Biological Time Equivalence in Vertebrates: Thermodynamic Framework, Comparative Tests, and Clade-Specific Deviations

Abstract

The product of resting heart rate and maximum lifespan is approximately constant across adult warm-blooded vertebrates, $N^\star = f_H L \approx 10^9$ cardiac cycles, a regularity documented since Rubner (1908) but lacking a thermodynamic derivation. We derive $N^\star$ from the non-equilibrium second law by treating the adult organism as a metabolic non-equilibrium steady state (NESS) and introducing the closure $\dot{e}_p = σ_0 f$, linking entropy production rate to heart rate via a mass-specific parameter $σ_0 \propto M^0$. Integration yields a finite dissipative budget $Σ= σ_0 N^\star$, identifying $N^\star = Σ/σ_0$ as the correct primitive conserved quantity; lifetime energy per unit mass is a derived consequence valid only under simultaneous constancy of body temperature and $σ_0$. Phylogenetically independent contrasts on 112 endotherm species yield a $\log f_H$--$\log L$ slope of $-0.99 \pm 0.04$ ($p=0.84$ against $-1$); the West--Brown--Enquist null of zero inter-clade variation is rejected ($F=12.7$, $p<0.001$). A factored multiplier $Φ_C = Φ_{\mathrm{duty}} \cdot Φ_{\mathrm{thermal}} \cdot Φ_{\mathrm{mito}} \cdot Φ_{\mathrm{haz}}$, calibrated from independently measured physiology, accounts for longevity deviations across four warm-blooded clades. The integral of physiological frequency defines a biological proper time classifying longevity mechanisms as time dilation (reduce $f$) or budget expansion (reduce $σ_0$). The decisive test is calorimetric measurement of $σ_0 = P/(TfM)$ across three body-mass decades.

Figures (15)

• Figure 1: Lifespan versus heart rate across 230 vertebrate species. Log-log scatter plot of maximum recorded lifespan $L$ (years) against effective resting heart rate $f_H^{\rm eff}$ (beats per minute) for the full comparative dataset of 230 vertebrate species spanning eight taxonomic groups. Non-primate placentals (filled grey circles, $n=46$), marsupials and monotremes (green crosses, $n=19$), primates (orange triangles, $n=18$), birds (blue squares, $n=78$), bats (dark green diamonds, $n=31$), cetaceans (open purple circles, $n=12$), Arrhenius-corrected reptiles (red crosses, $n=17$), and Arrhenius-corrected amphibians (orange crosses, $n=9$) are shown. For bats, $f_H^{\rm eff}$ is the duty-cycle-corrected time-average heart rate (Section \ref{['sec:duty']}); for cetaceans, $f_H^{\rm eff}$ is the dive-corrected average (Section \ref{['sec:cetacean']}); for ectotherms, $f_H^{\rm eff}$ is Arrhenius-corrected to $T_{\rm ref}=310$ K (Section \ref{['sec:arrhenius']}). The solid black line is the OLS regression fitted to the $n=43$ non-primate placentals with directly measured (non-imputed) heart rates, yielding slope $\hat{\beta}=-0.90\pm0.06$ (s.e.), $R^2=0.86$. The dashed black line is the PBTE null of slope $\beta=-1$, anchored at the non-primate placental mean $\bar{\ell}_0=8.995$. Grey dotted diagonal lines are iso-$\ell$ contours at $\ell=8$, $9$, and $10$, where $\ell=\log_{10}(f_H^{\rm eff}\cdot L\cdot 525{,}960)$. The systematic elevation of primates, birds, and bats above the mammalian OLS line, and the near-coincidence of cetaceans with the baseline before duty-cycle correction, are both predicted by the PBTE multiplier framework (Section \ref{['sec:clade']}).
• Figure 2: Distribution of the lifetime cycle count $\ell$ across clades. Box-and-whisker plots of $\ell = \log_{10}(f_H^{\rm eff}\times L\times 525{,}960)$ for all eight taxonomic groups in the 230-species dataset. For each group: the central line shows the median; box edges show the interquartile range; whiskers extend to $1.5\times$ the interquartile range; points beyond the whiskers are individual outliers. The horizontal dashed line marks the non-primate placental mean $\bar{\ell}=8.995$ ($n=46$), which serves as the thermodynamic baseline $N_0\approx10^9$. Numerical clade means are annotated in colour at the base of each box. The double-dagger ($\ddagger$) above the Primate column marks Homo sapiens ($\ell=9.65$), which lies beyond $1.5\times\rm IQR$ of the primate distribution owing to the combination of a high neural power fraction ($\varphi=0.20$) and a large modern-medicine-supported hazard correction. Clades elevated significantly above the baseline (Primates $\Delta\ell=+0.38$, Birds $+0.53$, Bats $+0.55$; all $p<0.001$, Welch $t$-test) are predicted by the PBTE multiplier framework from independently measured physiology. Cetaceans appear close to the baseline at $\bar{\ell}=8.80$ because their duty-cycle correction has already been applied to $f_H^{\rm eff}$; the raw observed count without correction would place large mysticetes well below the baseline (Section \ref{['sec:cetacean']}). Arrhenius-corrected reptiles ($\bar{\ell}=8.93$) and amphibians ($\bar{\ell}=8.82$) approach but do not reach the mammalian mean, with a residual gap of 0.07--0.17 dex attributable to unmodelled ectotherm-specific physiology (Section \ref{['sec:domain']}). [Data from Extended Data Tables 1--8; plotting script available from corresponding author.]
• Figure 3: Mammalian supertree schematic. Schematic of the pruned mammalian supertree bininda2007 showing the 112 endotherm species included in the primary phylogenetically independent contrasts (PIC) analysis, coloured by clade: non-primate placentals (grey); marsupials and monotremes (dark green); bats (green); primates (orange); birds (blue); cetaceans (purple). The tree was pruned from the Bininda-Emonds mammal supertree using the ape package in R 4.3 felsenstein1985bininda2007. Branch lengths are in millions of years. The topology shown is schematic for clarity; the full pruned topology with branch lengths is provided in Supplementary Data 1. The broad phylogenetic distribution of species across the tree confirms that the $f_H$--$L$ relationship tested by PIC regression (Extended Data Figure 1b) spans all major mammalian and avian lineages and is not driven by clustering within any single clade.
• Figure 4: Phylogenetically independent contrasts (PIC) regression. Scatter plot of PIC contrasts in $\log_{10}L$ against PIC contrasts in $\log_{10}f_H$ for 112 endotherm species (111 internal node contrasts plotted), computed by the Felsenstein felsenstein1985 method using the ape::pic() function in R 4.3 on the Bininda-Emonds supertree bininda2007. The OLS line is fitted through the origin, as required by the PIC method felsenstein1985, and has slope $-0.98\pm0.04$ (95% CI $[-1.07,-0.91]$), $R^2=0.96$, $F$-test $p=0.84$ against $\beta=-1$. Each point represents one phylogenetically independent contrast computed at an internal node of the tree; the tight linear clustering with $R^2=0.96$ demonstrates that the $f_H$--$L$ relationship is not a statistical artefact of shared phylogenetic ancestry. The PIC result---slope $-0.99\pm0.04$, $p=0.84$ against the PBTE null--- is the methodologically preferred test for the PBTE invariant because it accounts for the non-independence of species data; the OLS result on the 43-species non-primate placental subset ($p=0.09$) is a preliminary consistency check within this phylogenetically corrected framework. The near-unity slope confirms that the PBTE invariant is a genuine cross-species regularity and not an artefact of allometric body-mass scaling.
• Figure 5: Partial regression controlling for body mass. Scatter plot of residual $\log_{10}L$ (after regressing out $\log_{10}M$) against residual $\log_{10}f_H$ (after regressing out $\log_{10}M$), both computed from PIC contrasts to remove phylogenetic signal. The partial slope of $-0.95\pm0.05$ is statistically indistinguishable from $-1$ ($p = 0.32$), confirming that the $f_H$--$L$ relationship is not simply a consequence of the common allometric dependence of both variables on body mass. This analysis rules out the alternative hypothesis that the apparent $f_H$--$L$ association is a spurious by-product of the shared $M^{-1/4}$ and $M^{+1/4}$ scalings: even after removing all variance attributable to body mass, resting heart rate retains strong negative predictive power for maximum lifespan. The residual scatter ($R^2\approx0.90$ in the partial regression) confirms that the cardiac-longevity relationship carries substantial information beyond what is encoded in body size alone.

Theorems & Definitions (4)

• Corollary 1: Lifetime extension requires reduced entropy per cycle
• Corollary 2: The mammalian baseline
• Proposition 1: PBTE invariant
• proof : Consistency argument