Bell-Model 2D Membrane Adhesion Kinetics for T4 Phage Adsorption

Bell-Model 2D Membrane Adhesion Kinetics for T4 Phage Adsorption

Computational Verification Report

Hypothesis: E1-H4 from session 2026-04-15-scout-028

Title: ITC-Derived Per-Contact Kd Fed into Bell-Model 2D Membrane

Adhesion Kinetics Predicts Minimum OmpC Density for T4 Productive

Adsorption, With Baseplate Trigger Modeled as Cooperative Sequential Capture

Verdict: PARTIALLY_CONFIRMED

Verdict Detail: The Bell model framework is physically appropriate for

phage adsorption kinetics, and the corrected formulas produce biologically

plausible predictions. However, the original hypothesis contained three

mathematical/factual errors (binomial formula, dimensional conversion,

gp37 residue numbering) that, if used as stated, would give qualitatively

wrong results. After correction, sensitivity analysis reveals a key finding:

at any Kd below ~10 uM, biological OmpC density (~1,700-5,000/um^2) is

orders of magnitude above the critical threshold, meaning adsorption is

trivially efficient. The model becomes discriminating only for very weak

binders (Kd > 1 uM) or severely OmpC-depleted cells.

Error Corrections

1. Binomial Formula (Critical)

Original: P_trigger = C(6,3) p^3 (1-p)^3 (exactly 3 of 6 contacts)

Corrected: P_trigger = sum(k=3 to 6) C(6,k) p^k (1-p)^(6-k)

(at least 3 of 6 contacts)

The original formula gives the probability of exactly 3 contacts. It

peaks at p=0.5 (value = 0.3125) and approaches 0 as p approaches 1.

This is physically nonsensical: if every fiber binds (p=1), the

probability of triggering should be 1, not 0.

The corrected cumulative binomial gives P=1 when p=1, P=0.6563 at p=0.5,

and is monotonically increasing. The relative error between formulas

exceeds 10% for p in [0.123, 1.000]

and exceeds 50% for p in [0.482, 1.000].

2. Dimensional Conversion (Critical)

Original: k_on_2D = k_on_3D * h_eff

Corrected: k_on_2D = k_on_3D / h_eff

k_on_3D has units M^-1 s^-1 = 10^-3 m^3 mol^-1 s^-1. Multiplying by

h_eff (m) gives m^4 mol^-1 s^-1, which is not a 2D rate constant.

Dividing by h_eff gives m^2 mol^-1 s^-1, correct for 2D kinetics.

For the equilibrium Bell model, sigma_critical = Kd_2D, where

Kd_2D = Kd_3D (in concentration) * h_eff. The wrong formula inverts

the h_eff dependence, leading to orders-of-magnitude errors.

3. gp37 RBD Residue (Minor)

Original: Residues 850-1026

Corrected: RBD starts at residue 785 per PDB 2XGF

This is a factual error but does not affect the mathematical model.

Sensitivity Analysis Results

Using the corrected formulas with the Bell model:

• sigma_crit = Kd_2D, where Kd_2D = Kd_3D_conc * h_eff
• p_bind = sigma / (sigma + sigma_crit) per fiber
• P_trigger = cumulative binomial (at least 3 of 6)

Parameter Sensitivity (Tornado Analysis)

The 50% efficiency density (sigma_50) is most sensitive to Kd, followed

by h_eff. At baseline (Kd=100 nM, h_eff=20 nm, A_c=2500 nm^2):

• sigma_50 = 1 molecules/um^2

Critical insight: sigma_50 is extremely small compared to biological

OmpC density. Even at Kd = 10 uM (a very weak interaction), sigma_50

is only ~88 molecules/um^2, while biological OmpC density is

1667-5000 molecules/um^2 -- a factor of ~20-60x above threshold.

This means OmpC density is NOT the limiting factor for T4 adsorption

under normal growth conditions.

Biological Reality Check

E. coli OmpC copy number: 10,000-30,000 per cell

Cell surface area: ~6 um^2 (rod, 2 x 0.8 um)

OmpC density: 1667-5000 molecules/um^2

Key finding: for the corrected Bell model, OmpC is so abundant that

adsorption is efficient across nearly the entire Kd range:

This is a meaningful result: it shows the Bell model predicts that

T4 adsorption is NOT density-limited under normal conditions, consistent

with the observation that T4 is a highly efficient phage. The model

becomes informative in two regimes: (1) very weak binders (Kd >> 1 uM),

and (2) cells with severely reduced OmpC (e.g., porin-deficient mutants

or extreme osmotic stress).

Cooperativity Analysis

For the T4 trigger threshold (k_min = 3 of 6 LTFs):

• At Kd = 100 nM, all k_min thresholds give near-perfect efficiency

at biological densities (even k_min=6 gives eta > 0.99)

• The cooperativity threshold only becomes discriminating at Kd ~ 1 uM,

where sigma_50 ranges from 1 (k_min=1) to 98 (k_min=6) molecules/um^2

• Even at k_min=6, sigma_50 remains far below biological OmpC density

This means the 3-fiber threshold is not an evolved response to OmpC

scarcity. Rather, it likely serves a different function: ensuring

geometric alignment of the baseplate for tail sheath contraction,

or providing a kinetic checkpoint against non-specific binding.

Figures

1. fig1_binomial_error: Comparison of exactly-3 vs cumulative at-least-3

binomial formulas, all k_min variants, and relative error quantification

1. fig2_dimensional_error: Unit analysis and sigma_critical vs h_eff
2. fig3_heatmap_kd_vs_density: 2D heatmap of adsorption efficiency
3. fig4_sigmoidal_curves: Sigmoidal adsorption curves for 3 representative Kd values
4. fig5_sensitivity_tornado: Tornado chart of parameter sensitivity
5. fig6_phase_diagram: Phase diagram with biological OmpC range highlighted
6. fig7_cooperativity: Cooperativity threshold analysis

Verdict: PARTIALLY_CONFIRMED

Confirmed aspects:

• Bell model is the correct physical framework for 2D receptor-ligand kinetics
• The corrected math (cumulative binomial + correct dimensional conversion) is sound
• The framework makes quantitative, testable predictions
• An ITC-measured Kd would provide a genuinely useful input to the model

Key quantitative finding:

• At biological OmpC densities (~1,700-5,000/um^2), T4 adsorption is

predicted to be trivially efficient for any Kd below ~10 uM

• sigma_critical is orders of magnitude below biological OmpC density
• The model is most informative for weak binders or OmpC-depleted cells
• The cooperativity threshold k_min=3 is not density-driven; it likely

serves geometric alignment rather than receptor-scarcity compensation

Refuted/corrected aspects:

• Binomial formula was qualitatively wrong (exactly-3 vs at-least-3)
• Dimensional conversion was wrong (multiplication vs division by h_eff)
• gp37 RBD residue numbering was incorrect (785, not 850)
• The original errors would produce nonsensical predictions

Remaining uncertainties:

• gp37-OmpC Kd is unmeasured (the entire model is parametric)
• OmpC clustering on the membrane could alter effective encounter rates
• h_eff is poorly constrained (5-50 nm range spans an order of magnitude)
• The Bell model assumes equilibrium; kinetic trapping may dominate in vivo

Bottom line: The hypothesis identifies the correct physical framework

(Bell model + cooperative binomial trigger) for phage adsorption. After

correcting three errors, the model reveals that OmpC density is NOT the

limiting factor for T4 adsorption under normal conditions -- a non-obvious

prediction that is itself testable. The framework becomes experimentally

discriminating in the regime of OmpC-depleted cells or engineered phages

with weakened receptor binding.