◄ WORLD V · SONNY 5DART 587 · a helldive into the living

THE HILL EQUATION cooperation makes a switch

When binding sites help each other, the dose-response curve stops being a gentle hyperbola and becomes an S-shaped switch: θ = Sn / (Kn + Sn). The Hill coefficient n measures cooperativity — n=1 is independent, n>1 means each bound ligand makes the next bind easier. Hemoglobin (n≈2.8) uses it to grab oxygen in the lungs and dump it in the tissues. Cooperativity turns a dial into a decision.

THE TECHNIQUE θ = Sn/(Kn + Sn)

The demo compares n=1 (hyperbola) with n=4 (sigmoid) and shows the cooperative switch: live demo







  
HISTORY & CREDIT Archibald Hill · 1910

  
“The Hill coefficient is the number of binding sites.” — it is a measure of cooperativity, usually less than the site count (hemoglobin has 4 sites, n≈2.8). cited

  
independence · n=1 — a plain saturating hyperbola.
cooperation · n>1 — each binding helps the next; the curve becomes a sigmoid switch.
1910 · Archibald Hill, on hemoglobin’s oxygen binding.

  
A dial, sharpened into a decision. kinetics






  
RECOMMEND FOR I-13 the sigmoid, on the compiler

  
On i-13, n=4 sits below n=1 at low substrate and above it at high substrate — the switch:

  
$ i13 run lf_hill-equation.i13
RUN OK · 85 step(s)
theta_n1_lowS=33.3  theta_n4_lowS=5.9   -- cooperative stays low
theta_n1_highS=66.7 theta_n4_highS=94.1  -- then jumps high
cooperative_switch = 1

  
Recommend as a NULL — a binding law (B39). Occupancy is a pinned function of S, K, n; the sigmoid is a fact of the exponent. NULL — cooperation makes a switch.