SME
Cell & Molecular Biology · Problem type
Cooperative binding: the Hill equation
Use this for “With K = 26 and n = 2.8, how saturated is the protein at 40?”
Chart
θ(40) ≈ 0.7696
Half bound at L = K = 26: θ(K) = 0.5, half of 1
This is an example. Type your own values to start a new problem.
θ
=
[L]
K
+
[L]
r
Bound-to-free ratio
worked out (not typed)
Formulas
r
=
([L] ÷
K
)
n
_H
3.341
=
(
40
26
)
2.8
θ
=
r
1 +
r
0.76962
=
3.341
1 + 3.341
Assumptions
•
θ = [L]ⁿ ÷ (Kⁿ + [L]ⁿ), with [L] and K in one unit: nM for a hormone, mmHg of O₂ for hemoglobin.
•
K is the level at half saturation; nH above 1 means binding at one site helps the next.
•
The bound-to-free ratio θ ÷ (1 − θ) equals ([L] ÷ K)ⁿ, which the steps work first.
Step-by-step
WE KNOW
[L] = 40, K = 26, nH = 2.8
FIND
bound-to-free ratio (r), fraction bound (θ)
Step 1 · Find bound-to-free ratio (r)
Use r = ([L] ÷ K)ⁿH
Divide
the
ligand
level
by
K
,
then
raise
it
to
the
Hill
coefficient.
r
=
(
40
26
)
2.8
r
=
1.5385
2.8
r
=
3.3407
Step 2 · Find fraction bound (θ)
Use θ = r ÷ (1 + r)
Bound over bound plus free: the ratio over 1 plus the ratio.
θ
=
3.3407
1 + 3.3407
θ
=
3.3407
4.3407
θ
=
0.76962
Check
Put the numbers back in. Both sides should match.
3.3407
=
(
40
26
)
2.8
✓
0.76962
=
3.3407
1 + 3.3407
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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