4. Attractors — how many fates does a cell have? · Dr. Seok's Cancer-Cell Digital Twin, First Steps

Here is an argument that is very simple and very strong.

The states are finite (2ⁿ of them).
Time is infinite.
The next state is completely determined by the present one.
→ therefore you must eventually meet a state you have already passed through.
→ and from that moment on you repeat the same road forever.

That repeating stretch is called an attractor. There are two kinds.

Kind	Shape	Biological meaning
Fixed point (point attractor)	Stops on one state and never moves	A stable cell type
Cycle (cyclic attractor)	Goes round a few states periodically	The cell cycle, an oscillating signal

The trick is storing which time step it was in seen. If you only store whether you visited, you cannot recover where the cycle begins.

Python Runs in your browser

import bn

net = bn.mini()
s0 = net.state(DDX54=1, p53=0)

# The algorithm: write down every state you pass, and stop when you meet one again
seen, path = {}, []
cur = s0
while cur not in seen:
    seen[cur] = len(path)
    path.append(cur)
    cur = net.step(cur)

start = seen[cur]                       # the point where we met it again
print("states passed through")
for t, s in enumerate(path):
    mark = "  <- the attractor starts here" if t == start else ""
    print(f"  t={t}  {net.show(s)}{mark}")

cycle = path[start:]
print()
print(f"attractor length {len(cycle)}  ->", "fixed point" if len(cycle) == 1 else f"cycle of period {len(cycle)}")

Three starting points went to three different destinations.

Python Runs in your browser

import bn

net = bn.mini()

# bn does the same job in one line
for label, s0 in [("DDX54 on + p53 broken", net.state(DDX54=1, p53=0)),
                  ("DDX54 on + p53 intact", net.state(DDX54=1, p53=1)),
                  ("DDX54 off",             net.state(DDX54=0, p53=0))]:
    a = net.attractor(s0)
    kind = "fixed point" if len(a) == 1 else f"period {len(a)}"
    print(f"{label:<24} -> {kind}   ON: {', '.join(net.on_genes(a[0])) or '(none)'}")

bn.canonical takes the lexicographically smallest rotation as the representative.

Python Runs in your browser

import bn

net = bn.mini()

# The trap with cyclic attractors: the same loop written from a different starting point
cycle_a = ((0, 1), (1, 1), (1, 0))
cycle_b = ((1, 1), (1, 0), (0, 1))          # the same loop, rotated by one

print("at face value:", cycle_a == cycle_b)
print("after normalising:", bn.canonical(cycle_a) == bn.canonical(cycle_b))
print()
print("Skip the normalising step and your attractor count is inflated.")
print("It is a genuinely common bug — one oscillation gets counted as four.")

🎮 Attractor bench

Pick a starting point and watch the trajectory get sucked into an attractor.

The tail (transient) and the loop (attractor) are colour-coded apart.

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🤔 Check yourself

Why must a Boolean network always have an attractor?

🤔 Check yourself

Why record the states you have seen in a dictionary rather than a set?

🤔 Check yourself

What does the view 'cancer cells and normal cells use the same genome' imply?