Are the excess molecular-clock violations in protein-structure evolution driven by positive (diversifying) selection or by relaxation of purifying selection? A calibrated attribution problem

Pascual-García, Arenas and Bastolla (2019) introduced the Significant Clock Violations (SCV) test and showed, in three large CATH superfamilies, that violations of the molecular clock are correlated between protein sequence and protein structure but tend to be larger and more significant for structural divergence measures, with about one third of significant violations visible only in structure. The authors noted that their data are consistent with two competing explanations - positive (diversifying) selection favoring structural change versus relaxation of purifying (negative) selection on structure conservation - and are not conclusive. They further identified the calibration obstacles blocking a conclusive answer: the intrinsic-fluctuation exponent $\alpha$ and significance thresholds for structural divergence cannot be determined without a bona fide control set or validated simulations of structure evolution (neither exists), and the two candidate structural divergence measures trade sensitivity against consistency. This problem asks for a conclusive, calibrated attribution of the excess structural clock violations, robust to the choice between the two structural divergence measures used in the source. The question remains open: no later work discriminates the two selective explanations, and recent reviews still treat the clock-likeness of structures as an open, representation-dependent question.

Background

The molecular clock hypothesis states that substitutions accumulate in protein sequences at an approximately constant rate through time; real substitution processes are 'overdispersed', i.e. more variable across lineages than a Poisson clock (Gillespie 1989). Such clock violations are routinely used as hints of changing selective regimes. Protein three-dimensional structures also diverge during evolution, but more slowly than sequences on average (Chothia and Lesk 1986), raising the question of whether an approximate clock also holds for structure and what its violations mean.

Pascual-García, Arenas and Bastolla (2019) quantified clock violations for both sequences and structures as follows. For a pair of proteins $A$,$B$ that are sisters in a phylogeny, the clock violation is $CV(A,B)=\frac{(1/n)\sum_C (D(A,C)-D(B,C))}{D(A,B)}$, the mean over all available outgroups $C$ of the difference of divergences, normalized by the pairwise divergence $D(A,B)$, where $D$ is a pairwise evolutionary divergence measure; the triangle inequality implies $|CV| \le 1$, and because of this normalization the values of CV depend little on the scale of the evolutionary divergence, so that CVs of sequence and of structural divergences can be meaningfully compared. Under a strict clock the mean outgroup difference is zero apart from stochastic fluctuations, which are assumed to scale with divergence as $D(A,B)^{\alpha}$; the significance score is $SCV_{\alpha}=\frac{(1/n)\sum_C (D(A,C)-D(B,C))}{D(A,B)^{\alpha}}$, so that $CV = SCV_{\alpha=1}$, and a violation is declared significant when this score exceeds a threshold together with its standard error over outgroups. Unlike the classical Tajima (1993) test of the molecular clock, the SCV test applies to any divergence measure, including structural ones. Sequence divergence was measured by the Tajima-Nei distance, a pairwise substitution distance corrected for unequal amino-acid frequencies; structure divergence by two measures: contact divergence (CD), based on the overlap of residue-residue contact maps and requiring no spatial superimposition, and TM-score divergence $D_{TM}=-\log(TM)$, built on the TM-score - the fraction of residues that can be structurally superimposed under an optimal rotation (Zhang and Skolnick 2004). The method was validated with simulations of neutral sequence evolution under folding-stability constraints (which produce an overdispersed clock), yielding the fluctuation exponent $\alpha \approx 0.65$ and a significance threshold for sequences. Applied to three large superfamilies from the CATH structural classification database (NAD(P)-binding Rossmann-like domains, P-loop NTP hydrolases, and Globins), the study found: clock violations correlated between sequence and structure (average correlation about 0.74); systematically larger and more significant violations in structure than in sequence (except Globins); larger violations for protein pairs whose function annotation (manually curated Gene Ontology or InterPro terms, two curated function-annotation databases) changed; and about one third of significant violations significant in structure but not in sequence.

Two features of the framework remain uncalibrated. First, since there is no empirical control set of proteins known a priori to evolve under the same clock, and no validated simulator of protein-structure evolution beyond single-mutation elastic-network response models, which predict the structural deformation caused by a single mutation (Echave 2008), the exponent $\alpha$ and thresholds for structural divergences cannot be pinned down; real data support $1 \le \alpha \le 1.5$ for all three divergence measures including sequence (the source's discussion considers the wider range $0.5 < \alpha < 2$). Second, the TM-based measure shows larger clock violations (higher sensitivity) but also larger triangle-inequality violations, because optimal superimposition adds noise, whereas contact divergence is coarser but more consistent; which measure better reflects biologically meaningful structural divergence was left open.

The central interpretive question follows. Elevated substitution rates can arise from positive (diversifying) selection favoring changes, or from relaxation of purifying (negative) selection, i.e. weakened removal of deleterious variants, for instance after a decrease in effective population size; both produce larger pairwise divergences and larger CV values, so pairwise divergence rates alone cannot separate them. The authors argued qualitatively that structure is under stronger purifying selection than sequence, making stronger positive selection on structure-changing substitutions the more plausible cause of the excess structural clock violations, but stated explicitly that their data are not conclusive. Settling the attribution matters because structural clock violations are proposed as signals of accelerated evolution and functional change; whether they indicate adaptation or merely eroded constraint changes what such signals mean, and the same ambiguity now faces the rapidly growing field of structural phylogenetics, which reconstructs evolutionary trees from structural data (often predicted by AlphaFold, an artificial-intelligence method that computes protein structures from amino-acid sequences) and uses clock-likeness as a quality criterion.

Problem Statement

Determine, with a calibrated statistical framework, whether the excess of molecular-clock violations in protein-structure evolution - clock violations (CV, the mean over outgroups of the difference of divergences $D(A,C)-D(B,C)$, normalized by the pairwise divergence $D(A,B)$) that are systematically larger and more significant for structural divergence measures than for sequence divergence in the same protein families - is attributable primarily to positive (diversifying) selection acting on structural change or to relaxation of purifying (negative) selection on structure conservation.

The source study finds both explanations consistent with its data and identifies the obstacles below as the reasons its own analysis was not conclusive; an attribution answer must overcome all three obstacles, while a demonstration that the excess is not a genuine biological signal must satisfy only obstacle 1 within the same calibrated framework:

1. A calibrated significance assessment for structural clock violations: an intrinsic-fluctuation exponent $\alpha$ and significance thresholds for structural divergence measures replacing the currently borrowed, unvalidated calibration (sequence simulations give $\alpha \approx 0.65$; real data support $1 \le \alpha \le 1.5$ for all three divergence measures, and the source's discussion considers $0.5 < \alpha < 2$). The source identifies two routes to such a calibration - validated simulations of protein-structure evolution under selective constraints, or an empirical control set known a priori to evolve under the same clock - and notes that neither currently exists; building either is itself a hard open problem. Whichever route is used, the calibration must be validated against evidence independent of the selective regime being tested - in the source's own logic, the correct exponent makes the significance score independent of the pairwise divergence on average for proteins evolving under the same clock - so that $\alpha$ and the thresholds are not chosen in a way that favours either explanation, and the resulting false-positive rate must be quantified against the corresponding same-clock null.
2. Discrimination between the two selective explanations with evidence beyond pairwise divergence rates that separates them.
3. Robustness of the conclusion with respect to the choice between the two structural divergence measures used in the source - TM-score-based and contact-based divergence: the attribution must hold for both, and their documented sensitivity-versus-consistency trade-off, which the source itself leaves as an open question, must be addressed rather than assumed away. Post-source reports that structural-alphabet representations (which encode local backbone conformations as discrete symbols rather than atomic coordinates) show the opposite clock-likeness direction (Moi et al. 2025) are relevant background that keeps this measure-dependence a live methodological issue, but reconciling structural-alphabet representations with the coordinate- and contact-based measures is not itself a required deliverable of the attribution.

A conclusive answer must identify which selective regime primarily accounts for the excess; if instead the calibrated framework establishes that the apparent excess is not a genuine biological signal - for instance an artefact of measure inconsistency, superimposition noise, or estimator bias - then demonstrating this within the same calibrated framework constitutes a conclusive resolution of the problem rather than a failure to answer it.

'Excess' is understood comparatively, as in the source: relative to sequence-divergence clock violations in the same superfamilies under the same test, not against an absolute numerical threshold.

References

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