Energetic slack in ATP synthase

### PDF page 12 Murray - ATP synthase energetic slack Page 9 Figure 5. Model-comparison workflow. M1 must earn its complexity through the non-circular q_eff x D_env or q_eff x demand-class interaction. Given the small and clustered c-ring dataset, a global phylogenetic generalized least-squares test may be underpowered. This should be stated rather than hidden. If the available data do not support a global regression, the strongest empirical route becomes controlled engineering, within-lineage comparisons, or a carefully labelled exploratory dataset. 9. Interpretation and limits This framework does not require intelligent-design language, nor does it imply literal backward causation. The scientifically useful claim is narrower: observed molecular machines are survivorship-filtered members of an energy-admissible set. Lineages that fail energy closure disappear from the observable record. This can make molecular machines look design-like, but the mechanism is constraint-filtered persistence, not intention. The manuscript also does not claim a completed empirical proof. It is a framework and analysis plan anchored by selected cases. Its strongest form will be reached only when the extraction table is populated and M1 is tested against M0. Negative results must be reported. In particular, if phylogeny absorbs the signal, if L cannot be bounded, or if the interaction term fails, the framework should be revised or narrowed. The most important limitation is sample size. Experimentally measured c-ring stoichiometries remain sparse and clustered. A strong first empirical paper may therefore need to focus on controlled engineering, within-lineage comparisons, or a labelled exploratory review rather than a sweeping cross-life regression. 10. Conclusion ATP synthase does not reveal a universal molecular optimum. It reveals a conditional physical boundary. When that boundary binds, coupling geometry should matter strongly. When slack exists, evolution can preserve inherited architectures, tolerate drift, or trade thermodynamic efficiency for other system-level objectives. This binding/slack distinction is the contribution: it replaces both naive optimality and naive complexity arguments with a falsifiable phase-space claim. ---

c-Ring Stoichiometry, Energetic Slack, and the Limits of Molecular Optimality Hypothesis and retrospective-analysis framework Daniel J. Murray Independent Researcher, Melbourne, Australia Submitted to: Biochimica et Biophysica Acta (BBA) - General Subjects circularity, environmental pressure is expressed as a demand index, D_env = (Delta G_ATP + L_proxy)/(F Delta_mu_ion), which does not contain the organism’s actual coupling solution. Admissibility then requires q_eff >= D_env, and slack is Lambda = q_eff - D_env. The discriminating prediction is an interaction: q_eff should improve prediction most strongly when D_env is high or when pre-assigned categorical evidence indicates boundary binding. The loss term L is bounded through pre-specified proxy classes rather than fitted after the fact. Published cases from alkaliphilic bacteria and engineered FoF1 support boundary-shift behaviour, whereas chloroplast and cyanobacterial cases show why simple optimality fails. The manuscript is offered as a testable hypothesis and retrospective-analysis framework, not as a completed empirical proof. Keywords ATP synthase; c-ring stoichiometry; energetic slack; proton motive force; sodium motive force; chemiosmosis; bioenergetics; evolutionary constraint; viability kernel; chloroplast ATP synthase; alkaliphiles Abbreviations

geometry that is less efficient by one thermodynamic metric may be better for whole-system performance. The framework proposed here is conditional. It does not claim that ATP synthase is always optimized, nor that c- ring stoichiometry can be predicted from habitat pH alone. It claims that the effect of coupling geometry should strengthen when environmental demand approaches the ATP-synthesis boundary. The central test is therefore an interaction, not a main effect: q_eff should matter most when the demand imposed by cellular and environmental conditions is high. Because experimentally determined c-ring stoichiometries are sparse and phylogenetically clustered, the

Lambda_r = q_eff / D_env This formulation cleanly separates demand from solution. D_env describes what the environment and cell require; q_eff describes the enzyme’s coupling geometry. The framework predicts that selection on q_eff should be strongest when D_env is high or when q_eff is close to D_env. Table 1. Regimes defined by demand and slack Regime Condition Meaning Prediction Non-admissible q_eff < D_env ATP synthesis cannot be sustained under the specified energetic context. No viable ATP-synthesis phenotype unless conditions or architecture shift. Slack q_eff > D_env with margin ATP synthesis is possible without tight optimization of coupling ratio. q_eff should be weakly predictive; phylogeny, drift, kinetics, and regulation may dominate. Trade-off slack q_eff > D_env but another objective binds

it. Treating both as if they should occupy the same optimum is the error the binding/slack framework is designed to avoid. 4. Preventing L from becoming unfalsifiable The loss term L is the main danger point in the framework. If L is adjusted freely after the fact, the model can explain any anomaly and becomes scientifically weak. Therefore L must not be a free fitted rescue parameter. It should be decomposed into pre-specified proxy classes: L_proxy = L_leak + L_slip + L_kinetic + L_membrane + L_homeostasis

systems without pooling. 7. The discriminating prediction The central empirical prediction is not that q_eff has a universal main effect. Ordinary bioenergetics already predicts that coupling ratio affects the ATP-synthesis threshold. The additional claim is conditional: q_eff should add the most predictive value when environmental demand is high or when pre-assigned evidence indicates boundary binding. The non-circular statistical form is: outcome ~ q_eff + D_env + q_eff x D_env + L_proxies + phylogeny + sequence + system_class When continuous D_env cannot be estimated, the categorical version is: outcome ~ q_eff + demand_class + q_eff x demand_class + L_proxies + phylogeny + sequence + system_class Suitable outcomes include ATP-synthesis threshold, growth under boundary conditions, measured coupling efficiency, or stoichiometry when performance data are unavailable. The interaction is the hypothesis. A simple main effect of q_eff is not enough. A negative result would be clear: q_eff fails to improve prediction even in high-demand or boundary-binding cases, or apparent improvement disappears once phylogeny and independent transitions are accounted for.

Page 8 Figure 4. Non-circular interaction prediction. q_eff is predicted to matter most when D_env is high. D_env is derived from environmental and cellular demand, not from the organism’s actual q_eff. 8. Retrospective model-comparison plan The retrospective analysis should compare a conservative null model against the binding/slack model. M0 includes phylogeny, sequence similarity or sequence complexity, enzyme family, and broad system class. M1 misleading when c-ring data are sparse and phylogenetically clustered. Table 5. Model comparison and failure modes Model Predictors Support pattern Weakening pattern M0: null phylogeny + sequence similarity/complexity + enzyme family + system class Baseline comparison only. If M0 explains as much as M1, the M0 + q_eff + D_env/demand_class + L proxies + interaction Improved prediction driven by the interaction, not by post-hoc L fitting. No interaction; L proxies dominate without measurement; coding Engineering test controlled q_eff alteration in same background Boundary shifts in predicted direction under low pmf. Altered q_eff fails to shift threshold when losses are controlled. Slack/trade-off

inherited architectures, tolerate drift, or trade thermodynamic efficiency for other system-level objectives. This binding/slack distinction is the contribution: it replaces both naive optimality and naive complexity arguments with a falsifiable phase-space claim.