Spanning binding sites on allosteric proteins with polymer-linked ligand dimers
Kramer RH, Karpen JW. Nature 1998; 395(6703):710–713. DOI: 10.1038/27227 PMID: 9790193
TL;DR
First quantitative measurement of effective concentration (Ceff) for a synthetic bivalent ligand targeting a protein with two binding sites. Polymer-linked cGMP dimers (barbell of two cGMPs connected by PEG of variable length) activate cyclic-nucleotide-gated (CNG) channels up to 1000× more potently than monovalent cGMP. Optimal potency occurs when PEG length matches inter-site distance; Ceff is calculated from tether geometry and directly used to predict dissociation kinetics.
Key finding
The Ceff formula (from paper Fig. 1b legend):
C_eff = 1,000 / (N_A × (2/3) × π × r^3)
where r = r.m.s. end-to-end length of the PEG tether in cm, N_A = Avogadro’s number.
This models the tethered cGMP as uniformly distributed within a hemisphere of radius r after one end binds. The factor (2/3)π r^3 = hemisphere volume; 1,000 converts mL to L.
The dissociation rate for the bivalent complex is:
k_off(PLD) = 2 × k_off × K_d / (C_eff + K_d)
When Ceff >> K_d, k_off(PLD) approaches 2 k_off × K_d / C_eff — orders of magnitude slower than monovalent off-rate.
Numbers that matter
| Quantity | Value | Conditions |
|---|---|---|
| Ceff for 3400PEG tether (r = 39 Å) | 13.4 mM | Olfactory CNG channel, 21–23°C |
| K_1/2 for cGMP (monovalent), olfactory CNG | 3.1 μM | Inside-out patch, rat olfactory α-subunit |
| K_1/2 for 3400PEG-(cGMP)_2, olfactory CNG | 12 nM | 260-fold improvement over monovalent |
| K_1/2 for cGMP (monovalent), rod CNG | 72 μM | Bovine rod photoreceptor α-subunit |
| K_1/2 for 3400PEG-(cGMP)_2, rod CNG | 4.7 μM | 15-fold improvement |
| K_1/2 for cGMP (monovalent), PKG | 120 nM | cGMP-dependent protein kinase type Iα |
| K_1/2 for 282PEG-(cGMP)_2, PKG | 4 nM | 30-fold improvement; optimal PLD for PKG |
| Maximum overall potency improvement | up to 1000× | For optimal PLD length matched to inter-site distance |
| Dissociation rate slowing | 3400-fold slower | 2000PEG-(cGMP)_2 vs cGMP on olfactory CNG; matches Ceff/K_d ratio |
| Optimal PLD length for olfactory CNG | 2000PEG, r = 39 Å | |
| Optimal PLD length for rod CNG | 1200PEG, r = 30 Å | |
| Optimal PLD length for PKG | 282PEG, r = 15 Å | |
| PEG r.m.s. length range tested | 15–123 Å | Mr 282–20,000 Da PEG |
| High cGMP concentration needed to compete with PLD | <1 mM | Free cGMP must compete with Ceff of several mM from bound PLD |
Ceff range from the data: The paper shows maximum affinity enhancement at r = 39 Å (Ceff = 13.4 mM) and reduced enhancement at r = 123 Å (20000PEG), consistent with lower Ceff for longer tethers. The Ceff for flexible polymer tethers falls in the mM range at optimal matching — exactly the value cited in avidity-and-dimers.
Relevance to h26
This paper provides:
- The Ceff formula used in all h26 avidity calculations.
- Direct evidence that Ceff for flexible PEG tethers spans 0.1–13.4 mM range.
- Empirical validation that bivalent binding dissociation is slowed by Ceff/K_d — the predicted 3400-fold slowing matched observed kinetics.
- Demonstration that linker length is critical: sub-optimal length gives only 2× improvement; optimal gives 260-1000×.
Limitations
- Used homotetrameric channels (α-subunit only), not native heterotetramers.
- PEG is highly flexible; rigid linkers would give sharper length-dependence.
- Ceff formula assumes uniform distribution in hemisphere — valid only for flexible polymer where all conformations have equal probability.
- Specific tether lengths that match STRC homodimer inter-site distance are not known and cannot be predicted from AF3 ipTM alone.
Connections
[source]STRC Engineered Homodimer Avidity — Ceff = 13.4 mM value used in h26 avidity calculations[source]avidity-and-dimers — row in parameter table: Ceff typical range mM for flexible polymer tethers[applies]STRC Engineered Homodimer Avidity — bivalent enhancement formula applies to h26 homodimer scenario[see-also]1981-jencks-binding-energies-additivity-pnas — Jencks 1981 theoretical framework that Kramer implements[see-also]1998-mammen-polyvalent-interactions-angew — Mammen 1998 review contextualizes these results within general polyvalency theory