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The waste-product theory of aging: Transformation to unlimited growth in cell cultures
Experimental Gerontology, Voi. 24, pp. 97-112, 1989 Printed in the USA. All fights reserved.
0531-5565/89$3.00 + .00 Copyright © 1989PergamonPress pie
THE WASTE-PRODUCT THEORY OF AGING: TRANSFORMATION TO UNLIMITED GROWTH IN CELL CULTURES H . R . HIRSCH, 1 J.A. COOMES 1 a n d M. WITTEN 2 ~Department of Physiology and Biophysics, College of Medicine, University of Kentucky, Lexington, Kentucky 40536-0084 and 2Control Data Corporation, 8100 34th Avenue South, Minneapolis, Minnesota 55440-4700
Abstract - - A differential equation governing intracellular waste content is solved numerically to determine the circumstances under which the growth of an in vitro cell population is limited. Parameter values derived from data on human glial cell cultures are employed. It is assumed that a) waste accumulation depresses the rate of cellular reproduction and b) intracellular waste is diluted by cell division, but is not otherwise eliminated. Population size depends upon two parameters: the rate of waste production and the rate of cell division in the absence of waste. If the rate of waste production is sufficient, the population size approaches an asymptote as in phase III growth in vitro. If a lower rate of waste production allows the cells to outmultiply the waste, growth is unlimited as in a transformed cell population. The asymptotic population size and the threshold for unlimited growth are remarkably sensitive to small changes in the values of the two rate parameters unless the ratio of their values is constant. This suggests that there may be a cellular mechanism that relates the waste production and cell division rates.
Key Words: aging, waste-product theory, transformation, growth dynamics
INTRODUCTION THROUGHOUT MUCH of this century it was believed that somatic cells in culture are capable of unlimited growth. Consequently, explanations of metazoan aging focussed on interactions between cells that, as individuals, were held to be potentially immortal. Experimental evidence accumulated in the 1960s (Hayflick and Moorhead, 1961; Hayflick, 1965) showed that growth of normal diploid fibroblast cultures is intrinsically limited and that the growth limitation is a manifestation of cellular aging. The observation that normal cells age made it mandatory to shift the emphasis of theories of aging to the cellular level.
Correspondence to: H.R. Hirsch (Received 19 October 1987;Accepted 9 June 1988) 97
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A plausible cellular theory of aging can be based on thermodynamic restrictions which make it impossible for an organism to survive in an environment that consists of its own wastes. A form of this waste-product theory based on a very simple model was presented by Sheldrake (1974) and analyzed by Hirsch (1978). The theory proposes that wastes are produced and that cells divide at specific rates. Once produced, the waste material cannot be destroyed and cannot be eliminated by transport across the cell membrane. The cellular waste content increases until, at the time of cell division, the accumulation is apportioned between the daughters. Each daughter receives half of the waste if the division is symmetric. Thus, the waste is diluted by cell division. The waste content in the steady state is determined by the relative rates of waste production and cell division. If cell division ceases, the waste content increases without limit. A first order differential equation governing the waste content per cell was derived with the use of this model (Hirsch, 1978). Witten (1984) extended the model to density-dependent growth with the use of a form of the logistic equation. In Hirsch's (1978) treatment, various assumptions were made concerning the waste production and cell reproduction rates, and solutions of the differential equation for waste content were obtained corresponding to each set of assumptions. Population size was calculated by integrating the reproduction rate. In one instance it was assumed that the waste production rate was constant and that the reproduction rate varied inversely as a power of the intracellular waste content. Asymptotic solutions valid in the limits of large time and large waste content fell into one of two qualitatively different classes: Either a) population size was limited, approaching a plateau value, as in phase III of a normal diploid cell population, or b) population size increased indefinitely as in a transformed cell population (Hayflick and Moorhead, 1961; Hayflick, 1965). The nature of the solution depended on the values of parameters appearing in the differential equation. The asymptotic solutions were obtained analytically and are of interest because they are in qualitative agreement with the growth patterns observed in in vitro cell populations. Quantitative solutions could have been obtained numerically if the function relating cell division rate to waste content had been known. Unfortunately there is no cell population for which this information has been published. However the cell division function can be calculated from data on the fraction of sterile cells in a population by methods which have recently been described (Hirsch, 1986). In the present article, the calculated cell division function and the differential equation governing waste content are used to explore the effects of changes in the waste production and cell division rates on the dynamics of a cell population. Conditions under which transformation from limited to unlimited growth takes place are investigated. The relation of this growth transformation to the more general phase III phenomenon is discussed. METHODS
Dynamic equations The waste content per cell, w'(t), is governed by a differential equation which is essentially an expression of the principle of conservation of matter (Hirsch, 1978): dw
i
d-----i-+ kbw' = rw.
(1)
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I f w ' is given in volume units, for example tt 3, and time, t, in hours, the rate of waste production, rw, is expressed in/z3/hour, and the cell division rate, kb, in hour -1. Both of the rate parameters, rw and kb, may, in general, be functions o f w ' and t. Thus equation (1) may be nonlinear, and its solutions may display rather complex behavior. It will be assumed that cell death is negligible (Hirsch, 1986). Then the relation between the population size, n(t), and the cell division rate is
kb
--
1 dn n dt
(2)
•
If the passage level, P, is defined as the number of times the population size increases by a factor of M,
p=
log[n/n(O)] log M
'
(3)
where n(0) is the population size at time t = 0 (Hirsch and Curtis, 1973). If, for example, M = 2, P is the number of population doublings. If equation (3) referred to a population undergoing serial subcultivation, M would be the split ratio. However it is essential to note that equations (1) and (2) do not describe the step-like growth patterns associated with serial subcultivation. They treat the population as a single culture which grows continuously from time zero. Moreover, w'(t) and n(t) are population averages. Statistical fluctuations in waste content and population size are not treated. Calculated results are valid only in the limit in which the number of cells is large. Thus the condition n(0) = 1 may be adopted in the interest of mathematical simplicity, but it should not be inferred that the corresponding population has been cloned from a single cell. Time can be eliminated between equations (1) and (2) by solving each of them for dt, equating the results, and integrating to obtain w~
ln[
- I rwk°dw kbw' wP(O)
where w'(0) is the initial value of the waste content per cell. Equation (4) is convenient for the interpretation of certain results that will be presented below, but it is not as simple for purposes of numerical computation as equation (1) and (2) from which it was derived. If rw and kb are constants, the integration on the right side of equation (4) can be performed analytically. However, the results are cumbersome and are not presented here.
Waste production rate In order to apply equation (1), it is necessary to specify the waste production rate, rw, and the cell division function, kb(w'). The only data which have been published in quantitative form appropriate for this purpose describe lipofuscin accumulation in cultures of human glial cells of the U-787CG strain. The results reported by Collins and
H.R. HIRSCH et al.
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Brunk (1978) as interpreted by Hirsch (1986) indicate that rw is constant, that is, independent of time and waste content, and that a typical or normal value for the U-787CG strain is 2.033/~3/cell/hour. Cell division f u n c t i o n s
It is convenient to express cell division functions in the form where -kb(w ') = kbo/h (w '),
(5)
h ( w ' ) is a monotonically increasing function such that h(0) = 1, and kbo = kb(O) is the value of the cell division rate in the absence of waste. The requirement that h ( w ' ) be an
increasing function is motivated by the postulate that the cell division rate should decrease as the waste content increases. It is assumed that kb(W') is independent of time. The normalized cell division function is by definition the reciprocal of h(w'): kdkbo = 1/h(w').
(6)
Given that the generation time of human glial cells in culture is 24 hours (Ponten et al., 1969), kbo = (In 2)/24hr -1 = .028881 hr -1. It will be useful in a later discussion of the solutions to equation (1) to replace kb with the form given in equation (5): dw r
dt
- rw - kboW'/h(w').
(1')
D a t a - d e r i v e d f u n c t i o n . Direct measurements of the dependence of cell division rate on
intracellular waste content are possible in principle, but no such data appears in the literature. However the cell division function, kb(w'), can be calculated from data on the fraction of sterile cells found in a population at various passage levels (Hirsch, 1986). Data on the U-787CG strain reported by Blomquist et al. (1980) were used to obtain the cell division function shown in Fig. l(a). This function will be called the data-derived cell division function in order to distinguish it from other, more arbitrary functions to be described below. The values of the data-derived function will be regarded as typical or normal for U-787CG cultures. Quadratic f u n c t i o n . Unexpected results reported here were obtained with the use of the
data-derived cell division function. The question arose whether these results were peculiar to the particular cell division function that was employed. Two other forms, the quadratic and exponential functions, were arbitrarily chosen as alternatives for investigation. Both are special cases of the general form kn =
kbo
j 1 + ~, (w'/wO i
(7)
i=1
where the denominator of equation (7) represents the function h(w'). The requirement that the cell division rate be a monotonically decreasing function of waste content is
CELL TRANSFORMATION BY WASTE PRODUCTS
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k b/ kbo I.O
(a)
.8 .6 .4 .2 0 I 0
I I00
I 200
I 300
I 400
W'ljj 3) FIG. 1. Normalized cells division functions. Ratio of cell division rates kb(w')/kbo vs. waste content w'. (a) Data-derived function. See text. (b) Quadratic function, equation (8), w'2 = 138.03. (c) Exponential function, equation (9), w~ = 187.79.
satisfied if the coefficients, wi, of all non-zero terms in the summation in equation (7) are positive. It has been shown (Hirsch, 1978) that growth levels off at a plateau value i f j > 1 in equation (7). The simplest function for which this is the case is the quadratic:
kn =
kbo
1 + (w'/w2) 2
(8)
I f the value of the p a r a m e t e r w2 is adjusted such that the passage level reaches a plateau at the same Value, P = 42.2 doublings, which was obtained with the dataderived cell division function, the result is w2 = 138.03/z a. The quadratic cell division function c o m p u t e d with the use of this value is shown in Fig. l(b). If, in equation (7), wi= i!/(wx) i and j ---> ~, the cell division function is a declining exponential:
Exponential function.
ka = kooe -w'lw~ •
(9)
I f w~ is adjusted such that P = 42.2 doublings, its value is 187.79. The exponential cell
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P(DOUBLINGS) 40 30 20 I0
0 0
I I000
I 2000
I 3000
I 4000
TIME (HR) FIG. 2. PassagelevelP(t) vs. time. Effectof proportional changesin the rate parameters. (a) Normal parameter values doubled. (b) Normal parameter values. (c) Normal parameter values halved. division function corresponding to this parameter value is plotted in Fig. l(c).
RESULTS The work reported here consists of computer simulations or experiments based on equations (1), (2), and (3) and performed with the use of the IBM Continuous Systems Modelling Program (CSMP). The independent variables are the waste production rate, rw, the cell division rate in the absence of waste, kbo, and the values of the cell division function, kb(w'). These inputs are subject to experimental control. The dependent variables which serve as ob.servable outputs or experimental results are the number of population doublings, P(t), the waste content per cell, w'(t), and the cell division rate, kb(t), all functions of time. Results corresponding to normal or typical culture conditions are presented first. They can be regarded as controls. They are contrasted with other results that represent progressively less normal conditions, for example, transformed populations and growth patterns arising from hypothetical quadratic and exponential cell division functions which have not been observed in nature.
Data-derived cell division function Normal growth. The condition which matches the growth of normal U-787CG cultures
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1500~.. w, (p3)
(a) l
(b)
I000
500 (c) 0 I~o
I
I
I
I
.-
I000
2000
3000
4000
"
TIME (HR) FIG. 3. Waste contentw'(t) vs. time. Effectof proportional changesin the rate parameters. (a) Normalparameter values doubled. (b) Normalparameter values. (c) Normal parameter values halved. most closely is that in which the cell division rate function is data-derived and the rate parameters have the typical values cited previously, namely rw = 2.033/z3/hour and kbo = .028881 hour -1. Plots of P, w', and kb are shown in Figs. 2(b), 3(b), and 4(b), respectively. The population grows exponentially for several hundred hours as indicated by a linear increase in P(t), after which it approaches an asymptote or plateau value of 42.2 doublings. The waste content, after a brief period of rapid growth, levels off at a value of approximately 100 /za during the interval in which the population size increases exponentially. When the population growth slows, the waste content resumes its rapid rate of increase. The cell division rate declines almost linearly with time for the first 1500 h and thereafter approaches zero asymptotically.
Proportional changes in the normal rate parameters. The ratio of the normal rate parameter values, kbo/rw, is .014206. It is clear from equation (4) that changes in the individual parameters that are in the same proportion and which therefore leave this ratio unchanged have no influence on the plateau value of P. However inspection of equation (1) indicates that the rate of increase o f w ' is proportional to the values of the rate parameters. It follows from equation (6) that the rate of decline of the normalized cell division function is also proportional to the rate parameters.
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kb ( HR -I)
.o41.02
b) (c)
0~
0
I000
2000
3000
4000
TIME (HR) FtG. 4. Cell division rate, kdt) vs. time. Effect of proportional changes in the rate parameters. (a) Normal rate parameters doubled. (b) Normal rate parameters. (c) Normal rate parameters halved.
These observations are illustrated by examples shown in Figs. 2, 3, and 4. In each figure, plots (b), (a), and (c) correspond, respectively, to the normal rate-parameter values, to double the normal values, and to half the normal values. All three passage level plots shown in Fig. 2 approach an asymptote at 42.2 doublings, but the approach is twice as rapid in graph (a) as in graph (b), and twice as rapid in graph (b) as in graph (c). By the same token, all three waste content plots in Fig. 3 have the same shape, but the time scale doubles from graph (a) to graph (b) and doubles again from graph (b) to graph (c). The same time scale ratios apply to the cell division rate plots shown in Fig. 4. However it should be noted that the cell division rates are not normalized. Consequently the kb plots become identical only if they are scaled by ratios of 2:1 vertically as well as 1:2 horizontally. Independent changes in the rate parameters. The preceding results show that large but proportional changes in the rate parameters, for example, changes of the order of 100%,
105
CELL TRANSFORMATION BY WASTE PRODUCTS
PIDOUBLiNGS) I00
+4.59%
80 60
+1.14%
0%
40
- 2.29%
20 Oil "
0
I
I
I
I
iooo
2ooo
3000
4000
h~ V
TIME(HR) FIG. 5. Passage level P(t) vs. time. Effect of small changes in the cell division rate parameter kno. Percentages adjacent to curves indicate changes in kbo from its normal value. Limited (normal) growth, -2.29%, 0%, + 1.14%. Unlimited (transformed) growth, +2.29%, +4.59%.
have no influence on the shapes of the passage level or waste content curves predicted by the waste-product model. These changes affect only the time scale with which the dependent variables mature. However the results are quite different if independent changes in kb(w') or rw are allowed. Small independent changes in the values of the input rate parameters can completely change the functional forms of the observable variables P(t) and w'(t) and will, at the least, yield disproportionately large changes in their magnitudes. These changes are illustrated in Figs. 5, 6, and 7. Figure 5 shows that a decrease in koo of 2.29% from its normal value lowers the asymptotic value of population growth from 42.2 to 28.7 doublings, while an increase in kbo of the same amount leads to unlimited growth. Corresponding changes in waste content are evident in Fig. 6. The large rise in waste content which accompanies normal growth takes place much sooner when kbo is reduced. When growth is unlimited, waste content remains indefinitely at a low plateau value. Figure 7 shows that the cell division rate, as expected, remains high in the case of unlimited growth but drops to zero if growth is limited. Results produced by a given percentage change in kbo can, to a very good approximation, be produced by an equal but opposite percentage change in rw. Thus, the results shown in Figs. 5, 6, and 7 would apply to changes in rw rather than kbo if the signs of the percentages with which the curves are labelled were reversed.
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w'(jJ') 50C
÷1.14%
400 500 200 I00
+2.29% +4.59%
0
0
I000
2000 TIME (HR)
I 5000
I 4000
FIG. 6. Waste content w '(t) vs. time. Effect of small changes in the cell division rate parameter kb,,. Percentages adjacent to the curves indicate changes in kbo from its normal value. Limited (normal)growth, -2.29%, 0%, + 1.14%. Unlimited (transformed)growth, +2.29%, +4.59%.
It follows that the solutions of equation (1) are extraordinarily sensitive to changes in either kbo or rw. In effect there is a bifurcation of solutions corresponding to the transformation from limited to unlimited growth.
Arbitrary cell division functions It was surprising to discover that the behavior of the waste-product model is so sensitive to changes in the waste production rate and in the waste-free cell division rate. Since cell populations in the laboratory display a lesser degree of variability in their growth patterns in response to changes in experimental conditions, it became desirable to re-examine the cell division function. The cell division function, kb(w'), provides the only input to the model apart from the values of rw and k~o, so it was natural to ask whether there is something peculiar to the shape of the data-derived function that might lead to the observed sensitivity of the model results to the two rate parameters. In order to investigate the dependence of the results on the shape of the cell division function, it became necessary to repeat the computer simulation experiments with the use of cell division functions having shapes that differ significantly from the dataderived function shown in Fig. l(a). Since no data were available on which to base
CELL TRANSFORMATION BY WASTE PRODUCTS
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kb(HR-I) +4.59%
+ 2.29 .02
+1.14%
.01 - 2 . 2 9 °h
0 1
0
I000
2000
3000
4000
V
TIME(HR) FIG. 7. Cell division rate kb(t) vs. time. Effect of small changes in the cell division rate parameter k~o. Percentages adjacent to the curves indicate changes in kbo from its normal value. Limited (normal) growth, -2.29%, 0%, + 1.14%. Unlimited (transformed) growth, +2.29%, +4.59%.
functions of alternative shapes, the quadratic function, equation (8), Fig. l(b), and the exponential function, equation (9), Fig. l(c), were arbitrarily selected for this purpose. Their shapes are markedly different from each other and from the data-derived function, and their only relation to the data used to obtain the latter function is through the single-parameter adjustment to the plateau value of P which was previously described. Results obtained with the use of the quadratic and exponential cell division functions differed quantitatively from one another and from those obtained from the data-derived function. However the bifurcation of the solutions to equation (1) and the high degree of sensitivity of the model behavior to small changes in the rate parameters persisted. In the quadratic case, an increase in kbo or a decrease in rw of 2.00% produced a change from growth limited at the normal value of 42.2 doublings to unlimited growth. Rateparameter changes of 2.00% in the opposite directions lead to growth limitation at 28.5 doublings. In the exponential case, similar results were produced by rate-parameter
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changes of 1.90%. These results show that the sensitivity of population size and waste content to changes in the rate parameters cannot be attributed to the shape of the cell division function. DISCUSSION Aging in cells that have not undergone transformation leads first to the cessation of reproduction and, eventually, to metabolic death. Only the former characteristic of the aging process is considered here. The waste-product theory in the general form embodied in equation (1) is not specific to any particular kind of waste (Hirsch, 1978). The waste substance need only be indestructable, incapable of transport across the cell membrane, and capable of depressing the cell division rate. Since the pigment lipofuscin gradually accumulates in a wide variety of cells (Strehler, 1977), it is the waste substance most often associated with intracellular aging. However, the theory itself is applicable to entities as far removed from lipofuscin as the repressor molecules which, according to the codon-restriction theory (Strehler et al., 1971), are found on the surface of a cell in an aging clone. The results reported here are much more limited in scope; they apply directly only to the accumulation of lipofuscin in cultures of human glial cells. However it is not unreasonable to suppose that similar results would be obtained with cultures of other mammalian cells, for example, human fibroblasts. It is equally reasonable to assume that experimental findings based on fibroblast cultures are applicable to glial cells. Thus, studies which show that the intracellular chronometer is situated in the nucleus (Hayflick, 1984) must be reconciled with the cytoplasmic location of the lipofuscin. It must therefore be assumed that lipofuscin production, if relevant to cellular aging, is under nuclear control. Conversely, the modulation of the cell division rate by the waste content, if, in fact, it occurs, implies that cytoplasmic events influence the nuclear program. The genetic apparatus and the waste can be regarded as components of a closedloop feedback system. Cause and effect are difficuk to distinguish in the absence of experimental techniques which would allow the feedback loop to be opened. It would, for example, be informative but not easy to change intracellular waste content with the use of methods which, in themselves, have no effect on the cellular reproduction rate. No such results are available at this time. Thus it is clear that cell division influences waste accumulation through the dilution mechanism (Collins and Brunk, 1976, 1978) but it is uncertain whether waste content exerts a reciprocal influence on the cell division rate (Deamer and Gonzales, 1974). The question whether lipofuscin affects cellular performance and what its role may be remains open (Davies and Fotheringham, 1981). However it is possible that the influence of lipofuscin on cellular reproduction has been underestimated. If cellular reproduction were to display great sensitivity to waste content, the small amount of waste required to affect the cell division rate might remain undetected. Examination of the data-derived cell division function (Fig. l(a)) suggests (Hirsch, 1986) that the amount of lipofuscin required to depress the cell division rate significantly may be much smaller than the amount which a functional nondividing cell can tolerate. A waste content of 131 /x3 is associated with a 50% reduction in the cell division rate relative to the rate which obtains when there is no waste. Nondividing cells survive waste levels many times this value (Strehler, 1977; Collins and Brunk, 1978). The corresponding widths at half height of the quadratic (Fig. l(b)) and exponential
CELL TRANSFORMATION BY WASTE PRODUCTS
d
.5
IL
109
dw'/dt (p3/HR)
--
II
.4
(a)
(b)
.5 .2-
(c)'l 0
I o
I I w max
ioo
I zoo
I
3o0
v
w'(p
FIG. 8. Rate of change of waste content dw'/dt vs. waste content w'. Effect of small changes in the waste production rate parameter r~. (a) rw greater than normal by 2.29%. (b) rw normal. (c) r~ less than normal by 4.59%.
(Fig. l(c)) cell division functions are 138/z ~ and 130/z 3, respectively. The degree of agreement between the three values of width is remarkable in view of a) the differences in the shapes of the functions and b) the differences between the way in which the data-derived function was obtained and the way in which the adjustable parameters in the other cell division functions were fitted. The agreement among the widths provides additional support for the earlier conclusion (Hirsch, 1986) that relatively low waste levels may have large effects on cell division rates.
Transformation Hayflick and Moorhead (1961) list 11 differential characteristics for human cell lines and cell strains. The results presented here relate to only one of these characteristics, namely, the unlimited growth of the transformed lines, which stands in contrast with the limited growth of the normal strains.
1 10
H.R. HIRSCHet
al.
The bifurcation of the solutions to equation (1), that is, the existence of two classes of solution, one corresponding to unlimited and the other to limited growth, can most easily be understood by examining equation (1'). Assuming that the cells are waste-free to start with, the initial waste buildup, dw'(O)/dt, is at the maximum possible rate, rw. As w' increases, dw'/dt declines. Initially the decline is linear with w ' and takes place at the rate kbo, but the rate slows as h(w') increases. As shown in Fig. 8, there are two possibilities, corresponding to two classes of solution to equation (1') or, what is equivalent, to equation (1): 1) Figs. 8(a) and (b). The function h(w') increases sufficiently rapidly with w' that the decline in dw'/dt stops before the condition dw'/dt = 0 obtains, that is, dw'/dt > 0 for all w'. The dw'/dt vs. w' function passes through a minimum and asymptotically approaches its initial level, rw, at large values of w'. Only the minimum, and not the large initial and asymptotic, values, are displayed in Fig. 8 because of scale limitations. Since dw'/dt is always positive, waste accumulates indefinitely. The cell division rate consequently decreases to zero, and growth is limited, as it would be in a normal cell strain. 2) Fig. 8(c). The increase of h(w') with w' is inadequate to avert the condition dw'/dt = 0. For some value of w' W'max, waste accumulation ceases. Cell division continues indefinitely at the rate kb(w'max). The population grows without limit as it would in a transformed cell line. The existence of these two classes of solution explains the separation of the population growth, waste content, and cell division functions, illustrated in Figs. 5, 6, and 7, respectively, into qualitatively different normal and transformed sets. Each set is characterized by the shape of the dw/dt vs. w' function with which it is associated. Functions displaying minima, for example, Figs. 8(a) and (b), give rise to normal behavior, while functions that terminate, for example, Fig. 8(c), describe transformation.
Sensitivity to rate parameters The great sensitivity of the model behavior to small changes in the rate parameters rw and kbo can be explained by noting that the denominator of the integrand on the righthand side of equation (4) is identical to the dw'/dt vs. w' function given in equation (1) and discussed above in connection with transformed and normal growth. The integrand becomes large for values ofw' at which its denominator, that is, dw/dt, is small. Thus, a small difference in one of the rate parameters can be responsible for a large difference in the population size by influencing how closely the minimum of the dw/dt vs. w' function approaches zero. These considerations are illustrated in Fig. 8. Curves (a) and (b) appear closely similar even though the vertical scale is enlarged to such an extent that the upper 80% of each has been omitted. Both were generated by the same cell-division function and differ only in that the waste rate parameter for curve (a) is greater by 2.29% than that for curve (b). This small difference is, nevertheless, of great importance because the minimum value of curve (b) is half as great as the minimum of curve (a). The maximum of the integrand is approximately twice as great, and, as a result, 42.2, rather than 28.7 population doublings occur. In the case illustrated in curve (c), in which the minimum value ofdw'/dt is zero, rw is less than in curve (b) by 4.59%, but the integral in equation (4) diverges, and the population increases without limit. The extraordinary sensitivity of the population size and waste content to changes in
CELL TRANSFORMATION BY WASTE PRODUCTS
111
the rate parameters can be circumvented only if the rate parameters undergo changes which are approximately proportionate. Perfect proportionality is unnecessary. Small deviations from exact maintenance o f the kbo/rw ratio would allow modelling of the transformation process and of changes in the asymptotic doubling level attained in phase III. H o w e v e r , the sensitivity of the model outputs to independent changes in kbo and rw cannot be reconciled with the random changes which must occur in these parameters due to the normal variability of biological processes. With such great sensitivity, modest fluctuations in the rate parameters would be translated into extremely erratic variations in population size and waste content. One or more of the following considerations may serve to explain this inconsistency: 1. Lipofuscin accumulation may be a consequence rather than a primary cause o f cellular aging (Hayflick, 1985), that is, the waste-product theory may be incorrect. 2. N o n e o f the three cell division functions which were studied is directly based on data. If the validity of the waste product theory is ever to be determined, data on waste accumulation in dividing populations is badly needed. 3. The rate of waste production and the rate o f cell division in the absence of waste may be controlled in c o m m o n by a mechanism which maintains their ratio approximately constant. An investigation of a metabolic model embodying this concept is in progress. Relevant data would be of great interest. The relationship between the cellular metabolic rate and waste accumulation is neither simple nor linear. If waste accumulation were directly proportional to metabolic rate or time, waste product theories would be o f little value because the weight o f available evidence (Hayflick, 1977) shows that in vitro senescence is governed by the number of population doublings that the cell population has experienced. The metabolic model to which reference was made above contains reciprocal nonlinear interactions among variables representing cell division rate, waste accumulation, and metabolic rate. Ongoing research on the model indicates that the waste product theory in the form presented here is consistent with models of senescence based on either metabolic time or on a cell division counter but does not provide explicit support for one or the other.
REFERENCES BLOMQUIST, E., WESTERMARK, B., and PONTI~N, J. Aging of human giial cells in culture: Increase in the fraction of non-dividers as demonstrated by a microcloningtechnique. Mech. Ageing Dev. 12, 173-182, 1980. COLLINS, V.P. and BRUNK, U. Characterization of residual bodies formed in phase II cultivated human glial cells. Mech. Ageing Dev. 5, 193-207, 1976. COLLINS, V.P. and BRUNK, U. Quantitation of residual bodies in cultured human glial cells during stationary and logarithmic growth phases. Mech. Ageing Dev. 8, 139-152, 1978. DAVIES, I. and FOTHERINGHAM, A.P. Lipofuscin -- Does it affect cellular performance?Exp. Gerontol. 16, 119-125, 1981. DEAMER, D.W. and GONZALES, J. Autofluorescent structures in cultured WI-38 cells. Arch. Biochem. Biophys. 165, 421-426, 1974. HAYFLICK, L. The limited in vitro lifetime of human diploid cell strains. Exp. Cell Res. 37, 614--636, 1965. HAYFLICK, L. The cellular basis for biologicalaging. In: Handbook of the Biology of Aging, 1st ed., Finch, C.E. and Hayflick, L. (Editors), pp. 159-186, van Nostrand Reinhold, New York, NY, 1977.
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HAYFLICK, L. Intracellular determinants of cell aging. Mech. Ageing Dev. 28, 177-185, 1984. HAYFLICK, L. Theories of biological aging. Exp. Gerontol. 20, 145-159, 1985. HAYFLICK, L. and MOORHEAD, P.S. The serial cultivation of human diploid cell strains. Exp. Cell Res. 25, 585-621, 1961. HIRSCH, H.R. The waste-product theory of aging: Waste dilution by cell division. Mech. Ageing Dev. 8, 51-62, 1978. HIRSCH, H.R. The waste-product theory of aging: Cell division rate as a function of waste volume. Mech. Ageing Dev. 36, 95-107, 1986. HIRSCH, H.R. and CURTIS, H.J. Dynamics of growth in mammalian diploid tissue cultures. J. Theor. Biol. 42, 227-244, 1973. PONTI~N, J., WESTERMARK, B., and HUGOSSON, R. Regulation of proliferation and movement of human glia-like cells in culture. Exp. Cell Res. 58, 393-400, 1969. SHELDRAKE, A.R. The ageing, growth, and death of cells. Nature 250, 381-385, 1974. STREHLER, B.L. Time, Cells, and Aging, 2nd ed. Academic, New York, NY, 1977. STREHLER, B.L., HIRSCH, G., GUSSECK, D., JOHNSON, R., and BICK, M. Codon-restriction theory of aging and development. J. Theor. Biol. 33, 429--474, 1971. WlTTEN, M. Mathematics of molecular aging mechanisms. In: Molecular Basis o f Aging, Roy, A.K. and Chatterjee, B. (Editors), pp. 355-393, Academic Press, Orlando, FL, 1984.
0531-5565/89$3.00 + .00 Copyright © 1989PergamonPress pie
THE WASTE-PRODUCT THEORY OF AGING: TRANSFORMATION TO UNLIMITED GROWTH IN CELL CULTURES H . R . HIRSCH, 1 J.A. COOMES 1 a n d M. WITTEN 2 ~Department of Physiology and Biophysics, College of Medicine, University of Kentucky, Lexington, Kentucky 40536-0084 and 2Control Data Corporation, 8100 34th Avenue South, Minneapolis, Minnesota 55440-4700
Abstract - - A differential equation governing intracellular waste content is solved numerically to determine the circumstances under which the growth of an in vitro cell population is limited. Parameter values derived from data on human glial cell cultures are employed. It is assumed that a) waste accumulation depresses the rate of cellular reproduction and b) intracellular waste is diluted by cell division, but is not otherwise eliminated. Population size depends upon two parameters: the rate of waste production and the rate of cell division in the absence of waste. If the rate of waste production is sufficient, the population size approaches an asymptote as in phase III growth in vitro. If a lower rate of waste production allows the cells to outmultiply the waste, growth is unlimited as in a transformed cell population. The asymptotic population size and the threshold for unlimited growth are remarkably sensitive to small changes in the values of the two rate parameters unless the ratio of their values is constant. This suggests that there may be a cellular mechanism that relates the waste production and cell division rates.
Key Words: aging, waste-product theory, transformation, growth dynamics
INTRODUCTION THROUGHOUT MUCH of this century it was believed that somatic cells in culture are capable of unlimited growth. Consequently, explanations of metazoan aging focussed on interactions between cells that, as individuals, were held to be potentially immortal. Experimental evidence accumulated in the 1960s (Hayflick and Moorhead, 1961; Hayflick, 1965) showed that growth of normal diploid fibroblast cultures is intrinsically limited and that the growth limitation is a manifestation of cellular aging. The observation that normal cells age made it mandatory to shift the emphasis of theories of aging to the cellular level.
Correspondence to: H.R. Hirsch (Received 19 October 1987;Accepted 9 June 1988) 97
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A plausible cellular theory of aging can be based on thermodynamic restrictions which make it impossible for an organism to survive in an environment that consists of its own wastes. A form of this waste-product theory based on a very simple model was presented by Sheldrake (1974) and analyzed by Hirsch (1978). The theory proposes that wastes are produced and that cells divide at specific rates. Once produced, the waste material cannot be destroyed and cannot be eliminated by transport across the cell membrane. The cellular waste content increases until, at the time of cell division, the accumulation is apportioned between the daughters. Each daughter receives half of the waste if the division is symmetric. Thus, the waste is diluted by cell division. The waste content in the steady state is determined by the relative rates of waste production and cell division. If cell division ceases, the waste content increases without limit. A first order differential equation governing the waste content per cell was derived with the use of this model (Hirsch, 1978). Witten (1984) extended the model to density-dependent growth with the use of a form of the logistic equation. In Hirsch's (1978) treatment, various assumptions were made concerning the waste production and cell reproduction rates, and solutions of the differential equation for waste content were obtained corresponding to each set of assumptions. Population size was calculated by integrating the reproduction rate. In one instance it was assumed that the waste production rate was constant and that the reproduction rate varied inversely as a power of the intracellular waste content. Asymptotic solutions valid in the limits of large time and large waste content fell into one of two qualitatively different classes: Either a) population size was limited, approaching a plateau value, as in phase III of a normal diploid cell population, or b) population size increased indefinitely as in a transformed cell population (Hayflick and Moorhead, 1961; Hayflick, 1965). The nature of the solution depended on the values of parameters appearing in the differential equation. The asymptotic solutions were obtained analytically and are of interest because they are in qualitative agreement with the growth patterns observed in in vitro cell populations. Quantitative solutions could have been obtained numerically if the function relating cell division rate to waste content had been known. Unfortunately there is no cell population for which this information has been published. However the cell division function can be calculated from data on the fraction of sterile cells in a population by methods which have recently been described (Hirsch, 1986). In the present article, the calculated cell division function and the differential equation governing waste content are used to explore the effects of changes in the waste production and cell division rates on the dynamics of a cell population. Conditions under which transformation from limited to unlimited growth takes place are investigated. The relation of this growth transformation to the more general phase III phenomenon is discussed. METHODS
Dynamic equations The waste content per cell, w'(t), is governed by a differential equation which is essentially an expression of the principle of conservation of matter (Hirsch, 1978): dw
i
d-----i-+ kbw' = rw.
(1)
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I f w ' is given in volume units, for example tt 3, and time, t, in hours, the rate of waste production, rw, is expressed in/z3/hour, and the cell division rate, kb, in hour -1. Both of the rate parameters, rw and kb, may, in general, be functions o f w ' and t. Thus equation (1) may be nonlinear, and its solutions may display rather complex behavior. It will be assumed that cell death is negligible (Hirsch, 1986). Then the relation between the population size, n(t), and the cell division rate is
kb
--
1 dn n dt
(2)
•
If the passage level, P, is defined as the number of times the population size increases by a factor of M,
p=
log[n/n(O)] log M
'
(3)
where n(0) is the population size at time t = 0 (Hirsch and Curtis, 1973). If, for example, M = 2, P is the number of population doublings. If equation (3) referred to a population undergoing serial subcultivation, M would be the split ratio. However it is essential to note that equations (1) and (2) do not describe the step-like growth patterns associated with serial subcultivation. They treat the population as a single culture which grows continuously from time zero. Moreover, w'(t) and n(t) are population averages. Statistical fluctuations in waste content and population size are not treated. Calculated results are valid only in the limit in which the number of cells is large. Thus the condition n(0) = 1 may be adopted in the interest of mathematical simplicity, but it should not be inferred that the corresponding population has been cloned from a single cell. Time can be eliminated between equations (1) and (2) by solving each of them for dt, equating the results, and integrating to obtain w~
ln[
- I rwk°dw kbw' wP(O)
where w'(0) is the initial value of the waste content per cell. Equation (4) is convenient for the interpretation of certain results that will be presented below, but it is not as simple for purposes of numerical computation as equation (1) and (2) from which it was derived. If rw and kb are constants, the integration on the right side of equation (4) can be performed analytically. However, the results are cumbersome and are not presented here.
Waste production rate In order to apply equation (1), it is necessary to specify the waste production rate, rw, and the cell division function, kb(w'). The only data which have been published in quantitative form appropriate for this purpose describe lipofuscin accumulation in cultures of human glial cells of the U-787CG strain. The results reported by Collins and
H.R. HIRSCH et al.
100
Brunk (1978) as interpreted by Hirsch (1986) indicate that rw is constant, that is, independent of time and waste content, and that a typical or normal value for the U-787CG strain is 2.033/~3/cell/hour. Cell division f u n c t i o n s
It is convenient to express cell division functions in the form where -kb(w ') = kbo/h (w '),
(5)
h ( w ' ) is a monotonically increasing function such that h(0) = 1, and kbo = kb(O) is the value of the cell division rate in the absence of waste. The requirement that h ( w ' ) be an
increasing function is motivated by the postulate that the cell division rate should decrease as the waste content increases. It is assumed that kb(W') is independent of time. The normalized cell division function is by definition the reciprocal of h(w'): kdkbo = 1/h(w').
(6)
Given that the generation time of human glial cells in culture is 24 hours (Ponten et al., 1969), kbo = (In 2)/24hr -1 = .028881 hr -1. It will be useful in a later discussion of the solutions to equation (1) to replace kb with the form given in equation (5): dw r
dt
- rw - kboW'/h(w').
(1')
D a t a - d e r i v e d f u n c t i o n . Direct measurements of the dependence of cell division rate on
intracellular waste content are possible in principle, but no such data appears in the literature. However the cell division function, kb(w'), can be calculated from data on the fraction of sterile cells found in a population at various passage levels (Hirsch, 1986). Data on the U-787CG strain reported by Blomquist et al. (1980) were used to obtain the cell division function shown in Fig. l(a). This function will be called the data-derived cell division function in order to distinguish it from other, more arbitrary functions to be described below. The values of the data-derived function will be regarded as typical or normal for U-787CG cultures. Quadratic f u n c t i o n . Unexpected results reported here were obtained with the use of the
data-derived cell division function. The question arose whether these results were peculiar to the particular cell division function that was employed. Two other forms, the quadratic and exponential functions, were arbitrarily chosen as alternatives for investigation. Both are special cases of the general form kn =
kbo
j 1 + ~, (w'/wO i
(7)
i=1
where the denominator of equation (7) represents the function h(w'). The requirement that the cell division rate be a monotonically decreasing function of waste content is
CELL TRANSFORMATION BY WASTE PRODUCTS
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k b/ kbo I.O
(a)
.8 .6 .4 .2 0 I 0
I I00
I 200
I 300
I 400
W'ljj 3) FIG. 1. Normalized cells division functions. Ratio of cell division rates kb(w')/kbo vs. waste content w'. (a) Data-derived function. See text. (b) Quadratic function, equation (8), w'2 = 138.03. (c) Exponential function, equation (9), w~ = 187.79.
satisfied if the coefficients, wi, of all non-zero terms in the summation in equation (7) are positive. It has been shown (Hirsch, 1978) that growth levels off at a plateau value i f j > 1 in equation (7). The simplest function for which this is the case is the quadratic:
kn =
kbo
1 + (w'/w2) 2
(8)
I f the value of the p a r a m e t e r w2 is adjusted such that the passage level reaches a plateau at the same Value, P = 42.2 doublings, which was obtained with the dataderived cell division function, the result is w2 = 138.03/z a. The quadratic cell division function c o m p u t e d with the use of this value is shown in Fig. l(b). If, in equation (7), wi= i!/(wx) i and j ---> ~, the cell division function is a declining exponential:
Exponential function.
ka = kooe -w'lw~ •
(9)
I f w~ is adjusted such that P = 42.2 doublings, its value is 187.79. The exponential cell
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P(DOUBLINGS) 40 30 20 I0
0 0
I I000
I 2000
I 3000
I 4000
TIME (HR) FIG. 2. PassagelevelP(t) vs. time. Effectof proportional changesin the rate parameters. (a) Normal parameter values doubled. (b) Normal parameter values. (c) Normal parameter values halved. division function corresponding to this parameter value is plotted in Fig. l(c).
RESULTS The work reported here consists of computer simulations or experiments based on equations (1), (2), and (3) and performed with the use of the IBM Continuous Systems Modelling Program (CSMP). The independent variables are the waste production rate, rw, the cell division rate in the absence of waste, kbo, and the values of the cell division function, kb(w'). These inputs are subject to experimental control. The dependent variables which serve as ob.servable outputs or experimental results are the number of population doublings, P(t), the waste content per cell, w'(t), and the cell division rate, kb(t), all functions of time. Results corresponding to normal or typical culture conditions are presented first. They can be regarded as controls. They are contrasted with other results that represent progressively less normal conditions, for example, transformed populations and growth patterns arising from hypothetical quadratic and exponential cell division functions which have not been observed in nature.
Data-derived cell division function Normal growth. The condition which matches the growth of normal U-787CG cultures
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1500~.. w, (p3)
(a) l
(b)
I000
500 (c) 0 I~o
I
I
I
I
.-
I000
2000
3000
4000
"
TIME (HR) FIG. 3. Waste contentw'(t) vs. time. Effectof proportional changesin the rate parameters. (a) Normalparameter values doubled. (b) Normalparameter values. (c) Normal parameter values halved. most closely is that in which the cell division rate function is data-derived and the rate parameters have the typical values cited previously, namely rw = 2.033/z3/hour and kbo = .028881 hour -1. Plots of P, w', and kb are shown in Figs. 2(b), 3(b), and 4(b), respectively. The population grows exponentially for several hundred hours as indicated by a linear increase in P(t), after which it approaches an asymptote or plateau value of 42.2 doublings. The waste content, after a brief period of rapid growth, levels off at a value of approximately 100 /za during the interval in which the population size increases exponentially. When the population growth slows, the waste content resumes its rapid rate of increase. The cell division rate declines almost linearly with time for the first 1500 h and thereafter approaches zero asymptotically.
Proportional changes in the normal rate parameters. The ratio of the normal rate parameter values, kbo/rw, is .014206. It is clear from equation (4) that changes in the individual parameters that are in the same proportion and which therefore leave this ratio unchanged have no influence on the plateau value of P. However inspection of equation (1) indicates that the rate of increase o f w ' is proportional to the values of the rate parameters. It follows from equation (6) that the rate of decline of the normalized cell division function is also proportional to the rate parameters.
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kb ( HR -I)
.o41.02
b) (c)
0~
0
I000
2000
3000
4000
TIME (HR) FtG. 4. Cell division rate, kdt) vs. time. Effect of proportional changes in the rate parameters. (a) Normal rate parameters doubled. (b) Normal rate parameters. (c) Normal rate parameters halved.
These observations are illustrated by examples shown in Figs. 2, 3, and 4. In each figure, plots (b), (a), and (c) correspond, respectively, to the normal rate-parameter values, to double the normal values, and to half the normal values. All three passage level plots shown in Fig. 2 approach an asymptote at 42.2 doublings, but the approach is twice as rapid in graph (a) as in graph (b), and twice as rapid in graph (b) as in graph (c). By the same token, all three waste content plots in Fig. 3 have the same shape, but the time scale doubles from graph (a) to graph (b) and doubles again from graph (b) to graph (c). The same time scale ratios apply to the cell division rate plots shown in Fig. 4. However it should be noted that the cell division rates are not normalized. Consequently the kb plots become identical only if they are scaled by ratios of 2:1 vertically as well as 1:2 horizontally. Independent changes in the rate parameters. The preceding results show that large but proportional changes in the rate parameters, for example, changes of the order of 100%,
105
CELL TRANSFORMATION BY WASTE PRODUCTS
PIDOUBLiNGS) I00
+4.59%
80 60
+1.14%
0%
40
- 2.29%
20 Oil "
0
I
I
I
I
iooo
2ooo
3000
4000
h~ V
TIME(HR) FIG. 5. Passage level P(t) vs. time. Effect of small changes in the cell division rate parameter kno. Percentages adjacent to curves indicate changes in kbo from its normal value. Limited (normal) growth, -2.29%, 0%, + 1.14%. Unlimited (transformed) growth, +2.29%, +4.59%.
have no influence on the shapes of the passage level or waste content curves predicted by the waste-product model. These changes affect only the time scale with which the dependent variables mature. However the results are quite different if independent changes in kb(w') or rw are allowed. Small independent changes in the values of the input rate parameters can completely change the functional forms of the observable variables P(t) and w'(t) and will, at the least, yield disproportionately large changes in their magnitudes. These changes are illustrated in Figs. 5, 6, and 7. Figure 5 shows that a decrease in koo of 2.29% from its normal value lowers the asymptotic value of population growth from 42.2 to 28.7 doublings, while an increase in kbo of the same amount leads to unlimited growth. Corresponding changes in waste content are evident in Fig. 6. The large rise in waste content which accompanies normal growth takes place much sooner when kbo is reduced. When growth is unlimited, waste content remains indefinitely at a low plateau value. Figure 7 shows that the cell division rate, as expected, remains high in the case of unlimited growth but drops to zero if growth is limited. Results produced by a given percentage change in kbo can, to a very good approximation, be produced by an equal but opposite percentage change in rw. Thus, the results shown in Figs. 5, 6, and 7 would apply to changes in rw rather than kbo if the signs of the percentages with which the curves are labelled were reversed.
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w'(jJ') 50C
÷1.14%
400 500 200 I00
+2.29% +4.59%
0
0
I000
2000 TIME (HR)
I 5000
I 4000
FIG. 6. Waste content w '(t) vs. time. Effect of small changes in the cell division rate parameter kb,,. Percentages adjacent to the curves indicate changes in kbo from its normal value. Limited (normal)growth, -2.29%, 0%, + 1.14%. Unlimited (transformed)growth, +2.29%, +4.59%.
It follows that the solutions of equation (1) are extraordinarily sensitive to changes in either kbo or rw. In effect there is a bifurcation of solutions corresponding to the transformation from limited to unlimited growth.
Arbitrary cell division functions It was surprising to discover that the behavior of the waste-product model is so sensitive to changes in the waste production rate and in the waste-free cell division rate. Since cell populations in the laboratory display a lesser degree of variability in their growth patterns in response to changes in experimental conditions, it became desirable to re-examine the cell division function. The cell division function, kb(w'), provides the only input to the model apart from the values of rw and k~o, so it was natural to ask whether there is something peculiar to the shape of the data-derived function that might lead to the observed sensitivity of the model results to the two rate parameters. In order to investigate the dependence of the results on the shape of the cell division function, it became necessary to repeat the computer simulation experiments with the use of cell division functions having shapes that differ significantly from the dataderived function shown in Fig. l(a). Since no data were available on which to base
CELL TRANSFORMATION BY WASTE PRODUCTS
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kb(HR-I) +4.59%
+ 2.29 .02
+1.14%
.01 - 2 . 2 9 °h
0 1
0
I000
2000
3000
4000
V
TIME(HR) FIG. 7. Cell division rate kb(t) vs. time. Effect of small changes in the cell division rate parameter k~o. Percentages adjacent to the curves indicate changes in kbo from its normal value. Limited (normal) growth, -2.29%, 0%, + 1.14%. Unlimited (transformed) growth, +2.29%, +4.59%.
functions of alternative shapes, the quadratic function, equation (8), Fig. l(b), and the exponential function, equation (9), Fig. l(c), were arbitrarily selected for this purpose. Their shapes are markedly different from each other and from the data-derived function, and their only relation to the data used to obtain the latter function is through the single-parameter adjustment to the plateau value of P which was previously described. Results obtained with the use of the quadratic and exponential cell division functions differed quantitatively from one another and from those obtained from the data-derived function. However the bifurcation of the solutions to equation (1) and the high degree of sensitivity of the model behavior to small changes in the rate parameters persisted. In the quadratic case, an increase in kbo or a decrease in rw of 2.00% produced a change from growth limited at the normal value of 42.2 doublings to unlimited growth. Rateparameter changes of 2.00% in the opposite directions lead to growth limitation at 28.5 doublings. In the exponential case, similar results were produced by rate-parameter
108
H.R. HIRSCH et al.
changes of 1.90%. These results show that the sensitivity of population size and waste content to changes in the rate parameters cannot be attributed to the shape of the cell division function. DISCUSSION Aging in cells that have not undergone transformation leads first to the cessation of reproduction and, eventually, to metabolic death. Only the former characteristic of the aging process is considered here. The waste-product theory in the general form embodied in equation (1) is not specific to any particular kind of waste (Hirsch, 1978). The waste substance need only be indestructable, incapable of transport across the cell membrane, and capable of depressing the cell division rate. Since the pigment lipofuscin gradually accumulates in a wide variety of cells (Strehler, 1977), it is the waste substance most often associated with intracellular aging. However, the theory itself is applicable to entities as far removed from lipofuscin as the repressor molecules which, according to the codon-restriction theory (Strehler et al., 1971), are found on the surface of a cell in an aging clone. The results reported here are much more limited in scope; they apply directly only to the accumulation of lipofuscin in cultures of human glial cells. However it is not unreasonable to suppose that similar results would be obtained with cultures of other mammalian cells, for example, human fibroblasts. It is equally reasonable to assume that experimental findings based on fibroblast cultures are applicable to glial cells. Thus, studies which show that the intracellular chronometer is situated in the nucleus (Hayflick, 1984) must be reconciled with the cytoplasmic location of the lipofuscin. It must therefore be assumed that lipofuscin production, if relevant to cellular aging, is under nuclear control. Conversely, the modulation of the cell division rate by the waste content, if, in fact, it occurs, implies that cytoplasmic events influence the nuclear program. The genetic apparatus and the waste can be regarded as components of a closedloop feedback system. Cause and effect are difficuk to distinguish in the absence of experimental techniques which would allow the feedback loop to be opened. It would, for example, be informative but not easy to change intracellular waste content with the use of methods which, in themselves, have no effect on the cellular reproduction rate. No such results are available at this time. Thus it is clear that cell division influences waste accumulation through the dilution mechanism (Collins and Brunk, 1976, 1978) but it is uncertain whether waste content exerts a reciprocal influence on the cell division rate (Deamer and Gonzales, 1974). The question whether lipofuscin affects cellular performance and what its role may be remains open (Davies and Fotheringham, 1981). However it is possible that the influence of lipofuscin on cellular reproduction has been underestimated. If cellular reproduction were to display great sensitivity to waste content, the small amount of waste required to affect the cell division rate might remain undetected. Examination of the data-derived cell division function (Fig. l(a)) suggests (Hirsch, 1986) that the amount of lipofuscin required to depress the cell division rate significantly may be much smaller than the amount which a functional nondividing cell can tolerate. A waste content of 131 /x3 is associated with a 50% reduction in the cell division rate relative to the rate which obtains when there is no waste. Nondividing cells survive waste levels many times this value (Strehler, 1977; Collins and Brunk, 1978). The corresponding widths at half height of the quadratic (Fig. l(b)) and exponential
CELL TRANSFORMATION BY WASTE PRODUCTS
d
.5
IL
109
dw'/dt (p3/HR)
--
II
.4
(a)
(b)
.5 .2-
(c)'l 0
I o
I I w max
ioo
I zoo
I
3o0
v
w'(p
FIG. 8. Rate of change of waste content dw'/dt vs. waste content w'. Effect of small changes in the waste production rate parameter r~. (a) rw greater than normal by 2.29%. (b) rw normal. (c) r~ less than normal by 4.59%.
(Fig. l(c)) cell division functions are 138/z ~ and 130/z 3, respectively. The degree of agreement between the three values of width is remarkable in view of a) the differences in the shapes of the functions and b) the differences between the way in which the data-derived function was obtained and the way in which the adjustable parameters in the other cell division functions were fitted. The agreement among the widths provides additional support for the earlier conclusion (Hirsch, 1986) that relatively low waste levels may have large effects on cell division rates.
Transformation Hayflick and Moorhead (1961) list 11 differential characteristics for human cell lines and cell strains. The results presented here relate to only one of these characteristics, namely, the unlimited growth of the transformed lines, which stands in contrast with the limited growth of the normal strains.
1 10
H.R. HIRSCHet
al.
The bifurcation of the solutions to equation (1), that is, the existence of two classes of solution, one corresponding to unlimited and the other to limited growth, can most easily be understood by examining equation (1'). Assuming that the cells are waste-free to start with, the initial waste buildup, dw'(O)/dt, is at the maximum possible rate, rw. As w' increases, dw'/dt declines. Initially the decline is linear with w ' and takes place at the rate kbo, but the rate slows as h(w') increases. As shown in Fig. 8, there are two possibilities, corresponding to two classes of solution to equation (1') or, what is equivalent, to equation (1): 1) Figs. 8(a) and (b). The function h(w') increases sufficiently rapidly with w' that the decline in dw'/dt stops before the condition dw'/dt = 0 obtains, that is, dw'/dt > 0 for all w'. The dw'/dt vs. w' function passes through a minimum and asymptotically approaches its initial level, rw, at large values of w'. Only the minimum, and not the large initial and asymptotic, values, are displayed in Fig. 8 because of scale limitations. Since dw'/dt is always positive, waste accumulates indefinitely. The cell division rate consequently decreases to zero, and growth is limited, as it would be in a normal cell strain. 2) Fig. 8(c). The increase of h(w') with w' is inadequate to avert the condition dw'/dt = 0. For some value of w' W'max, waste accumulation ceases. Cell division continues indefinitely at the rate kb(w'max). The population grows without limit as it would in a transformed cell line. The existence of these two classes of solution explains the separation of the population growth, waste content, and cell division functions, illustrated in Figs. 5, 6, and 7, respectively, into qualitatively different normal and transformed sets. Each set is characterized by the shape of the dw/dt vs. w' function with which it is associated. Functions displaying minima, for example, Figs. 8(a) and (b), give rise to normal behavior, while functions that terminate, for example, Fig. 8(c), describe transformation.
Sensitivity to rate parameters The great sensitivity of the model behavior to small changes in the rate parameters rw and kbo can be explained by noting that the denominator of the integrand on the righthand side of equation (4) is identical to the dw'/dt vs. w' function given in equation (1) and discussed above in connection with transformed and normal growth. The integrand becomes large for values ofw' at which its denominator, that is, dw/dt, is small. Thus, a small difference in one of the rate parameters can be responsible for a large difference in the population size by influencing how closely the minimum of the dw/dt vs. w' function approaches zero. These considerations are illustrated in Fig. 8. Curves (a) and (b) appear closely similar even though the vertical scale is enlarged to such an extent that the upper 80% of each has been omitted. Both were generated by the same cell-division function and differ only in that the waste rate parameter for curve (a) is greater by 2.29% than that for curve (b). This small difference is, nevertheless, of great importance because the minimum value of curve (b) is half as great as the minimum of curve (a). The maximum of the integrand is approximately twice as great, and, as a result, 42.2, rather than 28.7 population doublings occur. In the case illustrated in curve (c), in which the minimum value ofdw'/dt is zero, rw is less than in curve (b) by 4.59%, but the integral in equation (4) diverges, and the population increases without limit. The extraordinary sensitivity of the population size and waste content to changes in
CELL TRANSFORMATION BY WASTE PRODUCTS
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the rate parameters can be circumvented only if the rate parameters undergo changes which are approximately proportionate. Perfect proportionality is unnecessary. Small deviations from exact maintenance o f the kbo/rw ratio would allow modelling of the transformation process and of changes in the asymptotic doubling level attained in phase III. H o w e v e r , the sensitivity of the model outputs to independent changes in kbo and rw cannot be reconciled with the random changes which must occur in these parameters due to the normal variability of biological processes. With such great sensitivity, modest fluctuations in the rate parameters would be translated into extremely erratic variations in population size and waste content. One or more of the following considerations may serve to explain this inconsistency: 1. Lipofuscin accumulation may be a consequence rather than a primary cause o f cellular aging (Hayflick, 1985), that is, the waste-product theory may be incorrect. 2. N o n e o f the three cell division functions which were studied is directly based on data. If the validity of the waste product theory is ever to be determined, data on waste accumulation in dividing populations is badly needed. 3. The rate of waste production and the rate o f cell division in the absence of waste may be controlled in c o m m o n by a mechanism which maintains their ratio approximately constant. An investigation of a metabolic model embodying this concept is in progress. Relevant data would be of great interest. The relationship between the cellular metabolic rate and waste accumulation is neither simple nor linear. If waste accumulation were directly proportional to metabolic rate or time, waste product theories would be o f little value because the weight o f available evidence (Hayflick, 1977) shows that in vitro senescence is governed by the number of population doublings that the cell population has experienced. The metabolic model to which reference was made above contains reciprocal nonlinear interactions among variables representing cell division rate, waste accumulation, and metabolic rate. Ongoing research on the model indicates that the waste product theory in the form presented here is consistent with models of senescence based on either metabolic time or on a cell division counter but does not provide explicit support for one or the other.
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