How could the Gompertz–Makeham law evolve

ARTICLE IN PRESS Journal of Theoretical Biology 258 (2009) 1–17

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How could the Gompertz–Makeham law evolve A. Golubev  Institute of Experimental Medicine, 12 Akademika Pavlova Str., Saint-Petersburg 197376, Russia

a r t i c l e in fo

abstract

Article history: Received 28 January 2008 Received in revised form 11 December 2008 Accepted 15 January 2009 Available online 21 January 2009

In line with the origin of life from the chemical world, biological mortality kinetics is suggested to originate from chemical decomposition kinetics described by the Arrhenius equation k ¼ A*exp(E/RT). Another chemical legacy of living bodies is that, by using the appropriate properties of their constituent molecules, they incorporate all their potencies, including adverse ones. In early evolution, acquiring an ability to use new molecules to increase disintegration barrier E might be associated with new adverse interactions, yielding products that might accumulate in organisms and compromise their viability. Thus, the main variable of the Arrhenius equation changed from T in chemistry to E in biology; mortality turned to rise exponentially as E declined with increasing age; and survivorship patterns turned to feature slow initial and fast late descent making the bulk of each finite cohort to expire within a short final period of its lifespan. Numerical modelling shows that such acquisition of new functions associated with faster functional decline may increase the efficiency of investing resources into progeny, in line with the antagonistic pleiotropy theory of ageing. Any evolved time trajectories of functional changes were translated into changes in mortality through exponent according to the generalised Gompertz–Makeham law m ¼ C(t)+L*exp[E(t)], which is reduced to the conventional form when E(t) ¼ E0gt and C is constant. The proposed model explains the origin of the linear mid-age functional decline followed by its deceleration at later ages and the positive correlation between the initial vitality and the rate of ageing. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Ageing Mortality Survivorship Lifespan Evolution Metabolism Antagonistic pleiotropy Epistemology

‘‘But a real problem that remains is what constitutes an ‘adequate explanation’. Can this ever be more than a matter of individual taste?’’ P. Nurse. The ends of understanding. Nature, vol. 387, 12 June 1997, p. 657

1. Introduction

dimensions. However, the accelerating mode of increase in mortality needs explanation in its turn. Therefore, the problem arises to define the point where such series of explanations should end. The next problem is that such point, which has no further explanations behind it, may seem satisfactory only as far as it fits one’s routine experience or is somehow analogous to it, that is, conforms to one’s personal taste. A widely accepted consensus concerning this situation is that metaphors are found, in the final analysis, at the base of every concept (Lakoff and Johnson, 1980; Brown, 2003).

1.1 Biogerontology forms an interface between natural sciences and the everlasting questions of life and death and thus is too closely associated with general outlooks to let one be satisfied with purely utilitarian explanations. One explanation of speciesspecific lifespans is that the age-dependent death rates rise in an accelerated fashion and eventually become high enough to exhaust the bulk of each finite cohort within a narrow time interval. This explanation is fundamentally different from the assumption of some lifespan limit (a sort of size in time by analogy to size in space), which is achievable under optimal conditions and features some variance similarly to all biological  Tel.: +7 812 301 07 79.

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1.2 This matters a lot for biogerontology because, perhaps, no other scientific field is treated by specialists in so many disciplines and, thus, with so many backgrounds and predispositions. This is partly because people, including prominent scientists, use to develop interest in biogerontology late enough in their lives to feel by themselves the problems of direct gerontological relevance. Such neophytes are burdened with their own experience, use their own language, and find it fascinating to use the words that can be said to reduce the gerontological problems to the spheres where no one feels better than they do. Endocrinologists may find their causae finale in the central neuroendocrine mechanisms, cytologists in the Hayflick limit, molecular biologists in telomere attrition, biochemists in free radicals, evolutionists in the age-dependent

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attenuation of selection pressure, physicists in the second law of thermodynamics, and demographers in an equation that fits more survivorship curves than any other can. Altogether, this makes it largely futile and even risky to communicate one’s general ideas about ageing to someone else whose outlooks are based on a different background. Are there ways to manage this Babelish situation? 1.3 One possible way is to direct explorations to exploitations rather than to explanations. After all, successful applications speak for themselves, do not they? It is surely a great success and more practical than any general discussion to find out that resveratrol, which is contained in dietary products implicated in increased longevity in humans, increases the lifespans of yeasts, worms, flies, fishes, and obese mice by mechanisms amenable to detailed studies (Bass et al., 2007; Baur et al., 2006). Well now, what does it increase? Moreover, what about increasing of cohort lifespan by increasing of cohort size? Indeed, 30% of the maximal lifespan increase observed in Sweden over the last 250 years is accounted for by the increase in population (Wilmoth et al., 2000). So what does lifespan measure and how should it be measured taken as a species-specific biological parameter rather than the difference between two dates on a tombstone or in lab records? 1.4 The ambiguity of this fundamental term is recognised as may be inferred from the increasing usage of survivorship curves instead of or complementary to numeric parameters supposed to be lifespan measures. This transition is symptomatic in pointing at the growing awareness that survivorship and mortality kinetics is what really matters. Thus, there emerges the problem of how is one to represent such kinetics and to interpret its parameters. 1.5 The most popular model of the age-dependent increase in mortality rate is the exponent first suggested by Gompertz, 1825 and known as the Gompertz law (reviewed in Gavrilov and Gavrilova, 1991; Carnes et al., 1996; Olshansky and Carnes, 1997): dn 1   ¼ mðtÞ ¼ l  egt dt nðtÞ

(1)

or ln mðtÞ ¼ ln l þ g  t

(1a)

A better fit to mortality data may be achieved, as Makeham (1860) suggested, by adding a constant to Eq. (1) thus making Eq. (2) (the Gompertz–Makeham law, GM law):

mðtÞ ¼ C þ l  egt

(2)

The parameter l is usually interpreted as the initial (t ¼ 0) frailty (vulnerability to causes of death), the parameter g, as the rate of ageing, the memberl  egt , as the age-dependent (intrinsic) mortality, and the parameter C, as the age-independent (extrinsic, accidental, or background) mortality. In the pure Gompertz law, the dependence of ln m on t is linear. C distorts this linearity, especially at smaller t; however, within their middle segments, the plots of ln m vs. t approximate linearity so closely, as far as C apparently decreases in human history and in model animals transferred to laboratory conditions, that this linear segment has ever been intriguing the ones concerned.

1.6 The Gompertz model for the age-dependent increase in mortality rate has a long history of controversies around it (Carnes et al., 1996; Olshansky and Carnes, 1997), especially with regard to whether it is rooted in some natural laws and thus is itself a law or is but a ‘‘useful tool’’ to treat mortality data. The former view was favoured by earlier authors (e.g., Gompertz, 1825, Makeham, 1860; Brody, 1924; Sacher and Trucco, 1956; Strehler and Midwan, 1960). The latter view seems to prevail today: ‘‘There is no theory of aging that requires mortality rate to follow Gompertz, logistic, or any other particular model’’ (Wilson, 1998); ‘‘But why that exponential increase suggested by Benjamin Gompertz? Essentially, that riddle of the Gompertzian function remains unsolved. What Gompertz called a law, remains a law without explanation’’ (Bonneux, 2003); ‘‘There was never any specific biological reasoning behind the Gompertz equation’’ (Rauser et al., 2006). 1.7 The sceptical attitude toward GM law is reinforced by the observations that increases in mortality rate slow down at advanced ages and thus deviate from the exponent (see Horiuchi, 2003; Rauser et al., 2006; Yashin et al., 1994). Thus, the whole problem has split into two at least: the midlife linear increase in log mortality rate and its subsequent deceleration. The latter is rated as ‘‘a revolution for aging research’’ (Rose et al., 2006) and shifts the focus of current concern from the former. 1.8 The disregard for GM law as a genuine law of nature is common in demography where the principal objective is to find better analytical approximations for the observed mortality patterns as they are influenced by a lot of factors, including the inherent and acquired heterogeneity of study populations (compositional effects; Barbi et al., 2003; Carnes and Olshansky, 2001; Yashin et al., 1994) and deviations from stationary conditions (tempo effects; see Wilmoth, 2005). Advances in accounting of possibly more deviations make the impression that more cumbersome models are more adequate to objects as complex as living systems. GM law may indeed seem conspicuously simple. However, all fundamental laws of nature are simple syntactically, even though their semantics may be challenging, models built on them sophisticated, and applications complex. Calculating of a satellite orbit requires much more maths than it is needed for the laws used in the calculations. 1.9 Of course, the reasons to treat a model as a law of Nature much depend of one’s attitude toward what the law should be like. In this regard, the present author is absolutely solidary with the following discussion of the Gompertz law: ‘‘1. The law should have been observed over a period of years and in different environments. 2. It should be usable for predictive purposes. 3. It should be consistent with other bodies of information, so that it is plausible from the point of view of such other disciplines and sciences as may be applicable. 4. There should be enough logical analysis behind it so that we can feel confident that we understand what influences would cause the crucial parameters to change. We should be able to judge when changes in the environment would necessitate changes in the parameters of the equation.’’ (Tenenbein and Vanderhoof, 1980).

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Based on a series of log mortality vs. age plots, the authors being cited concluded: ‘‘The above examples appear to provide adequate justification for the claim that Gompertz law passes the first stated test—it has wide applicability, not only for humans but for other life forms. The second criterion was the use of the law for prediction. Since it has been used for this purpose in the construction of annuity tables over many years, and many companies have succeeded by putting their money on its correctness, this criterion would seem to be satisfied also. The crucial remaining criteria, and they are ones that do not seem to have been handled satisfactorily over the years, are those having to do with plausibility, consistency with other bodies of knowledge, and enough understanding so that modifications could be made when needed.’’ 1.10 With regard to items 3 and 4 of the above, the objectives of the present paper are to suggest a generalised form of GM law upon the assumption that the mortality kinetics of the primordial living objects originated from the disintegration kinetics of the precursory macromolecular entities, which conformed to the Arrhenius equation, and to explore the relationships of the generalised GM law with the existing views on the evolution of ageing and on its basic mechanisms. 1.11 This approach, which was advocated for gerontology in Golubev (1997) and Pletcher and Neuhauser (2000), represents an alternative to practical outcomes or crucial experiments as a criteria for choosing between different concepts (or metaphors they are based on; Section 1.1). Checking of concept compatibility with possibly more other concepts related to the phenomenon of interest may be especially helpful when each aspect of the phenomenon is described by competing concepts, and the empirical evidence is never sufficient to choose between them and, in fact, adds confusion merely by its amount, as is true for gerontology, which is heavily overabundant in both, facts and theories. 1.12 Numerous as they are, they all may be referred to a few major aspects: (1) the generic principles of ageing applicable to all organisms (e.g, stochastic damage vs. evolved program); (2) the manifestations of the generic principles in the organisms of special interest (e.g., the roles of active oxygen and carbonyl species in ageing, the mechanisms of menopause and late-onset diseases, etc.); (3) the behaviour of populations of ageing organisms, which includes mortality kinetics and lifespan distributions (GM vs. Weibull and a host of models usable in demography); (4) the evolution of all of the above (e.g., late-acting germline mutations accumulation vs. antagonistic pleiotropy).

3

for the populations of gas particles is apparently violated at high pressures because, in its canonical form, it ignores particle size, which becomes important at high particle concentrations. Therefore, it is natural, in the most general sense, to see deviations from GM law at the extremes of age-at-death distributions, each deviation requiring a separate analysis, which is largely beyond the scope of the present discussion.

2. The origin of exponent in mortality rate increase 2.1 The first attempt to explain the exponential mode of the dependence of mortality on age was made by Gompertz (1825) who suggested that mortality is inversely proportional to the ‘power to oppose destruction’, which was assumed to loose a constant fraction of what has remained by the time of the loss, i.e., to decline exponentially. The idea to derive death rate increase by inversing of functional decline was developed by Brody (1924) who treated the decline as an exponential decay by analogy with the first-order chemical processes. Among other arguments, Brody used references to early experiments reporting that the dependence of mortality of simple organisms (Drosophila melanogaster) upon temperature conforms to Van’t-Hoff’s rule (about three-fold decrease upon 10 1C downward), which is consistent with the Arrhenius equation (Eq. (3) below) under moderate ambient conditions. The next step could be to examine, at least conceptually, how it would be upon a linear decline of vitality (see Section 2.6). However, the initial assumption was that vitality decline is exponential. Another idea is that somatic deterioration is autocatalytic (Orgel, 1963) and thus increases exponentially, whereas death rates may linearly depend on the degree of the deterioration. 2.2 However, the current consensus holds that the age-dependent decline of biological functions within the middle age span is close to linear. Thus, the need is recognised to reconcile the linear functional decline with the exponential increase in death rate. Reconciling theories were suggested in Sacher and Trucco (1956, 1962) and Strehler and Mildvan (1960). The former authors derived exponentially increasing mortality from a presumed distribution of the internal physiological conditions, which are subject to random fluctuations. The latter authors pointed out that this model requires an unrealistically high rate of the mean agedependent physiological decline and suggested instead that the mode of the increase in mortality results from the distribution of the strength of the external stresses experienced by organisms. Stress frequency was postulated to decrease exponentially as stress strength increases, so the assumption that deaths result from stresses that exceed the abilities of organisms to resist them immediately leads to the exponential age-dependent increase in mortality rate, provided that the age-dependent decline in stress resistance is linear.

1.13 2.3 The compatibility of the generalised GM law with the above aspects will be examined here upon two caveats. First, the care to be reasonable for different backgrounds (see Section 1.2) necessitates addressing issues that may seem redundant or primitive in specific contexts. Second, the absolute conformance to the basic laws of nature is never observed in the reality, but is the property of abstract ideal objects, the most evident deviations from them being seen in the real world at extremes. For example, Boyle’s law

The heuristic roots of this idea may be traced to statistical physics, i.e., to Maxwell–Boltzmann distribution, and the key role in the conceptual transition from it to the Gompertz law, as Strehler (2000) acknowledged, belonged to the Arrhenius Eq. (3) for the dependence of chemical reaction rate upon temperature: k ¼ A  eEa =RT

(3)

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where k is the rate constant, A is the pre-exponent, which is not temperature dependent, R is the universal gas constant, T is the absolute temperature (K), and Ea is the activation energy. 2.4 Importantly, chemical literature is overfull with reports about deviations from the Arrhenius equation; however, they are regarded as not the grounds to refute the equation but only as the incentives to search for additional parameters that may account of side reactions, actual heterogeneity of activation energy, etc. Such deviations are more evident with complex molecules, including proteins. Nevertheless, temperature dependence of disintegration kinetics of objects as complex as phages is still consistent with the Arrhenius equation (de Paepe and Taddei, 2006). 2.5 It will be argued here that, as far as living beings are likely to have evolved from some chemical entities, the exponential increase in mortality known as the GM law has evolved from the chemical law expressed by the Arrhenius equation. The chemical essence of the Arrhenius equation is that a molecule disintegrates or otherwise looses its identity by transformation into other molecule(s) when its energy acquired due to random collisions exceeds its activation energy, and the chance to acquire an amount of energy decreases exponentially as the amount increases. Although the equation was derived by S. Arrhenius theoretically from the Maxwell–Boltzmann distribution, this work was motivated by van’t-Hoff’s empirical observation that, under certain conditions, chemical reaction rates increase in apparently exponential ways with increasing temperature measured using Celsius scale. At present, chemical students can draw up their term papers by measuring, e.g., acetylsalicylic acid decomposition rate vs. absolute temperature (K) to make sure that the dependence of k upon 1/T is exponential indicating at the mode of distribution of molecules over the strength of stresses causing their decomposition. If the modern techniques used by the modern students were available to van’t Hoff, the Arrhenius equation could have been obtained in a purely empirical way. 2.6 Now, let us perform a mental experiment to discover the generalised GM law in such an empirical way. Let there be a cohort of some identical objects that exist under uniform and constant conditions, accumulate damage in a linear fashion, and collapse because of random stresses. The stresses arise by coincidences of different forces, so it may be expected that the more of the forces combine to produce a stronger stress, the less is the incidence of such combinations. We want to know exactly how stress incidence depends upon stress strength. It is reasonable to assume, at the first approximation, that stress intolerance of the objects is proportional to the amount of damage they accumulate, i.e., the minimal stress able to destroy such objects decreases linearly as this amount increases. With these provisos, all we need is to determine how the rate of objects collapsing depends on objects age. An adequate embodiment of this mental experiment would be to study liposome disintegration under controlled peroxidation conditions. In the closest analogue of such study found in the literature, cultured neurons were reported to accumulate peroxidative damage and to die out in an accelerating fashion, which was approximated with an exponent (Aksenova et al., 1999). In experiments with phages, ‘‘their probability of dying does not

increase with time, which is probably linked to their absence of metabolism, repair, and redundancy’’ (de Paepe and Taddei, 2006). It would be interesting to study phage mortality under forced deterioration of their capsid proteins, e.g., with an oxidative or proteolytic agent. Back to the above mental experiment, by finding that the dependence of interest is exponential, one will discover the law of distribution of objects over the strength of stresses experienced by them (importantly: not the law of objects disintegration rate vs. age, which is but the observable manifestation of the basic law, age being a linear correlate of objects’ inability to resist stresses). 2.7 This is what may be thought of the Gompertz law. By studying human mortality within the age interval where human functional decline happened to be close to linear, Gompertz discovered the law of distribution of living objects over the strength of stresses experienced by them. The law translates changes in stress resistance E into changes in mortality m. When E decreases linearly, i.e., E(t) ¼ E0gt, mortality rate increases exponentially. However, generally speaking, E(t) may be any function, even increasing within some intervals of t (e.g., during development); thus, the generalised Gompertz law will be m ¼ L  eEðtÞ , which is reduced to the conventional form when E(t) is linear. This is somewhat similar to what has been suggested in Vaupel et al. (1979): mortality at age t is the product of the initial mortality (scale factor) and any appropriate f(t) (shape factor). Herein, the point is that f(t) is essentially exp[E(t)]. In addition, it has been argued in Golubev (2004) and will be below (Section 4.4) that the essence of C is not that it is constant but that it relates to mortality caused by inherently irresistible stresses, which may change, e.g., because ageing organisms may prefer more protected environments and/or less risky behaviours. Therefore, the generalised GM law for homogenous populations will be

mðtÞ ¼ CðtÞ þ L  eEðtÞ

(4)

where the irresistible stresses are captured by C(t), and the resistible stresses are captured by L  eEðtÞ. Further elaborations may include, e.g., the assumptions that L is somehow agedependent and/or the parameters of Eq. (4) are interdependent (see Section 5.4). Population heterogeneity may be introduced by replacing the parameters of the core GM law with appropriate distributions. In the above mental experiment (Section 2.6), the law was discovered in much the same empirical way as, e.g., gas laws were discovered by Boyle and Charles. Irrespective of whether gas laws may be derived from more general premises, i.e., thermodynamics and statistical mechanics, they are still the laws of nature even if discovered empirically. 2.8 Living organisms are believed to have evolved from molecular conglomerates, such as coacervates, microspheres, etc. The key point in the present reasoning is that the origin of life was associated with that the primordial conglomerates, at difference from their molecular components, were able to change without loosing their identity. Such changes might include damage accumulation and the resulting decrease in the minimal stress required for conglomerate disintegration (identity loss), which is equivalent to decreasing Ea. Its linear decrease will produce exponentially increasing mortality (Fig. 1).

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about free will and all that? Ironically, the utmost free human will is manifested in a behaviour, which is not determined by anything, including any previous behaviour, and thus is ordered not more than Brownian motion. As a matter or fact, human survivorship curves, regardless of their analytical approximation, are most notably distorted when human free will is most severely violated, e.g., by different mass movements, including major wars, or by epidemics. 2.12 Fig. 1. Transition from disintegration kinetics of molecular entities featuring fixed identity, as in chemistry, to mortality kinetics of molecular aggregates that may deteriorate within the limits of their expanded identity, as in biology.

2.9 It was repeatedly argued (e.g., in Tenenbein and Vanderhoof, 1980) that putting an exponent at the base of derivation of the Gompertz law is tautology rather than explanation for the exponential increase in mortality. It is thus implied that the exponent must be derived from some other function by computations. However, since any computation will yield an exponent only by starting from the appropriate initial function, the exponent is also introduced arbitrarily, albeit less apparently, in any such case. Therefore, the above objection is the matter of convention about the amount of computations needed to make a sound conclusion. The starting premises require substantial leap of faith anyway. However, no leap of faith may be proved more or less rigorously, it may only be justified more or less persuasively, depending on the personal experience of the one to be persuaded. Therefore, the objective of this section is to suggest verbal reasons rather than formal proofs. With reference to Sections 1.2 and 1.3 for criteria of reasonability, it is suggested here that the biological regularity under consideration originates from the respective chemical one in line with the origin of life from the chemical world. Another important point concerning the required leap of faith is the following: it is not that the unobserved and thus postulated (let it be based on some metaphoric transfer) exponential stress distribution confirm GM law, but rather GM law, as far as it is observed to be approximated by mortality upon removing of possibly more interferences, is indicative of the exponential stress distribution. 2.10 With regard to the role of scientific background in preferences for reasons, it is remarkable that Strehler, whose approach, although turned upside down, makes the basis for the present reasoning, was physicist by his education, which might make it natural to treat any aggregates in a similar kinetic fashion, regardless of whether they consist of living beings or physical particles. The approach to mortality as to a stochastic Markovian process (e.g, Steinsaltz and Wachter, 2006) may underlie such kinetic treatment of mortality, just as statistical physics underlies gas laws. 2.11 The kinetic Arrhenius metaphor applied to living objects is likely to be judged as marginally acceptable with regard to microscopic worms flickering in a flask, questionable concerning mice scurrying about in their cages, and inconceivable and even humiliating when applied to humans because, after all, what

Another approach to mortality kinetics is from reliability theory (Gavrilov and Gavrilova, 2001) devised to treat artefact failure rates. In particular, the use of the Weibull distribution and its elaborations is justified with its origin from reliability theory. For the proponents of the reliability metaphor it looks natural, perhaps according to their experience, to consider systems under analysis as different combinations of parallel and consecutive series of elements, some combinations exhibiting, under specific assumptions, failure rates similar to those observed in living objects. An important caveat related to this approach is that artefacts, unlike living beings, are not self-maintaining and selfrepairing (Steinsaltz and Goldwasser, 2006). Also, the adequacy of the postulated structural relationships between the elements to the organisation of living organisms is questionable. However, using the inverted logic, such as that applied in Sections 2.6 and 2.7 to the Arrhenius-type stress distribution metaphor, one may ask whether the apparent mortality patterns prompt some hidden aspects of the organisation of living beings rather than some unobserved aspects of their relationships with their environment. Interestingly, the conceivability order of the reliability metaphor is opposite to that of the metaphor that likens living objects to particles. For biologists or physicians, at difference from physicists, it may be more comfortable to think of humans as of systems of elements arranged in some manner rather than as of material points. 2.13 Although each particular human may seem to be the less adequate object for being treated as a particle, the unmatched numerical strength of human populations makes them most adequate, compared with any experimental animal group, for being regarded as aggregates where only the global properties matter. The essential epistemological difference between the two approaches is that the reliabilistic one implies that separate complex entities are observed and counted to derive the global parameters of their aggregates, whereas the kinetic one implies the direct determination of the global parameters, such as the concentrations of unit entities. Attitudes to the human race could be more kinetic if it were possible to label human cohorts with something like mirrors and to find out, by observing the cohorts from the Moon with some technique like light scattering, that more respect to human rights will result in better fit of human mortality to GM law.

3. The origin of antagonistic pleiotropy: the parametabolic theory of ageing 3.1 The first thing the primordial living objects had to do in their environment exposing them to deadly stresses was to exist there perpetually. The perpetual existence of the objects, i.e., of the sort

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of things represented by them, might be possible due to stress resistance, de novo emergence, or self-reproduction. In the real world, the first two options are represented by, e.g., pebbles and raindrops. The durability of the former and the continual emergence of the latter require enough simplicity. The primordial living objects were likely to be not as hard as pebbles and not as simple as raindrops. The steady existence of their sort of things was possible due to their ability to reproduce themselves. Because of objects turnover, an increase in the efficiency of selfreproduction would lead to an increase in the predominance of the better self-reproducing objects. The self-reproductive capacity of the objects might be increased by (1) increasing the rate or output of reproduction events, which is more consistent with simplicity, or (2) by decreasing the vulnerability of the objects to environmental stresses, which will elevate the proportion of the objects that remain able to contribute to reproduction. The second option might be realised through an increase in strength or the development of an activity, such as escaping from stresses, repairing the damage they produce or other ways of making stronger stresses required to disintegrate the objects; however, all this comes at the expense of slower reproduction because of more difficulties in producing more strong or complex objects, as has been demonstrated for phages in De Paepe and Taddei (2006). The first option confined evolution to unicellular organisms that multiply by division at the expense of their individual identity to counterbalance their high vulnerability to environmental stresses. The second option resulted in budding unicellular (see Ackermann et al., 2007) and in multicellular organisms that are able to produce offspring without loosing their identity. The resulting distinction between parents and offspring is thought to be a prerequisite for ageing to emerge (Partridge and Barton, 1993).

because DNA is more resistant to hydrolysis. Another example is glucose selected out of all hexoses to occupy its place in metabolism presumably because it favours the circular form of sugar molecules more than their linear carbonylic form (Bunn and Higgins, 1981), which can adversely modify proteins and DNA. 3.5

It is important in the present context that the evolutionary aquisition of new functions occurs in a way consistent with the observation (Jacob, 1977) that evolution ‘‘works like a tinkerer— a tinkerer who does not know exactly what he is going to produce but uses whatever he finds around him, be it pieces of string, fragments of wood, or old cardboardsy. For the engineer, the realisation of his task depends on his having the raw materials and the tools that exactly fit his projecty. In contrast with the engineer’s tools, those of the tinkerer’s cannot be defined by project. What these objects have in common is ‘it might well be of some use’. For what? That depends on the opportunities’’.

However, nothing of chemistry disappears in biology even if becomes inapparent being counteracted by some other chemistry according to the biological design. Although Nature seems to select possibly better available molecules for the design of living beings and to exploit the appropriate properties of the molecules in a possibly better way by enzymic catalysis, the living beings incorporate the molecules with all their chemical potencies, including needless and even adverse ones (Bartosz, 1981; Golubev, 1996). These potencies are expressed as interactions that are not catalysed by enzymes, yet usually accompany enzymatic metabolic pathways and, therefore, have been labeled as parametabolic (Golubev, 1996). Oxygen and its reactive species (see Finkel and Holbrook, 2000) show the best known and, at least for the aerobic forms of life, the most important examples of parametabolic phenomena. However, oxygen is not unique in this sense. For example, methylglyoxal, which is a parametabolic by-product of glycolysis, is formed and can modify proteins and DNA without any involvement of aerobic processes. Much more examples are provided and suggested in Golubev (1996) to compose the pleiotropic effects envisioned by Williams (1957) in the antagonistic pleiotropy theory of ageing. In this sense, virtually no genes lack such effects, including genes that are absolutely necessary for viability. For example, any gene encoding an enzyme contributing to glyceraldehyde-3-phospate formation through glycolysis has antagonistic pleiotropic effects realised though parametabolic methylglyoxal generation from glyceraldehyde3-phospate. Less remote from genes are their antagonistic pleiotropic effects resulting from the propensity of their protein products for misfolding and self-aggregation, especially in the form of amyloid. (Proper folding is facilitated by chaperons in a way similar to enzymic catalysis). The propensity for amyloid formation is most expressed in gene products whose direct effects involve hydrophobic interactions, such as spanning of membranes (e.g., amyloid precursor protein, APP) or binding of hydrophobic molecules (e.g., transthyretin). Interestingly, the amyloidogenic potential of transthyretin increased in the course of vertebrate evolution presumably to better match thyroxin partition between increasing lipid-rich brain environment and the rest of organism (Schreiber and Richardson, 1997). Therefore, the greater amyloidogenic potential of APP fragments or amylin in humans in comparison with mice may also be the other side of some functional benefits. More remote from genes are their antagonistic pleiotropic effects manifested as the accumulation of senescent cells in cell populations. Cell senescence results from a complex interplay of genes involved in cell cycle control and genome integrity maintenance and, in cell populations, is associated with the gradual accumulation of senescent cells rather than with reaching of some predefined limit (Golubev et al., 2003).

3.4

3.6

Any function is brought about by specific properties of molecules implicated in its realisation; therefore, available molecules with increasingly optimal properties for the functions they are involved in were selected. For example, the transition from RNA to DNA for genetic storage occurred presumably

Anyway, all this antagonistic pleiotropy is not late-acting, contrary to the common view (see Section 6.2). Actually, the parametabolic interactions, which are responsible for the antagonistic pleiotropic effects of the genes coding for the enzymes involved in the respective metabolic pathways, come to action

3.2 All properties of living objects contributing to their reproduction either directly (e.g., via the frequency of reproduction events or the productivity of each such event) or through stress resistance (which increases the chance to survive to the time when conditions are favourable for reproductive success) will be regarded here as biological functions. Altogether, such functions constitute what is known as fitness in population biology. 3.3

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once the genes start to act, but the antagonistic by-effects of the genes accumulate, at difference from the direct effects, which, however, provide time for the accumulation (Section 6.10 and Fig. 6). Therefore, it is more accurate to speak of cumulative rather than late-acting antagonistic pleiotropic effects of genes.

and given qj); Ej is object’s disintegration barrier or resistance to jth type stresses, and qj is the environmental harshness related to the jth type of stresses (by analogy with temperature T in the Arrhenius equation). Under constant global environmental conditions and constant objects’ properties, objects’ mortality A will be

3.7

A¼

j¼i X

eEj =qj

j¼1

The parametabolic theory of ageing proposed here claims that any biochemical system is burdened with potencies for interactions that are not controlled by the means of this system. The addition of new control means and respective components will expand the possibilities for still other uncontrolled interactions. Therefore, ageing is not a result of some unfortunate quirk in biochemical evolution, a sort of frozen accident, which was spared by natural selection for some reasons, but is an outcome of the indispensable chemical properties of biomolecules, with which evolution has to cope, especially when it comes to benefits afforded by unrenewable structures, such as those present in the brain. Here it is tempting to recall Go¨del’s incompleteness theorem (see Smith, 2007; Franzen, 2006), which posits that any formal logical system (system of interacting elements) contains means for formulation of statements (possibilities for interactions) that cannot be proved or disproved (controlled) by the other means of this system, and any new postulates (means) intended to manage this problem provide for still other ‘‘uncontrolled’’ statements (interactions). In this sense, ageing is the manifestation of the validity of Go¨del’s theorem applied to the set of interacting elements composing a biochemical system. Any system of interacting elements provides for interactions that are not controlled by others, and so by the whole system. This not only results in limitations for the control of parametabolic product generation and accumulation but, also, makes the source of internal fluctuations, which are able, at their extremes, to destroy the whole system. In this sense, the internal milieu of an organism is a source of deadly stresses of no less importance than that of the stresses produced by the external environment. Therefore, the entire set of biological functions includes those that, on one hand, protect living bodies from the internally generated hazards and on the other hand, add new possibilities for uncontrolled interactions. It is also suggested in Section 5.2 that they are important in the linearisation of functional decline.

4. Meeting of antagonistic pleiotropy with exponent to produce the generalised GM law 4.1 Let there be an ideal cohort of identical objects of the type defined in Section 2.6. The behaviour of such cohort under changes in the parameters chosen to describe it will be explored here using numerical modelling and compared with the reality in order to see whether the initial premises should be revised radically or only supplemented with superimposed parameters and relationships. 4.2 Let the objects be exposed to i different types of stresses, e.g., osmotic, thermal, mechanical, chemical etc. Each jth type stresses (1pjpi) will produce kj ¼ eEj =qj where kj is the rate of object disintegration resulting from jth type stresses (kj turns to be the disintegration constant upon constant Ej

and the cohorts of the objects will die out by a first-order process: 

dn ¼ A  nðtÞ; dt 

nðtÞ ¼ n0  e

Rt 0

so A dt

¼ n0  eAt

4.3 In this case, the mean lifespan is defined as tmean ¼ 1/A. As to the maximal lifespan, the objects are potentially immortal, but their cohorts are countable and finite. Their counts less than 1 make no physical (biological) sense, so it reasonable to assume that, after the time when n(t) ¼ 1 is reached, a cohort no longer exists. Therefore, t(n ¼ 1) defines the maximal cohort lifespan. A cohort initially comprised of n0 nonaging objects will decrease to 1 at tðnÞ ¼ ln n0 =A. Defined in the above way, the maximal cohort lifespan depends on n0 in any case of mortality kinetics, as it does really, e.g., in humans (Wilmoth et al., 2000). A more rigorous treatment of the maximal lifespan problem with references to extreme values theory (Gumbel, 1954) may be found, e.g., in Barbi et al. (2003), the conclusion being that tmax distribution at increasing n0 becomes increasingly concentrated around t(n ¼ 1). 4.4 Now, let the objects become better adapted to their environment by acquiring of a function enabling them to withstand the lth sort of stresses, i.e., the respective disintegration barrier E increases. The resulting mortality rate will be

m ¼ A  eEl =ql þ eEL =ql ¼ C þ eEL =ql , where C ¼ A  eEl =q and EL4El, so moA. In this sense, C captures the mortality caused by the stresses that result from the coincidences that the objects inherently cannot resist. Any encounter of any object of the given type with any such coincidence will result in object’s death. Therefore, C does not depend on changes in the objects but only on changes in their environment. However, the possibility is not ruled out that the environment may change over the lifespans of the objects; e.g., the objects may loose their motility as they age and so may dwell within an increasingly narrow range of conditions. The semantics of C has always been challenging, as discussed, e.g., in Carnes et al. (2006) and Williams and Day (2003). Most often, C is interpreted as ‘‘extrinsic’’ or ‘‘accidental’’ mortality. However, accidents may be very different in the types of their irresistibility. High-flying aircraft wreck is an irresistible stress. Car accident may incur a trauma that a young victim can survive, whereas an old one cannot, and the distribution of the incidence of such traumas over the quantity of damage they produce may be roughly exponential. However, the circumstances of a car accident may make the quality of the resulting trauma be incompatible with life irrespective of age. There may be very different coincidences that can kill anyone of a given species. It is hardly

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possible to classify such odd cases, each of them being rare, so no resistance to it could evolve. However, they in aggregate may be numerous enough to make C considerable even under the most protected conditions.

Whatever is the dependence of E upon t , any reasonable functions put into the exponent of the generalised GM law will produce survivorship curves, which appear strikingly similar to that for the canonical GM law and may almost match each other at the appropriate values of their parameters as shown in Fig. 2.

4.5 4.6 The rate of damage accumulation and of the resulting decrease in stress resistance must depend on the rate of damaging factors production (D) and on the amount of the nonrenewable essential material (M) vulnerable to damage, i.e., still undamaged by time t. Let the contribution of the material to E be proportional to M and the coefficient of proportionality be 1 for simplicity, so E ¼ M. As to D, two extreme cases may be envisioned: D may be constant, being produced by some renewable components upstream of M, which will make dEðtÞ=dt proportional to E(t), or D may be proportional to M and so decrease with decreasing M (e.g., may depend directly on M or on something downstream of M), which will make dEðtÞ=dt proportional to [E(t)]2. These two cases will yield (1) EðtÞ ¼ E0  eat for constant D, and (2) EðtÞ ¼ E0 =E0  a  t þ 1 for D proportional to E(t).

Because of the similarity of the shapes of survivorship curves for different embodiments of the generalised GM law, the evolutionary transition from nonageing to ageing survivorship pattens may be explored using any specific case. This will be exemplified with the likely primary (see Section 5.2) mode of changes in stress resistance, i.e., with its exponential decay: EðtÞ ¼ E0  eat ;

at

so m ¼ C þ eEL0 e

=ql

Let us assume that n0 ¼ 1000, q ¼ 1, A ¼ 0.02, and the newly evolved function protects from the stresses responsible for 0.25 of the initial mortality and reduces it 1000-fold initially at the expense of subsequent increase caused by damage accumulation occurring at a ¼ 0.005. This corresponds to C ¼ 0.015 and m0 ¼ 0.015+5  106, so E0 ¼ ln(5  106). Fig. 3 shows that this results in an increase in mean lifespan (from ca. 50 to 63) as

Fig. 2. Embodiments of the generalised GM law at different modes of decrease in stress resistance. It is assumed that n0 ¼ 1000 and E0 ¼ 10 in each case. Dotted lines: the canonical GM law: linear decrease in E, E(t) ¼ E0at, a ¼ 0.03. Solid lines: case 1 of 4.5 (see text): exponential decay of E, EðtÞ ¼ E0  eat , a ¼ 0.0045. Dashed lines: case 2 in 4.5: EðtÞ ¼ E0 =E0  a  t þ 1, a ¼ 0.0007. Noteworthy features: The shapes of survivorship curves (middle and lower panels on the right) are similar in spite of differences in the time trajectories of functional decline (upper right panel). Cohorts become exhausted earlier (upper right panel) than the functional reserves (upper left panel) of their constituent organisms (discussed in Sections 5.1.3 and 5.2.2). The middle segments of the semilogarithmic mortality plots, which are curved by C in case of the canonical GM law, and linearised by C in two other cases thus producing sigmoid shapes, which are characteristic of mortality increase featuring late-age deceleration (compare middle and lower panels on the left). However, the two latter cases were found to be inferior to GM law in fitting of available mortality data (not shown).

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9

1·103

1000

100 500 10

1

0 0

100

200

0

300

100

200

t

300

t

Fig. 3. Transition from nonageing to ageing survival patterns (see text in Section 4.6). A: linear plots. B: semilogarithmic plots. Bold dots (a): initial nonageing cohort. Thin dots (b): cohort with a 1000-fold decrease in the rate of mortality from causes responsible for 0.25 of the initial mortality. Solid line (c): (b) supplemented with the agedependent decline of the function responsible for the decrease in mortality. Dash-point (d): (c) further modified with C ¼ 0 (e.g., the same cohort is transferred to laboratory conditions).

2000

1000

20

1000 0

100

200 0

0

100

100

200

200

Fig. 4. Gains in progeny of ageing cohorts featuring different modes of reproductive decline. Solid lines: linear reproductive decline. Dashed lines: exponential reproductive decline at rate constant equal to 0.9 of that of the overall functional decline. The gains are plotted (a) against cohort age when reproduction is possible throughout lifespan or (b, c) against the time of the onset of reproduction when it is possible (b) through the rest of lifespan or (c) during 1 time unit.

reflected by the areas under the respective survivorship curves (Fig. 3A), although the maximal cohort lifespan decreases (from ca. 345 to 285), as shown by the intersections of the respective semilogarithmic plots with n(t) ¼ 1 (Fig. 3B). Similar patterns were shown in Golubev (2004) for the linear decrease in stress resistance. 4.7 Now, the above situation will be explored with regard to progeny, the amount of which is defined as Z t2 FðtÞnðtÞ dt t1

where F(t) (fecundity) is the mean amount of progeny produced per unit time. F(t) is assumed to be constant in the original nonageing organisms. In the organisms with the newly evolved function, F(t) is assumed to decline linearly to reach 0 at tmax or to decline exponentially, in the same way as E(t) declines, i.e., FðtÞ ¼ F 0  eat . Three variants are considered: progeny generation is possible (a) throughout the whole lifespan, (b) after some time point, or (c) within a certain period (assumed to be 1 time unit). Fig. 4 shows that, in case of the linear decline of fecundity, the newly evolved function affords reproductive benefit in (a), is beneficial in (b), if the reproductive period begins within ca. 13 of cohort lifespan, and is beneficial in (c), if the possibility window for reproduction occurs within ca. 12 of cohort lifespan. In case of the exponential decline of fecundity occurring at the same rate as that assumed for the overall functional decline, there is no

reproductive benefit (not shown). However, if the constant of fecundity decline is decreased as slightly as to 0.9 of the constant for the overall functional decline, there is a clear reproductive benefit in (a), virtually no benefit in (b), and a time-dependent benefit in (c) when reproductive effort is possible within the first half of cohort lifespan, optimum being at about 14 of cohort lifespan. The possibility that the reproductive functions decline at slower rates in comparison with the overall ageing is shown in Broussard et al. (2005) and discussed here in Section 5.3.2. On the whole, a function newly evolved in the above way may increase the efficiency of investing resources into progeny (in spite of resulting ageing and reduced cohort lifespan) by decreasing the resources wasted by the objects that die before having their chance to contribute to reproduction.

5. Further evolution of the generalised GM law and its compatibility with other aspects of ageing 5.1. Linear midlife functional decline 5.1.1 The functions that protect living beings from the endogenous damage resulting from parametabolic processes are likely to be as vulnerable to the same damage as other functions. There is ample evidence of age-dependent decreases in DNA repair, control of metabolic by-product generation, detoxication, protection from reactive oxygen and carbonyl species, etc. (see Kirkwood and Austad, 2000). These decreases are often interpreted as the causes

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of ageing. However, for damage to accumulate, it is sufficient that the initial efficiency of these mechanisms is below 100%, and no further decrease is required. With all that, the decrease is suggested here to be essential for ageing, albeit not for it to exist at all, but for its specific time course. 5.1.2 Because protection from the endogenous damage is a part of the overall functional capacity, it must correlate with this capacity and thus with E(t). On the other hand, the rate of accumulation of the endogenous damage must negatively correlate with the abilities to protect from it. This may be represented as 

dE ¼ aEðtÞ=bEðtÞ dt

(5)

which will yield dE ¼ a=b  dt and so EðtÞ ¼ E0  ða=bÞ  t ¼ E0  g  t

(6)

Even without absolute coherence between the overall functional decline and the decline of the ability to slow it down, they will still tend to linearise each other. Ideally, at t approaching E0/g, E(t) approaches zero. This means the complete expiration of the functional reserve of the body and thus may be indicative of a ‘‘biological age-limit’’. This also means, formally, that dE/dt ¼ 0/0 at this point. The resulting indeterminacy may be resolved as usual by the consideration that, in the series of aE(t)/bE(t) at t-E0  b/a, any member will be equal to a/b ¼ g making the member at t ¼ E0/g be the same. Thereafter, E becomeso0, which makes no physical (biological) sense just as no1 in a cohort makes no sense. 5.1.3 Remarkable in this regard is the relationship between the two time points when the two biological senses expire: the parameters of GM law seem to be tuned in evolution so that cohorts usually become exhausted earlier than the functional capacities of their constituent organisms become expired (see Fig. 2, upper row). Exceptions are qualified as genetic diseases. For example, renal damage accumulation in Alport disease patients terminates their lives when the functional capacity of their kidneys completely deteriorates, which happens long before the exhaustion of the respective birth cohort of the general population. The same is true, e.g., for functional deteriorations caused by amyloid deposits in the brains of patients suffering from hereditary earlyonset forms of cerebral amyloidosis. 5.1.4 Of course, the above relates to populations or cohorts with their averaged global parameters. In a single body, a nearly parallel decrease in the above dividend and divisor will produce, besides a trend to the rectification of their time trajectories within the middle age span, an increasing instability of the respective quotients at later ages. 5.2. Late-life deceleration of ageing-dependent increase in mortality rate 5.2.1 In case of a single function, the above suggests that its ageing will stop after t ¼ E0  g because there will be nothing left to age in it, so the subsequent death rate will be determined by the constant sum of C and mL achieved by that time. This will be manifested as the late-life deceleration of mortality increase. The

increase in m will also slow down in the case of EðtÞ ¼ E0  eat used in Section 4.6 to model the evolutionary origin of ageing. Further evolution from this possibly primary embodiment of the generalised GM law might be associated with acquisition of means of protection from the endogenous damage and thus might transform the primordial exponential decay of functions into the present-time almost linear mid-age functional decline, which is followed by its deceleration at advanced ages. Indeed, most data related to the advanced ages of the existing organisms suggest discontinuous patterns, i.e., a roughly linear increase in ln m is followed by not a smooth transition but, rather, by a bent to a reduced rate of increase in m, and the subsequent patterns are not derivable from the preceeding courses and may look quite bizarre (Vaupel et al., 1998; Horiuchi, 2003). 5.2.2 A single biological function is an obvious oversimplification; however, no matter how numerous they are, the reasoning used in Section 4.4 is applicable to any of them. Therefore, the emergence of each of them will result in

m ¼ A  ðeEl =ql þ eEm =qm þ   Þ þ ðeEL =ql þ eEM =qm þ   Þ A linear decline of each biological function will produce the respective exponential increase in mortality related to the stresses counteracted by the function. More generally, any decline of stress resistance will result in a respectively exponentiated increase in mortality. The declines need not be synchronous, although Williams (1957) provided reasons to expect an evolutionary pressure toward their synchronisation. In addition, because of high integrity of living beings, impairing of any biological function will compromise virtually all others, which may be another factor of synchronisation of their age-dependent declines. Indeed, mortalities produced by the main age-dependent causes of death increase in roughly parallel manners in human populations (e.g., Dolejs, 1997). However, this does not rule out departures from synchronicity in any individual body. A living body thus looks as a sort of cohort of functions, each function being associated with protection from some cause of death and each declining at some rate. Every such decline is associated with an increase in the respective partial death rate, which is defined by exponentiation according to the generalised GM law, so the shapes of the trajectories of the partial mortalities will be similar (Section 4.5). Altogether, this will smoothen the transition of the cohort of functions represented by an organism through the age period comprising the moments t ¼ E0  b/a related to the expiration of each function. Within this period, a single organism will be loosing more and more of its functions and becoming unprotected against more and more stresses and diseases. As far as the onset of a disease may mean the failure of the function that protects from it, the increasing numbers of co-morbidities in ageing organisms may indicate this loss of more and more functions. Probably, the first events of this sort make what may be recognised as the onset of senility. In a cohort, a gradual transition to almost completely disabled and so virtually nonageing organisms will occur. Indeed, extremely old humans have been shown to die out by a first-order process (Suematsu and Kohno, 1999). However, it should be noted once again that the parameters of GM law seem to be tuned in evolution so that cohorts usually become exhausted earlier than the functional capacities of their constituent organisms completely expire (Section 5.1.3). 5.3. Specific patterns of partial ageing-dependent functional declines 5.3.1 Another oversimplification is to regard biological functions as independent from each other. Several levels and modes of their interplay may be envisioned.

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5.3.2 Numerous feedbacks exist between biological functions and can make their actual age-related changes highly dependent upon specific patterns of their interplay. The most striking examples of this may be found in the neuroendocrine system. The maximal capacities of steroidogenic glands markedly decline with increasing age. This functional deterioration is likely to be a by-result of the evolutionary extension of steroidogenic pathways to further final products, which are better suited for specific signaling functions and, at the same time, are capable of adverse parametabolic interactions with some upstream enzymes of steroidogenesis (Hornsby, 1986; Quinn and Payne, 1985) as reviewed in Golubev (1989) and Golubev (1996). The maximal capacities to produce polypeptides, including trophic hormones, decline too. However, since some of the latter are involved in the negative feedback regulation of the former, the actual production of the latter may increase due to feedback regulation and to upregulate the impaired steroidogenesis, as exemplified by a marked follicle stimulating hormone increase in ageing mammals (Bribiescas, 2006; Wu et al., 2005). Another possible contributor to this increase is functional decline of the hypothalamic centres through which the negative feedback is realised. This decline is likely dependent on the parametabolic autooxidation reactions of catecholamine neuromediators in these centers (reviewed in Golubev, 1989, 1996). As a result, the increase in trophic hormones is exaggerated making steroid production to decline slower than it would be otherwise. As a final and probably beneficial result, the decline of reproductive functions, which depend on sex steroids, may be retarded (see Section 4.7) until their final collapse, which occurs in some mammals, including humans, long before the cessation of other functions. Thus, the interplay of gradual partial functional declines may lead to highly age-specific patterns of functional changes making the impression of programmed phenomena. However this quasiprogram of ageing still makes it quite different from any truly programmed biological process. 5.3.3 The manifestations of the above and other factors may be found in numerous experimental and epidemiological studies showing different quirks in mortality patterns. These findings should not make the incentive to refute GM law in its generalised form but rather should prompt to elaborate it by specific analysis of each interesting case. The analysis of experimental results in terms of mortality kinetics was so far attempted using mainly the pure Gompertz model. Even in this more simple situation, the small sizes of experimental groups hampered the applicability of such analysis (de Magalhaes et al., 2005). However, this problem is practical rather than conceptual. 5.4. The Strehler–Mildvan correlation 5.4.1 An enigmatic feature of mortality vs. age dependencies observed in human populations is the so-called Strehler–Mildvan (SM) correlation (Strehler and Mildvan, 1960) or compensational effect of mortality (Gavrilov and Gavrilova, 1991), i.e., lower initial mortality (l) is associated with accelerated ageing (g). Some authors claim that any general theory of ageing has to suggest an explanation for this feature (Pletcher and Neuhauser, 2000; Milne, 2008). 5.4.2 It was argued (Gavrilov and Gavrilova, 1991; Golubev, 2004) that SM correlation is partly an artefact of neglecting C, i.e., of

11

treating data conforming to GM law as if they conform to the pure Gompertz model. Clearly, this artefact is more significant at greater values and ranges of C. 5.4.3 However, ln l and g show an astonishingly high correlation even in a comparative 2000–2004 period analysis of data on mortality in 18 developed countries, which was performed with the complete GM model suggesting that, in this case, C may be regarded as negligibly small. This and subsequent analyses carried out herein employed TableCurve2D software (SYSTAT Software Inc.) with its more than 3000 inbuilt functions including y ¼ a+b exp(x/c), which corresponds to GM law. Mortality data were obtained from Human Mortality Database (http://mortality.org). The age interval 25–80 years was found to be the best suitable for the analysis according to the complete GM law. Fig. 5A shows the resulting ln l vs. g correlation for females, which may be fitted with linear regression at R2 ¼ 0.9576. Male mortality features a similar pattern, although it is less expanded and demonstrative (R2 ¼ 0.9006). Thus, GM correlation remains a challenge for gerontological theory even though its demographic manifestations may seem quantitatively insignificant. 5.4.4 The assumption that GM law originates from the Arrheniustype kinetics suggests a link between l and g, which seems, at the first glance, to be the one that prompted SM correlation to Strehler and Mildvan (1960). Indeed, combining Eqs. (2), (3), and (6) will yield

m ¼ C þ A eðE0 ða=bÞtÞ=q

(7)

so

l ¼ A eE0 =q and g ¼

a b



1 q

which provides a negative linear association between ln l and g via q. As to l, it may seem natural that q increases mortality, as far as q may relate to the resistible stresses. However, it does not seem natural at all that q slows down ageing (g is inversely proportional to q). This situation, which Strehler and Mildvan themselves regarded as ‘‘contrary to the intuitive notions’’, may be corrected by assuming that the partitioning of resources between stress resistance and somatic maintenance depends on q. 5.4.5 Let the partitioning of body resources R between stress resistance E and protection from parametabolic damage P be formalised as E/P ¼ k at E+P ¼ R, so E ¼ k  R/(k+1) and P ¼ R/(k+1). Let k to increase with increasing q, i.e., k ¼ d  q, where d captures the ability of an organism to reallocate resources to stress resistance according to environmental challenges. Because any usage of resources is associated with parametabolic effects (see Section 3.7), the overall parametabolic damage will be produced at the rate a  R and accumulated, according to Eq. (5), at the rate 

dR a  R að1 þ kÞ ¼ ¼ dt bP b

making RðtÞ ¼ R0 

að1 þ kÞ b

t

Substituting E for R will yield EðtÞðk þ 1Þ að1 þ kÞ ¼ R0  t k b

(8)

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-8

-8.5

-8.5

-9.0

-9

-9.5 -10.0 -10.5

0

lnλ (δ)

-8.0

lnλ (δ)

ln 

12

-9.5 -10

-20

-10.5

-11.0 0.08 0.09 0.1 0.11 0.12 0.13 0.14 γ

-11 0.08 0.09 0.1 0.11 0.12 0.13 0.14 γ (δ)

-40 0

10 γ (δ)

20

Fig. 5. Correlation between ln l and g. A: Data points are derived from GM analysis of data on women mortality averaged over the period 2000-2004 in 18 counties with the lowest C values (data source: Human Mortality Database, http://www.mortality.org). B: The data from panel A are approximated with the model according to Eqs. (10) at R0 ¼ 35.875, b/a ¼ 3.297, and q ¼ 1.05, and is d varied within the range making l and b typical of human populations. C: The same as B except for a much broader range of d.

Fig. 6. Pattern of convergence of lnm vs. t plots at C near zero according to the model described by Eqs. (10). The pattern is shown within the age ranges of 25–100 years (left panel) and 85–95 years (right panel), the range of d being within the limits shown in Fig. 5B.

far from linear (Fig. 5C). However, within the range of d typical of human populations (Fig. 5B), the similarity between the real correlation (Fig. 5A), which is commonly believed to be linear, and the model one is quite apparent.

so EðtÞ ¼

R0 k ak  t kþ1 b

The Gompertz member of GM law will be

m  C ¼ e½R0 k=kþ1ak=bt=q ¼ e½R0 dq=kþ1adq=bt=q Ro d=dqþ1

¼e

 ead=bt

(9)

This means that ln l ¼ 

R0  d dqþ1

and

g¼

ad b

(10)

In this way, ln l and g become correlated via d instead of q, but this correlation is nonlinear. Indeed, expressing of d as d ¼ gb/a and putting it into ln l will yield

ln l ¼ 

R0

gb a

gb þ1 a

(11)

q

Eq. (11) corresponds to y ¼ a  b  x/(a  c  x+1), which fits Fig. 5A data at R2 ¼ 0.961 (vs. R2 ¼ 0.958 for linear regression). R2 difference is small; however, when the above function plus 46 functions inbuilt into TableCurve2D (selected for having simple algebraic forms and not more than three parameters) are ranked according to their ability to fit Fig. 5A data, the linear regression gets rank 19 whereas the equivalent of Eq. (11) gets rank 9. The numerical estimates of the parameters of Eqs. (10) are q ¼ 1.05, R0 ¼ 34.88, b/a ¼ 3.30. At an expanded range of d, Eqs. (10) produce a correlation between ln l and g, which is generally very

5.4.6 The explanations to the correlation between ln l and g suggested in Strehler and Mildvan (1960) and, basing on the reliabilistic metaphor, in Gavrilov and Gavrilova (2001) imply that, at increasing t, all m vs. t plots must converge to make a single intersection point, which is invariant for the given species. Such strict pattern motivated a good deal of speculations about the bilogical sense of this point (Gavrilov and Gavrilova, 1991). The present model also implies that m vs. t plots must converge as t increases; however, the pattern of their convergence is loose: each trajectory featuring a higher d (lower l and higher g) must intersect all trajectories featuring increasingly lower d at points with, respectively, increasing coordinates. At realistic values of d, these points will fall within a relatively narrow range, but no invariant will emerge. Fig. 6 shows how it would be upon the exact fit of the data points in Fig. 5A to the line in Fig. 5B. 5.4.7 It follows from the above that the apparent negative correlation between l and g results from two unidirectional factors. The first one is the artefact (Section 5.4.2) of the mathematical treatment of mortality data. Its significance increases with increasing C and/or range of C. Importantly, changes in conditions occurring within time periods comparable with the cohort

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lifespans of study populations and the lack of absolute linearity in functional declines (see Section 5.3) must limit the ability of GM model to decompose mortality data into the Makeham and Gompertz components, because the complete decomposition requires the complete conformance of mortality data to GM law, whereas the conformance obviously becomes less strict under non-stationary conditions. All this makes it virtually impossible to get rid of the above artefactual effect, which is likely to be the main contributor to the first demonstration of the intercountry SM correlation (Strehler and Mildvan, 1960) and to the subsequent demonstrations of SM correlation in the so-called longitudinal Gompertzian analysis of mortality data that refer to the XX century (e.g., Riggs and Hobbs, 1998; Yashin et al., 2001) when significant changes in C occurred (Gavrilov and Gavrilova, 1991). The second component is the real negative correlation between l and g suggested by the present model. It is reasonable to expect that d may vary not only in space (in different coexisting populations), but also in time (in different cohorts or existence periods of the same population). For example, improving developmental conditions may result in progressively stouter adults. In particular, an unprecedented increase in human stature occurred in the XX century concomitantly with no less unprecedented increase in life expectancy (Harris, 1997). A puzzling accompanying phenomenon of these gains is the increasing rate of ageing revealed with Gompertzian (Yashin et al., 2001) and GM (Mamaev et al., 2004) analyses. A solution to this paradox may be suggested by the consideration that the partitioning of extra resources towards E is equivalent to increasing d, which, according to Eqs. (9) and (10), must lead to decreasing l and simultaneously increasing g.

6. Discussion

13

apparently silent early in life (as manifested by the gently sloping initial segments of survivorship curves, the slopes being attributed to the accidental mortality) and become activated later (steep declines of the final segments of survivorship curves, the declines being attributable to ageing). In addition, as mentioned in Orzack (2003), ‘‘ypresent theories implicitly assume that aging and life span evolve independently of one another. So, for example, the antagonistic pleiotropy theory predicts how aging evolves but it does so only given an arbitrarily fixed life spany’’ (it should be noted that the reproductive life span is more appropriate in this context). The suggestion that the survivorship of living objects is derived from mortality patterns that are rooted in Arrhenius-type kinetics and so are inherently exponential eliminates these inconsistencies by explaining how gradually accumulating adverse pleiotropic effects can lead to the apparent transition from gentle early to precipitous late decreases in survivorship, the late decreases accounting for the bulk of deaths and thus being the primary determinants of the limited lifespans of finite cohorts. 6.3 The late-acting germline mutation accumulation theory (Medawar, 1952) stands apart from the present view. Still, it is worth mentioning that this theory does not explain how a mutation can have only late manifestations. This difficulty is well recognised, e.g., ‘‘y it may be that ‘‘mutation-accumulation’’ alleles are rare. These alleles have bad effects but only at older ages; at younger ages, they are neutral and have no effects. Such alleles may be rare because it is difficult in practice for mutations to occur that produce effects that first start operating at older ages’’ (Vaupel, 1997). However, it is even more difficult in practice for a purely adverse mutation to occur and be spared if its adverse effects are distributed over the whole lifespan.

6.1 6.4 The approach to the GM law proposed herein intersects with several theories related to the evolutionary, populational, basic biochemical, and particular physiological aspects of ageing and may help to fill gaps and resolve apparent incompatibilities between them and to reduce confusion concerning important issues. 6.2 In particular, this relates to the common belief that the antagonistic pleiotropy theory of ageing assumes that the adverse pleiotropic effects of some useful genes are confined to the late periods of life. Here are but a few statements showing the current prevalence of such understanding: ‘‘Medawar supposed that deleterious mutations expressed at older ages would accumulate in populations and reduce the survival and reproductive success of older individuals. Williams extended this idea in a second theory by proposing the existence of antagonistically pleiotropic genes that have deleterious effects in old age’’ (Monaghan et al., 2008); ‘‘y it was proposed that mutant genes advantageous to development and reproduction are deleterious after the reproductive age and cause senescence, which may explain why all species have a limited life span’’ (He and Zhang, 2006). ‘‘The antagonistic pleiotropy theory posits that senescence evolves because at least some early-acting beneficial alleles favoured by selection will have deleterious pleiotropic effects late in life’’ (Promislow, 2004). Even if the antagonistic effects of genes are explicitly associated with gradually accumulating damage (e.g., Zwaan, 1999), this association by itself does not make it clear why these effects are

The disposable soma theory of ageing (DST) has many intersections with the parametabolic theory (PMT) proposed herein (Sections 3.1 to 3.7), so differences between them should be outlined. DST derives the less-than-100% efficiency of maintenance and repair from the need to allocate some of the available resources to reproduction (Kirkwood and Holliday, 1979; Drenos and Kirkwood, 2005), whereas PMT implies that no molecular processes can be 100% efficient in principle, especially at the subcellular scale, and also suggests that the incompleteness of self-maintenance and self-repair may be related to the validity of Godel’s incompleteness theorem applied to the system of metabolic interactions (Section 3.7). It follows from these premises that no allocation problem is needed for ageing to emerge; so DST theory may relate to the evolution of mortality when ageing is present as a default factor, but not to the origin of ageing and not at all to the origin of the specific (exponential) pattern of age-related mortality. In fact, GM law is taken for granted in mathematical modelling performed according to the DST paradigm (Drenos and Kirkwood, 2005). 6.5 In DST, damage accumulation is not consistently associated with antagonistic pleiotropy; for example, the two are clearly separated in the influential review (Kirkwood and Austad, 2000): ‘‘y 2. Ageing is not programmed but results largely from accumulation of somatic damage, owing to limited investments

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in maintenance and repair. Longevity is thus regulated by genes controlling levels of activities such as DNA repair and antioxidant defence. 3. In addition, there may be adverse gene actions at older ages arising either from purely deleterious genes that escape the force of natural selection or from pleiotropic genes that trade benefit at an early age against harm at older ages’’. 6.6 The only ways to cope with the parametabolic damage are to dilute it by cell division or growth and/or to keep it at a stationary level by complete self-renewal, either way being incompatible with a number of higher physiological functions. The fact that the coelenterate Hydra vulgaris shows no age-associated increase in mortality (Martinez, 1998) is probably caused by the ability of this organism to renew all its body components (which prevents hydra from having useful facilities, such as brain). Otherwise, the unrenewable components will accumulate parametabolic products and become progressively less optimal for functioning. This aspect of ageing is emphasised in waste or junk accumulation theories (Hirsch, 1986; Brunk and Terman, 2002.) 6.7 To illustrate the above case, the ability to metabolise aromatic aminoacids to catecholamines used for signalling is likely to have been acquired in evolution as early as by protists (Shpakov et al., 2004) and lower invertebrates, including coelenterates and evergrowing organisms (see Lacoste et al., 2001). The high propensity of catecholamines to produce cytotoxic parametabolic products (reviewed in Golubev, 1996) was probably not a problem for these organisms because such products might be kept at a low stationary level. Being inherited by multicellular organisms, catecholamines turned out to be a real inner hazard for postmitotic structures, which in mammals include hypothalamic nuclei participating in the central neuroendocrine control and thus imparting important features to ageing (see Section 5.3.2). 6.8 In the parametabolic perspective, the problem of the evolutionary origin of ageing seems to be quite specific: how could traits that are associated with the inability for total self-renewal (e.g., as required for brain functions) be spared by natural selection in spite of that this inability actualises the preexisting potential for cumulative damage caused by the parametabolic interactions of the molecules involved in the respective functions? 6.9 The view presented herein means that ageing ‘‘is molded’’ by the chemical legacy of biology, i.e., parametabolic reactions, which are responsible for the adverse cumulative pleiotropic effects of genes within living bodies, and the generalised GM law (as far as it may be rooted in chemical kinetics), which operates on populations to skew the adverse pleiotropic effects, expressed as their influence on mortality, towards later ages. Numerical modelling basing on these premises (Section 4.7) shows that acquisition of new functions prone with accelerated functional decline caused by parametabolic damage may enhance investing of resources into progeny and thus may be favoured by natural selection. This view is complementary to the evolutionary theories, which prove only that selection pressure becomes attenuated depending on decreases in the probability to survive to later ages and/or to contribute to reproduction at these ages (Hamilton,

1966, Charlesworth, 2000; Rose et al., 2007). However, this attenuation is not the same as increasing mortality rate. The transition to the latter from the former, although plausible, is not straightforward at all. The attempts to derive specific mortality patterns from evolutionary considerations proceeding from premises based on physiological ecology tradeoffs (Abrams and Ludwig, 1995; Cichon and Koz"owski, 2000) or population genetics (Baudisch, 2005) suggest very different patterns, even nonageing (Baudisch, 2005); so the exponentially increasing mortality makes but a specific case. That is, such mode of age-dependent mortality increase and, moreover, this increase in general is compatible with the evolutionary considerations (in fact, it was the apparent incompatibility, which motivated evolutionists’ enquiries into the issue), but it is not deducible from them. 6.10 It is also acknowledged that ‘‘yevolutionary theory has been less successful in predicting the specific classes of genes that should be associated with the ageing process’’ (Promislow, 2004), and ‘‘despite much empirical work over several decades, evidence for the availability of genes with the necessary age specific effects appears to be thin’’ (Seymour and Doncaster, 2007). Inquiries into the nature of the adverse pleiotropic effects have been focused on the consequences of their expression after some age (i.e., the first or the last reproduction effort) rather than on the gradual increase in their expression over the entire lifespan. Therefore, the whole issue looks as not how gradual ageing has evolved but, rather, how limited total or reproductive lifespan evolved. In this perspective, ageing may at best seem to result from the accumulation of the ever-increasing number of different deleterious effects rather that from the gradually increasing expression of the effects present from the very start of lifespan. The search for genes with adverse effects is directed to cases where mutations that attenuate the direct functions of a gene are associated with lifespan increase. It is no wonder that such cases are hard to find (Leroi et al., 2005). Another direction for search relates to genes associated with increased risks of late-onset diseases, such as Huntington chorea (Albin, 1993). Still another direction is protein–protein interactions resulting in interferences of some proteins with other proteins’ functions (Promislow, 2004; He and Zhang, 2006), in particular, involved in signal transduction (Leroi, 2001). Still another approach is to derive antagonistic pleiotropy from competition of candidate gene-dependent functions for limited resources (Bochdanovits and de Jong, 2004; Drenos and Kirkwood, 2005). It is not clear in any of the above cases why antagonistic pleiotropy would be specifically lateacting. Fig. 7 illustrates the relationships between the above modes of antagonism in gene action and the parametabolic antagonism proposed in this paper. 6.11 At the same time, there is abundant evidence (obtained with no a priori reference to any evolutionary theory) that ‘‘damage’’, e.g., induced by free radicals, is produced from the very start of someone’s existence, and thus is not ‘‘late-emerging’’, but cumulative. In Sections 3.3–3.7, such cumulative damage is treated as the consequence of the parametabolic interactions of biomolecules, which are generated by proteins encoded by respective genes, and is explicitly interpreted as the adverse pleiotropic effects of these genes. One conclusion from this interpretation is that such antagonistically pleiotropic genes are not rare, but are quite ubiquitous. Another conclusion is that the adverse parametabolic products may result from the effects of

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E) NC 1 A N T FITNESS IO SIS CT RE N FU ESS TR nonrenewable ., S g . structures (e

(e. FU g., NC FE TI CU ON ND 2 IT Y)

accumulation

ol adverse ite 1

by-product

m e 1 tab

oli

te

me

pro tei n2

pro tein 1

competiton for resources

oli te

adverse by-product 2

parametabolic interactions protein-protein interference

tab

2

3

in 3

ab

ote

et

pr

m

damage

RESOURCES

gene2

gene3

Fig. 7. The modes of pleiotropic antagonistic relationships between genes. Thin solid arrows represent the direct effects of genes. Dashed arrows represent their antagonistic pleiotropic effects. The effects of parametabolic product generation are cumulative and may influence cohort lifespan through the rate of decrease in stress resistance (ageing rate or g in GM law). The effects of competition for resources and of protein-protein interference are stationary and may influence cohort lifespan through the initial (l in GM law) or any current stress resistance. Also, they may reallocate resources from protection against parametabolic effects and thus may influence the rate of ageing in this more complex way (see 5.4). Protein-protein interference may be used in the active mechanisms of reallocation of resources between different functions (e.g. self-reproduction vs. self-protection vs. self-maintenance).

genes that are indispensable for vitality, so the functionattenuating mutations of antagonistically pleiotropic genes must be more commonly lethal rather than life-extending. Still another conclusion is that ageing is not a result of some unfortunate events in evolution, which were not eliminated by selection for the reasons outlined in the evolutionary theories, but is the inevitable outcome of the excessive chemical potencies of molecules involved in the biological functions of the vitally important components that have to be non-renewable for their optimal performance and so are prone to the accumulation of the adverse consequences of respective parametabolic interactions.

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parametabolic theory suggests that much of antagonistic pleiotropy, which constitutes the driving force of ageing, emerged in the biochemical evolution very early, when this force was dissipated by cell division and so did little if any harm and, because of that, was not under selection pressure (see also Section 6.7). Therefore, this driving force is buried in metabolic networks so deep that is not easily modifiable and demonstrable by modifications. This depth is less in more primitive organisms, such as C. elegans, in which case many of the known modifications of survival patterns involve stress resistance or/and ageing rate as reflected by the parameters l and g of GM law (Eq. (2)), and is greater in more advanced organisms, such as humans. This view does not rule out the evolutionary decrease in any or all of the parameters of GM law, which did occur when humans were evolving from hominids. However, with regard to the already evolved humans, this view is less optimistic than the views advocated in de Gray (2003) concerning the reversibility of ageing and in Longo et al. (2005) concerning the programmability of ageing. The apparent confinement of the adverse parametabolic effects, as they are manifested in increased mortality, to later ages is suggested in Sections 2.5–2.8 to result from the inherently exponential dependence of changes in mortality upon changes in stress resistance, as it is expressed by the generalised GM law (Eq. (4)), the decline of the resistance being caused by the accumulation of the adverse parametabolic effects rather than by their switching-on at later ages.

6.13 The attitude to the generalised GM model as to a biological law implies the biological inconsistency of mortality models whose cores are based on different functions. At the same time, such attitude is compatible with any non-linear changes in stress resistance, which are captured by the exponent in the generalised GM law to be translated into changes in mortality, and with the compositional explanations of mortality kinetics in heterogenous populations, in which case one may substitute the parameters of GM law with reasonable distributions or/and combine several GM law-based models. It is also compatible with mortality kinetics, rather than lifespan distribution, as the object of gerontological experiments and of the everlasting experiment carried out with living beings by Nature.

Acknowledgements 6.12 Thus, the parametabolic theory of ageing provides a conceptual framework for integration of very important, yet essentially descriptive, theories that are focused on the sources of endogenous damage (e.g., free radical theory) and on the accumulators of the damage (somatic mutations in DNA, cross-links in proteins, etc.). Each of these theories addresses a partial mechanism, not the general cause of ageing. Driving forces for ageing would emerge if there were no whatever contact of organisms with oxygen or even no such thing as free radicals. The strength of the free radical, cross-link, or other related theories is that they suggest specific targets to interfere with ageing. However, an experiment showing that an interference with the postulated cause does modify ageing shows only that the cause is accessible for interference and may be practically significant, but not necessarily that it is pivotal or primary. The

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