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On a mathematical relationship between growth rate and life span
Mechanisms of Ageing and Development, 13 (1980) 401-406 ©Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
401
ON A MATHEMATICAL RELATIONSHIP BETWEEN GROWTH RATE A N D L I F E SPAN
W. BEIER Institut far Biophysik, Karl-Mar-x-Universitdt,Liebigstrasse 27, 701 Leipzig (G.D.R.) (Received April 13, 1980; in revised form July 7, 1980)
SUMMARY Using the Principles of Similitude a relation between the life span of an organism and its growth rate, ageing rate and initial weight is derived. It is shown that considerations based only on dimensional homogeneity allow us to make non-trivial assertions about the form of the equation of state. Closely related to these considerations is the specification of dimensionless numbers which we defined in the Concept of Vitality.
INTRODUCTION Most of early theoretical approaches to senescence involved the idea of fluctuations superimposed upon basic physiological processes in such a way that a Gompertz relationship between age and probability of death is obtained. Typical of these are the SacherTrucco Theory [1] and the studies of Lohmann [2]. Sacher argues that organisms exhibiting more effective homeostasis would show a longer life span. Reviewing the data, he proposed an allometric expression of the form T = 26.3E°'~M+°'a4S - ° m
(1)
where T is life span in years, E is brain size, M is body weight and S is metabolic rate per gram. The existence of allometric relationships of this type is closely related to such notions as the Principle of Similitude and the D'Arcy Thomsontheory of biological transformations [3]. Considerations of dynamical similarity go back at least to Galileo and have been extensively developed since that time [4]. A first general consideration of the Principle of Similitude was probably given by Buckingham [5]. In our case we wished to obtain a relation between the life span'T of organisms and its basic parameters, the specific growth rate k' =k/M, the ageing rate/a, and the initial weight me, where k is the growth rate obtainable from the growth curve and where M is
402
the weight of the organism in the steady state (Gompertz asymptote). To apply this principle we would write an equation of state in the form ¢(T, k', U, too) = 0
(2)
If we tried to take k', la and mo as fundamental quantities and T as the derived quantity, we would very quickly find that there are not enough conditions to determine the exponents in the expression of dimensional homogeneity for eqn. 2. Let us demonstrate why this statement is true. From Buckingham's n-Theorem it can be shown that the dimensionless product of the quantities T,/a, k', and mo is Tak't~la~rnSo = 1
(3)
where a,/3, 7 and 6 are the exponents of dimensional homogeneity. Putting a = 1, one finds, with reference to the physical dimensions of T, #, k', and mol namely: T: k' : /a : mo :
time r 1/(time r'mass m) 1/(time r) mass m
that the equation r 1 r - t 3 m - # r - ~ m ~ = dimensionless
(4)
This is equivalent to 1-~-7=0
t3+~
0
f
(5)
It is not possible to calculate the three exponents/3, 7, and 6 from the two algebraic eqns. 5. As, however, /a itself is a derived quantity, just as T is, from the fundamental quantities k' and mo, then, in fact, instead of eqn. 3 we have uak'am~ = 1
(7)
where a,/3, and 7 are the new set of exponents. With reference to eqn. 4 we fred r -1 F - a m - a m 7 = dimensionless
(Ta)
This is equivalent to
-1-t3=o 1 -t3 + ~ = o
(8)
from which it follows that /~ = - 1; 7 = - 1
(8a)
and we can shOW from eqn. 7 that u k ' - 1 m o 1 = *rl or
(9)
403
~k,_imol - ~ _
k'mo
I~
(k/M)mo
-~k-leo I =Iri
(10)
where M is the Gompertz asymptote of the organism, k the growth rate, and eo = mo/M is the relative birth weight. Irl is the symbol for the dimensionless value of eqn. 9. Similarly, the case in which T is a derived quantity from the fundamental quantities k' and mo leads to Tk'mo = T(k/M)mo
= Tkeo = ~2
(I I)
It follows that the equation of state (eqn. I) can be expressed in terms of the two dimensionless quantities(eqns. I0 and I I) in the form ¢(Irl, Ir2)= 0
(12)
Note that this equation can be written as (13)
r~2 = V/(lh) = Tkeo = ~Ikeo)
or solving for T T = -keo
0
(14)
"
From the concept of vitality [6] we can obtain the expression (see Appendix) I
In eo
T --
(15)
which can be written in the form T=
keo
-eolneo+
'
(16)
Iz
So we see that dimensional arguments are of considerable power, even in the case of an ageing organism. From the experimental data (k, eo and T) given by Laird [7] we obtain the product Tkeo and therefore the following values for the function ~(t~/keo) (see Table I), and in TABLE I SOME VALUES OF ~
~/ke O)AND :~/keo)
Mouse Hamster Rat Guinea pig Rabbit Goat Man Cow
5.09 1.03 0.92 4.32 1.29 8.02 0.23 2.95
~, :years - l ) 0.38 0.41 0.29 0.23 0.17 0.07 0.017 0.05
404
connection with eqn. 16 we can determine the ageing rate,/a. Moreover, since there are two derived quantities, the principle of similitude can be applied. In particular, let us suppose that T, k, #, e0 represent a definite organism. Let us change the fundamental quantities (k, %) to (k~, eo~). Then the principle of similitude says that the transformation of the derived quantities from (T,/a) to (TI, ttx) is given by TI = T(keo/kxeoO
(17)
t~l = ta (kleol/keo) I
First, we see that if/a = 0, then/al = 0; i.e. no ageing system can correspond to a non-ageing system. Second, if a set of Tand/a is known for a certain system and also the life span for other systems, then it is possible to determine the ageing rate of those systems (see Table II). From the concept of vitality it follows for the ageing rate of human beings that/~ = 0.017 years-1. Together with the life span of 72 years, we obtain from eqn. 17 /a = 1.22/T
(18)
TABLE II LIFE SPAN T A N D AGEING R A T E t~
Mouse Hamster Rat Guinea pig Rabbit Goat Man Cow
T (years)
~ (years-1)
3.25 2.5 3.5 4.5 6.0 13.5 72 22.5
0.38 0.49 0.35 0.27 0.20 0.09 0.017 0.05
A statistical analysis to compare the similarity of the ta values as computed by the two methods and shown in Tables I and II by means of a linear least.squares analysis of la versus la leads to /aO) = 1.12it(2 ) +4.11 X 10- a with r 2 = 0.98 So we see that it is reasonable to suppose, on various grounds, that the mechanism of senescence is similar in similar organisms. APPENDIX
The concept of vitality [6] is expressed by the equation t
v(t) = e(t) - la f o
e(t) dt
(At)
405
where v(t) is a dimensionless measure for the vitality of an organism, e(t) is given by
m(t) , •
(A2)
M
where m (t) is the weight of the organism at age t and M the Gompertz asymptote of the weight. The ageing rate /a is obtained from clinical observations to 0.017 years-1. We assume that v(T) = 0
(A3)
where T is the life span of the organism. With this assumption we obtain from eqn. AI T
0 = e(T) - I~ f
e(t) dt
(A4)
o
e(t) can be given by Bertalanffy's, Verhulst-Pearl's or Gompertz's equations of growth [8]. Using Verhulst-Pearl's equation Co
e(t) =
(As)
eo + (1 - e o ) e - ~ t
where k is the growth rate (for man 0.25 years-1) and eo = mo/M is the relative birth weight, we obtain from eqn. A4 o = e(r)-
_ EOe k T ~ ~
k
e(T)
(A6)
Putting m (T) = M, eqn. A6 is then of the form 0 = 1 - - y- In
k
(eoekT)
(AT)
From eqn. A7 we obtain the equation T-
1
In eo
which is used in the text (eqn. 15).
REFERENCES 1 G. Sacher and E. Trucco, The stochastic theory of mortality. Ann. N. Y. Acad. $ci., 96 (1962) 985. 2 W. Lohmann, Altersindex und Interpretation der Absterbelmrven. Z. Alternsforsch., 32 (1977) 461-466. 3 R. Rosen, Dynamical similarity and the theory of biological transformations. Bull. Math. Biol., 4 0 (1978) 549-579.
406 4 B. Giinther, On theoriesof biological similarity. Fortschr. Exp. Theor. Biophys., 19 (1975). 5 E. Buckingham, On physically similar systems. Phys. Rev., 4 (1915) 345-370. 6 W. Beier, K.-H. Brehme and D. Wiegel, Biophysikalische Aspekte des Alterns multizellul/irer Systeme. Fortschr. Exp. Theor. Biophys., 16 (1973). 7 A. K. Laird, Postnatal growth of birds and mammals. Growth, 30 (1966) 349. 8 W. Beier, in W. Beier and R. Rose'n, Biophysikalische Grundlagen der Medizin - Beitriige zur Theorienbildung, Fischer, Sttlttgart, p. 502 if, 1980.
401
ON A MATHEMATICAL RELATIONSHIP BETWEEN GROWTH RATE A N D L I F E SPAN
W. BEIER Institut far Biophysik, Karl-Mar-x-Universitdt,Liebigstrasse 27, 701 Leipzig (G.D.R.) (Received April 13, 1980; in revised form July 7, 1980)
SUMMARY Using the Principles of Similitude a relation between the life span of an organism and its growth rate, ageing rate and initial weight is derived. It is shown that considerations based only on dimensional homogeneity allow us to make non-trivial assertions about the form of the equation of state. Closely related to these considerations is the specification of dimensionless numbers which we defined in the Concept of Vitality.
INTRODUCTION Most of early theoretical approaches to senescence involved the idea of fluctuations superimposed upon basic physiological processes in such a way that a Gompertz relationship between age and probability of death is obtained. Typical of these are the SacherTrucco Theory [1] and the studies of Lohmann [2]. Sacher argues that organisms exhibiting more effective homeostasis would show a longer life span. Reviewing the data, he proposed an allometric expression of the form T = 26.3E°'~M+°'a4S - ° m
(1)
where T is life span in years, E is brain size, M is body weight and S is metabolic rate per gram. The existence of allometric relationships of this type is closely related to such notions as the Principle of Similitude and the D'Arcy Thomsontheory of biological transformations [3]. Considerations of dynamical similarity go back at least to Galileo and have been extensively developed since that time [4]. A first general consideration of the Principle of Similitude was probably given by Buckingham [5]. In our case we wished to obtain a relation between the life span'T of organisms and its basic parameters, the specific growth rate k' =k/M, the ageing rate/a, and the initial weight me, where k is the growth rate obtainable from the growth curve and where M is
402
the weight of the organism in the steady state (Gompertz asymptote). To apply this principle we would write an equation of state in the form ¢(T, k', U, too) = 0
(2)
If we tried to take k', la and mo as fundamental quantities and T as the derived quantity, we would very quickly find that there are not enough conditions to determine the exponents in the expression of dimensional homogeneity for eqn. 2. Let us demonstrate why this statement is true. From Buckingham's n-Theorem it can be shown that the dimensionless product of the quantities T,/a, k', and mo is Tak't~la~rnSo = 1
(3)
where a,/3, 7 and 6 are the exponents of dimensional homogeneity. Putting a = 1, one finds, with reference to the physical dimensions of T, #, k', and mol namely: T: k' : /a : mo :
time r 1/(time r'mass m) 1/(time r) mass m
that the equation r 1 r - t 3 m - # r - ~ m ~ = dimensionless
(4)
This is equivalent to 1-~-7=0
t3+~
0
f
(5)
It is not possible to calculate the three exponents/3, 7, and 6 from the two algebraic eqns. 5. As, however, /a itself is a derived quantity, just as T is, from the fundamental quantities k' and mo, then, in fact, instead of eqn. 3 we have uak'am~ = 1
(7)
where a,/3, and 7 are the new set of exponents. With reference to eqn. 4 we fred r -1 F - a m - a m 7 = dimensionless
(Ta)
This is equivalent to
-1-t3=o 1 -t3 + ~ = o
(8)
from which it follows that /~ = - 1; 7 = - 1
(8a)
and we can shOW from eqn. 7 that u k ' - 1 m o 1 = *rl or
(9)
403
~k,_imol - ~ _
k'mo
I~
(k/M)mo
-~k-leo I =Iri
(10)
where M is the Gompertz asymptote of the organism, k the growth rate, and eo = mo/M is the relative birth weight. Irl is the symbol for the dimensionless value of eqn. 9. Similarly, the case in which T is a derived quantity from the fundamental quantities k' and mo leads to Tk'mo = T(k/M)mo
= Tkeo = ~2
(I I)
It follows that the equation of state (eqn. I) can be expressed in terms of the two dimensionless quantities(eqns. I0 and I I) in the form ¢(Irl, Ir2)= 0
(12)
Note that this equation can be written as (13)
r~2 = V/(lh) = Tkeo = ~Ikeo)
or solving for T T = -keo
0
(14)
"
From the concept of vitality [6] we can obtain the expression (see Appendix) I
In eo
T --
(15)
which can be written in the form T=
keo
-eolneo+
'
(16)
Iz
So we see that dimensional arguments are of considerable power, even in the case of an ageing organism. From the experimental data (k, eo and T) given by Laird [7] we obtain the product Tkeo and therefore the following values for the function ~(t~/keo) (see Table I), and in TABLE I SOME VALUES OF ~
~/ke O)AND :~/keo)
Mouse Hamster Rat Guinea pig Rabbit Goat Man Cow
5.09 1.03 0.92 4.32 1.29 8.02 0.23 2.95
~, :years - l ) 0.38 0.41 0.29 0.23 0.17 0.07 0.017 0.05
404
connection with eqn. 16 we can determine the ageing rate,/a. Moreover, since there are two derived quantities, the principle of similitude can be applied. In particular, let us suppose that T, k, #, e0 represent a definite organism. Let us change the fundamental quantities (k, %) to (k~, eo~). Then the principle of similitude says that the transformation of the derived quantities from (T,/a) to (TI, ttx) is given by TI = T(keo/kxeoO
(17)
t~l = ta (kleol/keo) I
First, we see that if/a = 0, then/al = 0; i.e. no ageing system can correspond to a non-ageing system. Second, if a set of Tand/a is known for a certain system and also the life span for other systems, then it is possible to determine the ageing rate of those systems (see Table II). From the concept of vitality it follows for the ageing rate of human beings that/~ = 0.017 years-1. Together with the life span of 72 years, we obtain from eqn. 17 /a = 1.22/T
(18)
TABLE II LIFE SPAN T A N D AGEING R A T E t~
Mouse Hamster Rat Guinea pig Rabbit Goat Man Cow
T (years)
~ (years-1)
3.25 2.5 3.5 4.5 6.0 13.5 72 22.5
0.38 0.49 0.35 0.27 0.20 0.09 0.017 0.05
A statistical analysis to compare the similarity of the ta values as computed by the two methods and shown in Tables I and II by means of a linear least.squares analysis of la versus la leads to /aO) = 1.12it(2 ) +4.11 X 10- a with r 2 = 0.98 So we see that it is reasonable to suppose, on various grounds, that the mechanism of senescence is similar in similar organisms. APPENDIX
The concept of vitality [6] is expressed by the equation t
v(t) = e(t) - la f o
e(t) dt
(At)
405
where v(t) is a dimensionless measure for the vitality of an organism, e(t) is given by
m(t) , •
(A2)
M
where m (t) is the weight of the organism at age t and M the Gompertz asymptote of the weight. The ageing rate /a is obtained from clinical observations to 0.017 years-1. We assume that v(T) = 0
(A3)
where T is the life span of the organism. With this assumption we obtain from eqn. AI T
0 = e(T) - I~ f
e(t) dt
(A4)
o
e(t) can be given by Bertalanffy's, Verhulst-Pearl's or Gompertz's equations of growth [8]. Using Verhulst-Pearl's equation Co
e(t) =
(As)
eo + (1 - e o ) e - ~ t
where k is the growth rate (for man 0.25 years-1) and eo = mo/M is the relative birth weight, we obtain from eqn. A4 o = e(r)-
_ EOe k T ~ ~
k
e(T)
(A6)
Putting m (T) = M, eqn. A6 is then of the form 0 = 1 - - y- In
k
(eoekT)
(AT)
From eqn. A7 we obtain the equation T-
1
In eo
which is used in the text (eqn. 15).
REFERENCES 1 G. Sacher and E. Trucco, The stochastic theory of mortality. Ann. N. Y. Acad. $ci., 96 (1962) 985. 2 W. Lohmann, Altersindex und Interpretation der Absterbelmrven. Z. Alternsforsch., 32 (1977) 461-466. 3 R. Rosen, Dynamical similarity and the theory of biological transformations. Bull. Math. Biol., 4 0 (1978) 549-579.
406 4 B. Giinther, On theoriesof biological similarity. Fortschr. Exp. Theor. Biophys., 19 (1975). 5 E. Buckingham, On physically similar systems. Phys. Rev., 4 (1915) 345-370. 6 W. Beier, K.-H. Brehme and D. Wiegel, Biophysikalische Aspekte des Alterns multizellul/irer Systeme. Fortschr. Exp. Theor. Biophys., 16 (1973). 7 A. K. Laird, Postnatal growth of birds and mammals. Growth, 30 (1966) 349. 8 W. Beier, in W. Beier and R. Rose'n, Biophysikalische Grundlagen der Medizin - Beitriige zur Theorienbildung, Fischer, Sttlttgart, p. 502 if, 1980.