Slowed Gompertzian ageing in long-lived C. elegans results from expansion of decrepitude, not decelerated ageing

Domain: biology
Published: 2026-04-13
Editorial outcome: minor_revision

Abstract
In populations of many animal species, including humans, mortality rates increase exponentially with advancing age. The scale and rate of increase can be set by two parameters, α and β, respectively, of the Gompertz equation. Interventions that extend lifespan can reduce either or both parameters. A long-standing supposition is that β corresponds to biological ageing rate, and α to ageing-independent causes of mortality. Here, we investigate the biological basis of α and β using the nematode Caenorhabditis elegans, through the combined study in populations and individuals of effects of life-extending interventions on mortality and age-changes in health. We demonstrate that reductions in β arise not from slowed biological ageing, but rather from expansion of decrepitude (gerospan) in longer-lived population members. In contrast, reductions in α better reflect healthspan expansion, an indicator of slowed biological ageing. Thus, our investigation presents a new, empirical understanding of the Gompertz parameters that inverts their traditional interpretations.

Introduction
Ageing (senescence) is the main cause of disease and death in the world today1. Studies of the biology of ageing often employ short-lived animal models such as the nematode C. elegans. Though biological ageing affects individuals, researchers investigating it frequently use demographic (population) ageing as a metric2,3. However, it is difficult to link demographic metrics (e.g. changes in mean lifespan) to determinative biological mechanisms.

This is true of a salient and enigmatic feature of demographic ageing: the exponential increase in all-cause mortality rate during adulthood, that is common (though not universal) among animal species4,5. This was described in human populations two centuries ago by Benjamin Gompertz, and his eponymous equation6,7

(1) 𝜇(𝑥) = 𝛼𝑒𝛽𝑥

Here, mortality rate μ(x) at age x is a function of a scale parameter α and rate parameter β, with the latter specifying the mortality rate acceleration with age. Accordingly, α and β together determine survival, and mortality rate and frequency functions (Fig. 1a).

But to what do these demographic parameters correspond in terms of biological ageing, within individuals? Because α quantifies the initial mortality rate prior to ageing (x=0, usually defined as in early adulthood), it is widely interpreted as relating to ageing-independent mortality, arising e.g. from extrinsic hazards or intrinsic vulnerabilities8,9. By contrast, β describes the rate of mortality rate increase, and because mortality rate increase has been traditionally defined as ageing, β has thus become widely interpreted as capturing the rate of ageing. Here, equivalence between demographic ageing (mortality rate increase) and biological ageing (rate of ageing) is assumed. However, these standard biological interpretations of α and β have been challenged10-17, and to our knowledge have yet to be empirically validated by direct comparison of demographic and biological ageing. While the relationship between lifespan and late-life health has been investigated previously, including in C. elegans18-27, the specific question of the biological basis of the Gompertz parameters has not been directly and empirically examined.

Here we explore the biological meaning of α and β in terms of age changes in health at the individual level. To this end, we simultaneously quantified biological and demographic ageing by means of longitudinal analysis of individual C. elegans subjected to life-extending interventions. Specifically, we investigated: (1) how life-extension affects the duration of healthy versus decrepit life, (2) whether such changes in health/decrepitude correlate with changes in the Gompertz parameters, (3) whether these correlations can be explained causally, and (4) whether the Gompertz parameters can be causally related to specific ageing-related pathologies. We will present our results in the order of these aims. In brief, we find that β is a measure of inter-individual variability in gerospan (late-life decrepitude) rather than biological ageing rate, and α of biological ageing rate rather than ageing-independent processes.

Results
We examined the effects on biological and demographic ageing of three types of intervention that increase C. elegans lifespan: (i) reduction of ambient temperature from 25˚C to 20˚C and 15˚C, which has both thermodynamic and gene-mediated effects upon nematode ageing2,28,29, (ii) prevention of late-life infection (by dietary E. coli) using antibiotics30, and (iii) reduction-of-function (rf) mutation of the daf-2 insulin/IGF-1 receptor3,31, specifically the class 1 alleles daf-2(m577) and daf-2(e1368), and the class 2 (more pleiotropic) allele daf-2(e1370).

Lifespan was measured for all 24 combinations of conditions (3 temperatures, ± carbenicillin, 4 genotypes) in 6 successive trials. To relate the Gompertz parameters to individual-level biological ageing features, in 4/6 trials, all animals were tracked individually and their locomotory health assessed every 2–3 days until death. Adapting an established protocol32,33, animals were scored as either youthful (healthy, sinusoidal locomotion) or decrepit (non-sinusoidal/uncoordinated locomotion or immotility) following controlled physical stimulation with a platinum wire; this induces an escape response that reveals true locomotory capacity rather than behavioural locomotory preference19. We will refer to absolute and relative healthspan (H-spanabs, H-spanrel) and gerospan (G-spanabs, G-spanrel) to describe, respectively, the total number of days or proportion of life spent in youthfulness or decrepitude. In other words, for every individual, H-spanabs + G-spanabs = lifespan, and H-spanrel + G-spanrel = 1. Necropsies 23 were also performed to determine the mode of death of animals in these 4 trials.

Unless indicated otherwise, we will characterise the relationship between demographic and biological ageing using the combined pool of these 4 trials described above (n=116–179/cohort), as they collected both lifespan and health data. Mean lifespan and the Gompertz parameters (calculated by maximum likelihood estimation34) were stable between these 4 trials (Supplementary Fig. 1a, Supplementary Table 1), and the Gompertz model provided an excellent fit to their combined pooling, predicting mean, median, and first and third quartile lifespans across the 24 cohorts with R2=0.98–1.00 (Supplementary Fig. 2a–b, e–h). To confirm that sample sizes are sufficient for model fitting, and that these Gompertz parameters are representative of these cohorts, we ran two additional larger trials (measuring lifespan only), increasing total pooled sample sizes by 2–3-fold to 326–396/cohort (Supplementary Table 1). Notably, the Gompertz fit still predicted the above lifespan measures, with R2=0.97–0.99 (Supplementary Fig. 2c–d, e–h), and the Gompertz parameters were near-identical to those from the pool of the 4 smaller trials (Supplementary Fig. 1b). Anderson-Darling goodness-of-fit tests also supported these Gompertzian fits in most cohorts (21–22/24) despite localised deviations within some (Supplementary Table 2).

These data are therefore consistent between trials and robustly fitted by the Gompertz model. Although other mortality models can better fit many C. elegans populations20,35-37, our aim is not to find the very-best-fitting model, but to understand the biological basis of the mathematically parsimonious and widely employed Gompertz model.

i) Three life-extending intervention classes disproportionately extend gerospan

From the 24 cohorts, we focused on a total of 46 treatments of interest: 16 lower temperature treatments (20˚C and 15˚C, vs 25˚C, across antibiotic and genotype backgrounds) + 12 antibiotic treatments (carbenicillin vs no carbenicillin, across temperature and genotype backgrounds) + 18 daf-2(rf) treatments (3 daf-2(rf) alleles vs the N2 wild-type, across temperature and antibiotic backgrounds). As expected2,3,30, almost all (44/46) treatments increased mean lifespan (average class effects for lower temperature, antibiotic and daf-2(rf) treatments: +42.0%, +58.9%, +77.1%, respectively; Supplementary Fig. 3, Supplementary Table 1).

To begin relating lifespan to the biological ageing process, we asked if these life-extending treatments affect G-spanrel (proportion of life in gerospan). All (12/12) carbenicillin treatments increased G-spanrel (Fig. 1c), in line with a prior report of antibiotic-mediated morbidity expansion38. Somewhat against expectation, 12/16 lower temperature treatments also increased G-spanrel (Fig. 1b). 9/18 daf-2(rf) treatments also increased G-spanrel while only 2/18 decreased it (Fig. 1d). This is broadly consistent with previous reports of morbidity expansion in daf-2(rf) mutants16,18,38, including the finding that this expansion results from greater resistance to life-limiting infection (by the E. coli food source) than wild-type animals26,38. Indeed, at the standard culture temperature of 20˚C, daf-2(e1370) increased G-spanrel, and this increase was largely suppressed by carbenicillin (Fig. 1d). However, among other daf-2(rf) allele/temperature combinations, G-spanrel patterns varied markedly, revealing more complex allelic and environmental effects of daf-2(rf) on the relationship between health and lifespan. This condition-dependency may have contributed to disagreements in the literature about daf-2(rf) effects on relative morbidity19,26.

Overall, the longevity interventions tested here mostly increase the proportion of life in poor (aged) health. Indeed, direct regression of mean G-spanrel against mean lifespan for the 24 cohorts revealed a strong positive relationship (Fig. 1e). While gerospan here is necessarily simplified to a single portion of life, its definition (non-sinusoidal locomotion or immotility, despite stimulation) captures well-characterised stages in nematode ageing and is a physiologically integrative measure of overall health32,33. Accordingly, the age of this gerospan onset provides one measure of the overall rate of change in health, or equivalently, a simple estimate of an individual’s overall rate of ageing. Notably, the G-spanrel increases observed across these life-extending treatments argue against a simple deceleration of ageing, which would be expected to stretch H-spanabs and G-spanabs proportionally, leaving G-spanrel unchanged.

ii) Correspondence between reduced β and increased relative gerospan

Next, we considered the central question of this study: the relationship between biological and demographic ageing. To this end, we examined the relationship between effects on G-spanrel and the Gompertz parameters by the 46 life-extending treatments; we start with the rate parameter β, reduction of which is typically equated with decelerated biological ageing.

First, we considered effects of reducing temperature. On carbenicillin, lower temperature decreased β in all genotypes (Fig. 1f), which would be consistent with a general slowing of the rate of living. However, without carbenicillin, β was unchanged in wild-type and even increased in daf-2 mutants, suggesting complex effects of genotype and temperature on host-E. coli interactions. Surprisingly, β reduction by lowered temperature (on carbenicillin) consistently co-occurred with G-spanrel increase (Fig. 1f, right), suggesting that β reduction here reflects neither simply slowed ageing rate nor improved overall quality of life.

We similarly evaluated effects of the antibiotic treatments. Against the expectation that preventing infection (a largely extrinsic insult) might reduce α alone, carbenicillin decreased β in all genotypes at 15˚C, and in wild-type and daf-2(e1370) at 20˚C (Fig. 1g). In each case, G-spanrel was again increased. In the remaining carbenicillin treatments, both β and G-spanrel were increased, suggesting that β reduction may require G-spanrel increase, but not vice versa.

Finally, we assessed the daf-2(rf) mutations. daf-2(e1370), the severest (and longest-lived) allele, also lowered β while increasing G-spanrel, both off and on carbenicillin, at 15˚C and 20˚C (Fig. 1h). At 25˚C, β was decreased without change in G-spanrel, as an exception to the pattern; similarly, only 3/7 class 1 daf-2(rf) treatments (daf-2(m577) and daf-2(e1368)) reducing β increased G-spanrel. Despite these exceptions, 21/27 (78%) of all treatments that significantly reduced β simultaneously increased G-spanrel. Thus, amongst these life-extending interventions, reduction of β largely reflects expanded decrepitude rather than a simple deceleration of biological ageing. Of course, this does not preclude occurrence of more complex forms of ageing deceleration, as demonstrated in Results section (iv).

iii) Reduced β reflects inter-individually variable gerospan expansion

A notable feature of the demography of ageing is the high degree of inter-individual variability in lifespan. This is true of C. elegans, despite their being isogenic and maintained under identical conditions43. An often-overlooked mathematical property of the Gompertz rate parameter β is its inverse relation to lifespan variation, which in comparison is only marginally affected by the α parameter42 (Fig. 1a, fourth panel). Indeed, amongst our 24 treatments, β showed a strong inverse correlation with lifespan standard deviation (Fig. 2a).

Given this, we wondered whether this increased lifespan variation in small-β cohorts could result from increased inter-individual variation in the ageing process. Interestingly, it was previously estimated that G-span differences account for most (~67%) inter-individual lifespan variation in C. elegans21. Thus, the correlation between reduced β and increased G-spanrel in our cohorts may arise from inter-individually variable G-span expansion. To investigate this, we assessed H-spanabs and G-spanabs for all individuals in the 21 treatments displaying an inverse β–G-spanrel relationship, as explained next.

To visualise the variation in H-spanabs and G-spanabs in each cohort population, we regressed individuals’ H-spanabs and G-spanabs values over their survival proportion (Fig. 2b–g, top panels in each); here, the x-axis orders individuals by survival order (left: shorter-lived individuals, right: longer-lived individuals), therefore allowing H-span and G-span of individuals from different cohorts to be compared, akin to the y-axis of survival curves which allow comparison of lifespans between cohorts for a given survival proportion (e.g. 0.5 for median lifespan).

We first used these plots to assess the 8 reduced temperature treatments that decreased β and increased G-spanrel (Fig. 2b). Most (5/8: 20˚C & 15˚C in wild-type and daf-2(m577), and 20˚C in daf-2(e1370)) increased G-spanabs disproportionately more than H-spanabs in longer-lived population members (Fig. 2b top, Supplementary Fig. 4a to p and middle). As a result, G-spanrel was increased in these longer-lived individuals (Fig. 2b bottom, Supplementary Fig. 4a bottom). This shows that life-extending interventions can act by introducing or amplifying the extended “twilight”21 (greater G-spanrel) of longer-lived individuals within a population, which we will refer to as extended twilight longevity (ETL). A critical implication of this ETL is that the β reduction and associated extension of the survival curve tail arises from the disproportionate expansion of decrepitude in longer-lived population members.

Similarly, in all 6 carbenicillin treatments yielding the inverse β–G-spanrel relationship, G-spanabs again increased disproportionately more than H-spanabs in longer-lived individuals, thus increasing their G-spanrel (Fig. 2c–d, Supplementary Fig. 4b). Therefore, carbenicillin too is an ETL treatment that lowers β via variable G-span expansion. Finally, 4/7 of the daf-2(rf) treatments that decreased β and increased G-spanrel also did so through ETL (Fig. 2e–g, Supplementary Fig. 4c). These 4 treatments were daf-2(e1370) at 20˚C ± carbenicillin and at 15˚C without carbenicillin, and daf-2(m577) at 25˚C with carbenicillin.

Therefore, β reduction reflected ETL in 5/8 reduced temperature, 6/6 antibiotic, and 4/7 daf-2(rf) treatments (15/21 or 71%, in total) where β reduction co-occurred with G-spanrel increase. In the 6 non-ETL exceptions (Fig. 2b, e–g), G-spanabs still increased mostly in longer-lived population members (i.e. reducing β), in particular 15˚C treatment in daf-2(e1368) and daf-2(e1370) (Fig. 2b) and daf-2(e1370) treatment at 15˚C on carbenicillin (Fig. 2e). However, this was also largely true of H-spanabs; thus, in this minority of treatments, β reduction reflects inter-individually variable expansion of both G-span and H-span.

As a further test for whether β reduction arises from ETL, we compared contributions of H-spanabs and G-spanabs changes to β reduction, and to lifespan variation (standard deviation) increase (i.e. β reduction), in the 15 ETL treatments. Indeed, G-spanabs changes decreased β more than H-spanabs changes in 13/15 treatments (compared to 0/6 non-ETL exceptions) (Supplementary Table 3), and increased lifespan variation more in 14/15 interventions (compared to 3/6 non-ETL exceptions) (Supplementary Table 4). Consistent with these findings, across the 24 cohorts G-span variation increased more rapidly with mean G-span than H-span variation with mean H-span, while mean length and variability of G-span had wider ranges than that of H-span (Fig. 2h). This shows that across these cohorts, inter-individually variable expansion of G-span rather than H-span (i.e. ETL), is the primary mode of life extension.

These findings argue that expanded decrepitude not only correlates with a lower β, but causes it. That is, gerospan increase in longer-lived population members further increases their lifespan, thus extending the survival curve tail in the manner characteristic of a smaller β value (Fig. 1a, third panel). Therefore, β here is not a measure of intra-individual, biological ageing rate as commonly understood, but rather of inter-individual heterogeneity in late-life decrepitude.

iv) Reduced α reflects slowed biological ageing rate

Because healthspan expansion may be considered a reliable indicator of slowed biological ageing (given more time to G-span onset), we wondered if β reduction could at least indirectly reflect decelerated ageing, should G-spanabs and H-spanabs expansions occur together. Indeed, mean H-spanabs was increased in 9/15 ETL treatments (Fig. 3a–c). We therefore examined the effect of population-wide H-spanabs expansion on the Gompertz parameters in these 9 treatments. H-spanabs expansion significantly decreased β in 4/9 treatments but, strikingly, significantly decreased α in 8/9 (Fig. 3d, Supplementary Table 5a–c). This suggests that α, though not traditionally equated with biological ageing, may here be a better measure of it than β.

To assess if this unexpected finding is idiosyncratic to ETL, we also examined the non-ETL treatments, of which 27/31 significantly extended H-spanabs (Fig. 3a–c). However, again, H-spanabs expansion decreased α in 22/27 (81%) of these non-ETL treatments, but decreased β in only 13/27 (48%) (Fig. 3d, Supplementary Tables 5a–c). Additionally, H-spanabs expansion increased β in 9/27 treatments, including in all carbenicillin treatments; thus, slowing biological ageing can even increase β. Considering all treatments (ETL plus non-ETL), H-spanabs expansion decreased α in 30/36 (83%) cases, decreased β in 16/36 (44%), and increased β in 12/36 (33%). This suggests that α should better predict H-spanabs than β amongst the 24 cohorts, and this indeed proved to be the case (Fig. 3e).

In summary, healthspan expansion, a plausible metric of slowed biological ageing, more consistently reduces α than β, and even increases β. Therefore, overall, our empirically-based findings invert traditional interpretations of the biological meaning of the two Gompertz parameters.

v) α and β describe inter-individual heterogeneity in age-related infection

To further understand the biological mechanisms underpinning the Gompertz parameters, we performed necropsies on all corpses, building upon established mortality deconvolution methodology23. We scored for ageing-related bacterial colonisation of the pharynx and intestine (Fig. 4a, Supplementary Fig. 5a–b; in non-carbenicillin cohorts). We noted that almost all corpses with a swollen, infected pharynx (P or “big P”, as opposed to p or “small p” with an atrophied, uninfected pharynx)23 also had intestinal colonisation by E. coli, but not vice versa. Thus, P corpses are largely a subset of those with intestinal colonisation (Supplementary Fig. 5c). Death type was accordingly used to define three biologically-distinct subpopulations: P, pIC (p with intestinal colonisation), and pnIC (p with no intestinal colonisation).

Consistent with prior findings23, lifespan was shorter for P than p (pIC and pnIC) subpopulations for most treatments (Fig. 4b, Supplementary Table 6). Lifespan was also shorter for pIC than pnIC populations (at 15˚C and 25˚C, but not 20˚C). Thus, the three subpopulations exhibit distinct ageing trajectories.

We then asked how this inter-subpopulation heterogeneity affects β. Notably, β was consistently smaller in whole populations than in subpopulations (Supplementary Table 7). This provides empirical evidence of how β can underestimate demographic ageing rate in the presence of hidden subpopulation heterogeneity11. Moreover, the traditional interpretation of β would paradoxically predict that subpopulations age biologically faster than their combination (whole population). This implies that even for isogenic individuals in identical conditions, β can be a measure of inter-individual heterogeneity in ageing, rather than of biological ageing rate itself.

We next examined how life-extending treatments (here low temperature and daf-2(rf), without carbenicillin) affect subpopulation prevalence and lifespan. While both treatment classes increased lifespan of most subpopulations, increases were generally greater for P by low temperature, and pIC by daf-2(rf) (Fig. 4c–d; Supplementary Table 6). Lower temperature modestly decreased P prevalence in all genotypes except daf-2(e1370), suggesting reduced bacterial pathogenicity and/or enhanced host immunity, and allele-specific temperature sensitivity in daf-2(e1370)44 (Fig. 4e). Interestingly, however, low temperature strongly increased pIC prevalence (and thereby decreasing pnIC prevalence). Thus, reducing temperature acts antagonistically on terminal infection, respectively decreasing and increasing pharyngeal and intestinal colonisation by pathogenic E. coli. Meanwhile, daf-2(rf) treatment also decreased P prevalence as previously seen44 and affected pIC prevalence in an allele- and temperature-specific manner (Fig. 4f), largely consistent with known daf-2(rf) infection resistance38.

Finally, we determined the relative contribution of these changes in subpopulation lifespan and prevalence to the Gompertz parameters. Effects of low temperature on β proved to be determined mainly by changes in pnIC lifespan, followed closely by the increase in pIC to pnIC ratio in daf-2(m577) (Fig. 5a, Supplementary Fig. 6a). In daf-2(rf) treatment, changes in β were again primarily determined by changes in pnIC and/or pIC lifespan, rather than in P lifespan or subpopulation prevalence (Fig. 5b, Supplementary Fig. 6b). Thus, β is largely a function of p lifespan; specifically, increases and decreases in β typically result, respectively, from decreased and increased p (pIC and/or pnIC) longevity. This further demonstrates that β is less a measure of biological ageing rate than inter-individual heterogeneity, here as subpopulation-specific responses to longevity interventions.

We similarly deconvolved determinants of α. In low temperature treatment, determinants of α were genotype-specific: α was wholly determined by P lifespan in wild-type, mainly by pnIC lifespan in daf-2(m577) and daf-2(e1370), and by all determinants except pIC lifespan in daf-2(e1368) (Fig. 5c, Supplementary Fig. 6a). In daf-2(rf) treatment, α was primarily determined by P lifespan at 25˚C and 20˚C, while at 15˚C, several determinants were important; in common, the increase in pnIC to P ratio (Fig. 5d, Supplementary Fig. 6b). Therefore, unlike β, α is largely a function of P lifespan and subpopulation prevalence, but like β, reflects subpopulation-specific responses to life-extending interventions.

Taken together, these findings provide further evidence that being demographic measures, the Gompertz parameters can often reflect inter-individually variable biological changes whose complexity is unlikely to be captured by the traditional, theoretical interpretations of the parameters.

Methods
C. elegans culture and strains

C. elegans were maintained at 20˚C using standard protocols54, on Nematode Growth Medium (NGM) plates seeded 2 days before use with a bacterial food source (Escherichia coli OP50). Floxuridine (5-fluoro-2-deoxyuridine), sometimes used to block progeny production, was not used in this study.

Nematode strains used were: N2 (wild-type, hermaphrodite stock55), GA1959 daf-2(m577) III, GA1960 daf-2(e1368) III, and GA1928 daf-2(e1370) III. All strains were raised from egg at 20˚C on live E. coli, and transferred at L4 stage to the appropriate experimental conditions (15˚C, 20˚C or 25˚C; with or without carbenicillin). Carbenicillin solution was added topically to plates one day before adding animals (further details below).

Lifespan-only trials (Trials 1-2)

Nematodes were cultured throughout life in 60 mm Petri dishes (containing 10 mL of NGM) seeded with approximately 80 μL of E. coli, and where relevant, treated with 80 μL of 500 mM carbenicillin (Fisher Scientific Ltd, catalogue no. 12737149). At L4 stage (time 0 in all analyses), 30–40 animals were placed on each plate, with three plates per condition. Animals were transferred every two days during the reproductive period, and approximately every seven days thereafter. Scoring of survival was performed every two days. Animals showing no movement were gently touched with a platinum wire (worm pick) on the head and/or tail; those that showed no movement at all in response were scored as dead. Animals that died due to desiccation on the Petri dish wall, internal hatching of larvae, or rupture of internal tissues through the vulva, or that became contaminated by non-E. coli bacteria or fungi, or could not be found, were censored.

Lifespan and healthspan trials (Trials 3-6)

Prior to the end of egg laying, nematodes were handled as for the lifespan-only trials, but with two plates containing 25 animals each, per condition. Following the end of egg laying, animals were transferred to individual wells of 24-well tissue culture plates, containing 2 mL of NGM and seeded with 3.5 μL of E. coli OP50, and where relevant, treated with 16 μL of 500 mM carbenicillin. Animals were subsequently transferred to fresh plates monthly, before media desiccation (plates were sealed with parafilm to delay desiccation, and to prevent bacterial/fungal contamination). Scoring of survival and censors were performed as above every 2–3 days, alongside scoring of locomotory class, and necropsy at death.

Quantification of locomotory decline with age

Locomotory health class (belonging to H-span or G-span) was scored by classifying individuals into one of three classes, adapted from earlier systems32,33: A – sinusoidal locomotion; B – non-sinusoidal locomotion; C – no locomotion. To accurately determine locomotory class, animals were gently touched on the tail with a platinum wire worm pick for up to 20 seconds to induce an escape response that reveals movement capacity, and additionally on the head as a final check. The duration spent in A class was defined as H-span, and the summed duration spent in B and C classes as G-span. Here, B and C classes were summed to improve data tractability, and to provide a definition of G-span that captures both early and late-stage functional declines.

Necropsy analysis

Necropsy to define patterns of E. coli-associated pathology was performed by examining fresh corpses under a Leica MZ8 stereomicroscope (50x magnification). Scoring of swollen, bacterially-infected pharynxes (P), and uninfected, atrophied pharynxes (p) was performed as previously described23. Intestinal colonisation (IC) with E. coli was scored where severe bacterial accumulation was observed in the anterior and/or posterior intestine. Such colonisation presented as extreme lumenal distension by proliferating bacteria and/or colonisation of the intestine beyond the lumenal barrier, with concomitant intestinal tissue degeneration and atrophy. Consistent across pharyngeal and intestinal tissues, sites of bacterial colonisation exhibit a yellowish-brown colour (as that of the E. coli lawn and colocalising with RFP-labelled E. coli), translucent and uniform texture (loss of healthy tissue structures that otherwise appear dark, granular and opaque), and swollen/distended morphology (extensive proliferation of live E. coli). Images of representative examples of the three corpse subpopulations (P, pIC and pnIC) are presented in Fig. 4a and Supplementary Fig. 5b.

Microscopy

Microscopy slides were prepared by placing individual nematode corpses in a small drop of M9 buffer on 2% agar pads, under glass coverslips. Brightfield images were captured using an ApoTome.2 Zeiss microscope with a Hamamatsu digital camera C13440 ORCA-Flash4.0 V3 and Zen software, at 100x total magnification, with 125 ms exposure time and 1.1 V illumination intensity. The presence of E. coli OP50-RFP in the pharynx was assayed using the mRF12 channel (excitation: 577–604 nm; emission: 612 nm) at the same magnification, with 750 ms exposure time and 75% LED intensity. Brightness and contrast were adjusted equally across the entire image, and where applicable applied equally to controls. Brightfield and RFP epifluorescence necropsy images were overlaid in ImageJ and backgrounds removed with Adobe Express (online tool). The maximum intensity threshold of RFP channel images was adjusted in ImageJ from 255 to 70 for all images.

Mortality deconvolution (subpopulation) analyses

Age-specific survival proportions for full (not deconvolved into subpopulations) populations were obtained by conventional Kaplan-Meier analysis (including censors), from which age-specific mortality pseudofrequencies (which sum to 1) were calculated. For each age, these pseudofrequencies were partitioned into subpopulation pseudofrequencies, weighted by the proportion of total mortality at that age belonging to each subpopulation. Standard survival and mortality analyses of subpopulations were then performed utilising these pseudofrequencies (in place of conventional mortality frequency), to enable unbiased inclusion of censor data in subpopulation analyses. Survival analyses in Fig. 4b–d were performed in this manner. Contributions of subpopulation changes to the Gompertz parameters in Fig. 5a–d were similarly performed on simulated (mortality pseudofrequency-derived) survival data.

Simulated survival data was generated for each component change (in subpopulation prevalence or lifespan), by accordingly combining simulated control and treatment subpopulations. Specifically, changes in subpopulation prevalence were simulated by combining treatment cohort subpopulation prevalence with control subpopulation lifespans (based on principles of parsimony, Fig. 4e–f), and changes in lifespan were simulated by combining treatment cohort subpopulation lifespan with control subpopulation prevalence, for each subpopulation at a time. The sum of these component changes closely predicts the true change between control and treatment cohorts (Supplementary Fig. 7a). Contributions of component changes to α and β were then estimated as the change in, respectively, lifespan of the 10% shortest-lived individuals and lifespan variance, which strongly predict α and β across these 12 non-antibiotic cohorts (Supplementary Fig. 7b–c) and whose component sum of these changes predicts their true change between control and treatment cohorts (Supplementary Fig. 7d–e).

Statistics, software and data handling

Statistical tests were performed using JMP Pro (SAS Institute, Inc.), except for Gompertz parameter estimation and assessment of statistical differences between them, which were performed using WinModest34. Right censors were included in all Kaplan-Meier and WinModest analyses. Specific statistical tests and associated methodological details are described in the respective figure/table captions. The statistical significance threshold used in this study is α = 0.05. Necropsy image capture, processing and editing was performed using Zen software and ImageJ.

Analyses of treatment effects on lifespan (Supplementary Fig. 3, Supplementary Table 1) were performed on the pool of all 6 trials, while analyses relating to locomotory healthspan and gerospan (Fig. 1–3, Supplementary Fig. 4, Supplementary Tables 3–5) were performed on the pool of 4 trials (Trials 3–6; see Methods above), in which locomotory senescence was quantified. Subpopulation analyses utilising necropsy data (Fig. 4d–i, 5a–d, Supplementary Fig. 5a, 6–7, Supplementary Tables 6–7) were similarly performed on this pool of 4 trials (or 3 trials, Trials 4–6, for N2 cohorts), in which necropsy was performed. Further information and summary statistics for individual trials are provided in Supplementary Table 1, and analyses confirming the statistical suitability of these trial poolings and the Gompertz fit are provided in Supplementary Fig. 1–2 and Supplementary Table 2.

Data Availability

Data generated in this study on the lifespan, locomotory healthspan and gerospan, and necropsy subpopulation type of all individuals from the 6 trials (n=8,830), where measured, are provided in the Source Data file.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This study reinterprets Gompertz mortality parameters in C. elegans. Contrary to convention, it argues that reductions in β reflect heterogeneity and extended late-life decrepitude (gerospan), while reductions in α better track healthspan extension and slower biological ageing. Linking longitudinal locomotor health to survival, the authors propose α, not β, captures ageing pace.\nThere is little doubt that this paper merits publication. In fact, it reads almost as two papers in one: a hierarchy in which one layer generates evidence for reinterpretation of the Gompertz parameters, and another delves deeply into the biology of ageing. The breadth of the experiments and the strength of the data are remarkable. Yet unless a study of this complexity is presented with exceptional clarity, its impact beyond the inner sanctum of cell and molecular biology, reaching, for example, demographers and population biologists, will remain limited. Demographers in particular, are unlikely to fully understand or even engage with the results without clear guidance through the visual and conceptual density.\nIn addition to the need to make the paper more accessible (outlined later), I have two conceptual concerns that I would like the authors to address more clearly in the manuscript.\n1. Justification for temperature as treatment. While reduced temperature reliably extends C. elegans lifespan, in my view its interpretive value for ageing is limited. Lower temperature slows virtually all biological processes including development, reproduction, and decline through thermodynamic effects, akin to redefining a “day” as 12 hours instead of 24. The clock runs more slowly, so individuals live longer in calendar time, but nothing has fundamentally changed about the mechanisms of ageing. By contrast, interventions such as dietary restriction or daf-2 mutations alter how long worms remain healthy relative to their developmental and reproductive schedules. These pathway-level manipulations reflect genuine changes in ageing pace, whereas temperature effects reflect environmental pacing. Thus, cold exposure demonstrates lifespan plasticity but is a weaker test of mechanistic theories of ageing. The authors cite no papers on temperature as a life-extending method that situates it as an slowed aging concept.\n2. Frailty versus gerospan. The authors reinterpret β as a marker of heterogeneity in gerospan rather than as the rate of biological aging. Although they cite Vaupel’s 1979 frailty model, they do not engage with its central implication that β still represents the rate of aging within sub-cohorts, with the pooled β emerging from their mixture. By contrast, their gerospan framework discards this interpretation altogether, treating β solely as a metric of variation in decrepitude across individuals. Thus, both models place heterogeneity at the center of demographic aging, but they diverge sharply on whether β should be viewed as a rate of aging or a measure of dispersion. The authors need to make this distinction clear so readers can understand how their reinterpretation departs from established demographic theory.\nI see two levels of clarity that need to be improved in the manuscript. The first concerns the overarching hypothesis or framework. With 46 treatments and a bewildering number of “moving parts” (graphs; sub-graphs; sub-sub-graphs), the big experimental picture is extremely difficult to grasp. Readers would benefit from a clear, anchoring framework, something akin to a guiding syllogism (“if premises n are met, then X, Y, Z”), that makes the logical structure of the argument explicit. Such a framework would help orient the reader as they work their way through the results.\nThe second concerns the details (i.e., Results). The figures in this study demand readers process patterns at several levels: within plots (curvature of survival curves, divergence of lines), between subfigures (temperature vs. antibiotic vs. daf-2 effects), and across figures (gerospan expansion, variability, α vs. β, necropsy subpopulations). They must parse directional shifts (α vs. β), proportional changes in healthspan vs. gerospan, cross-condition contrasts, subpopulation divergences, statistical signals, and conceptual overlays like “extended twilight longevity.” The cumulative expectation is that readers track 46 treatments, distinguish absolute from relative measures, reinterpret α and β, connect demographic curves with individual outcomes, and integrate diverse statistical modes—spread across 5 figures with ~58 distinct plots. In sum, this is a bewildering interpretive task requiring alignment of scales, metrics, and interventions. For this reason, I strongly recommend that the authors revisit the Results and consider a clearer, more structured way of presenting both the figures and the accompanying narrative, so that readers outside of the immediate field can engage with and appreciate the significance of their findings. I understand this will be no easy task, but without a clearer and more structured presentation, many readers will be left unable to follow the argument and may simply defer to the authors’ authority rather than genuinely engaging with the evidence.\nPossible syllogism:\nPremise 1 (health vs. decrepitude): If locomotor health can be separated into two phases, H-span and G-span, then we can track healthspan and decrepitude separately across lifespan interventions.\nPremise 2 (effects of interventions): If interventions such as lower temperature, antibiotics, and daf-2 mutations extend mean lifespan, then we should observe whether this added life accrues to H-span (healthy years) or to G-span (decrepit years).\nPremise 3 (empirical finding): If most longevity interventions increase both mean lifespan and the fraction of life spent in decrepitude (G-span_rel), then lifespan extension is not purely “healthy” but often involves expanded late-life frailty.\nPremise 4 (parameter association): If β consistently decreases when G-span_rel increases, then β reductions largely reflect heterogeneity in decrepitude (extended twilight longevity, ETL), not uniformly slower ageing.\nPremise 5 (healthspan correlation): If α consistently decreases when absolute H-span increases, then α more directly indexes the timing of decrepitude onset and thus the “pace” of biological ageing.\nPremise 6 (subpopulation effects): If β is always lower for the pooled cohort than for pathology-defined subpopulations, then β is sensitive to hidden heterogeneity in survival distributions (frailty mixture effect).\nThen: (1) β should be reinterpreted as a proxy for demographic heterogeneity in late-life decrepitude (ETL) rather than the biological ageing rate; (2) α more closely reflects the pace of ageing, since it tracks expansion of healthy lifespan; and (3) therefore, the traditional reading of Gompertz parameters is inverted: α ≈ biological ageing pace, β ≈ survival heterogeneity.", "reviewer_2": "Zhang and Gems investigate, in C. elegans, the relationship between lifespan distributions and the relative proportions of healthspan and gerospan, in their manuscript entitled “Slow Gompertzian ageing in long-lived C. elegans results from expansion of decrepitude, not decelerated ageing.”\nBy measuring lifespan, healthspan, gerospan, and tissue-specific infections under interventions including temperature shifts, antibiotics, and daf-2 mutations, they report several noteworthy findings:\n1. Relative gerospan is positively correlated with lifespan and negatively correlated with the Gompertz timescale.\n2. The previously described “extended twilight” phenomenon—where longer-lived individuals tend to have disproportionately longer gerospans—was amplified by interventions associated with reduced Gompertz aging rate (β, inverse of Gompertz timescale).\n3. Healthspan correlates more strongly with the Gompertz shape parameter (α) than with the scale parameter (1/β).\n4. Lifespan distribution shifts correspond with changes in both prevalence and age of tissue-specific infections.\nFrom these observations, the authors question the traditional interpretation of Gompertz β as the sole demographic proxy of ageing rate and instead propose that α may be a more informative parameter.\nWhile I remain unconvinced by some of the conclusions, the manuscript brings valuable data to an important question that is worth examining carefully. Understanding whether demographic mortality patterns truly reflect underlying biological ageing, or instead the dynamics of specific morbidities, is central to biodemography. The authors provide exactly the kind of individual-level measurements—lifespan, healthspan, and morbidity—that are needed to move the field forward.\nMajor comments\n1. Parametric fitting and goodness-of-fit.\nThe reliance on Gompertz parametric fits throughout the analysis makes it essential that goodness-of-fit (GoF) be demonstrated rigorously. The Extended Data Fig. 2 suggests fits are strong when evaluated by R², but visual inspection alone and R² are not fully appropriate for survival distributions, which are constrained. Tests such as Kolmogorov–Smirnov or Anderson–Darling would provide a more robust evaluation. This point is important not as a technical detail, but because deviations from Gompertz have been repeatedly reported in many species, including C. elegans. Stronger statistical validation of fit would bolster confidence in the parametric comparisons presented.\n2. Interpretation of β.\nThe manuscript makes a bold claim that reductions in Gompertz beta arise from expanded decrepitude rather than slowed ageing. The data clearly show a correlation between relative gerospan and β, but the conclusion that decrepitude expansion causes lower beta may be overstated.\n3. Overall coherence.\nThe breadth of data is impressive, but the narrative sometimes risks overwhelming the reader. Much of the results section walks carefully through cohort-specific outcomes, yet the integrative message becomes harder to follow. A clearer synthesis—perhaps through schematic models or summary figures—could help the reader see how the many detailed findings converge on the authors’ central proposal.\nLine-by-line comments\n- Line 80-89: There are quite significant methodological issues with the survival analysis described above. In addition, the Weibull model the authors cited in line 87-88 is not more complex than Gompertz (both have 2 parameters) but at least for some C. elegans survival curves generate better GoF. While I sympathize with the author’s goals of finding biological basis for simple mortality models, the GoF presented here and in Extended Data Fig. 2 are not convincing.\n- Line 151-168, Fig. 1f-h: The authors here is trying to discuss the trends of gerospan and correlations with aging rate (parameter beta) in the Gompertz model, specifically arguing against a model of overall slowed pace of aging. While I can sympathize the idea that the data is much more complex than simple temporal scaling, the method and data interpretation is flawed: beta is not an independent variable, given that alpha is not held constant. If we look at the data itself (Extended Data Fig. 2) without the Gompertz model, for example at the first row (N2), it is hard to argue that lower temperature increased or held constant the rate of aging (stated on line 152 or shown in the 1st row of Fig. 1f).\n- Line 173-176: The authors define inter-individual variability as variance/standard deviation in lifespan, whereas in reference 29 in which similar ideas are discussed, lifespan variability is measured in dimensionless, for example in CV.\n- Line 178-180: I am confused by the logic here.\n- Line 241-243, Line 251-253: The causal here is unwarranted. When concluding that beta reflect the inter interindividual variation, the authors are ignoring beta’s effect on mean lifespan.\n- Figure 2a: Negative correlation between beta and standard deviation follows from the definition of Gompertz law, I am not fully understanding what this panel is supposed to tell us.\n- Figure 2: Many panels showing data in a nonstandard way, and only shows trends with no raw data.\n- Line 300-305: this should be shown in a figure. The corresponding Figure 4 shows a lot of data but leave with the reader no tool to interpret that data.\n- Line 316-322, Line 352-357: again, I don’t agree with isolating of one demographic parameter, and interpret its correlation with health as causal relations.\n- Line 540-542: raw data (n=8830) is not provided, only summary statistics. Critically, H-spans & G-spans of individual worms should be provided in some form to be consistent with the data availability statement here.", "reviewer_3": "Summary: In the provided paper, Zhang and Gemes investigate the demographic parameters alpha and beta of the Gompertz equation to identify their biological correlatives and to provide new insights into their interpretation. To quantify aging independent of the Gompertz parameters, they individually tracked the locomotory activity of each worm and scored their health based on the locomotion pattern. These quantifications enable them to determine the absolute and relative health and gerospan of the animals. They then measure health and gerospan across different lifespan-extending interventions, including three daf-2 mutations, wild-type animals, carbonite ceiling, different temperatures, and combinations thereof. Their data in Figures 1–3 provide the following conclusions: 1) Low beta values, traditionally associated with a lower aging rate, prolong the gerospan. 2) Beta is associated with a wider inter-individual variability of the gerospan. 3) Conversely, a low alpha is a (somewhat) better parameter to quantify absolute health span. In figures four and five, they then investigate how beta as a value relates to the prevalence of pathological subpopulations and analyze the different modes of death and how beta relates to their distribution across different lifespan extensions.\nIn general, the paper is well done. The studies are sufficiently powered, and the work conducted is extensive. This is a high-quality and informative study that sheds light on one of the most widely used mathematical models of aging. It empirically quantifies the relation of alpha and beta to the rate of aging.\nMajor criticisms: There is a lot to like in this study, and it is a significant step forward in biologically interpreting the parameters of the Gompertz equation. However, the authors do not explain how the original interpretation of beta as the rate of aging came about and why their updated explanation differs so much. The reason why the beta parameter in the Gompertz equation was equated with the rate of aging was that aging has been traditionally defined as the rate of all-cause mortality. Gompertz developed his equation to model mortality, with the original interpretation of beta as the rate of mortality with age. Based on the definition that aging is the rate of mortality, beta remains the rate of aging. However, what this older interpretation of aging implied—a belief that is unequivocally refuted by the current work—was that a lower rate of aging is associated with greater health. In this manuscript, the authors start out by redefining aging through the measure of locomotion. They then quantify health span and gero-span based on this new measure that is no longer based on all-cause mortality but on a measure of organismal health. It is the redefinition of aging based on health parameters rather than mortality that consequently disrupts the association of beta with aging. Redefining aging by including various health parameters is a perfectly reasonable approach and one that has become increasingly popular in the aging field, and a view that probably supplanted all-cause mortality as the key defining feature of aging. However, the authors should clearly state that the definition of beta as a measure of the rate of aging is dependent on one’s definition of aging. By the definition of all-cause mortality, beta remains a measure of the rate of aging. Conversely, defining aging based on declining health—a more current interpretation of how the field views aging—changes the interpretation of beta, and it becomes a measure of inter-organismal variability and a measure of the time spent in decrepitude. It is crucial to make clear to readers who are less familiar with the intricacies of the Gompertz equation that the precise interpretation of beta inherently depends on the definition of aging.\nMinor criticisms:\n1) In Figure 2, the unit for absolute lifespan is missing and should be indicated, which is presumably days. See y-axis Figure 2,b c, d, e, f & g, Y-axis: absolute health-span and gero-span. The unit is missing, which is probably days.\n2) In Figure 4C, panel daf-2(e1370) big P @ 25°C. The lifespan curve seems to be missing. Is that because there are no big P deaths at 25°, or is that an omission? Please clarify."}, "author_response": "Author response: We are thankful for the reviewer’s acknowledgment of the work’s data quality and interdisciplinary impact, and have endeavoured to improve its accessibility for biologists and demographers alike, as detailed below.\nIn addition, our results in fact indicate that temperature controls not only the rate but mechanisms of ageing (which we did not expect). For instance, in 12/16 lower temperature treatments, relative G-span was increased, while lowering temperature changed the prevalence of end-of-life pathologies (decreasing and increasing pharyngeal and intestinal bacterial colonisation, respectively). This argues that temperature has modified the mechanistic trajectory of ageing. We have now also cited two other studies that view temperature as a modulator of C. elegans ageing beyond thermodynamic effects, at first introduction of temperature as a treatment of interest:\n1. Xiao et al. (2013) “A genetic program promotes C. elegans longevity at cold temperatures via a thermosensitive TRP channel”. Cell, 152:806–817.\n2. Vakkayil and Hoppe (2022) “Temperature-dependent regulation of proteostasis and longevity”. Frontiers in Aging, 3:853588.\n\nAuthor response: This is indeed a key distinction between our new view of β and existing views of β from frailty/heterogeneity models. We are glad the reviewer has noted this subtle but critical detail and alerted us to its need for greater clarity. We have now stated this explicitly in the discussion, as follows:\n“Notably, our reinterpretation of β differs from that of existing heterogeneity-based models, e.g. Vaupel et al. (1979), which assume that at least within subpopulations, β does reflect biological ageing rate. In contrast, our data suggest that β better reflects heterogeneity than biological ageing rate between and within subpopulations.”\n“Specifically, we investigated: (1) how life-extension affects the duration of healthy versus decrepit life, (2) whether such changes in health/decrepitude correlate with changes in the Gompertz parameters, (3) whether these correlations can be explained causally, and (4) whether the Gompertz parameters can be causally related to specific age-related pathologies. We will present our results (summarised in Fig. 5e) in the order of these aims.”\n\nAuthor response: The reviewer accurately describes the complexity of our dataset and results, which stem primarily from their multidimensionality and inter-relatedness. In presenting our results, we have aimed to strike a careful balance between simplicity and transparency. For instance, we have considered replacing the various multi-panel figures for the 3 treatment classes (low temperature, antibiotic, daf-2 mutation) with single figures portraying the overall pattern (e.g. most common effect of life-extension on relative gerospan, rather than displaying the pattern for each individual treatment), and/or moving the more complex multi-panel figures to the supplement. However, we realised that this could discourage readers from actively engaging with the data, particularly if its full details were in the supplement. This study has been possible through careful re-evaluation and re-definition of existing ideas, which required an understanding of the boundaries of processes and concepts, and importantly, their exceptions. We therefore feel that the precision preserved within our multi-panel figures is necessary for the interested reader to engage scientifically with the data.\nWe have therefore employed a hybrid strategy with several ways of improving readability of the Results:\n(1) The now improved accompanying narrative (see previous author response) we believe significantly improves structure and clarity.\n(2) The accompanying narrative now always summarises the overall patterns from the multi-panel figures (e.g. “44/46 treatments increased mean lifespan”).\n(3) The multi-panel figures always appear in the same general format.\n(4) We have removed repetitive/redundant y-axis labels in Fig. 1b–h and 2c–g to reduce visual clutter, and colour-coded the two y-axis labels (β and Relative G-span) in Fig. 1f–h for reader convenience.\n(5) “ETL” (extended twilight longevity) vs “non-ETL” treatments have been annotated in Fig. 2b–g, to help readers connect between the main text and figure.\n(6) The term “inverted ETL” has been removed from Results section (iii) to reduce the number of new terms, and its explanation rewritten to be simpler (yet still precise).\n\nAuthor response: We thank the reviewer for their constructive feedback, and are grateful for their appreciation of the work’s relevance for advancing the field.\n“Anderson-Darling goodness-of-fit tests also supported these Gompertzian fits in most cohorts (21–22/24) despite localised deviations within some (Supplementary Table 2).”\n\nAuthor response: We appreciate that the distinction between correlation and causation must be navigated carefully here. In Results section (ii) (Fig. 1f–g) we present the correlation between increased gerospan and decreased β, and in section (iii) (Fig. 2) we explain how this correlation arises (i.e. can we demonstrate causality?). We show in several ways that lifespan extension in longer-lived population members (in/closer to the survival curve tail) results primarily from gerospan extension (rather than healthspan extension). This means that the ‘stretching out’ (i.e. β reduction) of these survival curves is in fact caused by the gerospan expansion event. Here, causation is understood in a life-course/mathematical way (changes in lifespan are caused by additive changes in healthspan and/or gerospan). We summarise this idea visually in Fig. 5e.\nBut we do understand that there can be other levels of causality. For instance, there are also more biologically-mechanistic causes of β reduction, such as the molecular/cellular/systemic processes that directly determine lifespan and therefore β. However, as we understand it, this is a different hypothesis and one that we have been careful not to claim as our focus.\n\nAuthor response: We have now improved the readability and clarity of our central findings. In particular, Fig 5e provides a clear summary schematic of our central findings, which we refer readers to at the end of the introduction.\nWe list here other improvements to this end:\n(1) A list of key aims (same order as results section) is presented at the end of the introduction, and the results section openings have been adjusted to match this.\n(2) The accompanying narrative now always summarises the overall patterns from the multi-panel figures (e.g. “44/46 treatments increased mean lifespan”).\n(3) The multi-panel figures always appear in the same general format.\n(4) We have removed repetitive/redundant y-axis labels in Fig. 1b–h and 2c–g to reduce visual clutter, and colour-coded the two y-axis labels (β and Relative G-span) in Fig. 1f–h for reader convenience.\n(5) “ETL” (extended twilight longevity) vs “non-ETL” treatments have been annotated in Fig. 2b–g, to help readers connect between the main text and figure.\n(6) The term “inverted ETL” has been removed from Results section iii to reduce the number of new terms, and its explanation rewritten to be simpler (yet still precise).\n(7) More detailed cohort-specific descriptions have been streamlined to focus attention on the integrative message.\nRegarding Vanfleteren et al. (1998), we have cited this in reference to the three-parameter logistic model that was used (in addition to two-parameter logistic and Weibull models). However, to improve precision, we have amended our sentence as follows:\n“Although other mortality models can better fit…”, instead of “Although more complex mortality models can better fit…”\n\nAuthor response: This is a complex concern to which we will respond with care. We apologise for the long response.\nIn the Results section (ii) we presented how each life-extending treatment affects β and relative gerospan (e.g. Fig. 1f–h) to ask a simple question: when β changes, what happens to relative gerospan? We are interested in the latter because it is one readout of the biological ageing process: assuming a simple slowing of the overall ageing process, relative gerospan (proportion of life spent in gerospan) should not change, since all life stages would be stretched in proportion. Therefore, our finding that life-extending treatments simultaneously decrease β and increase relative gerospan suggests that these β reductions do not reflect a simple slowing of biological ageing rate.\nHere there are two separate claims about biological ageing rate. The observed relative gerospan expansion suggests that (1) ageing rate is not simply slowed as though stretching an elastic string, and (2) β does not reflect such a mode of slowed ageing (which is what it is often thought to reflect). Here we are essentially playing Devil’s advocate, to test our hypothesis that the traditional interpretation of β is too simple to explain real data. Our finding of gerospan expansion suggests a more complex form of ameliorated ageing, that increases lifespan more than healthspan.\nWe appreciate that a single demographic change in lifespan (and therefore the ageing process) shapes both α and β, such that it can be difficult or seemingly fallacious to determine the relative contribution to each parameter. However, assuming correct model fitting, by breaking down the ageing process into biologically-distinct components (i.e. healthspan vs. gerospan, or infected vs non-infected individuals), it becomes possible to start exploring how these components might interact (in an additive-like manner) to produce the observed Gompertz parameters. Indeed, we find relationships between α and mean healthspan, and between β and gerospan variation, and these relationships do not depend on each other (e.g. if a treatment changed only one parameter, the relationship would still be true for this parameter). We depict this idea in summary Fig. 5e, showing a treatment that decreases both α and β, reflecting respectively, inter-individually invariant healthspan expansion combined with inter-individually variable gerospan expansion.\nAdditionally, traditional biological interpretation of the Gompertz parameters (α: ageing-independent life-limiting processes and β: ageing rate-determining processes) assume independency of their mechanisms, and these are indeed interpreted independently (e.g. there are simultaneously increased ageing-independent processes and decreased ageing-rate processes). In order to test these views and re-interpret the Gompertz parameters, we follow the template of α=X, β=Y. We are, however, aware that the Gompertz parameters are just a tool, and the biological processes (X, Y) can readily be reassigned to the parameters of any other well-fitting mortality model.\nRegarding Line 152, it states that β (not ageing rate) is increased or held constant by lowered temperature (first column rather than row of Fig. 1f).\n\nAuthor response: That is correct, we have quantified absolute rather than relative (dimensionless) variation. The two measure slightly different things: absolute variation (e.g. standard deviation) directly measures variation, whereas relative variation (e.g. coefficient of variation) measures the variation relative to the mean. The latter assumes biological relevance of some interaction between variation and mean. In our study, we preferred the biologically simpler and intuitive measure of absolute variation, which importantly enabled us to relate the Gompertz parameters to inter-individual variation in healthspan/gerospan in a tractable way.\n\nAuthor response: We have rewritten this section to be clearer:\n“An often-overlooked mathematical property of the Gompertz rate parameter β is its inverse relation to lifespan variation, which in comparison is only marginally affected by the α parameter42 (Fig. 1a, fourth panel). Indeed, amongst our 24 treatments, β showed a strong inverse correlation with lifespan standard deviation (Fig. 2a).\nGiven this, we wondered whether this increased lifespan variation in small-β cohorts could result from increased inter-individual variation in the ageing process. Interestingly, it was previously estimated that G-span differences account for most (~67%) inter-individual lifespan variation in C. elegans21. Thus, the correlation between reduced β and increased G-spanrel in our cohorts may arise from inter-individually variable G-span expansion.”\nIn our model of β reflecting inter-individual gerospan variation, we do account for β’s effect on mean lifespan, because our model proposes that inter-individually variable expansion of gerospan decreases β. Here, mean lifespan is increased due to the inter-individually variable gerospan expansion event. This idea is summarised visually in Fig. 5e.\n\nAuthor response: We include this figure to illustrate this mathematical relationship between β and lifespan variation for those readers from less mathematical/demographic backgrounds. The relationship between β and lifespan variation may not be immediately obvious to all within the diverse field of ageing research. It also helps establish our motivation to focus on inter-individual variation, in not only lifespan but the preceding ageing process.\nAdditionally, this plot provides a reassuring sanity check for the reader that the data and model are behaving as expected, before delving into the more complex analyses.\n\nAuthor response: We have presented these data in non-standard ways due to the unconventional questions that we have asked. We chose modes of data presentation that most easily communicate our findings to the reader. The multidimensional dataset (all combinations of temperature, antibiotic status and genotype) also poses a challenge for data presentation, which accounts for the large number of related panels. To help readers connect between the main text and figure, “ETL” (extended twilight longevity) vs “non-ETL” treatments have been annotated in panels b–g.\nWe provide a table of raw data sufficient to reproduce these figures, which includes healthspan and gerospan values for each individual where scored. We have remembered to attach this file in our resubmission.\n\nAuthor response: This paragraph describes the β values of the whole populations and their component subpopulations. We feel that within the formatting constraints, these β values are most effectively communicated in table format (Supplementary Table 7).\nTo our knowledge, this paragraph does not directly correspond to Figure 4.\n\nAuthor response: We thank the reviewer for drawing out attention to this. We have now remembered to attach the raw data, which can be viewed in the resubmission. It includes lifespan, healthspan, gerospan and necropsy corpse type for all individuals where those traits were measured.\n“Though biological ageing affects individuals, researchers investigating it frequently use demographic (population) ageing as a metric2,3. However, demographic metrics (e.g. changes in mean lifespan) are difficult to link to determinative biological mechanisms.\nThis is true of a salient and enigmatic feature of demographic ageing: the exponential increase in all-cause mortality rate during adulthood, that is common (though not universal) among animal species4,5….\nBut to what do these demographic parameters correspond in terms of biological ageing, within individuals? … β describes the rate of mortality rate increase, and because mortality rate increase has been traditionally defined as ageing, β has thus become widely interpreted as capturing the rate of ageing. Here, equivalence between demographic ageing (mortality rate increase) and biological ageing (rate of ageing) is assumed.”\nWe have also ensured to specify “biological ageing” and “demographic ageing”, rather than just “ageing”, throughout the manuscript.\n\nAuthor response: We have remade these y-axes to include the units (days).\n\nAuthor response: That is correct, there were no daf-2(e1370) P deaths at 25˚C. We have now added the following to the figure caption to make this immediately clear:\n“At 25˚C, no daf-2(e1370) P deaths were observed, as previously noted(Zhao et al., 2021).”\n“Therefore, β reduction reflected ETL in 5/8 reduced temperature, 6/6 antibiotic, and 4/7 daf-2(rf) treatments (15/21 or 71%, in total) where β reduction co-occurred with G-spanrel increase. In the 6 non-ETL exceptions (Fig. 2b, e–g), G-spanabs still increased mostly in longer-lived population members (i.e. reducing β), in particular 15˚C treatment in daf-2(e1368) and daf-2(e1370) (Fig. 2b) and daf-2(e1370) treatment at 15˚C on carbenicillin (Fig. 2e). However, this was also largely true of H-spanabs; thus, in this minority of treatments, β reduction reflects inter-individually variable expansion of both G-span and H-span.”\n\nAuthor response: We appreciate the reviewer’s further suggestions regarding visual clarity and flow. We have worked to format the figures as intuitively as possible, while accommodating editorial formatting policies. In particular, we have used consistent and simple colour schemes to ensure clarity, a consistent layout of our multi-panel figures which readers will become familiar with throughout the study, and introduce figure panels in the same order that they appear in the text. However, we do understand that there will be an inevitable “reading curve” that accompanies this complex study, but for which we have done our best to lower."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I accept the author’s defence of temperature as a mechanistic modifier by showing it disproportionately expands relative gerospan in 12 of 16 treatments and alters specific pathological trajectories. Citing Xiao et al. (2013) and Vakkayil and Hoppe (2022) establishes that temperature regulates longevity through specific genetic and proteostatic programs. The authors also clearly distinguish their work from traditional (Vauple) frailty theory. While Vaupel suggests beta reflects the aging rate within hidden subpopulations, their study demonstrates that beta primarily measures inter-individual heterogeneity in late-life decrepitude (gerospan). Thus, beta is a metric of survival dispersion rather than biological aging pace.\nI also accept their arguments regarding reducing some of the complexity in the manuscript write up. But it will be a heavy lift for some readers.", "reviewer_2": "Upon reading the revised manuscript and the authors’ responses, I find that the manuscript has been substantially improved. In the original version, my concerns were largely technical and stemmed from difficulty in following the central conceptual picture and assessing the statistical adequacy of the numerous sub-claims within a complex set of experiments and data. The revised manuscript now presents the experiments and results in a much clearer and more coherent manner, and several of my methodological concerns have been addressed directly through additional analyses, sharing of original data and improved presentation.\nIn particular, the authors’ central proposal—that reductions in the Gompertz β parameter reflect heterogeneous expansion of gerospan more than extension of healthspan—is now articulated more transparently. The connection between the empirical findings and the reinterpretation of demographic aging parameters is clearer and better supported by the data.\nMajor points from previous review\n1. Parametric fitting and goodness-of-fit\nI requested more rigorous goodness-of-fit evaluation for the Gompertz modeling. The authors have now implemented Kolmogorov–Smirnov and Anderson–Darling tests and reported these results. While deviations remain in some cohorts (as expected), the additional statistical treatment is appropriate and sufficient for the purposes of the manuscript, which focuses on the biological interpretation of Gompertz parameters rather than on defending Gompertz as a universally exact statistical model.\n2. Interpretation of β\nI previously raised concerns that the manuscript risked overstating causality in linking reduced β to expanded decrepitude. The revised text now clarifies the level of causation being invoked (life-course/mathematical rather than mechanistic), and this distinction is adequately explained. The claim is more carefully framed and appropriately supported.\n3. Narrative coherence\nThe manuscript is now conceptually much clearer. The addition of a summarizing schematic (Fig. 5e) and restructuring of the Results section substantially improve readability and transparency of the authors’ argument. The central message—that demographic slowing (reduced β) in these experiments corresponds primarily to gerospan expansion rather than healthspan extension—is now presented in a coherent and integrated manner.\nRemaining points\nI still have the following minor reservations regarding aspects of presentation, but these are not central to the overall conclusions. At this stage, I do not believe further line-by-line revision is necessary.\n1. Scope of individual-level analyses (Fig. 2b–g; Results section iii).\nIndividual-level gerospan and healthspan analyses are presented for a subset of 21 interventions (8/16 reduced temperature, 6/12 carbenicillin treatments, 7/18 other interventions) out of the 46 total cohorts analyzed in the study. Moreover, the extended twilight longevity (ETL) pattern is evident in 15 of these 21. Thus, the claim that reduced β primarily reflects gerospan expansion is directly demonstrated for this subset rather than across all interventions. It may improve transparency to qualify this point more explicitly in the text.\n2. Figure organization and panel structure.\nI appreciate the substantial effort to reorganize the figures, and the manuscript is now much easier to follow. However, the ordering and placement of some sub-panels remain nonlinear and could still pose challenges for first-time readers. Further minor streamlining or clearer visual grouping of related panels might enhance accessibility.\nI believe the manuscript is now suitable for publication, pending minor editorial adjustments as deemed appropriate.", "reviewer_3": "The authors have thoroughly addressed my concerns, and I fully endorse publication of this paper. Their revisions were thoughtful, comprehensive, and clearly strengthened the overall quality and clarity of the work."}, "author_response": ""}]

Decision letter