On invariant property in population statistics

Mathematical Biosciences 226 (2010) 38–45

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On invariant property in population statistics Kazumi Suematsu * Institute of Mathematical Science, Ohkadai, 2-31-9, Yokkaichi, Mie 512-1216, Japan

a r t i c l e

i n f o

Article history: Received 13 October 2009 Received in revised form 12 March 2010 Accepted 25 March 2010 Available online 1 April 2010 Keywords: Advancement of aging Velocity of aging Hibernation Relative mean life span Relative metabolism

a b s t r a c t Concept of aging is developed to yield a relationship between life spans and the velocity of aging. The mathematical analysis shows that the mean extent of the advancement of aging throughout one’s life  A , is conis conserved, or equivalently, the product of the mean life span, T, and the mean rate of aging, v  A ¼ k. The result is in harmony with our experiences: it accounts for the unlimited replicabilstant, T  v ity of tumor cells, and predicts the prolonged life spans of hibernating hamsters according to the equation, T ffi T v¼0 11 v (Tv=0 is a constant and v denotes the total fraction of hibernation periods), in accordance with the Lyman and co-workers experiment. Comparing the present result and the empirical relationship between life spans of various mammals and basal metabolic rates, it is suggested that the mean rate of aging is intimately connected with the mean basal metabolic rate. With the help of this information, we inquire the reason of the difference in mean life spans between women and men, the result showing that the relative mean life span of women to men is Twomen/Tmen ffi 1.08 for various nations, which is close to the corresponding relative value of the basal metabolic rate. The present analysis suggests, however, that this relationship between life spans and basal metabolic rates must be treated with caution. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Aging is the oldest subject in human science. In spite of long and deep concern to aging phenomena, to date human cultures have not laid gerontology on a firm scientific foundation. The reason comes from the extreme complexity of an organism composed of a vast variety of molecular species, infinitely diversified functions and structures together with turn over mechanisms, which has rejected so far the traditional microscopic approach based on the Newtonian mechanics. This circumstance would be sufficient to discourage most theorists. Thus, it is natural that modern theorists who think much of mathematical rigor have not taken up seriously aging as a research subject. On the other hand, a great deal of observed data has been accumulated by experimental biologists which in turn induced countless conjectures about the nature of aging. An essential point is that most of them remain qualitative and explanatory arguments [1,2]. As a result, those propositions fail to predict new phenomena and new experiments. Thus far aging research has not yet been quantitative science. What is strange with aging is that it appears that the more deeply one observes an aged body, the more obscure the aging process becomes; for instance, we can correctly guess one’s age from his appearance, thus aging is recognizable in human-body level, while it is hard to guess one’s age by observing his single cell, and almost * Tel./fax: +81 (0) 593 26 8052. E-mail address: [email protected] 0025-5564/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2010.03.008

impossible by a single molecule, a single atom and so forth. It is evident that aging is a phenomenon characteristic of macroscopic systems; in microscopic levels, aging seems not to exist in the usual meaning. For this reason, we have pursued so far a macroscopic approach of aging. Not surprisingly, it is found that we can say many about the nature of aging without entering the complication of the biological machinery. According to Strehler, there is one effect of aging which can be accurately and quantitatively studied in light of the rigor of modern science. That is the probability of survival, P, as a function of age. It has been widely known that the survival curves in advanced societies have a common shape, the plateau in youth, the rapidly descending zone of middle age and the exponentially decreasing zone at higher age; so one can correctly estimate the death probability, Q = 1  P, between 0 to an arbitrary age [3]. Thus it will be reasonable for us to attempt to construct an aging theory on the basis of the well-established quantity, P. In this report, we push forward our arguments on the basis of the relationship, A = k log P, observed earlier [4], between the extent of the advancement of aging, A, and the probability of survival, P. We show that this relation leads us to an interesting consequence: the total extent of the advancement of aging throughout one’s life is conserved, suggesting that there is an upper bound of the advancement of aging beyond which one cannot survive on average. The result further leads us to a statement: the product  A , is constant, of the mean life span, T, and the mean rate of aging, v which is reminiscent of the empirical relationship [5–7],

39

K. Suematsu / Mathematical Biosciences 226 (2010) 38–45

 M  const:, between the mean life span and the mean basal T v  M , once discarded because of little evidence to metabolic rate, v support it and apparent contradiction in interspecies comparison. In this paper, we re-investigate these problems from a theoretical point of view.

ment of aging. It has been well established that the mortality function is an increasing function of age [12] and attains the maximum point, p0, beyond the critical age, tc [4,13–18]. Thus we may have the statement: The velocity of aging increases with advancing age, or

€ P 0; A

2. Theoretical background

ð7Þ

and approaches the maximum speed at higher age of t P tc Consider a society comprised of biologically identical n(t) members with equal chronological age, t. Consider a deterministic stochastic process where members can only disappear, which is an example of the familiar decay model of population [8]. Thus we focus our attention on the population statistics of cohorts.1 To treat aging phenomena mathematically, let us introduce the assumption: Assumption 1. The society is ideal, where members disappear (die) only due to aging, with no accidental deaths, no starvation, no infectious diseases, and so forth. Under this assumption, a relationship can be derived between the extent of the advancement of aging, A, and the continuation probability, P, of survival [4]: by Assumption 1, the probability, dn/n, of members disappearing at age t must directly be linked with the advancement of aging, A, per unit member. Thus it follows that

dAðtÞ ¼ wðdn=nÞ;

ð1Þ

where w represents a mathematical function. With w(0) = 0 in mind, expand Eq. (1) into the Taylor series

dAðtÞ ¼ kðdn=nÞ þ Oðdn=nÞ;

ð2Þ

where k is a constant. Integrate the resultant series from t = 0 to t to yield

AðtÞ  Að0Þ ¼ k log PðtÞ;

ð3Þ

with P(t) = n(t)/n(0) being the continuation probability of survival. If we set the boundary condition, A(0) = 0 for t = 0, Eq. (3) then reduces to

AðtÞ ¼ k log PðtÞ:

ð4Þ

Eq. (4) links the quantity, A, with the survival probability, P. There is a useful way to look at the biological aspect of Eq. (4). A partial advancement of aging from age t1 to t2 can be written as

DA21 ¼ k log Pðt 2 Þ=Pðt 1 Þ

ð8Þ

The above statement can be explained reasonably in mathematical terms through the Taylor expansion of l with respect to P; i.e.,

_ ¼ p0 þ p1 P þ p2 P2 þ    ; lðPÞ ¼ P=P

DA32 ¼ k log Pðt 3 Þ=Pðt 2 Þ: The overall advancement of aging from age t1 to t3 then becomes

where {pi} are coefficients. Observations have shown that l is continuous and finite for all P of 0 6 P 6 1, suggesting that Eq. (9) is generally convergent. Thus, since P ? 0 for sufficiently high age, we have l ? p0. The tc mentioned above therefore corresponds to an age above which l = p0 is approximately fulfilled (but not necessarily means a clear-cut critical age). For the higher age where l = p0 is satisfied, p0 can be equated with the reciprocal of the mean life span as discussed previously (Remark 1) [4]. Remark 1. We can assess the physical soundness of Eq. (4). For this purpose, in Fig. 1, A is plotted as a function of age, t, where P is assumed to follow the general sigmoid curve observed in advanced societies, but the scale is arbitrary since k is indeterminable at present. As is seen in Fig. 1, A increases rapidly at younger age, but linearly at higher age so that A / t (we know by the observations that P / exp(p0t) for higher age, hence the existence of the linear regime). It is widely accepted that the aging is a direct consequence of biological deterioration. If this is true, the biological function must decline progressively with advancing age, corresponding to the behavior of A. Recent observations have shown this trend: Wilson and co-workers could show that the cognitive ability declines more rapidly in older persons than in younger persons [19], in accord with the behavior of Eq. (4) (see Fig. 1). The physical meaning of the constant k of Eq. (4) can be speci_ satisfies the normalization confied. Note that the death rate, P, dition in the interval t = [t0, 1]:

Z

1

  P_ dt ¼ 1;

ð10Þ

where t0(P 0) is a point on a time axis that satisfies P(t0) = 1; the interval [0, t0] is therefore defined as a period in one’s life during which the extent of the advancement of aging remains 0, namely,

ð5Þ

The reproduction of offspring [9,10] is not considered here.

tio or rp ea lin

ð6Þ

where the symbol dot denotes the differentiation with respect to age t. The term (  ) of the right-hand-side is the familiar mortality function, l, of Gompertz. Thus the quantity A is equivalent to the cumulative force of mortality (see [11]). It turns out that the Gompertz function represents the measure of the rate of the advance1

n

It turns out that the total advancement of aging is equal to the sum of the partial quantities. The quantity, A, is additive in nature, in accord with our experiences. We show in the following that some new relationships can be derived from this equality. Let t be an age and differentiate Eq. (4) with respect to age t to yield:

  _ A_ ¼ k P=P ;

ð9Þ

t0

and the corresponding quantity from t2 to t3 is

DA31 ¼ k log Pðt 3 Þ=Pðt 1 Þ  DA21 þ DA32 :

€ ! 0: A

60

70

80

90

100

110

Fig. 1. Representation of the extent, A, of the advancement of aging as a function of age.

40

K. Suematsu / Mathematical Biosciences 226 (2010) 38–45

AðtÞ ¼ 0 for 0 6 t 6 t 0 :

ð11Þ

Thus it follows that t0 is not necessarily equal to 0 years. Let T(t) be the mean life span from age t. With Eq. (10), one immediately finds the general expression of the mean life span from age t0:

Tðt0 Þ ¼

Z

1

Z   ðt  t 0 Þ P_ dt ¼

t0

1

P dt:

ð12Þ

t0

Since P_ is normalized in the interval [0, 1] also, one can write

Tð0Þ ¼

Z

1

Z   t P_ dt ¼

0

1

P dt:

ð13Þ

0

The right-hand-side integral is separable, and

Tð0Þ ¼

Z

1

P dt ¼

0

Z

t0

P dt þ

0

Z

1

P dt:

ð14Þ

t0

According to the definition of t0, P = 1 for 0 6 t 6 t0, and one has the equality:

Tð0Þ ¼ t0 þ Tðt 0 Þ:

ð15Þ bA

Now let us derive some related functions of A. With P = e by Eq. (4), the mth moment of A is given by m

hA i ¼

Z

1

  A P_ dt; m

Fig. 2. Representation of the change of vA–t curves from an old to a young generation. The curves were drawn by reference to recent observed data of the mortality function, l.

(b = 1/k)

ð16Þ

T

hAi ¼ k: T

ð20Þ

The second term hAi/T of the left-hand-side represents the velocity, v A , of aging averaged out throughout one’s life span. Thus,

0

which since AmP ? 0 (m = 0, 1, 2, . . ., m) for both t ? 0 and 1, yields

hAm i ¼

m! : bm

ð17Þ

T  v A ¼ k;

ð21Þ

with k being the mean extent of the advancement of aging as mentioned above. Now one has the theorems:

The case of m = 1 is of special interest.

hAi ¼

1 ð¼ kÞ: b

ð170 Þ

Eq. (170 ) states that the mean extent of the advancement of aging throughout one’s life is constant; conversely, this may be interpreted as: one cannot continue to survive, on average, beyond the critical quantity, Ac = k, which, following the relationship of Eq. (4), corresponds to

P ¼ ebAc ¼ e1 ffi 0:37:

ð18Þ

This is the ideal limit of the mean survival probability based on the Ac concept. Let us apply this result to the P expansion of the mortality function l [4]. One has for sufficiently high age

pðtÞ ¼ expðp0 tÞ for t P tc ;

ð19Þ

where p(t) = P(t)/P(tc) is the normalized survival probability, and tc the critical age above which Eq. (19) applies. We can now transpose the aforementioned aging quantity to Eq. (19), and thus

A ¼ k log p for t P t c :

ð40 Þ

The mean extent of the advancement of aging for this regime is again hAi = k. And one has at A = hAi

pðTÞ ¼ e1 :

ð180 Þ

Comparing with Eqs. (180 ) and (19), one finds

T ¼ 1=p0

for t P t c ;

which is exactly the previous result [4]: the mean life span is constant for higher ages of t P tc. This is an example that the mean life span based on the aging concept coincides with T defined by Eq. (12). From Eq. (170 ) we get an implication that since one cannot survive, on average, beyond Ac, all one can do for longevity is only to delay the advancement of aging. The statement can be made clearer by transforming Eq. (170 ) as

1. total quantity of aging throughout one’s life is, on average, constant;  A is constant. 2. the product of T and v  A . The lesser the velocity of the T is in inverse proportion to v advancement of aging, the longer the life span. For the higher age  A is a constant, kp0, and independent of of t P tc,T = 1/p0, so that v age [20].2 The essence of the above statement is illustrated in Fig. 2 where the age-change of vA is shown for different two cohorts (a) and (b); the x-axis being t and the y-axis vA. The representation may be viewed as an example of annual change of the mortality function observed commonly in advanced societies; for instance, the curve (a) representing an old generation and (b) a younger generation. The areas enclosed by curve (a) x-axis–t1 and curve (b) xaxis–t2 stand for total quantities of aging, hAi, of the respective generations. According to the Theorem 1, the two areas are equal, so that t1 corresponds to the mean life span of the cohort (a) and t2 that of (b), respectively. Thus the areas of the two loops enclosed by (a)–(b) t1 and (b) t1x-axis–t2 must also be equal. It is seen that the cohort (b) has lower vA than (a) throughout the whole life span and therefore has a longer mean life span; i.e., t2 P t1. Through the shift from the generation (a) to (b), the mean life span increased by Dt = t2  t1. From a biological point of view, T in Eq. (21) should be equated with T(t0), since A = 0 for 0 6 t 6 t0 and aging cannot be defined for the interval, 0 6 t 6 t0 [4]; t0 is an age where aging starts to advance. Unfortunately, until today no theoretical and experimental verifications for t0 have not been put forward. Here we replace, for practical purposes, T(t0) with T(0). Rigorously, this replacement might be controversial, but useful for the purpose of testing the biological soundness of Eq. (21). 2 The recent works by Hwang, Krapivsky and Redner deal with the solution of the conservation equation for this regime with the aging concept consistent with the present work.

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K. Suematsu / Mathematical Biosciences 226 (2010) 38–45

Eq. (21) states that only one possible way to elongate one’s life  A , of aging. Note that Eq. (21) does span is to lower the velocity, v not specify a path leading to the critical quantity, Ac; i.e., it does  A . If that is true, not preclude the possibility that one can change v Eq. (21) may provide us with a possible strategy to elongate one’s life span. Then let us assess the applicability of Eq. (21).

2000

(a) 1500

1000

3. Comparison with experiments

500 3.1. Aging in cells Aging should be a result of the decline of biological functions. According to Assumption 1, the decline of the functions can be related to the disappearance of a member.3 Now let us focus our attention on the problem of cellular aging. It was found earlier that ordinary body cells show a form of aging at the cellular level when grown in culture; i.e., the replication rate of cells slows down with age. The first observations were made in 1961 by Hayflick and Moorhead [23], who laid the foundation of the cellular aging. It appears obvious that the decline of the cell replicability is closely related to the decline of the biological functions. It will therefore be reasonable to generalize the basic concept of aging to include the aging of cell divisions. It has been well established that cancer cells (typically the Hela cell) exhibit no appreciable aging in their replicability, in contrast to ordinary body cells [24]. It has been known that the period of the cell division of the Hela cell does not change with advancing chronological age; that shows no appreciable aging in the replication rate. This can be regarded as a typical example of vA ffi 0. Substituting this into Eq. (21), one has T ffi 1. The result is in good accord with our observations, the unlimited replicability of tumor cells. 3.2. Hibernating hamsters It has been suggested that hibernating hamsters live longer than those that do not hibernate. This phenomenon was first observed by Lyman et al. who coined the term ‘suspended animation’ [25]. Following Lyman et al., let us accept the notion of the suspended animation. Then the hibernation should directly lead to the lowering of the rate of aging. Let v be the fraction of the total hibernating period throughout a whole life span. Following the aforementioned additive law, Eq. (5), of aging, the mean critical extent of aging must be a sum of the mean extent of aging during the non-hibernating periods and that during the hibernating periods:

  Ac ¼ hAN i þ hAH i ¼ T v A;N ð1  vÞ þ v A;H v ;

ð22Þ

with the subscripts H and N denoting the hibernation and the nonhibernation, respectively. The general expression of the mean rate of aging is therefore given by

v A ¼ v A;N ð1  v þ gvÞ;

ð23Þ

 A;H =v A;N is the relative aging rate of the hibernation to where g ¼ v the non-hibernation. From Eqs. (21) and (23), we have

T ¼ T v¼0

1 ; 1  v þ gv

0 0

0.1

0.2

0.3

0.4

0.3

0.4

2000

(b) 1500

1000

500 0 0

0.1

0.2

Fig. 3. Mean life span (days) as a function of the fraction (v) of the total hibernating period. Solid line: Eq. (24)0 . Observed points: } and  by Lyman et al.; their original data were averaged out in the interval: Dv 6 0.005 for (a) and Dv = 0.05 for (b).

rived: by the assumption of the suspended animation, it follows that 0 6 g 6 1. But we are particularly interested in the case, g  1, then

v A ffi v A;N ð1  vÞ:

ð230 Þ

Eq. (24) then becomes

T ffi T v¼0

1 1v

for g  1:

ð240 Þ

Eq. (240 ) gives exact values for small v0 s. We can now compare the present theory with the Lyman et al.s’ data [25]. Prior to comparison, it is convenient to average out the experimental points every narrow interval of v (Dv 6 0.005 for Fig. 3(a); Dv = 0.05 for Fig. 3(b)). Numerical data (} and ) thus obtained are plotted in Fig. 3 together with the theoretical prediction (240 ) (solid line) to which the pre-factor Tv=0 = 750 was given. As one can see, agreement between the theory and the observation is excellent in support of the mathematical soundness of Eq. (21); the result giving a confirmation that the hibernation lowers the rate of aging.4 4. Mammalian aging

ð24Þ

the pre-factor T v¼0 ¼ ðk=v A;N Þ being experimentally determinable as a limiting case of v = 0. Unfortunately no rigorous calculation of g is available at present, so we cannot directly apply Eq. (24) to observed data. However, an approximate form of Eq. (24) can be de-

In 1780, Lavoisier and Laplace carried out the first quantitative investigation of mammalian activity. They measured heat evolved by a guinea pig placed in the calorimeter and compared it with the amount of oxygen consumed by the animal. After close investigation, they reached the conclusion that respiration is a slow com-

3 Replacing the ‘biological’ with ‘mechanical’ and the ‘disappearance of a member’ with ‘failure of products’, this definition of aging may be applied to more general cases including the aging of industrial materials [21,22].

4 Once Eq. (24) is confirmed, there are many ways to determine g. The simplest one is the gradient method at v = 0. Let s = T/Tv=0. Then from Eq. (24) we have ds/ dvjv=0 = 1  g, from which one can immediately determine the g value.

42

K. Suematsu / Mathematical Biosciences 226 (2010) 38–45

bustion [26,27]. As plenty of data have been accumulated since Lavoisier’s days, it became recognized that basal metabolic rates per unit mass (BMR), VM/W = vM, are intimately related with life spans of mammals. According to the findings, BMR is roughly in inverse proportion to life spans between different species of most mammals; that is to say, total amounts, M, of metabolism per unit mass throughout their lives are roughly constant irrespective of species, which leading to the empirical relationship:

T 90%  v M ffi constant;

ð25Þ

where T90% denotes, according to Cutler [7], an age when 90% mortality occurs. Although T90% has no direct connection with the above-defined quantity, T, it may be regarded roughly to represent the mean life span. The finding has lead biologists to the notion of the rate-of-living hypothesis [5,6,27]. The historical background can be found in the review articles by Heusner [28], and Holliday et al. [29]. Observed data from numerous authors were summarized by Cutler [7] (see Fig. 4) who compared BMR’s and T90% of various mammals over wide species. In Fig. 4, an example of the relationship (25) given a value of constant = 2  106 (solid line) is illustrated together with the observed points (}: primates; : African elephant, horses, red deer, and N: rodents). The horizontal axis indicates BMR (kJ kg1 year1) and the vertical axis the life span, T90% (year). Good agreement is found between the empirical relationship (25) and the observations, which may lead us to the conjecture:

v M () v A ;

ð26Þ

T  v M ¼ constant:

Eq. (26) implies that the mean basal metabolic rate is in inverse proportion to the mean life span as has been suggested earlier. It appears from Eqs. (21) and (26) that the mean aging rate correlates with the mean basal metabolic rate, and the total extent of the advancement of aging the total metabolism per unit mass. The advancement of aging appears intimately correlated with the basal metabolism. 4.1. Unit metabolism There has been long debate whether a universal relation exists between metabolic rates, VM, and body weights, W [30].

The answer is that a life must create an energy essential for maintaining the bio-chemical machinery which is proportional to the body weight, and the other energy equal to a heat loss dissipated through the body surface, namely, the scaling law: 2

V M ¼ aW þ bW 3 ;

ð27Þ

where a and b are coefficients. This important formula was first found by Bykhovsky [31]. The coefficient b must be proportional to the temperature difference between the body core and the surroundings, so that b = jDT, where j is still another coefficient. For heterotherms, this dissipation energy term will be negligible be: cause DT ¼ 0; hence b = 0 and VM / W, in accord with the observations for tropical snakes [32,33]. Our concern is the metabolism per individual molecular components; i.e., the metabolism per unit mass, namely 1 3

v M ¼ a þ bW  :

ð270 Þ

To date there are some evidences that the coefficients a and b are not universal constants. To clarify the biological detail of the coefficients is therefore an important future-subject in theoretical biology. 4.2. Reason of difference in life span between women and men Let us accept, for a while, the correlation between Eqs. (21) and (26), and test the applicability to observations. According to the relationship (26), the relative mean life span is expected to be inversely proportional to the relative mean metabolic rate. A rough correlation has been confirmed in the interspecies comparison, as mentioned above. According to Assumption 1, however, the relation (26) must be verified rigorously using biologically identical samples. The most reliable data will be those for humans: human BMR’s were measured extensively in the last century by Harris and Benedict [34,35]. Mean metabolic rates can readily be evaluated from their data to yield

(

v M ffi

1

98:5 kJ kg 91:2 kJ kg

1

1

for men;

1

for women;

day day

ð28Þ

respectively (Remark 2). According to the relationships (21) and (26), we expect that the women in question can live longer 1.08 times than the men can do. Now, we expect

T women 98:5 ¼ ffi 1:08: 91:2 T men

ð29Þ

Unfortunately, no information is available for the life spans of those women and men [34,35]. So, we assess Eq. (29) with recent observations based on population statistics.

80 60

90

40

80 20

70 0

0

100000

300000

500000

Fig. 4. Relation of the life span T90% (year) and basal metabolic rate v M ðkJ kg1 year1 Þ. BMR’s were evaluated as the quantity per year rather than day in conformity with the unit of mean life spans (year). Thus the total metabolism  M  55  100 is evaluated (from an example of humans) to be T 90%  v 1 365 ffi 2  106 kJ kg . Solid line: Eq. (25). Observed data (quoted from Cutler, 1983), }: primates (man, orangutan, gorilla, capuchin, rhesus, lemur, gibbon, owl monkey, marmoset, tree shrew, and tarsier); : African elephant, horses, and red deer; N: rodents (rat, gerbil, mouse, and pygmy shrew).

60 50 45 45 50

60

70

80

90

Fig. 5. Relative mean life span of women to men. Solid line: Eq. (29); h: observed data from 1920 to 1997 in Japan.

K. Suematsu / Mathematical Biosciences 226 (2010) 38–45

4.3. The case of rats

90

BMR’s of the laboratory rats, were measured by Benedict [35]. According to his paper, the ages of the rats investigated ranged from 180 to 660 days, and the measurement was made near 28 °C. The numerical values can be read from his figure (some uncertainty may therefore arise), which gives

80 70 60

v M;male  0:9: v M;female

50

Thus we expect the relative mean life span of female rats to males as

45 45 50

60

70

80

90

T female  0:9: T male

Fig. 6. Relative mean life span of women to men. Solid line: Eq. (29). Observed data: : Sweden; }: The Netherlands; : Switzerland; and 4: Norway, from 1967 to 1997.

The annual data of the mean life span of Japanese women and men from 1920 to 1997 [36] are plotted in Fig. 5. The solid line is the linear equation, Twomen ffi 1.08  Tmen, expected from the relative metabolic rate per unit mass. The symbols, h, denote the observed data of the relative mean life span of women to men in Japan. Agreement between the predicted line (29) and the observed data are remarkably good. To confirm this result, Eq. (29) is compared with statistics of other nations (: Sweden, }: The Netherlands, : Switzerland, 4: Norway) from 1967 to 1997 [36]. The results are shown in Fig. 6. Agreement between the observed data and the expected line is again very good.

R1   _ dt Z 1 A P_ dt R 1 AP 0 v A ¼ R 1   ¼ R0 1 v A qðtÞ dt; ¼ P dt 0 t P_ dt 0 0

ð30Þ

and thereby q(t) represents the probability of a member lying at age t. Likewise, v M can be written in the form:

Z

1

v M qðtÞ dt;

ð31Þ

0

v M can be evaluated by multiplying observed weight q(t) and summing over all age.

ð32Þ

ð33Þ

In contrast to the case of humans, male rats should live longer than females. Let us examine our expectation by observations. The relationship between the mean life spans of female and male rats have been investigated by Hursch et al. using Wister rats of an identical strain [38]. They showed

T female  0:85: T male

ð34Þ

Taking into account that their observations were made with small sample size (N = 24) so that the numerical values are not very reliable, agreement of Eq. (33) with Eq. (34) is satisfactory. In particular, the general trend, the inverse proportionality between the mean life spans and the mean metabolic rates,

T female v A;male v M;male ¼

 T male v A;female v M;female

ð35Þ

is clearly manifested (the symbol represents an assumed relationship); male rats live longer than females, as suggested from the relationship (35).

Remark 2. Note that

v M ¼

43

vM by the statistical

At first sight, the agreement between Eq. (29) and the observations is quite unexpected and even disturbing. The mean life span is at present rapidly increasing year by year. So, one’s immediate intuition will be that the relative mean life span should asymptotically approach to a fixed point, say 1.08, with advancing year. However, the results of Figs. 5 and 6 show that this is not the case; the relative mean life span seems invariant. One possible explanation for the remarkable agreement of Eq. (29) and the observations is that the mean rate of aging as biological phenomena is in fact now decreasing with year, while the relative value, v A;men =v A;women , remains invariant. This is by no means an unreasonable explanation. Recent observed data show that the decrease  A is, in fact, what is occurring [36]: the quantity, l, equivalent of v to vA is now rapidly decreasing with year over all generations from 0 to 100 years of age,5 consistent with the above results. 5 Some portion of the change of vA may come from the decrease of death due to essential diseases (excluding accidental death), since the death from the essential diseases approximately follows the Gompertz law, namely vA itself [37]. Those diseases are indistinguishable from the aging phenomena.

5. Discussion For the examples taken up in this paper, the theory agrees well with the experimental observations; no inconsistency can be found for Eqs. (4) and (21): Eq. (21) is in harmony with our observations for the unlimited replicability of tumor cells and the prolonged life spans of hibernating hamsters (Sections 3.1 and 3.2). Eq. (21) is consistent with the recent observations in population statistics, the increasing mean life span and the decreasing mean rate of aging (Fig. 2). Whereas Eq. (21) is on the firm mathematical foundation, the empirical relationship (26) remains conjecture. We discuss this point in the following. It appears that the empirical equation (26) explains well the difference in life span between male and female of mammals, but Eq. (21) implies that the relation (26) must be treated with caution. It cannot explain the unlimited replicability of tumor cells. Recall that T  1 for tumor cells, so if the relation (26) is true, they must  M  0, which contradicts our experihave zero metabolic rate, v ences. Moreover, it has been observed that the mean basal metabolic rate gradually decreases with age [35], while the rate of aging equivalent to the mortality function, l(t), increases steeply with advancing age [4,12,19]. It turns out that the rate of aging is not identical with the metabolic rate. Notwithstanding, the results of Section 4 appear to suggest that  M , are intimately connected  A and v the respective mean values, v with each other. It is possible that the relation (26) is true for the aging of a body such as the mammals. According to Eq. (21), the ratio of the mean rate of aging can be equated with the inverse ratio of the mean life span, namely, v T female ¼ v A;male , this ratio being conserved as an invariant quantity, T male

A;female

44

K. Suematsu / Mathematical Biosciences 226 (2010) 38–45

while the results of Figs. 5 and 6 suggest that this equality seems to be connected with the corresponding ratio of the mean basal metv abolic rate, v M;male . This leads us to the conjectured relationship M;female (35). Whether Eq. (35) is valid for the aging of a body remains conjecture. To prove this, it is necessary to examine whether the mean  M , is, in fact, decreasing with year in parallel with metabolic rate, v v A so that the relation (26) can account for the recent increase of  M is now the mean life span. It is by no means unlikely that v decreasing year by year, say in Japan, if we recall the rapid increase of the body size of Japanese people. Indeed, one example in USA seems to support this conjecture: in Table 1 and Fig. 7, the data in 1919 by Harris and Benedict is compared with their own data in 1928 [34,35]. It is found that during the decade from 1919 to 1928, the mean metabolic rate decreased to 90/100 for both men and women. It is worth noting that the decrease of vM occurs over all generations (see Fig. 7), while the mean life span in USA in the same period has been reported to have increased 1.06 times for men and 1.08 times for women, respectively [39]. These seem to conform to the relation (26), in support of the annual change of v M and the validity of the reciprocal relation (35). It is useful to look at this problem from another point of view. In the same way as for the calculation of Eq. (16), we can also define

Data in 1919 Data in 1928

1

ðkJ kg

day

1

Þ

108 98.5

v M;women

1

ðkJ kg

day

1

Þ

102 91.2

BMR (KJoules/Kg Day)

(a) 130 1919 (Male)

110 90

50

1928 (Male)

0

20

40

60

80

Age (Ye ars)

1

  M P_ dt:

ð36Þ

0

With M = 0 at t = 0 in mind, we can rewrite Eq. (36) as

hMi ¼

Z

1

_ dt ¼ T  v M ; MP

ð37Þ

0

that is,

T  v M ¼ hMi:

ð38Þ

Comparing Eq. (26) with Eq. (38), we find that the aforementioned ‘constant’ is exactly equal to the mean total metabolism, hMi, throughout the life span, as it should be. Remark 3. Mean total quantity of metabolism throughout the life span from age t0 can be estimated. By Eq. (37), we have

hMi ¼

Z

1

_ dt ¼ MP

Z

0

t0

_ dt þ MP

0

Z

1

_ dt MP

t0

¼ Mð0; t 0 Þ þ v M ðt0 Þ  Tðt0 Þ;

hMðt0 Þi ¼ v M ðt0 Þ  Tðt 0 Þ:

ð39Þ

T ¼ T v¼0

1 : 1  v þ gv

ð24Þ

To this end, it is essential to scrutinize the relation (35) by extensive experimental observations.

150

BMR (KJoules/Kg Day)

Z

Now our question is whether the quantity, hMi or hM(t0)i, is an invariant for a given species. To answer this question, we must scrutinize the annual change of vM with more extensive experimental data of population statistics. To date, no such experiment has been carried out. Whether the remarkable agreement seen in Figs. 5 and 6 is a natural consequence of the reciprocal relation (35) depends on whether the annual decrease of vM is in fact what is occurring. If this is refuted, then we must seek the reason of the mysterious value, 1.08, to the others. On the other hand, if this is confirmed, then Eq. (35) is valid, and so is the relation (26), and the aforementioned factor g can be equated with the relative mean metabolic rate of hibernation to non-hibernation. Then we will be able to accomplish a complete plot of the suspended animation according to the following general expression:

150

70

hMi ¼

where M(0, t0) represents the total quantity of metabolism from age 0 to t0. Thus the mean total quantity of metabolism from age t0 is

Table 1 Annual change of the basal metabolic rate, v M , in USA.

v M;men

the mean total metabolism per unit mass throughout the life span as

(b) 130

6. Conclusion

1919 (Female)

1. The equality between the extent of the advancement of aging and the survival probability,

110

AðtÞ ¼ k log PðtÞ;

90 70 50

1928 (Female)

0

20

40

60

80

Age (Ye ars) Fig. 7. Annual change of BMR (basal metabolic rate). The original data by Harris and Benedict were averaged out every interval of Dt 6 5, and plotted as against age t: (a) men () in 1919 and (}) in 1928; (b) women (N) in 1919 and (4) in 1928.

ð4Þ

leads to a consequence that the total advancement of aging throughout one’s life is conserved. From this, we have a theorem: the product of the mean life span, T, and the mean rate of aging, v A , is constant

T  v A ¼ k:

ð21Þ

The lesser the velocity, v A , of the advancement of aging, the longer the life span, T. The result is in harmony with our experiences: the  A ffi 0; the increasunlimited replicability of Tumor cells, T ffi 1 for v

K. Suematsu / Mathematical Biosciences 226 (2010) 38–45

ing mean life span and the decreasing mean rate of aging. If we accept the notion of the suspended animation, Eq. (21) can be generalized to account for the mean life span of hibernating animals

T ffi T v¼0

1 ; 1v

ð240 Þ

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