periodic DNA replication-dependent fluctuations (bottom).
Figure 2.
Two contradictions when assuming the p53 oscillation is generated by the feedback loop.
(A) Our mathematical analysis illustrates that the damping ratio of the p53 oscillation increases (more damped or less oscillatory) as the radiation dose is stronger. This indicates that when there is more DNA damage there will be less p53 oscillation, which is not consistent with the experimental data shown in Figure 1B. (B) It is also mathematically shown that the p53 oscillation period increases as the radiation dose is increased, in contrast to the experimental data that shows it is constant regardless of the radiation dose (Figure 1B).
Figure 3.
Disturbance rejection (fluctuation filtering).
(A) A system receives the input signal in the presence of an external disturbance (shown in blue). As a result, the system generates the output affected by both the input and external disturbance. (B) The external disturbance effects can be filtered (removed) by adding a negative feedback loop and a controller (shown in red) to the system. (C) It is shown that NUMB (input) activates p53 (system) in the presence of fluctuations exerted on p53. (D) MDM2 (controller) and a feedback loop are added to the NUMB-p53 network shown in (C). As a result, the NUMB signal (input) transmitted to p53 (system) is reflected on the output without being affected by fluctuations exerted on p53.
Figure 4.
Bode plot of simple gene regulation.
There are three plots with py = 0.02 min−1 and pxy = 10 (red), 1 (green), and 0.1 (blue) min−1. On the horizontal axis, [rad/min] is used for the unit of angular frequency ω and [min] is used for the unit of equivalent period T (T = 2π/ω). The vertical axis represents the magnitude by which the system amplifies or reduces the input. The magnitude is shown in both folds (M) and dB (20log10M). The figure illustrates that simple gene regulation can either amplify or reduce intrinsic noise, extrinsic noise, and periodic DNA replication-dependent oscillations depending on the value of pxy.
Figure 5.
Simple gene regulation with and without the degradation/dilution term of Eq. 1.
(A) The ODE, transfer function, and block diagram of simple gene regulation in which the second or degradation/dilution term is removed. (B) The ODE (equivalent to Eq. 1), transfer function, and block diagram of simple gene regulation with the second term intact. Note that the block diagram has a negative feedback component, which could be hardly revealed by examining Eq. 2. (C) The Bode plot for (A) and (B). The figure illustrates that low-frequency signals are increasingly amplified as their frequencies decrease in the case of (A). On the other hand, (B) exhibits a “plateau-like” curve (brown), indicating that the amplifying magnitude remains constant regardless of the frequency decrease. Note that their capabilities to filter intrinsic noise and other types of fluctuation are quite similar.
Figure 6.
There are four plots with pxy = 1 min−1 and py = 0.002 (blue), 0.02 (green), 0.2 (red), and 2 (yellow) min−1. On the horizontal axis, [rad/min] is used for the unit of angular frequency ω and [min] is used for the unit of equivalent period T (T = 2π/ω). The vertical axis represents the magnitude by which the system amplifies or reduces the input. The magnitude is shown in both folds (M) and dB (20log10M). The figure shows that positive autoregulation (blue) similarly amplifies extrinsic noise and periodic DNA replication-dependent oscillations compared to simple gene regulation (green), indicating that it cannot play a role in filtering the effects of such fluctuations. Negative autoregulation clearly decreases the magnitude of amplification of those fluctuations (red) or even reduces it (yellow). However, it is also shown that entire low-frequency signals are affected in a non-selective way.
Figure 7.
(A) Schematic drawing of the p53-network. (B) Block diagram representation. X(s), Y(s), and Z(s) are the Laplace transform of x(t), y(t), and z(t), respectively. O(s) is the Laplace transform of the output and E(s) is the transform of an error e(t), which is the difference between the input and output. D(s) is a disturbance representing fluctuations exerted on p53. (C) A network configuration discussed in Figure 3B. It has an equivalent oscillation filtering effect as the configuration shown in (B).
Figure 8.
Simple gene regulation cannot filter periodic DNA repair-related fluctuation.
A simulation result shows that simple gene regulation (x activating z or NUMB activating p53) can reduce intrinsic noise but amplify periodic DNA repair-related fluctuation. At top left, x (NUMB) is assumed to be constant (1000 molecules/cell). On the left of the figure, it is shown that intrinsic noise (period T = 1 min and amplitude = 100 molecules/cell) is filtered by simple gene regulation. However, periodic DNA repair-related fluctuation (shown on the right, period longer period T = 40 min) with the same amplitude (100 molecules/cell) is not reduced but amplified. In order to explicitly illustrate the frequency-dependence, sinusoidal waveforms are used as fluctuation signals.
Figure 9.
p53-MDM2 feedback loop can filter DNA repair-related oscillatory signals.
At top left, x is assumed to be constant ( = 1000 molecules/cell). On the left of the figure, it is shown that intrinsic noise (period T = 1 min and amplitude = 100 molecules/cell) is filtered by the feedback loop. Periodic DNA repair-related fluctuation (period T = 40 min) with the same amplitude (100 molecules/cell) is also filtered as shown on the right.
Figure 10.
The effects of the pyz strength.
(A) As pyz is decreased, periodic DNA repair-related fluctuation (T = 40 min and amplitude = 100 molecules/cell) is filtered less, so the p53 levels fluctuate more. Note that the period of the oscillation does not change as the pyz value is decreased. (B) Without periodic DNA repair-related fluctuation, the decrease in pyz will also decrease the p53 oscillatory behavior (more damping). The period of the oscillation increases as the pyz value is decreased. (C) Step responses of a second-order system with respect to the damping ratio ζ (the poles of the transfer function are shown as X on the complex plane) [19].