Diploid boundary identifiability

PDF page 10 Figure 4. Sampling design, with an increased-depth control. Left: mid-frequency sampling (blue) recovers dominance with quantified uncertainty; sampling the same process near the boundary (red) fails for alleles that remain rare. Right: usable recovery fraction (final sampled frequency > 0.5 for mid, > 0.2 for rare; 60 replicates per condition) for three conditions—mid-frequency at depth 150, rare at depth 150, and rare at depth 1500 (tenfold)—shown per dominance value. Mid-frequency sampling yields high; rare sampling collapses near the boundary, most severely for recessive variants, and the tenfold-depth bar shows that depth does not rescue it. Dominant alleles ( h ≥ 0.5) leave the boundary quickly and are recovered under all three. beyond diploid selection. Relation to existing work. Structural identifiability analysis asks when parameter combinations rather than parameters are determined by data (Walter & Pronzato 1997; Raue et al. 2009); Proposition 1 is a sharp, boundary-localised instance and Proposition 3 its resolution through the state-dependence coefficient. The phenomenon of individually ill-constrained parameters with well-determined combinations is the “sloppy models” programme (Gutenkunst et al. 2007); here the boundary ridge is the fixed- hs direction, and the information that breaks that ridge is localised in the coefficient a1. The question of where to sample is the subject of optimal experimental design, in which D- and A-optimal designs for logistic and Michaelis–Menten models place points away from the asymptotes toward maximum curvature; “sample the bend” is the dominance-specific dynamical case. In population genetics, the difficulty of jointly estimating s and h from allele- frequency trajectories is well documented in time-series inference under Wright–Fisher dynamics (Malaspinas et al. 2012; Tataru et al. 2017; Taus et al. 2017; Paris et al. 2019). Previous likelihood methods can estimate ( s, h) from sufficiently informative trajectories; the present result identifies the local coefficient that contains dominance and explains why early-frequency data fail even when hs is well estimated. The least-squares fit on log-odds used here is itself a simple full-trajectory method; more sophisticated HMM or diffusion-based inference would use the same curvature, and the inversion is offered to make explicit where that curvature lives. 8. Conclusion For diploid selection we prove that the rare-allele boundary identifies only the product hs, that dominance re-enters through the first state-dependence coefficient of the log-odds increment, and that the boundary value and that coefficient identify s and h in closed form. The design information makes the mechanism plain: near the boundary the data express only a0, the score directions are collinear (analytically, by Corollary 1), and dominance is unidentifiable; the bend expresses a1 and separates them. We demonstrate the inversion on noisy data, give its error PDF page 11 propagation, and show that the design implication—sample the bend, not the boundary—cannot be circumvented by sequencing depth. The rule is simple and practical: to estimate dominance, place samples across the frequency range where the trajectory bends. Methods Dynamics and propositions. Deterministic update from (1); Propositions 1–3, Corollary 1, the quadratic coefficient a2, the finite-interval expansion, and the sensitivity ∂h/∂a 1 verified symbolically (SymPy). Design information. Ft = C[∂sxt, ∂hxt]⊤[∂sxt, ∂hxt]/[xt(1 − xt)] summed over sampled times; Schur- complement information Fhh − F 2 hs/Fss; score-collinearity cosines (the cosine between ∂sxt and ∂hxt, as in Corollary 1) within frequency bands; sensitivities by central differences (∆ = 10−5, stable across 10−4–10−6). This is the expected binomial observation information along the deterministic trajectory, conditional on that trajectory; the Wright–Fisher simulations test survival of the prediction under process noise. Inversion on simulated data. Wright–Fisher drift (Ne = 5000), pool-seq depth 200, 120 generations from x0 = 0.02; Jeffreys pseudocount ˜x = (k + 1 2)/(C + 1); a0, a1 by least squares of per-generation log-odds increments on x over 0.02 < x < 0.6 (robustness: window x < 0.3 and weighted least squares with delta-method variances, Section 5); invert by Proposition 3; 200 replicates per h, estimates retained when finite and in [ −0.5, 1.5]. Recovery simulations. Ne = 2000, depth 150 (1500 for the increased-depth control), 80 generations sampled every 8, 60 replicates per h (all conditions, including the depth control); full-trajectory fit of (s, h, x0) by least squares on pseudocount log-odds, multistart over s0 ∈ {0.05, 0.1, 0.2}, h0 ∈ {0, 0.5, 1}, Nelder–Mead, tolerances 10 −4/10−8; usable = final sampled frequency > 0.5 (mid) or > 0.2 (rare); full retention in Table 1. True s = 0.1 throughout; initial frequency 0 .15 (mid) or 0 .02 (rare). Coordinate and null tests (Appendix A). Bending recomputed in probit Φ −1(x), arcsine-root arcsin √x, raw x; null trajectories: shifted logistic, power-law ( t/T )1.5, bounded random walk, all under the same bounds and binomial weight. Reproducibility. All figures and numerical values regenerate from a single self-contained script with fixed random seeds (Python 3, NumPy, SciPy, SymPy), provided as supplementary material with a reviewer copy at submission and to be archived with a permanent DOI on acceptance; it is written to be usable as a standalone sampling-depth calculator. A. Coordinate specificity and null dynamics Coordinate specificity. The information–bending alignment is read in the log-odds, the coordinate in which the update is additive. “Bending” is not coordinate-invariant, so this is explanatory geometry, not an identifiability theorem (which is Propositions 1–3). Recomputing the bending in coordinates that do not linearise the odds-multiplicative composition gives, for the same trajectory, correlations of information with bending of −0.24 (probit), 0.00 (arcsine-root), and 0 .66 (raw frequency), versus 0 .999 in the log-odds. Null dynamics. For the null trajectories the comparison uses the same binomial observation weight x(1 − x) and coordinate bending as an alignment diagnostic, not a dominance-information calculation, since the null flows contain no dominance parameter. Holding the bounds and the binomial weight fixed but replacing the selection trajectory with other bounded trajectories collapses the alignment: a shifted logistic gives −0.36, a power-law rise −0.72, a bounded random walk −0.12. A relationship forced by the coordinate alone could not be destroyed by changing the flow; the mechanism is that for the selection flow the bending region and the high-Fisher-weight region x(1 − x) co-locate at mid-frequency, while the null flows place them apart. Declaration of generative AI and AI-assisted technologies in the writing process During the preparation of this work the author used generative AI and AI-assisted tools to assist with drafting, figure preparation, and editorial revision. Symbolic and numerical checks were performed by author-directed scripts and independently reviewed by the author. After using these tools, the author reviewed and edited the content as needed, verified the mathematics, simulations, and references, and takes full responsibility for the content of the publication. No AI system is listed as an author.

This is the expected binomial observation information along a deterministic trajectory; we use it as a design diagnostic, and the Wright–Fisher simulations below test whether the same geometric prediction survives process noise. The relevant scalar is the information for h after profiling out s, the Schur complement I prof h = Fhh − F 2 hs/Fss. The design matrix treats the initial frequency x0 as fixed; the full-trajectory simulations below estimate x0 as a nuisance parameter, which does four (Figure 1, right; the horizontal axis is the maximum frequency reached, with F accumulated over all observations up to that point). The prediction localises directly. Defining the bending of the path as the second difference of the log-odds, κt = |ρt+1 − 2ρt + ρt−1|, the Fisher information about h and κt peak at the same frequency (x ≈ 0.49 for s = 0.1, h = 0.5), and essentially none of the total information lies where the path does not bend: less than 10 −4 of it sits where κt is below one percent of its maximum (Figure 2). This co-location is a property of the selection flow, not of the log-odds transform:

Corollary 1) within frequency bands; sensitivities by central differences (∆ = 10−5, stable across 10−4–10−6). This is the expected binomial observation information along the deterministic trajectory, conditional on that trajectory; the Wright–Fisher simulations test survival of the prediction under process noise. Inversion on simulated data. Wright–Fisher drift (Ne = 5000), pool-seq depth 200, 120 generations from x0 = 0.02; Jeffreys pseudocount ˜x = (k + 1 2)/(C + 1); a0, a1 by least squares of per-generation log-odds increments on x over 0.02 < x < 0.6 (robustness: window x < 0.3 and weighted least squares with delta-method variances,