8. Feedback loops — why do p53 and MDM2 oscillate? · Dr. Seok's Cancer-Cell Digital Twin, First Steps

Once a circuit has loops in it, new things happen. Ours has three.

Loop	Sign	Result
p53 → MDM2 ⊣ p53	Negative (one ⊣, odd)	Oscillation
DDX54 → MDM2 ⊣ p53 ⊣ DDX54	Positive (two ⊣, even)	Two locked branches (bistability)
Proliferation → T_Kill → Apoptosis ⊣ Proliferation	Negative (one ⊣)	Immune surveillance

The rule is simple. Walk once around the loop and count the inhibitory arrows (⊣): odd means negative feedback, even means positive feedback.

Negative feedback makes oscillation, positive feedback makes a fork in the road. Those two sentences are half of systems biology.

The p53-MDM2 oscillation is a phenomenon actually observed in cells: irradiated cells show it rising and falling over a few hours.

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import bn

net = bn.twin()

# Start with p53 alone switched on - a normal cell with DDX54 off
net.trace(net.state(p53=1), n=9, stop_when_stable=False)

print()
print("p53 and MDM2 chase each other around with a period of 4.")
print("Proliferation stays off throughout - a healthy cell on patrol.")

Drawn as a step plot the phase difference is unmistakable.

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import bn
import matplotlib.pyplot as plt

net = bn.twin()
path = net.run(net.state(p53=1), 13)

genes = ["p53", "MDM2", "DDX54", "Proliferation"]
fig, ax = plt.subplots(figsize=(7.4, 3.4))
for k, g in enumerate(genes):
    i = net.genes.index(g)
    y = [s[i] * 0.8 + (len(genes) - 1 - k) * 1.2 for s in path]
    ax.step(range(len(path)), y, where="post", lw=2.2, label=g)
ax.set_yticks([(len(genes) - 1 - k) * 1.2 + 0.4 for k in range(len(genes))])
ax.set_yticklabels(genes)
ax.set_xlabel("t")
ax.set_title("p53-MDM2 oscillation (period 4)")
ax.grid(axis="x", alpha=.3)
plt.tight_layout()

print("The top two traces rise and fall out of step with each other.")
print("That phase shift is the signature of negative feedback.")

p53-MDM2 is negative, so it oscillates; DDX54-MDM2-p53 is positive, so it creates a fork.

Python Runs in your browser

import bn

net = bn.twin()

# Find the feedback loops ourselves (length 2 and 3)
from itertools import permutations

edges = {(u, v): sg for u, v, sg in net.edges}
arrow = lambda sg: "->" if sg > 0 else "-|"

seen = set()
print(f"{'feedback loop':<48}{'inhibits':<10}sign")
print("-" * 70)
for L in (2, 3):
    for cyc in permutations(net.genes, L):
        pairs = [(cyc[i], cyc[(i + 1) % L]) for i in range(L)]
        if not all(e in edges for e in pairs):
            continue
        key = frozenset(cyc)                      # count each loop once
        if key in seen:
            continue
        seen.add(key)
        signs = [edges[e] for e in pairs]
        n_inhibit = sum(1 for sg in signs if sg < 0)
        path = cyc[0] + "".join(f" {arrow(signs[i])} {pairs[i][1]}" for i in range(L))
        kind = "negative -> oscillates" if n_inhibit % 2 else "positive -> forks"
        print(f"  {path:<46}{n_inhibit:<10}{kind}")

🎮 Oscillation bench

Flip the sign of a feedback loop and see what happens to the waveform.

Whether the number of inhibitory arrows is odd or even is the whole story.

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🤔 Check yourself

How do you tell whether a feedback loop is negative?

🤔 Check yourself

What is the period of the p53-MDM2 oscillation in our circuit?

🤔 Check yourself

What does a positive feedback loop create?