Metabolic response attenuation

### PDF page 12 It claims only this: in the first-order regime, molecular-to-functional attenuation of the adaptive arm is the reciprocal of the summed response coefficient; inserting this η into the parent peak model makes the amplitude term measurable; and observed amplitude is further shaped by a separate bounded-observable factor that yields the parameter-free prediction P6. The premise that makes this non-trivial — that induction and control are distinct axes — is consistent with genome-wide data. 12. Discussion The advance is a clean separation of layers usually conflated: bounded adaptive composition gives the curve shape (parent work); response coefficients give the transmission of the adaptive arm (this note); and a separate bounded-observable factor compresses what is seen near a limit. This clarifies several puzzles. Two agents can both be “multi-target” yet produce very different peaks, because target number is a crude proxy for the summed response coefficient. A large molecular induction can produce a small functional peak if the induced pathway has little control over the measured phenotype — exactly the genome-scale observation that most stress-induced genes are dispensable for surviving that stress. A strong network response can still produce a small measured peak near a boundary. The connection to functional genomics is the substantive empirical content. The long-standing observation that stress-induced genes are largely not the genes required to survive that stress has been treated as a puzzle about the purpose of stress-activated transcription. Within this framework it is the expected macroscopic signature of the induction–control separation that response-coefficient attenuation requires: induction is an elasticity-like axis, survival- requirement is a control-like axis, and the adaptive-arm amplitude follows control-weighted induction. The framework thus links two literatures — metabolic control analysis and stress functional genomics — that have not previously been connected, and does so in a way testable before any curve is fitted. 13. Conclusion The bounded-adaptive-systems framework explains why hormesis has its biphasic shape. This note explains why the adaptive arm of the peak differs in size: response-coefficient attenuation, η = (ΣᵢCᵢεᵢ)⁻¹, inserted into the parent peak model so that a heuristic constant becomes a measurable quantity. The single-chain heuristic η ≈ n is recovered for one linear pathway; the multi-site heuristic η ≈ n/m is recovered only under co-directional, comparable-control, comparable-elasticity conditions. The general result is stronger because it names the conditions under which multi-target agents produce large functional peaks, and the conditions under which they do not. Its central premise — that molecular induction and phenotypic control are distinct axes — is consistent with genome-wide functional-genomics data. The next empirical steps are direct: test the parameter-free baseline-scaling prediction against existing baseline-stratified data, and measure ΣCε in the same system to test whether it predicts adaptive-arm amplitude.

Response-Coefficient Attenuation Predicts Hormetic Peak Amplitude: A Metabolic-Control Extension of Bounded Adaptive Systems A theoretical note, with genome-scale support for the induction–control separation premise Daniel John Murray that induction and phenotypic control are distinct axes. Conclusion. Hormetic amplitude is a measurable control-transmission quantity, not a descriptive feature. The specific form is offered as a falsifiable prediction, testable by same- system measurement of control coefficients and elasticities, and by a conditional baseline- compression test. Keywords: hormesis; metabolic control analysis; response coefficient; control coefficient; elasticity; bounded adaptive systems; dose-response; attenuation; environmental stress

a multi-target amplitude advantage exists over a matched comparator when the effector's summed response coefficient exceeds the comparator's (Section 5). That is stricter and more falsifiable than the heuristic it replaces. 2. Relationship to the bounded-adaptive hormesis derivation This note does not replace the prior derivation and does not repeat its proof; its role is complementary. The parent bounded-adaptive hormesis paper (Murray, in press, Dose-Response) establishes attenuation factor η, that paper treated the single linear-pathway case (η ≈ n) as rigorous via the summation theorem of metabolic control analysis, and explicitly labelled the multi-target case (η < n) as a plausible heuristic, offered as a testable hypothesis rather than a derivation. Its leading-order peak expression has the schematic form below, in which η enters only through the functional adaptive amplitude A_func = A_mol/η: peak amplitude ≈ 1 + A_mol/η (with peak located near 2×D_a; zone width set by D_t/D_a). The present note supplies the missing derivation of η and, in doing so, replaces that heuristic

theorem supplies the output. It inherits the assumptions of metabolic control analysis. It is named to make clear that the empirical content lies not in the identity itself but in whether ΣCε varies across systems in ways that predict amplitude (Sections 8 and 10). 4.1 Derivation, with explicit normalisation For a single targeted site, the response coefficient theorem of MCA gives the fractional sensitivity of Y to m as the product of the site’s control coefficient over Y and its elasticity to m, R = Cᵢεᵢ (Kacser & Burns 1973). For an effector acting at several sites, the first-order total

The factor (1 − x₀²) is the derivative of the tanh chart x = tanh(ω); the bounded coordinate is a reparameterization, not a biological dynamics, so this factor is a property of that chart rather than a mechanistic claim, and prediction P6 below is therefore a conditional one: it holds if the bounded endpoint follows the tanh chart, and its failure would reject that mapping rather than the response-coefficient result. Stated this way it remains falsifiable, while making explicit that it tests the coordinate description and not a hyperbolic biological dynamics. To avoid the inconsistency of folding this factor into η, two distinct quantities are defined: the network attenuation η = 1/R (independent of baseline), and the observed amplitude, which additionally carries the bounded-observable factor (1 − x₀²). Comparisons of observed peaks across systems must therefore be adjusted for baseline x₀ before being attributed to network differences. This separation yields a parameter-free prediction (Section 10, P6). Equivalently, ΣᵢCᵢεᵢ can be read as the projection of the effector's induced perturbation onto the phenotype's control architecture. This is an interpretive aid only and adds no measurement beyond ΣCε; the inner-product reading is not used quantitatively, because the angle between control and elasticity is not a metric-invariant of the site space, whereas the scalar sum is. The

bounded-observable factor of Section 5.4. 7. Pharmacological consequence: target count is not target quality A direct pharmacological consequence follows, and it is conditionally predictive rather than prescriptive: the framework does not predict that multi-target agents are intrinsically superior, but that adaptive benefit scales with positive control-weighted induction. A drug or stressor that strongly induces many molecular markers may produce little functional benefit if those markers have low control over the measured phenotype; conversely, a narrower intervention may produce a larger hormetic peak if it acts on fewer but higher-control, higher-elasticity targets.

protective role against future stress — molecularly, acquired stress resistance — using a mild-pretreatment-then-severe-challenge (preconditioning, i.e. hormetic) design. The context- dependence predicted by Corollary 5.3 is also observed: the genes required to acquire H₂O₂ tolerance differ by mild pretreatment (Berry et al. 2011). The growth-rate/stress-tolerance trade- off — that inducing the defensive program is costly and traded against growth — has been confirmed in chemostat culture (Zakrzewska et al. 2011), matching the bounded, costly adaptive reserve the framework assumes. discriminate it from other control-weighted aggregates (non-linear combinations, threshold functions, multiplicative forms). I therefore treat Tier 1 as consistent with the data, and offer Tier 2 as the sharper, falsifiable prediction the premise points toward — in the same spirit that the parent paper offered its peak-location law as a prediction rather than a completed validation. The decisive test is the same-system measurement of Section 10. 9. Scope: static versus dynamic response coefficients The identity is a local, near-steady-state result. Hormesis experiments differ sharply in exposure timescale, and acute, intermittent, and chronic exposures do not interrogate the same control should be read as a leading-order estimate rather than an exact value. For chronic exposures, the structural expectation is that a slow controller re-centres the operating point, compressing or smoothing the peak; the operational consequence is concrete and is retained as a prediction (Section 10, P5): amplitude comparisons must stratify by exposure timescale, or η estimates will be confounded. I state this as a scope limitation, not a solved case. 10. Falsifiable predictions and proposed tests 10.1 Predictions Page 8 of 13 Dose Response

P1 — Response-coefficient stratification. Curves with larger positive ΣCε should show larger adaptive-arm amplitude than curves with smaller or mixed-sign ΣCε, after adjusting for baseline (P6) and timescale (P5). Falsified if high-ΣCε curves do not exceed low-ΣCε curves in matched systems. P2 — Elasticity-conditional multi-target advantage. Multi-target effectors should exceed a matched single-target comparator only when R_multi > R_single, i.e. when target elasticities are appreciable and co-directional. Falsified if target count predicts amplitude even when measured elasticities are near zero or mixed-sign. P3 — Mixed-sign cancellation. Agents acting on both adaptive and anti-adaptive targets should show reduced, broadened, or unstable peaks relative to co-directional agents. Falsified if mixed-sign architecture is indistinguishable from co-directional architecture in matched systems. P4 — Control redistribution at shared bottlenecks. Where putative parallel targets converge on a shared saturated step or pool, the measured full-network control coefficients should be small, redistributed, or sign-changed relative to naive per-site estimates. Falsified if independently measured Cᵢ match naive additive estimates despite a demonstrated shared bottleneck. P5 — Acute–chronic split. For matched architecture, the adaptive-arm amplitude should be larger in acute exposures that preserve the transient and smaller in chronic exposures that permit controller re-centring. To be falsifiable this must be made quantitative per system: specify whether the contrast is peak height at fixed dose, area under the stimulatory region, or peak-to- baseline ratio, and pre-register the metric. Falsified if the pre-registered metric does not differ by at least a system-justified minimal effect (for example, chronic amplitude ≤ 0.8× acute amplitude at matched dose) across timescales with matched architecture. P6 — Baseline-scaling prediction (conditional, parameter-free). This is the one prediction that requires no measurement of C or ε. Holding the effector, dose, and network fixed and varying only the baseline x₀, the observed adaptive amplitude should scale as (1 − x₀²). For two baseline states, the ratio of observed amplitudes should equal (1 − x₀,₁²)/(1 − x₀,₂²), with no free parameters. This is directly testable by re-analysing existing data stratified by baseline — for example the same hormetic agent and dose applied to young versus aged, or unstressed versus mildly preconditioned, systems. Falsified if observed amplitude does not scale with (1 − x₀²) after controlling for molecular induction and network class. This prediction is parameter-free only if the network response coefficient ΣCε and the molecular induction are held approximately fixed as baseline varies; it is therefore most cleanly testable where baseline is changed by an external parameter that does not rewire the network (for example mild temperature shifts without adaptation, or different initial densities), and a negative result where the baseline shift itself alters the Cᵢ or εᵢ (as ageing may) would not bear on the bounded-observable factor. Because it is a coordinate-level prediction and not a claim about hyperbolic dynamics, it is the cleanest available test of the bounded-coordinate layer. 10.2 Prospective same-system assay (the decisive test of the form) Page 9 of 13 Dose Response

2. Estimate control coefficients Cᵢ by independent perturbation of each target site while measuring phenotype Y. 3. Compute predicted attenuation η_pred = (ΣᵢCᵢεᵢ)⁻¹. 4. Measure the adaptive-arm amplitude. 5. Test whether predicted η explains observed amplitude better than target count, pathway label, or fitted curve parameters. 10.3 Operationalisation and proxies Full measurement of Cᵢ and εᵢ for every target of a stressor is a large undertaking, and most laboratories will rely on proxies. As a concrete worked template: for a heat-shock response in 10.4 Retrospective database test (with a blinding safeguard) A faster, weaker test classifies existing hormetic curves by response-coefficient class and asks whether that class predicts amplitude. This is valid only if the architectural classification is made without access to the amplitude being predicted; otherwise it is circular. Classification must therefore follow a pre-registered, amplitude-blind rule based solely on target architecture, control, and elasticity information. The minimal model is A_peak ~ α + b ₁·(estimated ΣCε) + b₂·(1 − x₀²) + b₃·timescale + b₄·mechanism class + study random effect, with the primary, pre-registered prediction b₁ > 0 and the secondary prediction b₂ > 0 (P6). 11. What this note does not claim This note does not claim that all multi-target agents are superior medicines, nor that any substance class is therapeutically privileged by geometry alone. It does not claim that η = n/m universally; that is one special case. It does not claim validity for arbitrarily large perturbations functional form — they support the premise (Tier 1) and are consistent with the form (Tier 2). It does not claim a hyperbolic or relativistic dynamics for hormesis; the bounded coordinate is a reparameterization whose only empirical bite is the baseline-scaling prediction P6. Page 10 of 13 Dose Response

arm is the reciprocal of the summed response coefficient; inserting this η into the parent peak model makes the amplitude term measurable; and observed amplitude is further shaped by a separate bounded-observable factor that yields the parameter-free prediction P6. The premise that makes this non-trivial — that induction and control are distinct axes — is consistent with genome-wide data. 12. Discussion The advance is a clean separation of layers usually conflated: bounded adaptive composition they do not. Its central premise — that molecular induction and phenotypic control are distinct axes — is consistent with genome-wide functional-genomics data. The next empirical steps are direct: test the parameter-free baseline-scaling prediction against existing baseline-stratified data, and measure ΣCε in the same system to test whether it predicts adaptive-arm amplitude. Statements and Declarations Page 11 of 13 Dose Response