J. theor. Biol. (1975) 55, 533-545
A Model for the Regulation of Growth in Mammalian Cells F. A. M. ALBERGHINA
Cattedra di Biochimica Comparata, Facolt~ di Scienze, UniversiM di Milano, Via G. Colombo 60, 20133 Milano, Italy (Received 24 September 1974, and in revisedform 19 March 1975) A dynamic model of cellular growth is presented which allows accurate prediction of the growth kinetics of both growing and resting mammalian cells (mouse fibroblasts), provided that it is supplied with the information on the values of the relevant cellular parameters. The relative rates of protein and of ribosomal RNA syntheses calculated from the model agree quite well with those experimentally determined in growing and resting fibroblasts. A hypothesis on the regulation of cellular growth is suggested by the model. The resting state is achieved when the rates of synthesis and of degradation of the proteins are balanced. The stimuli (hormones, growth factors, contact inhibition, etc.) which control cellular growth act because they modify the values of the parameters of the system. It is likely that they use the cell membrane as a decoder to feed appropriate signals to the cellular machineries which synthesize (or degrade) DNA, ribosomes and proteins. Biochemical experiments are suggested to test these predictions. The heuristic value of this integrated approach to the studies on the regulation of cellular growth is briefly discussed and compared to that of the reductionistic approach. 1. Introduction The control of cellular growth is a fundamental problem of modern biology which remains unsolved although a wealth of information has now been collected on several of its aspects. So one wonders whether the difficulties encountered in formulating a theory able to offer a framework for the available experimental observation may derive from the reductionistic perspective prevailing in these studies. That investigators are trying to move away from the simple reductionistic analysis toward a more integrated consideration of the various aspects of cellular growth is indicated by acceptance of the hypothesis of Hershko, Mamont, Shields & Tomkins (1971) which considers as a whole the biochemical reactions which respond in a co-ordinated way to modifications of the environment affecting cellular growth. They propose the 533
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term "pleiotypic response" for the entire regulatory programme which includes transport of molecules into the cell, protein and RNA synthesis, protein degradation and eventually DNA synthesis and suggest that the coordination of all these processes is mediated by a "pleiotypic mediator", afterwards tentatively identified as cyclic AMP (Kram, Mamont & Tomkins, 1973). According to this hypothesis the metabolically unrelated processes which are affected by growth changes do not need to be interconnected and co-ordinated among themselves but are just required to respond in a coordinated fashion to a change in the level of the mediator. No defined mathematical relationship is expected between the actual growth rate of the cell and the rates of these processes, which sometimes act in opposite directions as for instance the rates of synthesis and of degradation. A departure from this type of approach is found in a study in which the growth of microbial cells was analysed and a mathematical model of growth proposed (Alberghina, 1974). The growing cell is recognized as a system: i.e. "a set of physical components connected or related in such a manner to form and/or to act as an entire unit" (Di Stefano, Stubberud & Williams, 1967) whose defining components or state variables are DNA, ribosomes and proteins. Cellular growth is expressed by a closed loop in which positive as well as negative feedbacks are operational. The level of ribosomes and, indirectly, the level of proteins are stabilized around goal values by the action of negative feedbacks. The model has been successfully tested with experimental data for the exponential growth of Neurospora (Alberghina, 1974). In the present paper evidence is reported to show that system analysis helps also to develop a mathematical model which rationalizes most of the findings on mammalian cells growth and allows the proposal of a working hypothesis on the mode of action of several factors which control growth.
2. Quantitative Features of Mammalian Cell Growth Mammalian cells exist both in vivo and in tissue culture in two reversible growth states: a state of rapid proliferation (growing) and a viable state of non-proliferation (resting). Growing and resting cells have been shown to differ in the level of ribosomes and of proteins as well as in the rate of synthesis and of degradation of these macromolecules. The correlation between the growing state of eukaryotic cells and their increased content of RNA was first observed by Caspersson & Brachet (see Brachet, 1950). As about 80 % of the cellular R N A is ribosomal R N A (rRNA) the increased amount of RNA reflects the presence of a higher number of
GROWTH
R E G U L A T I O N IN M A M M A L I A N CELLS
535
ribosomes in growing ceils. In bacteria (Maaloe & Kjeldgaard, 1966) and in eukaryotic micro-organisms (Alberghina, Sturani & Gohlke, 1975) there is a correlation between the number of ribosomes per genome and the growth rate of the culture. In various kinds of mammalian cells, the growing ceils have been shown to contain more rRNA, per unit of DNA, than resting cells, between 1-4 and 2-8 fold, according to the type of cells (Becker, Stanners & Kudlow, 1971 ; Mauck & Green, 1973; Johnson, Abelson, Green & Penman, 1974). While in microbial cells with steady state growth the level of protein per genome is fairly constant and independent from the growth rate (Maaloe, 1969; AIberghina et al., 1975), growing mammalian ceils have been reported to have a higher (about 50 ~o more) protein content than resting ones (Mauck & Green, 1973). The rates of macromolecular syntheses have been determined in mammalian cells in both growth states. DNA synthesis occurs only in growing ceils; the resumption of DNA synthesis takes place only several hours after the transition from resting to growing state has been induced (Mauck & Green, 1973; Rudland, Seifert & Gospodarowicz, 1974). The measurements of the rates of synthesis of ribosomes and of proteins are usually done by determining the incorporation of radioactive precursors added to the culture medium. When comparing ceils in different growth conditions corrections have to be made for the variability of the intracellular specific activity of the radioactive precursors. The more so as a reduction of the transport across cell membrane has been shown to be associated with the transition from growing to resting ceils (Cunningham & Pardee, 1969; Weber & Edlin, 1971). Therefore the previous reports on the reduction of proteins and RNA synthesis in resting ceils (Levine, Becket, Boone & Eagle, 1965; Stanners & Becket, 1971) have been recently revised. In mouse fibroblasts the rate of protein synthesis has been found to be approximately three times higher in growing than in resting ceils (Rudland, 1974). That this may not be the case for all cell lines is indicated by a report which shows no difference between the rate of protein synthesis of growing and resting hamster kidney fibroblasts BHK 21/13 (Baenzinger, Jacobi & Thach, 1974). In these cells the maintenance of a constant protein level in the resting cells appears to be regulated not at the level of synthesis, but rather at the level of protein degradation or export. In a following section of this paper the functional equivalence of the two types of response will be discussed. The rates of synthesis of the various RNA species corrected for the change in the uptake of radioactive precursors have been determined for fibroblasts
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ALBERGHINA
(Rudland, 1974). The following results have been obtained: (a) the rate of synthesis of stable RNA, mainly rRNA, is three- to fourfold higher in growing than in resting cells; (b) the rate of synthesis of mRNA, determined as cytoplasmic poIy A-containing RNA, is about the same in growing and resting cells; (c) the rate of synthesis of stable RNA is equivalent to the rate of synthesis of mRNA, in growing cells; it is about one-third of the rate of synthesis of mRNA in resting cells. From these data the relative rates of total RNA synthesis can be estimated for both growing and resting cells and it can be concluded that the rate of synthesis of total RNA is reduced in the resting cells to about 70 % of that in the growing cells, in fairly good agreement with previously reported data (Weber & Edlin, 1971). Hence, while the rate of total RNA synthesis is only slightly diminished in resting cells, the rate of rRNA synthesis is reduced to one-third to one-fourth of the rate in growing cells. The turnovers of ribosomes and of proteins are larger in resting than in growing mammalian cells (Leob, Howell & Tomkins, 1965; Quincey & Wilson, 1969; Scornick, 1972). A protein turnover of 2 ~/hr in growing fibroblasts and of 3 ~o/hr in resting ones has been reported (Hershko, Mamont, Shields & Tomkins, 1971; Morhenn, Kram, Hershko & Tomkins, 1974). The rate of protein degradation generally follows exponential decay equations (Goldberg & Dice, 1974). So the amounts of proteins remaining after a time interval (t), starting from an initial .4 o amount, is: A t = Ao exp ( - - ; )
(1)
and the extent of degradation which occurs in the time interval t is:
S being the time constant of the negative exponential equation. Ribosomal RNA is completely stable in growing fibroblasts and unstable in resting ones, in the latter the half-life of 18 s RNA is about 72 hr and that of 28 s RNA is about 50 hr (Abelson, Johnson, Penman & Green, 1974). The two RNA species are present in the cytoplasm in equimolecular amounts and both originate from a single precursor molecule in the nucteolus, therefore the greater lability of 28 s RNA in the cytoplasm requires that some 18 s RNA be "wasted" before it can appear in the ribosomes. In order to maintain a balanced complement of rRNAs, resting cells need to have a rate of rRNA synthesis, considered at the level of precursor rRNA formation, so as to balance the turnover of the more labile molecule of rRNA.
GROWTH REGULATION IN MAMMALIAN CELLS
537
3. General Properties of the Model A dynamic model of growth for m a m m a l i a n cells is proposed in Fig. 1. A negative feedback maintains the ribosome level at the assigned goal value Pl (required ribosome level per DNA). Another feedback controls D N A formation so that a required protein level per D N A (P2) is maintained. When the content of protein, due to the activity of the existing ribosomes, increases to be more than that indicated by P2, more D N A is made to balance the increase, and this event in turn allows that more ribosomes are made. The rate of protein synthesis depends upon the level o f ribosomes and the efficiency of the translation machinery, K2, expressed as amino acids polymerized per min per ribosome. The levels of ribosomes and of proteins are replenished by the rates of synthesis and depleted by the rates of degradation, which are determined by the values of the inputs St (stability of ribosomes), and $2 (stability of proteins), the time constants of the corresponding negative exponential equations. The model given as block diagram in Fig. 1 can be also expressed by mathematical equations either as differential equations or in a form suitable
Ribosomes Degroded ribosomes ;e i Degraded proleins
S! Sz Kz
Proteins
P , ~
T
1
I ..........
]
FxG. 1. A model for the regulation of the cellular growth. A negative feedback loop maintains the ribosome level at the assigned goal value, px (required ribosome level per genom¢). Another negative feedback controls DNA formation so that a required protein level per genome (P2) determines the rate of DNA synthesis. The rate of protein synthesis depends upon the level of ribosomes and the efficiency of the translation machinery, K2, expressed as amino acids polymerized/min/ribosome. The levels of ribosomes and of proteins are replenished by the rates of synthesis and depleted by the rates of degradation. The velocity of the degradation is determined by the inputs Sz (stability of ribosomes) or Sz (stability of proteins) which are the time constants (min) of the corresponding negative exponential equations (see text).
538
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ALBERGHINA
for simulation solution. Using the latter type o f equations the level o f ribosomes (R), proteins (P) and D N A (D) can be c o m p u t e d at time j + 1 after a short time interval, At, starting from an initial condition at time j, as follows:
Ri+l = R j + K I ( p I D j - R j ) A t - R t Pj+I = P j + K 2 R ~ A t - P j
1-exp
[
1-exp
(
-
(3)
-
(4)
Dj+ 1 = Dj + K'I(Pj/P2 -- Dj)AI.
(5)
These equations state that the g r o w t h dynamics can be c o m p u t e d when the values o f the parameters Pl, P2, K~, K~, K2, $1 and 5'2 are known. N o w K2, $ I and $2 can be determined directly by using standard biochemical techniques and in a system in a steady state condition the values o f R / D and o f P/D give a close approximation o f respectively p~ and P2, if K 1 and K~ are set to 1 (rain-l).
TABLE l
Parameters used for the solution o f the equations describing the model Parameter Number of ribosomes, per genomet Number of amino acids into protein, per genome.~ Efficiency of the protein synthesis machinery (amino acids polymerized/min/ribosome)§ Stability of ribosomes (as $1, min)il Stability of proteins (as $2, min)J't
Value for Growing fibroblasts Resting fibroblasts 8-5 × 106
5"3 × 106
5.0 x 10~2
3"3 × 1012
650 oo 3000
310 4300 2000
t From data reported by Mauck & Green (1973) the mol. wt of the genome of 3T6 fibroblast was calculated. Then from the RNA/DNA values of growing and resting cells (Johnson et al., 1974) the numbers of ribosomes per genome were calculated considering 80% of total RNA to be rRNA, mol. wt 2-4 × 106 (Loening, 1968). :~ From protein/DNA values of growing fibroblasts (Bose & Zlotnick, 1973) assuming 100 as average reel. wt for amino acids and considering that the resting cells have to increase by 50% their protein content to become growing cells (Mauck & Green, 1973). § A rate of elongation of 800 amino acids/min and 80 % active ribosomes were estimated for growing fibroblast (Adamson, Howard & Herbert, 1969). For the resting cells the value was calculated from that of the growing cells taking into account the percent reduction of protein synthesis and the lower ribosome content (Rudland, 1974, Mauck & Green, 1973). IICalculated using equation (2) from the values of the half-lives of rRNA (Abelson et al., 1974) (see text). ~t Calculated using equation (2) from the rates of protein degradation (Hershko et al., 1971).
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539
IN M A M M A L I A N CELLS
At this point the validity of the model can be tested by challenging it to predict the growth dynamics of growing and resting fibroblasts, for which the values of the parameters were estimated on the basis of the findings discussed earlier and reported in Table 1. 4. Simulated Growths
A first simulation was run feeding as inputs the values proper to the growing fibroblasts. The growth dynamics obtained are represented in Fig. 2: an exponential growth is generated and the syntheses of DNA, ribosomes
5-0
6
2-0
o
t.0
0-5 I
I
I ....
5
IO
15
l
I
20 35 Time (hours)
I
I
I
30
35
40
FIG. 2. Simulated growth of growing fibroblasts. The simulation studies according to the model of cellular growth of Fig. 1 were performed using the equations (3), (4) and (5). The values assigned to the inputs (Pl, ,o2, $1, $2 and K2) were those reported in Table 1 for growing cells. K1 and K[ were set to 1/min (see text). As an initial condition that of one growing cell was taken. The calculations were done for I min intervals. The macromolecular levels were printed after every hour of the simulated growth and the values obtained in this way are reported in the figure on a semilogarithmic graph. D N A ( O ) is expressed as genome units, ribosomes (©) are expressed as number ( × I0-7), proteins are expressed ( A ) as polymerized amino acids ( × I 0 - l a ) .
and proteins are balanced. The computed constant of the exponential rate is: k = 4.6 x 10 -2 (hr-1), that is a doubling time of 15 hr, exactly that of growing fibroblasts (Johnson et al., 1974). A second simulation was run feeding the input values of the resting cells. As shown in Fig. 3 the dynamics of growth are profoundly different from that of Fig. 2: the system is in an equilibrated zero growth condition (expressed by an exponential equation with k = 0) as the resting cells are in. The results of the simulation and the data of Table 1 allow the rates of synthesis and of degradation for cells in the two growth conditions to be ealculated quite easily. T.B.
35
540
F. A. M. A L B E R G H I N A
5"0 m
2.0 o
~o
5
I0
15
20
25
30
35
40
Time (hours)
Fro. 3. Simulated growth of resting fibroblasts. The simulation studies were done as described in the legend of Fig. 2 giving to the inputs the values indicated in Table 1 for the resting cells. As an initial condition that of one resting cell was taken. The macromolecular levels DNA (O), ribosomes (©), proteins (A) were obtained and are expressed as indicated in Fig. 2. In resting cells: rate of protein synthesis = rate of protein degradation =3.3x1012[1-exp(-2d~)]
= 1.65 x109 amino acids/min,
rate of ribosome synthesis = rate of ribosome degradation = 5-3 x I06 [1 - e x p ( - 4 - 3 0 0 ) ] = 1250 ribosomes/min. In growing cells: rate of protein synthesis = rate of accumulation + rate of degradation = 5 x 10t2 [exp k t - 1] + 5 x 10'2 [ 1 - e x p ( - 3 0 - ~ ) = 5 x 10~2 [exp (4.6 x 10-2 : 6 0 ) - 1 ] + 5 x 1012 [ 1 - e x p ( - 3 0 ~ )
]
-- 5.5 × 109 amino acids/rain, rate of ribosome synthesis = rate of accumulation --- 8.5 x 106 [exp (4.6 x 10 -2 : 6 0 ) - 1 ] -- 6500 ribosomes/rain. So the model reproduces correctly the kinetics of growth of both growing
GROWTH
REGULATION
IN
MAMMALIAN
CELLS
541
and resting fibroblasts and, in stating that in resting cells the rate of protein synthesis is about one-third of that of growing cells and the rate of ribosome synthesis is about one-fifth of that of the growing ones, it is in fairly good agreement with the experimental observations recalled in section 2. In conclusion the model presented here is able to give a faithful representation of the dynamics of growth of mammalian cells and it rationalizes most of the features of its biochemical aspects.
5. Macromolecular Turnovers and the Resting Cell
The simulation experiment of Fig. 3 indicates that the condition of zero growth is achieved in a resting cell by a careful balance of the rate of synthesis and of the rate of degradation of ribosomes and of proteins, no further DNA synthesis taking place in the system when the level of proteins remains unchanged at the value indicated by P2. So a cell should enter in the resting state when it is programmed in such a way to have the rate of protein synthesis equivalent to the rate of protein degradation. Considering equation (4), it should be: and therefore:
This equation, expressed as a differential equation for At an infinitesimal time interval, becomes: P K2. R = - (6) $2 Equation (6) is therefore the equation of the resting state. If this is the condition that has to be satisfied for a cell to be in a resting state one can visualize that the transition from the growing to the resting condition does not require p e r s e a decrease of the rate of protein synthesis, it is enough that the balance between synthesis and degradation be obtained by adequately decreasing $2. From this point of view the results of Baenzinger et aL (1974) are not contradictory of those reported for mouse fibroblasts (Rudland, 1974), only refer to a different way to verify equation (6). On the other hand the equation of the growing state is: P
(7) $2 Therefore a transition from resting to growing state requires that the state variables assume values which verify equation (7). This object will be K 2 . R
> --.
542
F. A. M. A L B E R G H I N A
achieved if the values of one or more of the variables K2, R and $2 are raised with respect to the values they had during the resting state [in which equation (6) was verified]. A variety of growing states may be obtained if each of the state variables may increase independently one from the other. That the turnover of proteins may be a process closely associated to the regulation of cell proliferation is suggested also by the following considerations. Transformed cells do not enter the resting state in vitro when cultured under conditions (high cell density, serum deprivation) which induce the resting state in normal cells (Temin, 1972; Oey, Vogel & Pollack, 1974) and they maintain a low protein turnover in serum deprivation which causes increase of protein breakdown in normal cells (Hershko et aL, 1971). Following the logic suggested by the model this effect is not one among many other features of the phenotype of transformed cells but it is closely linked to the essential modification that brings the inability of the transformed cells to restrain from proliferation under environmental conditions which restrict growth of normal cells. Another point that can be discussed at this point is the significance of the resting condition for the cells of higher organisms. The word "resting" might be misleading because it suggests the idea of inactivity. It is of course correct if one considers the proliferation activity of the cell, but it is not true if one considers the activity of the specialized functions of the cell. For instance, a resting liver cell is carrying on a number of biochemical activities which are useful to the organism as a whole. Furthermore, the relatively high turnover of proteins allows change to the enzymic set-up of the cell according to the variations in the environment and therefore gives a higher degree of adaptation to the system. So the resting cell weaving a Penelope's cloth prevents unnecessary proliferation and assures in a versatile way the development of essential functions.
6. The Regulation of Growth: an Hypothesis
The model has been shown to be sufficiently valid to predict with accuracy the dynamics of growth of resting and growing cells and to give a reasonable explanation for a number of biochemical features of cellular growth especially on the levels and the rates of synthesis of the different macromolecular components of the cell. These results do not prove that the model reflects the true regulatory mechanisms of cellular growth. A critical test in this sense would be to identify at the molecular level the transducers, the signals and the comparators which are suggested by the model. The possibility of checking the predictions of the model by biochemical experiments and
G R O W T H R E G U L A T I O N IN M A M M A L I A N CELLS
543
improvement of the mathematical model in order to consider new experimental findings may allow the biological validity of the model to be tested. The hypothesis suggested in this paper is that the regulation of cellular growth is obtained by changing the values of the inputs Pl, P2, $1, $2 and K2 (see Fig. 4). Therefore the environmental stimuli hormones, contact with other cells, serum factors, and nutrients which affect cellular growth should
Disturbonce (tronsformotiott)
I
L..
Nulrients
Hormones
~p,
~,
Dynomics of DNA Growth foctors--~
Contoct inhibition-,,, Toxins
~'Pz"
--Sz
._p
~. ribosomes and protein occumulotions
(--Growth).
~'KI
P,
Fic. 4. Hypothetical model of the mechanism by which environmental conditions affect cellular growth. Positive (hormones, growth factors) or negative (contact inhibition, toxins) stimuli that affect growth are decoded by the cell membrane so that each one alone or a combination of them yields a set of inputs which determine, for the interlocked macromolecular syntheses (/MS) shown in Fig. I, the dynamics of growth. If the decoding ability of the cell membrane is altered, for instance by transformation, to one set of stimuli, it does not correspond any longer a normal set of inputs. In the case of transformation a set of higher inputs is fed to. IMS, therefore yielding growth under conditions that normally restrict it.
be expected to change the values of the inputs. Higher values would correspond to growth promoting stimuli and lower values to growth inhibiting stimuli. Insofar as most of the factors controlling growth have been shown to interact with the cell membrane (Cuatrecasas, 1972, 1974; Hollemberg & Cuatrecasas, 1973) it could be suggested that the membrane has a decoding function: it detects the stimuli coming from the outside of the cell and produces inside the cell signal(s) which yield the actual growth dynamics of the cell, when fed to the enzymic machineries that synthesize DNA, ribosomes and proteins and that degrade ribosomes and proteins (Fig. 4). Alterations in the pattern of regulation of cellular growth, as those observed in transformation, will ensue from modifications in the structure
544
F. A. M. ALBERGHINA
of the decoding membrane. Changes in the structure of the cell membrane in transformed cells have been reported by Aub, Sanford & Wang (1965), Kraemer (1966), Scott, Furcht & Kersey (1973) and Barnett, Furcht & Scott (1974). As for the signals that the cell membrane may produce in response to environmental stimuli, cyclic AMP and cyclic G M P are likely candidates. Adenyl cyclase and guanyl cyclase are proteins associated with the cell membrane (Bar & Hechter, 1969; Rudland, Gospodarowicz & Seifert, 1974), and they are modulated in their activity by factors controlling growth (Illiano & Cuatrecasas, 1972; Hepp & Renner, 1972; Illiano, Tell, Siegel & Cuatrecasas, 1973; Rudland e t al., 1974). Two patterns of response have so far been shown to be associated to growth promoting stimuli: stimulation of growth with a concomitant decrease of the cyclic A M P level and an increase of cyclic G M P level (Otten, Johnson & Pastan, 1972; Seifert & Rudland, 1974), or stimulation of growth while the cyclic AMP level remains unaffected and the cyclic G M P level increases (Hadden, Hadden, Haddox & Goldberg, 1972; Rudland, Gospodarowicz & Seifert, 1974). If the condition necessary for growth is that equation (7) be verified and if the value of any of the state variables of equation (7) are determined by the levels of either cyclic A M P or cyclic G M P a difference of these values is expected between the two conditions. Direct biochemical analysis may elucidate this point. In conclusion, the model, attractive for its logic simplicity, appears to put a structure in the studies on cellular growth and suggests experimental tests in order to ascertain its validity. Further studies which are now under way are required to assess its real biological significance.
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CUATRECASAS,P. (1974). A. Rev. Biochem. 43, 169. CUNNINGHAM,D. D. & PARDEE,A. B. (1969). Proc. hath. Acad. Sci. U.S.A. 64, 1049. Di STEFANO,J. J., III, STUBBERUD,A. R. & WILUAMS,I. J. (1967). Feedback and Control System, Schaum's Outline Series. New York: McGraw-Hill Book Co. GOLDBERG, A. L. & DICE, J. F. 0974). A. Rev. Biochem. 43, 835. HADDEN, J. W., HADDEN, E. M., HADDOX, M. K. & GOLDBERG,N, D. (1972). Proc. natn. Acad. Sci. U.S.A. 69, 3024. HEPP, K. D. & RENNER, R. (1972). F E B S Lett. 20, 191. HERSHKO, A., MAMONT,P., SHIELDS,R. & TOMKINS,G. M. (1971). Nature New Biol. 232, 206. HOLLEMBERG,N. D. & CUATRECASAS,P. (1973). Proc. hath. Acad. Sci. U.S.A. 70, 2964. ILUANO, G. & CUATRECASAS,P. (1972). Science, N. I:. 175, 906. ILLIANO,G., TELL, G. P. E., SIEGEL,M. I. & CUATRECAS~,P. (1973). Proc. natn./[cad. Sci. U.S.A. 70, 2443. JOHNSON, L. F., ABELSON,H. T., GREEN, H. & PENMAN,S. (1974). Cell 1, 95. KRAEMER,P. M. (1966). Y. cell. Physiol. 6"/, 23. KRAM,R., MAMONT,P. & TOMKINS,G. U. (1973). Proc. natn. Acad. Sei. U.S.A. 70, 1432. LEVINE, E. M., BECKER,J., BOONE,C. W. & EAGLE,H. (1965). Proc. natn. Acad. Sci. U.S.A. 53, 350. LOEB, J. N., HOWELL, R. R. & TOMKn~S, G. M. (1965). Science, N.Y. 149, 1093. LOENING, U. (1968). J. molec. Biol. 38, 355. MAAL~E,O. & KJELDGAARD,N. O. (1966). ControlofMacromolecular Synthesis. New York: W. A. Benjamin. MAUCK, J. C. & GREEN, H. (1973). Proc. natn. Acad. Sci. U.S.A. 70, 2819. MORHENN, V., KRAM, R., HERSHKO, A. & TOMKINS,G. M. (1974). Cell 1, 91. OEY, J., VOGEL, A. & POLLACK,R. (1974). Proc. hath. Acad. Sci. U.S.A. 71, 694. OTrEN, J., JOHNSON, G. S. & PASTAN,I. (1972). J. biol. Chem. 247, 7082. QUINCEY, R. V. & WILSON,S. H. (1969). Proc. hath. Acad. Sci. U.S.A. 64, 981. RUDLAND, P. S. (1974). Proc. natn. Acad. ScL U.S.A. 71, 750. RUDLAND, P. S., GOSPODAROWICZ,D. & SEt~RT, W. (1974). Nature, Lond. 250, 741. RUDLAND, P. S., SEIFERT,W. & GOSPODAROWICZ,D. (1974). Proc. ham. Acad. Sci. U.S.A. 71, 2600. SCORNICK, O. A. (1972). Biochem. biophys. Res. Commun. 47, 1063. SCOTT, R. E., FURCWr, L. T. & KERSEY, J. H. (1973). Proc. hath. Acad. ScL U.S.A. 70, 3631. SEIFERT, W. E. & RUDLAND,P. S. (1974). Nature, Lond. 248, 138. STANNERS,C. R. & BECKER, H. (1971). J. cell. Physiol. 71, 31. TEMIN, H. M. (1972). J. cellPhysiol. 78, 161. WEBER, M. J. & EDHN, G. (1971). J. biol. Chem. 246, 1828.
A Model for the Regulation of Growth in Mammalian Cells F. A. M. ALBERGHINA
Cattedra di Biochimica Comparata, Facolt~ di Scienze, UniversiM di Milano, Via G. Colombo 60, 20133 Milano, Italy (Received 24 September 1974, and in revisedform 19 March 1975) A dynamic model of cellular growth is presented which allows accurate prediction of the growth kinetics of both growing and resting mammalian cells (mouse fibroblasts), provided that it is supplied with the information on the values of the relevant cellular parameters. The relative rates of protein and of ribosomal RNA syntheses calculated from the model agree quite well with those experimentally determined in growing and resting fibroblasts. A hypothesis on the regulation of cellular growth is suggested by the model. The resting state is achieved when the rates of synthesis and of degradation of the proteins are balanced. The stimuli (hormones, growth factors, contact inhibition, etc.) which control cellular growth act because they modify the values of the parameters of the system. It is likely that they use the cell membrane as a decoder to feed appropriate signals to the cellular machineries which synthesize (or degrade) DNA, ribosomes and proteins. Biochemical experiments are suggested to test these predictions. The heuristic value of this integrated approach to the studies on the regulation of cellular growth is briefly discussed and compared to that of the reductionistic approach. 1. Introduction The control of cellular growth is a fundamental problem of modern biology which remains unsolved although a wealth of information has now been collected on several of its aspects. So one wonders whether the difficulties encountered in formulating a theory able to offer a framework for the available experimental observation may derive from the reductionistic perspective prevailing in these studies. That investigators are trying to move away from the simple reductionistic analysis toward a more integrated consideration of the various aspects of cellular growth is indicated by acceptance of the hypothesis of Hershko, Mamont, Shields & Tomkins (1971) which considers as a whole the biochemical reactions which respond in a co-ordinated way to modifications of the environment affecting cellular growth. They propose the 533
534
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term "pleiotypic response" for the entire regulatory programme which includes transport of molecules into the cell, protein and RNA synthesis, protein degradation and eventually DNA synthesis and suggest that the coordination of all these processes is mediated by a "pleiotypic mediator", afterwards tentatively identified as cyclic AMP (Kram, Mamont & Tomkins, 1973). According to this hypothesis the metabolically unrelated processes which are affected by growth changes do not need to be interconnected and co-ordinated among themselves but are just required to respond in a coordinated fashion to a change in the level of the mediator. No defined mathematical relationship is expected between the actual growth rate of the cell and the rates of these processes, which sometimes act in opposite directions as for instance the rates of synthesis and of degradation. A departure from this type of approach is found in a study in which the growth of microbial cells was analysed and a mathematical model of growth proposed (Alberghina, 1974). The growing cell is recognized as a system: i.e. "a set of physical components connected or related in such a manner to form and/or to act as an entire unit" (Di Stefano, Stubberud & Williams, 1967) whose defining components or state variables are DNA, ribosomes and proteins. Cellular growth is expressed by a closed loop in which positive as well as negative feedbacks are operational. The level of ribosomes and, indirectly, the level of proteins are stabilized around goal values by the action of negative feedbacks. The model has been successfully tested with experimental data for the exponential growth of Neurospora (Alberghina, 1974). In the present paper evidence is reported to show that system analysis helps also to develop a mathematical model which rationalizes most of the findings on mammalian cells growth and allows the proposal of a working hypothesis on the mode of action of several factors which control growth.
2. Quantitative Features of Mammalian Cell Growth Mammalian cells exist both in vivo and in tissue culture in two reversible growth states: a state of rapid proliferation (growing) and a viable state of non-proliferation (resting). Growing and resting cells have been shown to differ in the level of ribosomes and of proteins as well as in the rate of synthesis and of degradation of these macromolecules. The correlation between the growing state of eukaryotic cells and their increased content of RNA was first observed by Caspersson & Brachet (see Brachet, 1950). As about 80 % of the cellular R N A is ribosomal R N A (rRNA) the increased amount of RNA reflects the presence of a higher number of
GROWTH
R E G U L A T I O N IN M A M M A L I A N CELLS
535
ribosomes in growing ceils. In bacteria (Maaloe & Kjeldgaard, 1966) and in eukaryotic micro-organisms (Alberghina, Sturani & Gohlke, 1975) there is a correlation between the number of ribosomes per genome and the growth rate of the culture. In various kinds of mammalian cells, the growing ceils have been shown to contain more rRNA, per unit of DNA, than resting cells, between 1-4 and 2-8 fold, according to the type of cells (Becker, Stanners & Kudlow, 1971 ; Mauck & Green, 1973; Johnson, Abelson, Green & Penman, 1974). While in microbial cells with steady state growth the level of protein per genome is fairly constant and independent from the growth rate (Maaloe, 1969; AIberghina et al., 1975), growing mammalian ceils have been reported to have a higher (about 50 ~o more) protein content than resting ones (Mauck & Green, 1973). The rates of macromolecular syntheses have been determined in mammalian cells in both growth states. DNA synthesis occurs only in growing ceils; the resumption of DNA synthesis takes place only several hours after the transition from resting to growing state has been induced (Mauck & Green, 1973; Rudland, Seifert & Gospodarowicz, 1974). The measurements of the rates of synthesis of ribosomes and of proteins are usually done by determining the incorporation of radioactive precursors added to the culture medium. When comparing ceils in different growth conditions corrections have to be made for the variability of the intracellular specific activity of the radioactive precursors. The more so as a reduction of the transport across cell membrane has been shown to be associated with the transition from growing to resting ceils (Cunningham & Pardee, 1969; Weber & Edlin, 1971). Therefore the previous reports on the reduction of proteins and RNA synthesis in resting ceils (Levine, Becket, Boone & Eagle, 1965; Stanners & Becket, 1971) have been recently revised. In mouse fibroblasts the rate of protein synthesis has been found to be approximately three times higher in growing than in resting ceils (Rudland, 1974). That this may not be the case for all cell lines is indicated by a report which shows no difference between the rate of protein synthesis of growing and resting hamster kidney fibroblasts BHK 21/13 (Baenzinger, Jacobi & Thach, 1974). In these cells the maintenance of a constant protein level in the resting cells appears to be regulated not at the level of synthesis, but rather at the level of protein degradation or export. In a following section of this paper the functional equivalence of the two types of response will be discussed. The rates of synthesis of the various RNA species corrected for the change in the uptake of radioactive precursors have been determined for fibroblasts
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ALBERGHINA
(Rudland, 1974). The following results have been obtained: (a) the rate of synthesis of stable RNA, mainly rRNA, is three- to fourfold higher in growing than in resting cells; (b) the rate of synthesis of mRNA, determined as cytoplasmic poIy A-containing RNA, is about the same in growing and resting cells; (c) the rate of synthesis of stable RNA is equivalent to the rate of synthesis of mRNA, in growing cells; it is about one-third of the rate of synthesis of mRNA in resting cells. From these data the relative rates of total RNA synthesis can be estimated for both growing and resting cells and it can be concluded that the rate of synthesis of total RNA is reduced in the resting cells to about 70 % of that in the growing cells, in fairly good agreement with previously reported data (Weber & Edlin, 1971). Hence, while the rate of total RNA synthesis is only slightly diminished in resting cells, the rate of rRNA synthesis is reduced to one-third to one-fourth of the rate in growing cells. The turnovers of ribosomes and of proteins are larger in resting than in growing mammalian cells (Leob, Howell & Tomkins, 1965; Quincey & Wilson, 1969; Scornick, 1972). A protein turnover of 2 ~/hr in growing fibroblasts and of 3 ~o/hr in resting ones has been reported (Hershko, Mamont, Shields & Tomkins, 1971; Morhenn, Kram, Hershko & Tomkins, 1974). The rate of protein degradation generally follows exponential decay equations (Goldberg & Dice, 1974). So the amounts of proteins remaining after a time interval (t), starting from an initial .4 o amount, is: A t = Ao exp ( - - ; )
(1)
and the extent of degradation which occurs in the time interval t is:
S being the time constant of the negative exponential equation. Ribosomal RNA is completely stable in growing fibroblasts and unstable in resting ones, in the latter the half-life of 18 s RNA is about 72 hr and that of 28 s RNA is about 50 hr (Abelson, Johnson, Penman & Green, 1974). The two RNA species are present in the cytoplasm in equimolecular amounts and both originate from a single precursor molecule in the nucteolus, therefore the greater lability of 28 s RNA in the cytoplasm requires that some 18 s RNA be "wasted" before it can appear in the ribosomes. In order to maintain a balanced complement of rRNAs, resting cells need to have a rate of rRNA synthesis, considered at the level of precursor rRNA formation, so as to balance the turnover of the more labile molecule of rRNA.
GROWTH REGULATION IN MAMMALIAN CELLS
537
3. General Properties of the Model A dynamic model of growth for m a m m a l i a n cells is proposed in Fig. 1. A negative feedback maintains the ribosome level at the assigned goal value Pl (required ribosome level per DNA). Another feedback controls D N A formation so that a required protein level per D N A (P2) is maintained. When the content of protein, due to the activity of the existing ribosomes, increases to be more than that indicated by P2, more D N A is made to balance the increase, and this event in turn allows that more ribosomes are made. The rate of protein synthesis depends upon the level o f ribosomes and the efficiency of the translation machinery, K2, expressed as amino acids polymerized per min per ribosome. The levels of ribosomes and of proteins are replenished by the rates of synthesis and depleted by the rates of degradation, which are determined by the values of the inputs St (stability of ribosomes), and $2 (stability of proteins), the time constants of the corresponding negative exponential equations. The model given as block diagram in Fig. 1 can be also expressed by mathematical equations either as differential equations or in a form suitable
Ribosomes Degroded ribosomes ;e i Degraded proleins
S! Sz Kz
Proteins
P , ~
T
1
I ..........
]
FxG. 1. A model for the regulation of the cellular growth. A negative feedback loop maintains the ribosome level at the assigned goal value, px (required ribosome level per genom¢). Another negative feedback controls DNA formation so that a required protein level per genome (P2) determines the rate of DNA synthesis. The rate of protein synthesis depends upon the level of ribosomes and the efficiency of the translation machinery, K2, expressed as amino acids polymerized/min/ribosome. The levels of ribosomes and of proteins are replenished by the rates of synthesis and depleted by the rates of degradation. The velocity of the degradation is determined by the inputs Sz (stability of ribosomes) or Sz (stability of proteins) which are the time constants (min) of the corresponding negative exponential equations (see text).
538
F.A.M.
ALBERGHINA
for simulation solution. Using the latter type o f equations the level o f ribosomes (R), proteins (P) and D N A (D) can be c o m p u t e d at time j + 1 after a short time interval, At, starting from an initial condition at time j, as follows:
Ri+l = R j + K I ( p I D j - R j ) A t - R t Pj+I = P j + K 2 R ~ A t - P j
1-exp
[
1-exp
(
-
(3)
-
(4)
Dj+ 1 = Dj + K'I(Pj/P2 -- Dj)AI.
(5)
These equations state that the g r o w t h dynamics can be c o m p u t e d when the values o f the parameters Pl, P2, K~, K~, K2, $1 and 5'2 are known. N o w K2, $ I and $2 can be determined directly by using standard biochemical techniques and in a system in a steady state condition the values o f R / D and o f P/D give a close approximation o f respectively p~ and P2, if K 1 and K~ are set to 1 (rain-l).
TABLE l
Parameters used for the solution o f the equations describing the model Parameter Number of ribosomes, per genomet Number of amino acids into protein, per genome.~ Efficiency of the protein synthesis machinery (amino acids polymerized/min/ribosome)§ Stability of ribosomes (as $1, min)il Stability of proteins (as $2, min)J't
Value for Growing fibroblasts Resting fibroblasts 8-5 × 106
5"3 × 106
5.0 x 10~2
3"3 × 1012
650 oo 3000
310 4300 2000
t From data reported by Mauck & Green (1973) the mol. wt of the genome of 3T6 fibroblast was calculated. Then from the RNA/DNA values of growing and resting cells (Johnson et al., 1974) the numbers of ribosomes per genome were calculated considering 80% of total RNA to be rRNA, mol. wt 2-4 × 106 (Loening, 1968). :~ From protein/DNA values of growing fibroblasts (Bose & Zlotnick, 1973) assuming 100 as average reel. wt for amino acids and considering that the resting cells have to increase by 50% their protein content to become growing cells (Mauck & Green, 1973). § A rate of elongation of 800 amino acids/min and 80 % active ribosomes were estimated for growing fibroblast (Adamson, Howard & Herbert, 1969). For the resting cells the value was calculated from that of the growing cells taking into account the percent reduction of protein synthesis and the lower ribosome content (Rudland, 1974, Mauck & Green, 1973). IICalculated using equation (2) from the values of the half-lives of rRNA (Abelson et al., 1974) (see text). ~t Calculated using equation (2) from the rates of protein degradation (Hershko et al., 1971).
GROWTH REGULATION
539
IN M A M M A L I A N CELLS
At this point the validity of the model can be tested by challenging it to predict the growth dynamics of growing and resting fibroblasts, for which the values of the parameters were estimated on the basis of the findings discussed earlier and reported in Table 1. 4. Simulated Growths
A first simulation was run feeding as inputs the values proper to the growing fibroblasts. The growth dynamics obtained are represented in Fig. 2: an exponential growth is generated and the syntheses of DNA, ribosomes
5-0
6
2-0
o
t.0
0-5 I
I
I ....
5
IO
15
l
I
20 35 Time (hours)
I
I
I
30
35
40
FIG. 2. Simulated growth of growing fibroblasts. The simulation studies according to the model of cellular growth of Fig. 1 were performed using the equations (3), (4) and (5). The values assigned to the inputs (Pl, ,o2, $1, $2 and K2) were those reported in Table 1 for growing cells. K1 and K[ were set to 1/min (see text). As an initial condition that of one growing cell was taken. The calculations were done for I min intervals. The macromolecular levels were printed after every hour of the simulated growth and the values obtained in this way are reported in the figure on a semilogarithmic graph. D N A ( O ) is expressed as genome units, ribosomes (©) are expressed as number ( × I0-7), proteins are expressed ( A ) as polymerized amino acids ( × I 0 - l a ) .
and proteins are balanced. The computed constant of the exponential rate is: k = 4.6 x 10 -2 (hr-1), that is a doubling time of 15 hr, exactly that of growing fibroblasts (Johnson et al., 1974). A second simulation was run feeding the input values of the resting cells. As shown in Fig. 3 the dynamics of growth are profoundly different from that of Fig. 2: the system is in an equilibrated zero growth condition (expressed by an exponential equation with k = 0) as the resting cells are in. The results of the simulation and the data of Table 1 allow the rates of synthesis and of degradation for cells in the two growth conditions to be ealculated quite easily. T.B.
35
540
F. A. M. A L B E R G H I N A
5"0 m
2.0 o
~o
5
I0
15
20
25
30
35
40
Time (hours)
Fro. 3. Simulated growth of resting fibroblasts. The simulation studies were done as described in the legend of Fig. 2 giving to the inputs the values indicated in Table 1 for the resting cells. As an initial condition that of one resting cell was taken. The macromolecular levels DNA (O), ribosomes (©), proteins (A) were obtained and are expressed as indicated in Fig. 2. In resting cells: rate of protein synthesis = rate of protein degradation =3.3x1012[1-exp(-2d~)]
= 1.65 x109 amino acids/min,
rate of ribosome synthesis = rate of ribosome degradation = 5-3 x I06 [1 - e x p ( - 4 - 3 0 0 ) ] = 1250 ribosomes/min. In growing cells: rate of protein synthesis = rate of accumulation + rate of degradation = 5 x 10t2 [exp k t - 1] + 5 x 10'2 [ 1 - e x p ( - 3 0 - ~ ) = 5 x 10~2 [exp (4.6 x 10-2 : 6 0 ) - 1 ] + 5 x 1012 [ 1 - e x p ( - 3 0 ~ )
]
-- 5.5 × 109 amino acids/rain, rate of ribosome synthesis = rate of accumulation --- 8.5 x 106 [exp (4.6 x 10 -2 : 6 0 ) - 1 ] -- 6500 ribosomes/rain. So the model reproduces correctly the kinetics of growth of both growing
GROWTH
REGULATION
IN
MAMMALIAN
CELLS
541
and resting fibroblasts and, in stating that in resting cells the rate of protein synthesis is about one-third of that of growing cells and the rate of ribosome synthesis is about one-fifth of that of the growing ones, it is in fairly good agreement with the experimental observations recalled in section 2. In conclusion the model presented here is able to give a faithful representation of the dynamics of growth of mammalian cells and it rationalizes most of the features of its biochemical aspects.
5. Macromolecular Turnovers and the Resting Cell
The simulation experiment of Fig. 3 indicates that the condition of zero growth is achieved in a resting cell by a careful balance of the rate of synthesis and of the rate of degradation of ribosomes and of proteins, no further DNA synthesis taking place in the system when the level of proteins remains unchanged at the value indicated by P2. So a cell should enter in the resting state when it is programmed in such a way to have the rate of protein synthesis equivalent to the rate of protein degradation. Considering equation (4), it should be: and therefore:
This equation, expressed as a differential equation for At an infinitesimal time interval, becomes: P K2. R = - (6) $2 Equation (6) is therefore the equation of the resting state. If this is the condition that has to be satisfied for a cell to be in a resting state one can visualize that the transition from the growing to the resting condition does not require p e r s e a decrease of the rate of protein synthesis, it is enough that the balance between synthesis and degradation be obtained by adequately decreasing $2. From this point of view the results of Baenzinger et aL (1974) are not contradictory of those reported for mouse fibroblasts (Rudland, 1974), only refer to a different way to verify equation (6). On the other hand the equation of the growing state is: P
(7) $2 Therefore a transition from resting to growing state requires that the state variables assume values which verify equation (7). This object will be K 2 . R
> --.
542
F. A. M. A L B E R G H I N A
achieved if the values of one or more of the variables K2, R and $2 are raised with respect to the values they had during the resting state [in which equation (6) was verified]. A variety of growing states may be obtained if each of the state variables may increase independently one from the other. That the turnover of proteins may be a process closely associated to the regulation of cell proliferation is suggested also by the following considerations. Transformed cells do not enter the resting state in vitro when cultured under conditions (high cell density, serum deprivation) which induce the resting state in normal cells (Temin, 1972; Oey, Vogel & Pollack, 1974) and they maintain a low protein turnover in serum deprivation which causes increase of protein breakdown in normal cells (Hershko et aL, 1971). Following the logic suggested by the model this effect is not one among many other features of the phenotype of transformed cells but it is closely linked to the essential modification that brings the inability of the transformed cells to restrain from proliferation under environmental conditions which restrict growth of normal cells. Another point that can be discussed at this point is the significance of the resting condition for the cells of higher organisms. The word "resting" might be misleading because it suggests the idea of inactivity. It is of course correct if one considers the proliferation activity of the cell, but it is not true if one considers the activity of the specialized functions of the cell. For instance, a resting liver cell is carrying on a number of biochemical activities which are useful to the organism as a whole. Furthermore, the relatively high turnover of proteins allows change to the enzymic set-up of the cell according to the variations in the environment and therefore gives a higher degree of adaptation to the system. So the resting cell weaving a Penelope's cloth prevents unnecessary proliferation and assures in a versatile way the development of essential functions.
6. The Regulation of Growth: an Hypothesis
The model has been shown to be sufficiently valid to predict with accuracy the dynamics of growth of resting and growing cells and to give a reasonable explanation for a number of biochemical features of cellular growth especially on the levels and the rates of synthesis of the different macromolecular components of the cell. These results do not prove that the model reflects the true regulatory mechanisms of cellular growth. A critical test in this sense would be to identify at the molecular level the transducers, the signals and the comparators which are suggested by the model. The possibility of checking the predictions of the model by biochemical experiments and
G R O W T H R E G U L A T I O N IN M A M M A L I A N CELLS
543
improvement of the mathematical model in order to consider new experimental findings may allow the biological validity of the model to be tested. The hypothesis suggested in this paper is that the regulation of cellular growth is obtained by changing the values of the inputs Pl, P2, $1, $2 and K2 (see Fig. 4). Therefore the environmental stimuli hormones, contact with other cells, serum factors, and nutrients which affect cellular growth should
Disturbonce (tronsformotiott)
I
L..
Nulrients
Hormones
~p,
~,
Dynomics of DNA Growth foctors--~
Contoct inhibition-,,, Toxins
~'Pz"
--Sz
._p
~. ribosomes and protein occumulotions
(--Growth).
~'KI
P,
Fic. 4. Hypothetical model of the mechanism by which environmental conditions affect cellular growth. Positive (hormones, growth factors) or negative (contact inhibition, toxins) stimuli that affect growth are decoded by the cell membrane so that each one alone or a combination of them yields a set of inputs which determine, for the interlocked macromolecular syntheses (/MS) shown in Fig. I, the dynamics of growth. If the decoding ability of the cell membrane is altered, for instance by transformation, to one set of stimuli, it does not correspond any longer a normal set of inputs. In the case of transformation a set of higher inputs is fed to. IMS, therefore yielding growth under conditions that normally restrict it.
be expected to change the values of the inputs. Higher values would correspond to growth promoting stimuli and lower values to growth inhibiting stimuli. Insofar as most of the factors controlling growth have been shown to interact with the cell membrane (Cuatrecasas, 1972, 1974; Hollemberg & Cuatrecasas, 1973) it could be suggested that the membrane has a decoding function: it detects the stimuli coming from the outside of the cell and produces inside the cell signal(s) which yield the actual growth dynamics of the cell, when fed to the enzymic machineries that synthesize DNA, ribosomes and proteins and that degrade ribosomes and proteins (Fig. 4). Alterations in the pattern of regulation of cellular growth, as those observed in transformation, will ensue from modifications in the structure
544
F. A. M. ALBERGHINA
of the decoding membrane. Changes in the structure of the cell membrane in transformed cells have been reported by Aub, Sanford & Wang (1965), Kraemer (1966), Scott, Furcht & Kersey (1973) and Barnett, Furcht & Scott (1974). As for the signals that the cell membrane may produce in response to environmental stimuli, cyclic AMP and cyclic G M P are likely candidates. Adenyl cyclase and guanyl cyclase are proteins associated with the cell membrane (Bar & Hechter, 1969; Rudland, Gospodarowicz & Seifert, 1974), and they are modulated in their activity by factors controlling growth (Illiano & Cuatrecasas, 1972; Hepp & Renner, 1972; Illiano, Tell, Siegel & Cuatrecasas, 1973; Rudland e t al., 1974). Two patterns of response have so far been shown to be associated to growth promoting stimuli: stimulation of growth with a concomitant decrease of the cyclic A M P level and an increase of cyclic G M P level (Otten, Johnson & Pastan, 1972; Seifert & Rudland, 1974), or stimulation of growth while the cyclic AMP level remains unaffected and the cyclic G M P level increases (Hadden, Hadden, Haddox & Goldberg, 1972; Rudland, Gospodarowicz & Seifert, 1974). If the condition necessary for growth is that equation (7) be verified and if the value of any of the state variables of equation (7) are determined by the levels of either cyclic A M P or cyclic G M P a difference of these values is expected between the two conditions. Direct biochemical analysis may elucidate this point. In conclusion, the model, attractive for its logic simplicity, appears to put a structure in the studies on cellular growth and suggests experimental tests in order to ascertain its validity. Further studies which are now under way are required to assess its real biological significance.
REFERENCES ABELSON,H. T., JOHNSON,L. F., PENMAN,S. t~ GREEN,H. (1974). Cell 1, 161. ADAMSON,S. D., HOWARD,G. A. & HERBERT,E. (1969). Cold Spring Harb. Syrup. quant. Biol. 34, 547. ALBER~SHINA,F. A. M. (1974). Rend. Ace. no. 2 Lincei 56, 971. ALBERGHINA,F. A. M., STURANI,E. & GOHLKE,J. R. (1975). d. biol. Chem. 250, 4381. AUB, J. C., SANFORD,B. H. & WANGH,H. (1965). Proc. hath. Acad. Sci. U.S.A. 54, 400. BAENZtNGER,N. L., JACOm,C. H. & THACH,R. (1974). J. biol. Chem. 249, 3483. B.~R,H. P. & HECHTER,O. (1969). Proc. hath. Acad. Sci. U.S.A. 63, 350. BARNETT,R. E., FURCHT,L. T. & ScoTT, R. E. (1974). Proc. hath. Acad. 3ci. U.S.A. 71, 1992. BECKER,H., STANNERS,C. P. & KUDLOW,J. E. (1971). J. cell. Physiol. 77, 43. BOSE,S. K. & ZLOTNICK,B. J. (1973). Proc. natn. Aead. Sci. U.S.A. 70, 2374. BRACHET,J. (1950). Chemical Embriology. New York: Interscience Publishers. CUATRECASAS,P. (1972). Proc. hath. Acad. Sei. U.S.A. 69, 318.
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