Competing Risk Theorem for Protein Design

Competing Risk Theorem for Protein Design: Verification Report

MAGELLAN Session: 2026-04-05-scout-017, C1-H3

Composite Score: 8.25 (PASS)

Verification Date: 2026-04-05

Verdict: PARTIALLY CONFIRMED

Hypothesis Statement

Original Claim

When optimizing a designed therapeutic protein against a single dominant failure mode,

the CIF (cumulative incidence function) of at least one non-dominant mode NECESSARILY

increases.

Why the Original Claim Is False

GPT-5.4 identified a valid counterexample: if a modification reduces ALL hazards

simultaneously (e.g., a disulfide bond improving both aggregation resistance and

proteolytic stability), then non-dominant CIFs can decrease or remain unchanged.

Verification (Part 3): With a 5-mode system (h = [1.0, 0.3, 0.2, 0.1, 0.05]),

global 50% reduction of all hazards results in ZERO non-dominant CIF increases.

The cause probabilities are preserved exactly (proportional reduction), and at finite

timepoints all CIFs decrease because overall survival increases.

Corrected Claim (Ceteris-Paribus Theorem)

If a protein modification reduces exactly one cause-specific hazard h_{k*}(t) while

leaving all other h_j(t) unchanged, then CIF_j(t) increases for all j != k* at all

finite timepoints t > 0.

Why the Corrected Claim Is True

Analytical proof (Part 1):

For constant hazards, CIF_j(t) = (h_j / H)(1 - exp(-Ht)). Define g(H) = (1/H)(1 - exp(-Ht)).

Then dg/dH = (1/H^2)[u exp(-u) - (1 - exp(-u))] where u = Ht. The inequality

u exp(-u) < 1 - exp(-u) holds for all u > 0 (since exp(u) > 1 + u), giving dg/dH < 0.

Since reducing h_{k} reduces H, g(H) increases, and CIF_j = h_j g(H) increases for

all j != k* (where h_j is fixed).

• Key inequality verified numerically: u*exp(-u) < 1 - exp(-u) for all u in [0.001, 20] (min gap: 5.00e-07)
• Numerical derivative d(CIF_j)/d(h_{k*}) < 0 confirmed at all 500 test points
• CIF increase verified at finite timepoints t = 0.5, 1.0, 2.0, 5.0

Gap Formula Verification (Part 2)

Gemini's formula: gap_ratio = f / ((f + r)(1 + r))

where f = hazard reduction fraction, r = sum(h_other) / h_dominant.

Derivation verified algebraically. The ratio of actual half-life improvement

(under competing risks) to naive single-mode prediction simplifies exactly to this formula.

Test case (h1=1.0, h2=0.5, f=0.5):

• Empirical gap ratio: 0.333333
• Formula gap ratio: 0.333333
• Matches expected 1/3: True

Systematic verification: True across 100 random parameter combinations

(max error: 1.44e-15).

Protein-Calibrated Results (Part 4)

Five protein failure modes with realistic timescales:

Protease-resistance engineering (90% h_proteolysis reduction):

• Baseline total half-life: 1.576 h
• Engineered total half-life: 5.416 h
• Improvement: 243.7%
• Naive single-mode overestimates improvement by 369%
• Gap ratio: 0.2133 (you get only 21.3% of expected gain)

Key insight: Proteolysis strongly dominates (79.2% of failures at baseline).

Even so, the competing risk correction reduces the expected improvement by

79%. After engineering, unfolding becomes the new dominant

mode (45% of failures), fundamentally changing the degradation

profile.

Monte Carlo Validation (Part 6, N = 10,000)

• Empirical median survival matches theoretical within 0.001h (baseline)
• Max cause-probability error: 0.0041 (baseline), 0.0021 (engineered)
• Bootstrap 95% CIs confirm CIF increases are statistically significant: False
• Theory-Monte Carlo agreement: excellent across all failure modes and timepoints

Practical Significance (Part 7)

The theorem matters when failure modes are comparable in rate and becomes negligible

when one mode strongly dominates:

| Proteolysis | 3.713 | 0.5120 | YES |

| Aggregation | 0.034 | 0.0005 | YES |

| Unfolding | 0.151 | 0.0092 | YES |

| Oxidation | 0.043 | 0.0009 | YES |

| Immunogenicity | 0.007 | 0.0000 | YES |

Rule of thumb: The competing risk theorem is practically significant (>20% overestimation)

when the dominance ratio is below ~5-10 (depending on reduction fraction).

For our 5-mode protein:

• Optimizing proteolysis (dominant, ratio 3.75): gap ratio 0.512 at f=0.5 -- significant but manageable
• Optimizing aggregation (weakest, ratio 0.16): gap ratio very small -- theorem is critical

Verdict: PARTIALLY CONFIRMED

The hypothesis correctly identifies a real and mathematically rigorous constraint on

protein engineering, but the original formulation was too broad. The ceteris-paribus

qualification (from GPT-5.4 cross-model validation) is essential -- the theorem only

applies when exactly one hazard is modified. The Gemini gap formula provides a useful

quantitative tool for estimating the magnitude of the effect.

Testable Experimental Predictions

1. Protease-resistant antibody fragment: Engineer a VHH antibody with a protease-resistant

linker. Measure total serum half-life improvement. The competing risk theorem predicts

the improvement will be 21% of the naive single-mode prediction

(gap ratio = 0.213).

1. Successive optimization cycles: After fixing proteolysis, the next bottleneck (unfolding,

t_1/2 = 12h) has a much lower dominance ratio. Fixing it should show an even larger gap

between naive and actual improvement, quantifiable by the gap formula.

1. Global vs targeted stabilization: Two variants of the same protein -- one with a

targeted protease cleavage site mutation, one with a global stabilizing disulfide bond.

The theorem predicts CIF redistribution only for the targeted variant.

Figures

• Figure 1: Three modification scenarios (ceteris paribus / global / mixed)
• Figure 2: Protein CIF curves before and after protease resistance engineering
• Figure 3: Failure mode probability redistribution (pie charts)
• Figure 4: Gap ratio heatmap vs reduction fraction and dominance ratio
• Figure 5: Monte Carlo validation (N=10,000) vs theoretical CIF curves
• Figure 6: Practical significance analysis with protein failure mode markers