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Clonal Attenuation In Chick Embryo Fibroblasts Experimental Data, A Model and Computer Simulations

Clonal Attenuation In Chick Embryo Fibroblasts Experimental Data, A Model and Computer Simulations J. C. Angello and J. W. Prothero Department of Biological Structure, University o Washington, School o Medicine, Seattle, WA 98195, U.S.A. f f (Received 7 September 1983; revision accepted 8 August 1984) Abstract. When cells from mass cultures of chick embryo fibroblasts are grown at very low density, some cells yield large clones while others produce smaller clones, and some cells fail to divide a t all. The distribution of clone sizes is related to the number of population doublings which the donor mass culture has undergone: the more doublings which have occurred, the smaller the average clone size. In this report we describe a model which analyses this phenomenon, referred to as ‘clonal attenuation’, in detail. The model is based on the concept that a cell with hypothetically unlimited replicative potential-i.e. a ‘stem’ cell-can become ‘committed’ to a programme of limited replicative potential. This event is assumed to be stochastic and to have a fixed probability per stem cell division. The parameters of the model are: P,,the probability of commitment; N , the number of differentiative divisions; and T,, the cell-cycle times. By computer simulation, it is shown that P, increases roughly exponentially at each successive stem cell division. According to the model, when the daughter of a stem cell becomes committed, its progeny proceed through N obligatory divisions before becoming terminally differentiated (post-mitotic). The best-fit value of N was found to be seven. The simulations also reveal that the absolute number of stem cells in the total population increases for most of the lifespan of the culture. When P , becomes much greater than 0.5, the number of stem cells declines rapidly to zero, and the culture nears senescence. Sensitivity analysis shows that P, can assume only a limited range of values at each stem-cell division. Mass cultures of diploid fibroblastic cells undergo, on average. a constant number of population doublings (Hayflick & Moorhead, 1961). This behaviour is under active study in many laboratories as an in vitro expression of ‘cellular aging’ (Merz & Ross, 1969; Martin, Sprague & Epstein, 1970: Good & Smith, 1974: Smith & Hayflick, 1974: Absher & Absher. 1976; Schneider & Mitsui, 1976; Smith, Pereira-Smith & Schneider, 1978: Pereira-Smith & Smith, 1982). Two mathematically simple models of in vitro senescence have been proposed to account for the aggregate growth curves o f fibroblastic cultures. Both of these models Correspondence: Dr John Angello, Department of Biological Structure SM-20. University of Washington. School of Medicine. Seattle. WA 98195. U.S.A. J. C. Angello and J . W . Prothero invoke the notion that the transition between cell classes is probabilistic (stochastic), an idea first formalized by Kendall (1 948) and central to subsequent work modelling the behaviour of cell populations (e.g. Kubitschek. 1962; Till. McCulloch & Siminovitch, 1964: Vogel, Niewisch & Matioli. 1969: Mackey & Combs. 1974: Good, 1977; Shields, 1977; Gilbert, 1978: Brooks. Bennett & Smith. 1980; Edmunds & Adams, 1981). According to the model of Kirkwood & Holliday ( 1 9 7 9 , cells enter an ‘incubation period’ as a consequence of a probabilistic event termed ‘commitment’. The cells then undergo a substantial number of divisions prior to becoming post-mitotic. The ‘probability of commitment‘ ( P , ) is considered to be a constant in the current formulation of the kirk wood-Holliday model. A second model of senescence has been advanced by Shall & Stein (1979). These authors propose that cells ‘commit‘ with a probability that varies with the number of cell divisions undergone. Once committed. the cells are considered to become post-mitotic immediatelyi.e. there is no ‘incubation period‘. Aggregate growth curves do not provide a stringent test of the adequacy of models of in uifro senescence. More detailed information can be obtained by culturing cells at low (i.e. clonal) density. By growing cells at very low density. it is possible to determine the proportion of cells in the mass population which yield clones of specific sizes (i.e. 1,2,4 .... cells). A graph of the percentage of clones having a given minimum size is termed a ‘clone-size distribution curve‘. Smith, Pereira-Smith & Good ( 1977) plotted clone-size distribution (CSD) curves for human lung fibroblast cultures at different stages. or ‘passage’ levels in the life history of their cultures. They obtained a family of curves with negative slopes. Each C S D curve is cumulative. starting at 100%-i.e. all clones consist of at least one cell-and decaying, approximately linearly. to the right (indicating that a progressively smaller proportion of clones have larger numbers of cells). Curves derived from cells at later ‘passage’ levels have a more negative slope. meaning that the average clone size progressively decreases with increasing passage level. This has been termed ’clonal attenuation’ (Martin eta/., 1974). It is evident that there is much more information in a family of C S D curves tha? in a single aggregate growth curve. A satisfactory model of clonal attenuation needs to account for both the great heterogeneity in clone sizes at all stages in the life history of a culture and the decline in clone size with passage number. It has been shown (Prothero & Gallant. 1981) that neither the Kirkwood-Holliday nor the Shall-Stein model of it? uitro senescence is capable of giving a good account of the CSD data of Smith er nl. (1977). At the same time. it was shown that a hybrid model, which incorporates an increasing probability of commitment (as does the mortalization model of Shall and Stein) and a small number of incubation divisions (seven. in contrast to the large number in the Kirkwood-Holliday model) is able to give a satisfactory account of the data of Smith et ul. (1977). Using this hybrid niodel and appropriate computer implementations. it is possible, in principle. to analyse the kinetics of clonal attenuation and associated stochastic variation in substantial detail. Because of the significant effort required for this analysis. it was considered important to use our own data. the statistical variability of which we could monitor. We elected to use chick embryo fibroblasts-see also Ryan (1979) and Smith el a/. ( 1 9 8 0 t b e c a u s e senescence of the chick cell strain is complete in about thirty population doublings ( r , about sixty for human fibroblasts). Thus, the experiments can be carried out in a more reasonable period of time. We ha\se collected the data necessary to construct C S D curves for chick fibroblastic cells (Angello. Prothero & Gallant. 1981). At the same time. we modified our earlier model and Clonal attenuation wrote two new computer programs to make possible a more detailed analysis of clonal attenuation than has been reported hitherto. We first discuss briefly the experimental methods, then we present the C S D curves which were obtained. Before discussing our model of clonal attenuation, we make a few remarks about model building in order t o review our approach and to clarify some of the more pertinent terms. At that point, it is convenient to describe our model of clonal attenuation, the computer implementations of the model, and the results of an extensive series of computer simulations of the C S D curves. Then the results of a sensitivity analysis are reported. We conclude with a discussion of our findings. MATERIALS AND METHODS Patches of dorsal skin from day-9 chick embroys were dissociated with 0.25% trypsin (GIBCO No. 610-5090, Grand Island Biological Co., Grand Island, NY). Most of the epidermal cells were removed as a.ggregates by filtration through 48 pm Nitex (Tetko, Inc., Elmsford, NY) and the remaining single cells were cultured either at high density (mass cultures) or at very low density (clonal cultures). Primary mass cultures were established at 1-2 x 10' cells per 35-mm plastic culture dish (Corning No. 25000, Corning Glass Works, Corning, NY) in MEM (Eagles, GIBCO No. 410-1500, Grand Island Biological Co.) containing 10% fetal bovine serum (GIBCO No. 31N4004, Grand Island Biological Co.). After 24 hr, the number of attached cells was determined and this number was used to calculate an approximate initial plating efficiency. Sister cultures were passaged with STV (GIBCO No. 6 10-5400, Grand Island Biological Co.) just prior to confluency and the cell number determined in order to assess the number of population doublings the cells had undergone. Subsequent mass cultures were established at 0.4 x 10' cells per 60-mm plastic dish (Corning No. 25010, Corning Glass Works), and passaged, just prior to confluency, until the cell strain senesced. Periodically during the lifespan of the cell strain. cells were removed for clonal culture (50-100 cells per 60-mm plastic dish). The clones were grown in a 1:1 mixture of conditioned medium (CM) and fresh medium (FM) because a higher percentage of large clones (> eight doublings) was obtained with this medium than with F M alone (Angello et al., 198 1). After 2 weeks of undisturbed incubation, the clonal cultures were fixed with Bouin's, stained with haematoxylin and scored. The resulting data were expressed as CSD curves. RESULTS Experimental results Clonal analysis of early- to late-passage chick embryo fibroblasts ( P D L 6-22) shows that the clone-size distribution (CSD) does not change significantly between PDLs 6 and 12, and that there is a progressive decline in the slope of the CSD curves between PDLs 12 and 22 (Fig. I ) . Similar results have been reported both by Ryan ( I 979) and by Smith et al. ( 1980). Model building The potential value of model building in the study of cell kinetics is now widely appreciated. However, the steps followed in the creation of a model may merit a few words. The steps normally followed are outlined briefly in Table 1 and discussed in the legend. J . C. Atzgello and J . W .Prothero 80- PI .$ 0 , 60- Ai v, 40- 20- Clone size n r Q 22 128 256 Fig. I . Cells from mas5 cultures iccre plated at clonal density and grown undisturbed for 2 weeks (see Materials and Methods). The resulting clones were scored for the number of cells per clone and the ‘clone-size distribution’ (CSD) computed. The data derived from cells of PDL 12-22 are shown (PDL-12: A: PDL-15: 0: PDL-18: 0: PDL-22: 0): data obtained from cells of PDL = 6 are shown i n the insert. The solid lines represent the the compirrc~rsitmdntiom of the empirical data. Model of clonal attenuation According t o the schematic conception of Fig. 2. stem cells may divide t o give rise either to other stem cells or to ‘newly committed‘ cells. The probability of a stem cell dividing t o give rise t o another stem cell is termed the probability of recycling and is denoted by P,. Alternatively. a stem cell may divide and. with probability P,, give rise t o a newly committed cell. The probability of commitment ( P , ) is defined in terms of the probability of recycling-i.e. P, = 1 - P,.As in our earlier model (Prothero & Gallant, 1981). the ‘cells’. once committed. undergo a constant number ( R i ) of symmetrical divisions. which we have characterized as ‘differentiative’ divisions. The behaviour of the model schematized in Fig. 2 is dominated by the parameters P, (or Pr). N and thc cell-cycle times T,,,,,, and T,,,,,,,,, (for stem cell and committed cells, respectively). I n particular. the way in which P, varies effectively determines the maximum lifespan (in population doublings) of a cell population. It should be noted that the potential number of cell doubliirgs which occur in the lifespan of a cell strain is much larger and is not related in any simple way to the number of populatiorr doubliizgs (see Prothero & Gallant, 198 1). According to our model, approximately thirty-two cell doublings would be required to account for the nineteen P D s from P D L s 6-25-i.e. the number of cell doublings exceeds the number of population doublings by about 70%). Computer implementations of model The computer program used in our earlier studies (Prothero. 1980: Prothero & Gallant. I98 1) generated ‘cultures‘ from a single ’stem cell‘. The details of the CSD curves were simulated Clonal attenuation Table 1. Steps in model building Formulation of a hypothesis Symbolic representation Implementation Simulation Best-fit parameter values Sensitivity analysis New predictions Typically, model building begins with a verbal statement of a h.vpothesis. In the present instance, the major hypothesis is that clonal attenuation reflects a movement of cells from a stem-cell compartment with hypothetically unlimited replicative potential to a committed-cell compartment with severely limited replicative potential. This transition is assumed to be probabilistic, an idea invoked by many others (see Introduction) and also supported directly by the subcloning experiments of Martin et al. (1974), Martinez el a/. (1978) and Smith C Whitney (1980). These latter experiments show that the range in proliferative potential among clonally related sister cells can be as much as an order of magnitude, a result most easily explained by a stochastic commitment mechanism. Before a hypothesis can be rigorously tested it must be cast into symbolic (mathematical) form. In biology it is typically the case that the resultant equations are intractable or can be evaluated only with great difficulty. Here we take advantage of the ‘number crunching’ power of computers by i m p h e n t i n g the mathematical equation (i.e. the mathematical model) in the form of a computer program. Usually, the next step in model building is to carry out a simulation. This involves varying the parameters of the model to see whether it behaves in a manner which is formally analogous to the phenomenon under study. In our case, we already know (Prothero & Gallant, 1981)that the model is qualitatively capable of accounting for the phenomenon of clonal attenuation (in human fibroblasts). Once it has been shown that a reasonable simulation is possible one goes on to determine the values of the parameters which give the ‘best fit’ to the empirical data. This always involves a criterion of ‘best fit’. In this study we have used ‘mean per cent deviation’ between the simulation and the empirical data as a measure of best fit. After the best-fit values of the parameters have been determined, it is often useful to investgate how sensitive the goodness-of-fit is to changes in the parameter values. In the present study, we have carried out a sensitivity analysis with respect to the probability of commitment ( P , ) , the number of differentiative divisions ( N ) and the cell-cycle times (T‘). A good model will predict behaviour which has not previously been reported or studied. Confirmation. amendment or rejection of the model then follows the acquisition of new empirical data. Normally. this initiates another round of model building. By and large the process described above is simply a restatement of the scientific method where the power of computers is exploited to allow for the quantitative testing of hypotheses. simply by varying the probability of commitment (P,), once the number of differentiative divisions was set. The cell-cycle times were equal in all compartments and all cell divisions were synchronous. The CSD curves obtained with chick fibroblasts proved to be ‘S-shaped’. in contrast to the approximately linear curves obtained by Smith e al. (1 977) with human fibroblastic cells. f Similar results with chick cells have been reported both by Ryan (1979) and by Smith et al. (1980). Our attempts to simulate the S-shaped curves using our earlier computer programs proved to be unsatisfactory. We have therefore written two new computer programs which provide alternative implementations of the underlying model (Table 2). Both of the new programs allow us to start the simulations with an arbitrary distribution of cell types-i.e. arbitrary proportions of ‘stem’ cells, ‘committed’ cells and ‘terminally differentiated’ cells-rather than with a single cell. In this sense the current programs give a more realistic account of the experimental situation than did the earlier ones. In addition, each of the new programs allow us to vary the cell-cycle times of the different classes of dividing cells (stem cells and committed cells). The first, and simpler, new program, termed AVCLONE. is J . C . Angello and J . W. Prothero I . Newly. committed cell Committed cell N-1 N-2 h v d N= 7 probability of recycling: P,-the probability of commitment: A-the number of divisions remaining prior to terminal differentiation: TTren,-the stem-cell-cycle time ( h r ) : TLt,,,,,,-the cormvirred cell-cycle time (hr) (see text). Fig. 2. Pattern of cell proliferation postulated in the model: P,-the Table 2. Analytical features of the model and the computer implementations Variables in the computer implementations AVCLONE (Synchronous cell divisions: deterministic) CLONE (Asynchronous cell divisions: probabilistic) Parameters of the underlying model ,V is the maximum number of divisions undergone by newly committed cells. P, is the probability of commitment (in the program CLONE). F, is the prescribed fraction of cells and T,,,,,, hhich commit at each stem-cell division (in the program AVCLONE). Tc,,,,,,,, represent the duration of the committed- and stem-cell cycles, respectively. f,. and f, f, , represent the initial proportions ( i t . at the beginning of a computer run) of stem. committed and terminally differentiated cells. designed to simulate the average behaviour of an indefinitely large number of cells. In this program. as in the earlier program. the ‘cells’ all divide synchronously. Unlike the earlier program. however. the cell-cycle times can be varied from cell division to cell division. In AVCLONE a fixedfraction (FC. Table 2) of the ‘cells’ are ‘committed’ at each division, in see accordance with the probability of commitment. The behaviour of AVCLONE is completely deterministic-i.e. could be reduced to a set of algebraic formulae. Clonal attenuation We have found AVCLONE useful for making trial runs to simulate C S D curves (see below). Thus one can use AVCLONE to determine the number of committed cell compartments, cell-cycle times, and the way in which the probability of commitment is likely to vary. Since AVCLONE is a relatively small program, it is inexpensive t o run. However, in order to obtain some insight into the stochastic nature of commitment, we have written a more elaborate program, called CLONE, in which the ‘commitment events’ are in fact probabilistic-i.e. generated with a random number generator. In CLONE, a certain number of ‘cells’ (maximum of forty) can be ‘plated’ at random phases of the cell cycle. As with AVCLONE, the cell-cycle times can be varied arbitrarily from cell division to cell division. In CLONE. a complete history is maintained for each ‘cell’. For example, a complete genealogical record would be maintained for the following sequence of events: a cell starts out (i.e. is ‘plated’) as a ‘stem cell’, divides a few times and ‘commits’, the committed progeny then proceed through several ‘differentiative’ divisions and finally become terminally differentiated. In each run of CLONE. we assign a different ‘seed’ value to the random number generator. For each seed value, a different sequence of probability values is generated. Thus, slightly different proportions of stem cells will ‘commit’, even if all the parameters4.e. mean probability of commitment, cell-cycle times, number of committed cell compartments and proportions of cell types-are identical in each run. Thus, by using CLONE, we obtain a measure of the effects of simple stochastic variation. Because this program uses a good deal of memory and requires a moderate amount of execution time, it is fairly expensive to run. Computer simulations Once a model has been implemented in the form of a computer program, it is possible to carry out computer simulations. This is achieved by varying the parameters of the model so as to give a ‘best fit’ to the data. In our study we have used mean deviation between a simulation and the experimental data as a measure of best fit (see below). We elected to begin the simulations at PDL-6. As indicated above, our approach is to carry out exploratory simulations with AVCLONE. The initial proportions of cell types (stem, committed and terminally differentiated) were determined by simulation of the empirical data of thefirst C S D curve (PDL = 6). Then numerical values were assigned to N and T,. and the ‘probability of commitment’ (which specifies the proportion of stern cells committing at each successive division) was varied in order t o obtain the best fit possible to the next C S D curve (PDL = 12). Subsequently, new values were assigned to N and T,, and the process was repeated until the best fit to the PDL = 12 curve was achieved. Once a reasonably good fit to the CSD curve at P D L = 12 was achieved, the values of N and T , were held constant and the probability of commitment alone was further varied to generate the succeeding C S D curves-i.e. at PDLs 15, 18 and 22. Having roughly defined the parameters using AVCLONE, a detailed simulation was carried out using CLONE. Three to five simulations were made, each with a different seed value. for each C S D curve. The mean percentage of clones of each size, as well as the standard deviations. were then calculated. The variation associated with the simulated C S D curve for a particular PDL could then be compared with the variation in the empirical data. The comparisons show that the deviations within each simulation are smaller than those for the empirical data. (See Fig. 3 for an example comparing the simulated and empirical data obtained for cells at PDL-12.) Therefore factors other than P , may exist which cause scatter in the empirical data (see Discussion). J . C. Arigello arid J. W . Prothero Clone size Fig. 3. Comparison of the standard deviations associated with the experimental data and with the simulation of the P D L ~ CSD curies. The variation associated with the simulation is. in every instance. smaller than the variation 12 asociated irh the empirical data. Results and discussion of simulations The solid lines fitted to the data of Fig. I represent the results of the best-fit computer simulations. The mean fit at each P D L was calculated according to the formula: Percent deviation = -L 100 -y I Y,’ - Yi‘1 /.VF, where J,: = the percentage of clones >X for the ith simulated data point, Y: = the percentage of clones >Xfor the ith empirical data point. and ti = the total number of data points. For example, at PDL = 12, 77.9% of the clones were experimentally found to contain at least two cells (i.e. clone size 3 X = 2. first empirical data point). The ‘best fit’ to the PDL = 1 2 data was found when 83.4% of the clones contained at least two cells. Therefore, or 7. loo. At PDLs 12. 15. 18 and 22. the mean fit was 4.5. 9-5. 14.3 and 29.4%, respectively (Table 3). The computer simulations of the data are. on the whole, in excellent agreement with the data (see Fig. 1 and Table 3). It is possible that the greater ‘YO deviations’ at later PDLs are the consequences of some inadequacy in the current model. However, it is more likely that this progressi\e error is an artifact of using ‘% deviation’ as a measure of ‘goodness of fit’. Recall that the more negative slopes of the CSD curves from older cultures mean that these cultures yield very small numbers of large clones. Hence, even a small absolute difference between the empirical and stimulated data translates as a large ‘% deviation’. For example, at PDL = 18, 17.9% of the clones were empirically found t o contain at least sixty-four cells, while 21. 1% was determined by simulation. Hence. a difference of 3.2% (21.1 - 17.9) corresponds to an 1 8 O 6 deviation. Again. at P D L = 22. 1.2% of the clones were empirically Clonal attenuation Table 3. Sensitivity analysis: variations in N a n d T, ~~ Variations in N t $ : 'Best-fit' parameters*+$ PDL ~~~ ~ (%)§ ("/.I =6 N =8 ("6) Variations in TL*$: T,,,, = TL<,",,,, (?h) Mean %I deviation *N=l. + T,,,,, = 2/3 Tcomm. $ Values of P, as described by the 'best-fit' curve in Fig. 7. 5 Determined as the ' O h deviation' of the simulated clone-size distribution (CSD) curve from the corresponding empirically derived curve. The determinations at PDLs 12, 15 and 18 utilized seven data points/CSD curve, while the determinations at PDL-22 utilized six data points/CSD curve. .o 80* I I Fig. 4. The proportion of stem. committed and terminally differentiated cells as a function of lifespan of the culture. Curves are derived from computer simulations of the CSD curves. determined to contain at least sixty-four cells, whereas 0.3% was determined by simulation. In this case, a difference of 0.9% (1.2-0.3) represents a 75% deviation. In the course of simulating the data, we found that the best fit to the CSD curves was obtained when N (the number of 'differentiative' cell divisions) was set t o seven. This is the same number we (Prothero & Gallant, 1981) obtained in our earlier simulations of the human fibroblast data of Smith el al. (1977). We also found that a significant improvement in the fit to the CSD curves (of chick cells) was obtained when the stem-cell-cycle time (T,,,,,,)was shorter than that of committed cells (Tco,,,,,,). The simulations provide detailed information about the kinetics underlying clonal attenuation. In Fig. 4, the proportion of each cell type (stem, committed and terminally J. C. Aiigello arid J . W. Prothero Fig. 5. Tlir ribsolure fiumhcr. of stem. committed and terminally differentiated cells i n the total population. i.\prcsst.d :IS a funcrion of the llfespan of the culture. Curves were obtained hv computer simulation of the CSD c u r \ c \ . N o w especiall!. that the ahsolutc nurnher of stem cells increases throughout 80nh of the lifespan of the cult urc. differentiated) is illustrated at each stage of the lifespan. It can be seen that the relative number of stem cells a n d of committed cells declines at -40% and at -80% of lifespan, respectively. Of perhaps even more interest is the behaviour of the three cell populations in absolute ferrns (Fig. 5). The number of stem cells rises to a maximum at -8000 of lifespan and then falls dramatically to zero. The number of committed cells likewise increases. until -95% of lifespan and then declines precipitously as the strain senesces. The population of coininitred cells can be further analysed in terms of subpopulations. which are those cells having between one and seven remaining divisions. Figure 6 shows clearly that each subpopulation of comniitted cells reaches a maximum size and then declines to zero. The sequential order i n which this occurs is. of course. predetermined by the initial assumptions of >t'obligatory cell di\risions after commitment. However. the quantitative relationships among the subpopulations is solely dependent on the way in which P, changes. Sensitivity analysis I t is useful to k n o w how stringent the model parameters are. This can be assessed by investigating the effects on the 'cgoodness of fit' to the empirical data when simulations are carried out uith parameter \,dues different from the *best-fit' values. Using the program AVCLONE. we ha1.e varied each parameter individually (keeping the other parameters constant) and determined the mean '('6 deviation' of the resulting CSD curves from the empirically derived curves. The results are summarized in Tables 3 and 4. As may be expected. a different \.slue for any of the parameters will change the quantitative relationships among all of the cell types. Some changes will be more pronounced earlier in the lifespan of the culture. and some will be most noticeable in the later stages. I n the following section. we have given a short description of the inujor changes which occur (see also Table 5). Clonal attenuation I001 Fig. 6. The dynamic change in the relative sizes of the subpopulations of committed cells, expressed as a function of the lifespan of the culture. iV = 1-7 identifies each compartment of committed cells. For example. cells in compartment 1 have seben divisions remaining. Cells in compartment 2 have six divisions remaining. and so on. Cells in compartment 7 have one division remaining prior t o terminal differentiation. Table 4. Sensitivity analysis: variations in P,* P,” ‘Best-fit’ p, (Oh) P,’ (V”) PDL (%I)+ I2 15 18 22 Mean %deviation * ‘Best-fit’ values of N and TLare: N = 7: T,,,,,,, 213 T,,,,,,,,,,. = + See footnote (3) t o Table 3. P,: values of P , as described by the ‘best-fit’ curve in Fig. 7. P,”: values <: P, (see Fig. 7). P,’: values > Pc(see Fig. 7). Table 5. Summary: effects of using other parameter values N (obligatory divisions) TL PL N-8 Progeny from each newly committed cell Stem cells Maximum PDL N-6 T,,,,,, = T<<,,,,,,, Greater Lesser 256 High Too many High LOW Rate of accumulation not optimal Too few Low J. C. Angello and J. W. Prothero 1 0 20 Number of stem cell divisions Fig. 7. The change in Pi with successive stem-cell divisions: the 'best-fit' P , (-). The sensitivity analysis of P , (see rrxt) was performed with values of P, ivhich were either greater ( P c ' )or less (Pi') than the 'best-fit' values given by fL. Variations of N (see Table 3) I f N (the number of obligatory cell divisions after commitment) is either increased or decreased by I . the 'mean (yo deviation' of the set of simulated PDL curves is approximately 2-fold greater than if N = 7. When N is increased from 7 to 8. a larger number of obligatory divisions is required of newly committed cells. This reduces the proportion of terminally differentiated (post-mitotic) cells throughout most of the lifespan of the culture. However, the larger number of obligatory divisions eventually results in too many cells being produced (i.e. a maximum PDL which is too high). When N is decreased from 7 to 6, the smaller number of obligatory divisions by newly committed cells cannot produce a sufficient number of progeny to simulate the empirically determined maximum PDL. Variarions o T (see Table 3) f The relationship between T,,,,,, and Tconlm governs the relative number of stem cells in the mass population. If T,,,,, = Tc,,llInl. stem-cell population divides, on average, in parallel the with the committed-cell populations. However. if T,,,,, < Tcol,,ll,, the stem-cell population then divides more often than the committed-cell populations. Early in the lifespan of the culture, when P, is low. this difference in the two T,s results in an increased number of stem cells, and an increased proportion of stem cells. The 'best-fit' values of T, are T,,,,, = 2/3 T,.,,,,,,,,. In other nords. we have found that the best fit to the empirical data is achieved when stem cells cycle 30% faster than committed cells. If T,,,,, and T,.,,,,,,,, have the same value, the 'mean O h deviation' of the simulated CSD curves from the empirical curves is nearly 3 times greater, mainly because the number of stem cells does not increase rapidly enough. Varia(ions of P, (Table 4) The 'best-fit' of the simulated CSD curves to the empirically derived curves was obtained when P , increased progressively at successive stem-cell divisions. The shape of the resulting curve ( P , 1 ' . stem-cell division number) was approximately exponential (Fig. 7, solid line). In Clonal attenuation order to test the effect of different P,s at the same stem-cell division, parallel curves were constructed above and below the ‘best-fit’ P, curve (see Fig. 7, dashed lines). The resulting (greater and lesser) values of P, were read from the curves and used for the sensitivity analysis. When lesser values of P, (P,”) are employed, the fit to the empirical data is about 4 times worse through PDL = 18. More significantly, the fit to the last CSD curve (PDL = 22) is an order of magnitude worse (see Table 4). The primary reason for these discrepancies is that lower values of P, permit too many stem cells to recycle (recall, P , = 1 - P,). One consequence is that too many cells are generated-i.e. the maximum PDL is too high. When greater values of P, ( P , ’ ) are employed, the fit to the empirical data is reasonable through PDL = 15, but is 2-3 times worse at later PDLs (see Table 4). The main reason for this discrepancy is that too many stem cells commit early in the lifespan of the culture. As a result, the stem-cell population never achieves a size which can sustain the culture long enough-i.e. the maximum PDL achieved is too low. Stem-cell-derivedclones One of the advantages of the computer program, CLONE, is that it records the life history of each ‘cell’. As a result, the simulated clonal progeny of individual stem cells can be compared with each other and with experimental data. Because commitment is probabilistic in our model, there should be significant variation in the distributions of cell types which are generated by the initial stem cells, especially at the earlier PDLs when P, is relatively small. For example. if both daughters of the first clonal division commit, the maximum clone size would be 2 x 2’ = 256 cells, and 100% of the cells would be post-mitotic after eight cell divisions. At the other extreme, if none of the stem-cell progeny commit, 100% of the cells would be mitotically active after the time required for eight committed cell divisions and the clone size would be 256 cells (or greater if T,,,, is less than T,,,,). The characteristics of most of the clones naturally, will be intermediate between these two values. To document the predicted variability among progeny derived from a cohort of stem cells, the clonal progeny of stem cells from simuZated PDL-12 cultures were analysed for mitotically active and for post-mitotic ‘cells’ at a time when eight committed-cell divisions had occurred. An average of approximately 70% of the ‘cells’ were mitotically a c t i v e 4 . e . were either stem cells or committed cells-and the range among the individual clones was 44-100% (Table 6). Experimentally, 13-day-old clonal cultures of PDL-12 cells were labelled for 20 hr with [ -‘HITdR, and the largest clones (those > 256 cells) were autoradiographically assayed for ‘% labelled nuclei’. An average of about 60% of the cells within the clones were labelled, and the range was - 2 9 4 9 % (Table 6). The similarity of the experimental and simulated stem-cell data, especially the range of ‘% cycling cells/clone’, reinforces the hypothesis of a probabilistic basis for clonal attenuation. DISCUSSION We have shown that a simple model of clonal attenuation is capable of explaining the experimental results obtained with chick embryo fibroblasts. This model is essentially the same as that which we proposed to explain the human fibroblast clonal data of Smith et a/. (1 977). However, the computer program which we used to implement the model and simulate the Smith data was not as useful when we attempted to simulate the chick data. This was because the CSD curves obtained from the chick cells were more ‘S-shaped’ than were the CSD curves obtained from the human cells. At this time we presume that this difference is due J . C. Angello and J . W. Prothero Table 6. Stem-cell~derived clones Percent di\.iding cells Experimental data 0 1 Simiilation ( / I = 9 ) SD Range 5) 5p.2* 69.4: !I= 28.9-88.9 44-100 sample rire. * Labelled nuclei. t Stem cells plus committed cells. to a different distribution of cell t y p e s 4 . e . stem. committed and terminally differentiated-in the two tissues of origin. Therefore. we have written two new computer programs ( A V C L O N E and CLONE) which offer the flexibility of choosing the initial distribution of cell types. and with which we can simulate the chick data more satisfactorily. A major result of this study is the most detailed currently available characterization of the kinetics underlying clonal attenuation. Perhaps the most striking prediction of the model is that the n u t n b ~ r stem cells increases exponentially throughout 80% of the lifespan of the of strain (Fig. 5 ) . a feature which is not apparent when cell numbers are expressed on a percentage basis (as in Fig. 4). This prediction can be tested whenever a suitable marker for stem cells is developed. I t is scarcely necessary to emphasize that it is important to understand the dynamics of the stem-cell subpopulation in order to understand the behaviour of the total cell population. The model incorporates the idea that an orderly sequential process is involved in chick fibroblast clonal attenuation : committed cells which have N divisions remaining are the daughters of cell Lvhich previously had N + 1 divisions remaining (Fig. 6). This aspect of the model is consistent with the concept that the final steps of commitment are a deterministic process. genetically programmed and. to a large extent. irreversible. Similar ideas have been advanced for the myogenic (Kligman & Nameroff. 1980). the erythroid (Prothero. Starling & Rosse. 1978) and epidermal (Potten et 01.. 1952) cell lines. Computer simulation of our data leads to an estimate of seven obligatory cell divisions after the commitment ‘event‘. The simulations also suggest faster cell-cycle times for stem cells than for committed cells (T,,,,,, T,,, ,,“,). = 2/3 The concept of a few ‘differentiative’ divisions (seven) for fibroblastic cells has been proposed by us in an earlier report (Prothero & Gallant. 198 1). and utilized by Jones & Smith (1982) in the development of their model. The concept has also been proposed recently for a totally different proliferating system. the murine epidermis, by Potten et al. (1982). These authors. also drawing upon computer simulation. concluded that the most appropriate model for their system was one containing three ’transit proliferating subpopulations’. Furthermore, the proposed cycle times for stem cells and for ’transit‘ cells were different. In contrast to our results with fibroblasts. however. Potten ef al. (1982) deduced that the cycle time of epidermal stem cells u a s longer than that of ‘transit’ cells. It may be that the length of the stem-cell cycle relative to that of the committed daughters is cell type- or even tissue-specific. In our fibroblastic system. if N (the number of obligatory committed cell divisions) is either increased or decreased by 1. and the other parameters are held constant, the ‘goodness of fit‘ to the empirical data deteriorates significantly. If N = 8. the larger number of obligatory committed cell divisions reduces the proportion of terininalli*dflerenriafed cells in the early and midpassage cell populations. In addition. too many cells are ultimately produced (i.e. the Clonal atteiiuation maximum P D L is too great). If N = 6, the smaller number of obligatory divisions by committed cell produces too few cells (i.e. the maximum P D L is too low). We have also found that the best fit to the empirical data is achieved when stem cells cycle faster than do committed cells (T,,,, = 2/3 T,,,,,,). Unless this constraint is imposed, the size of the stem-cell population is too small to generate a sufficient number of newly committed cells at the earlier PDLs. As a result, the relative number of larger (64 and 128 cell) clones at the earlier PDLs is too low. When T,,,, < T,,,,, this also has a favourable consequence at later PDLs-it assures that the stem-cell population will begin to decline sooner in the lifespan of the culture than if T,,,, = T,,,,,,. This contributes to the dramatic fall in the CSD curve at PDL = 22. Once the values of N , T,,,, and T,,,, are established, variation in P , is the sole determinant of how rapidly the strain will senesce. We have made the simplest initial assumption-that the value of P, does not decrease-and have determined by computer simulation that the value of P, increases roughly exponentially with each successive stem-cell division (Fig. 7, solid line). A sensitivity analysis was designed to investigate how well determined the values of P, are. First, two curves parallel to the ‘best-fit’ P, curve were drawn (Fig. 7, dashed curves). One curve described values of P, greater than the best fit values (P,’) and the other curve described values which were less (P,”). Values of P, for each stern-cell division were then read from these curves and used to simulate the empirical CSD curves. Curves whose starting values of P, were only 20% different from the best-fit value of P, yielded simulations whose fit was 2-8 times worse (‘mean % deviation’, Table 4). Especially in the case of smaller values of P, (P,”), a difference from the best-fit simulations occurs early (by PDL = 12) and continues to get worse throughout the lifespan of the simulated culture. Quite simply. too many stem cells recycle when P, is too low and, consequently, too few cells become ‘newly committed’. The consequence of higher values of P, (P,’) is more subtle because simulations even up to PDL = 18 are reasonable. However, by this time P, is very near a value of 0.5. Further stem-cell divisions lead to such a dramatic decrease in the stem-cell population that, by P D L = 22 there are virtually no newly committed cells being produced, and the fit to the empirical data becomes quite unacceptable. For all of the perturbations tested. excepting the variations in T,, an important constraint which is not met is a simulation of the empirically determined maximum P D L of the cell strain. When N = 8, or when lower values of P, (P,”, Fig. 7) are used, the final simulated P D L is too large; when N = 6, or when higher values of P, (Pc’,Fig. 7) are used, the final P D L is too small. Both the value of N and the manner of variation of P, are crucial to obtaining the correct final PDL. While the ‘percent deviation’ from the average experimental data (Tables 3 and 4) is a reasonable method for evaluating the quality of the simulations, another aspect of the statistical analysis is the variation in both the experimental data and the simulations. We have determined that the deviations within each simulation are smaller than those for the experimental data (see Fig. 3 for comparison at PDL-12). Therefore. factors other than P, may exist which cause scatter in the experimental data. One possibility is the existence of a small ‘probability of terminal differentiation’ (PTD). According to the concept of PTD, committed cell is not obligated to proceed through all a seven differentiative divisions. Rather, the daughter of any committed cell may become terminally differentiated (post-mitotic). The probability of terminal differentiation may be very small. This idea would be consistent with data reported by Smith & Whitney (1980). We have tested the effect of varying P,, on the goodness of fit of the simulations and, so far, have 42 J. C. Ailgello and J . W. Prothero found no improvement. However. we do not dispute the results of Smith & Whitney (1980). We assume that the effects of varying P , are too subtle to be observed by our C S D analysis. , Another possible explanation of the scatter in the empirical data is that there are subpopulations of stem cells iir pica. each of which completes its first in uitro division with a different initial value of P,. With regard to this possibility. it should be noted that the current computer programs treat all stem cells as equivalent, which is, no doubt, an oversimplification. Identifying possible subsets of stem cells is another topic for future research. With the use of the program. CLONE. it was shown that the progeny of stem cells from midpassage cultures may form a variety of populations: the range among these populations is substantial (Table 6 .Experimental data to test this prediction were obtained by labelling ) clones with I'HITdR for the last 20 hr of a 2-week clonal growth period. The largest clones-i.e. those derived from stem cells-contained an average of -60% labelled nuclei with a range of about 3&90% (Table 6). Assuming that the cells which incorporate I 'HITdR are also capable of dividing. the experimental data are in good agreement with the predictions of the model. This is consistent with the hypothesis that commitment is a probabilistic event. In summary. a relatively simple model can simulate the experimental C S D curves obtained from chick embryo fibroblasts. The parameters of the model are: P,, the probability of commitment: N . the number of differentiative divisions: and T,. the cell-cycle times. The model offers the following explanation of the dynamics which govern clonal attenuation: each daughter of a stem-cell division is capable of being committed to a deterministic mitotic sequence. namely. seven divisions. culminating in terminal differentiation. Commitment is probabilistic and governed by the function P,. P, is relatively small during the first few in z,itro stem-cell divisions and thereafter increases roughly exponentially each time a stem-cell divides. The absolute number of stem cells increases throughout most of the lifespan of a fibroblastic culture. and then declines precipitously as P, becomes greater than 0.5. ACKNOWLEDGMENTS This work was supported in part by USPHS Grant No. 2 PO1 AG-01751 from the National Institutes of Health to Dr George M. Martin. We thank Drs Jonathan Gallant and George Martin for their comments on a draft of the manuscript. and Ms Doris Ringer for preparation of the manuscript. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Cell Proliferation Wiley

Clonal Attenuation In Chick Embryo Fibroblasts Experimental Data, A Model and Computer Simulations

Angello, J. C.; Prothero, J. W.
Cell Proliferation , Volume 18 (1) – Jan 1, 1985
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Publisher
Wiley
Copyright
1985 Blackwell Science Limited
ISSN
0960-7722
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1365-2184
DOI
10.1111/j.1365-2184.1985.tb00630.x
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Abstract

J. C. Angello and J. W. Prothero Department of Biological Structure, University o Washington, School o Medicine, Seattle, WA 98195, U.S.A. f f (Received 7 September 1983; revision accepted 8 August 1984) Abstract. When cells from mass cultures of chick embryo fibroblasts are grown at very low density, some cells yield large clones while others produce smaller clones, and some cells fail to divide a t all. The distribution of clone sizes is related to the number of population doublings which the donor mass culture has undergone: the more doublings which have occurred, the smaller the average clone size. In this report we describe a model which analyses this phenomenon, referred to as ‘clonal attenuation’, in detail. The model is based on the concept that a cell with hypothetically unlimited replicative potential-i.e. a ‘stem’ cell-can become ‘committed’ to a programme of limited replicative potential. This event is assumed to be stochastic and to have a fixed probability per stem cell division. The parameters of the model are: P,,the probability of commitment; N , the number of differentiative divisions; and T,, the cell-cycle times. By computer simulation, it is shown that P, increases roughly exponentially at each successive stem cell division. According to the model, when the daughter of a stem cell becomes committed, its progeny proceed through N obligatory divisions before becoming terminally differentiated (post-mitotic). The best-fit value of N was found to be seven. The simulations also reveal that the absolute number of stem cells in the total population increases for most of the lifespan of the culture. When P , becomes much greater than 0.5, the number of stem cells declines rapidly to zero, and the culture nears senescence. Sensitivity analysis shows that P, can assume only a limited range of values at each stem-cell division. Mass cultures of diploid fibroblastic cells undergo, on average. a constant number of population doublings (Hayflick & Moorhead, 1961). This behaviour is under active study in many laboratories as an in vitro expression of ‘cellular aging’ (Merz & Ross, 1969; Martin, Sprague & Epstein, 1970: Good & Smith, 1974: Smith & Hayflick, 1974: Absher & Absher. 1976; Schneider & Mitsui, 1976; Smith, Pereira-Smith & Schneider, 1978: Pereira-Smith & Smith, 1982). Two mathematically simple models of in vitro senescence have been proposed to account for the aggregate growth curves o f fibroblastic cultures. Both of these models Correspondence: Dr John Angello, Department of Biological Structure SM-20. University of Washington. School of Medicine. Seattle. WA 98195. U.S.A. J. C. Angello and J . W . Prothero invoke the notion that the transition between cell classes is probabilistic (stochastic), an idea first formalized by Kendall (1 948) and central to subsequent work modelling the behaviour of cell populations (e.g. Kubitschek. 1962; Till. McCulloch & Siminovitch, 1964: Vogel, Niewisch & Matioli. 1969: Mackey & Combs. 1974: Good, 1977; Shields, 1977; Gilbert, 1978: Brooks. Bennett & Smith. 1980; Edmunds & Adams, 1981). According to the model of Kirkwood & Holliday ( 1 9 7 9 , cells enter an ‘incubation period’ as a consequence of a probabilistic event termed ‘commitment’. The cells then undergo a substantial number of divisions prior to becoming post-mitotic. The ‘probability of commitment‘ ( P , ) is considered to be a constant in the current formulation of the kirk wood-Holliday model. A second model of senescence has been advanced by Shall & Stein (1979). These authors propose that cells ‘commit‘ with a probability that varies with the number of cell divisions undergone. Once committed. the cells are considered to become post-mitotic immediatelyi.e. there is no ‘incubation period‘. Aggregate growth curves do not provide a stringent test of the adequacy of models of in uifro senescence. More detailed information can be obtained by culturing cells at low (i.e. clonal) density. By growing cells at very low density. it is possible to determine the proportion of cells in the mass population which yield clones of specific sizes (i.e. 1,2,4 .... cells). A graph of the percentage of clones having a given minimum size is termed a ‘clone-size distribution curve‘. Smith, Pereira-Smith & Good ( 1977) plotted clone-size distribution (CSD) curves for human lung fibroblast cultures at different stages. or ‘passage’ levels in the life history of their cultures. They obtained a family of curves with negative slopes. Each C S D curve is cumulative. starting at 100%-i.e. all clones consist of at least one cell-and decaying, approximately linearly. to the right (indicating that a progressively smaller proportion of clones have larger numbers of cells). Curves derived from cells at later ‘passage’ levels have a more negative slope. meaning that the average clone size progressively decreases with increasing passage level. This has been termed ’clonal attenuation’ (Martin eta/., 1974). It is evident that there is much more information in a family of C S D curves tha? in a single aggregate growth curve. A satisfactory model of clonal attenuation needs to account for both the great heterogeneity in clone sizes at all stages in the life history of a culture and the decline in clone size with passage number. It has been shown (Prothero & Gallant. 1981) that neither the Kirkwood-Holliday nor the Shall-Stein model of it? uitro senescence is capable of giving a good account of the CSD data of Smith er nl. (1977). At the same time. it was shown that a hybrid model, which incorporates an increasing probability of commitment (as does the mortalization model of Shall and Stein) and a small number of incubation divisions (seven. in contrast to the large number in the Kirkwood-Holliday model) is able to give a satisfactory account of the data of Smith et ul. (1977). Using this hybrid niodel and appropriate computer implementations. it is possible, in principle. to analyse the kinetics of clonal attenuation and associated stochastic variation in substantial detail. Because of the significant effort required for this analysis. it was considered important to use our own data. the statistical variability of which we could monitor. We elected to use chick embryo fibroblasts-see also Ryan (1979) and Smith el a/. ( 1 9 8 0 t b e c a u s e senescence of the chick cell strain is complete in about thirty population doublings ( r , about sixty for human fibroblasts). Thus, the experiments can be carried out in a more reasonable period of time. We ha\se collected the data necessary to construct C S D curves for chick fibroblastic cells (Angello. Prothero & Gallant. 1981). At the same time. we modified our earlier model and Clonal attenuation wrote two new computer programs to make possible a more detailed analysis of clonal attenuation than has been reported hitherto. We first discuss briefly the experimental methods, then we present the C S D curves which were obtained. Before discussing our model of clonal attenuation, we make a few remarks about model building in order t o review our approach and to clarify some of the more pertinent terms. At that point, it is convenient to describe our model of clonal attenuation, the computer implementations of the model, and the results of an extensive series of computer simulations of the C S D curves. Then the results of a sensitivity analysis are reported. We conclude with a discussion of our findings. MATERIALS AND METHODS Patches of dorsal skin from day-9 chick embroys were dissociated with 0.25% trypsin (GIBCO No. 610-5090, Grand Island Biological Co., Grand Island, NY). Most of the epidermal cells were removed as a.ggregates by filtration through 48 pm Nitex (Tetko, Inc., Elmsford, NY) and the remaining single cells were cultured either at high density (mass cultures) or at very low density (clonal cultures). Primary mass cultures were established at 1-2 x 10' cells per 35-mm plastic culture dish (Corning No. 25000, Corning Glass Works, Corning, NY) in MEM (Eagles, GIBCO No. 410-1500, Grand Island Biological Co.) containing 10% fetal bovine serum (GIBCO No. 31N4004, Grand Island Biological Co.). After 24 hr, the number of attached cells was determined and this number was used to calculate an approximate initial plating efficiency. Sister cultures were passaged with STV (GIBCO No. 6 10-5400, Grand Island Biological Co.) just prior to confluency and the cell number determined in order to assess the number of population doublings the cells had undergone. Subsequent mass cultures were established at 0.4 x 10' cells per 60-mm plastic dish (Corning No. 25010, Corning Glass Works), and passaged, just prior to confluency, until the cell strain senesced. Periodically during the lifespan of the cell strain. cells were removed for clonal culture (50-100 cells per 60-mm plastic dish). The clones were grown in a 1:1 mixture of conditioned medium (CM) and fresh medium (FM) because a higher percentage of large clones (> eight doublings) was obtained with this medium than with F M alone (Angello et al., 198 1). After 2 weeks of undisturbed incubation, the clonal cultures were fixed with Bouin's, stained with haematoxylin and scored. The resulting data were expressed as CSD curves. RESULTS Experimental results Clonal analysis of early- to late-passage chick embryo fibroblasts ( P D L 6-22) shows that the clone-size distribution (CSD) does not change significantly between PDLs 6 and 12, and that there is a progressive decline in the slope of the CSD curves between PDLs 12 and 22 (Fig. I ) . Similar results have been reported both by Ryan ( I 979) and by Smith et al. ( 1980). Model building The potential value of model building in the study of cell kinetics is now widely appreciated. However, the steps followed in the creation of a model may merit a few words. The steps normally followed are outlined briefly in Table 1 and discussed in the legend. J . C. Atzgello and J . W .Prothero 80- PI .$ 0 , 60- Ai v, 40- 20- Clone size n r Q 22 128 256 Fig. I . Cells from mas5 cultures iccre plated at clonal density and grown undisturbed for 2 weeks (see Materials and Methods). The resulting clones were scored for the number of cells per clone and the ‘clone-size distribution’ (CSD) computed. The data derived from cells of PDL 12-22 are shown (PDL-12: A: PDL-15: 0: PDL-18: 0: PDL-22: 0): data obtained from cells of PDL = 6 are shown i n the insert. The solid lines represent the the compirrc~rsitmdntiom of the empirical data. Model of clonal attenuation According t o the schematic conception of Fig. 2. stem cells may divide t o give rise either to other stem cells or to ‘newly committed‘ cells. The probability of a stem cell dividing t o give rise t o another stem cell is termed the probability of recycling and is denoted by P,. Alternatively. a stem cell may divide and. with probability P,, give rise t o a newly committed cell. The probability of commitment ( P , ) is defined in terms of the probability of recycling-i.e. P, = 1 - P,.As in our earlier model (Prothero & Gallant, 1981). the ‘cells’. once committed. undergo a constant number ( R i ) of symmetrical divisions. which we have characterized as ‘differentiative’ divisions. The behaviour of the model schematized in Fig. 2 is dominated by the parameters P, (or Pr). N and thc cell-cycle times T,,,,,, and T,,,,,,,,, (for stem cell and committed cells, respectively). I n particular. the way in which P, varies effectively determines the maximum lifespan (in population doublings) of a cell population. It should be noted that the potential number of cell doubliirgs which occur in the lifespan of a cell strain is much larger and is not related in any simple way to the number of populatiorr doubliizgs (see Prothero & Gallant, 198 1). According to our model, approximately thirty-two cell doublings would be required to account for the nineteen P D s from P D L s 6-25-i.e. the number of cell doublings exceeds the number of population doublings by about 70%). Computer implementations of model The computer program used in our earlier studies (Prothero. 1980: Prothero & Gallant. I98 1) generated ‘cultures‘ from a single ’stem cell‘. The details of the CSD curves were simulated Clonal attenuation Table 1. Steps in model building Formulation of a hypothesis Symbolic representation Implementation Simulation Best-fit parameter values Sensitivity analysis New predictions Typically, model building begins with a verbal statement of a h.vpothesis. In the present instance, the major hypothesis is that clonal attenuation reflects a movement of cells from a stem-cell compartment with hypothetically unlimited replicative potential to a committed-cell compartment with severely limited replicative potential. This transition is assumed to be probabilistic, an idea invoked by many others (see Introduction) and also supported directly by the subcloning experiments of Martin et al. (1974), Martinez el a/. (1978) and Smith C Whitney (1980). These latter experiments show that the range in proliferative potential among clonally related sister cells can be as much as an order of magnitude, a result most easily explained by a stochastic commitment mechanism. Before a hypothesis can be rigorously tested it must be cast into symbolic (mathematical) form. In biology it is typically the case that the resultant equations are intractable or can be evaluated only with great difficulty. Here we take advantage of the ‘number crunching’ power of computers by i m p h e n t i n g the mathematical equation (i.e. the mathematical model) in the form of a computer program. Usually, the next step in model building is to carry out a simulation. This involves varying the parameters of the model to see whether it behaves in a manner which is formally analogous to the phenomenon under study. In our case, we already know (Prothero & Gallant, 1981)that the model is qualitatively capable of accounting for the phenomenon of clonal attenuation (in human fibroblasts). Once it has been shown that a reasonable simulation is possible one goes on to determine the values of the parameters which give the ‘best fit’ to the empirical data. This always involves a criterion of ‘best fit’. In this study we have used ‘mean per cent deviation’ between the simulation and the empirical data as a measure of best fit. After the best-fit values of the parameters have been determined, it is often useful to investgate how sensitive the goodness-of-fit is to changes in the parameter values. In the present study, we have carried out a sensitivity analysis with respect to the probability of commitment ( P , ) , the number of differentiative divisions ( N ) and the cell-cycle times (T‘). A good model will predict behaviour which has not previously been reported or studied. Confirmation. amendment or rejection of the model then follows the acquisition of new empirical data. Normally. this initiates another round of model building. By and large the process described above is simply a restatement of the scientific method where the power of computers is exploited to allow for the quantitative testing of hypotheses. simply by varying the probability of commitment (P,), once the number of differentiative divisions was set. The cell-cycle times were equal in all compartments and all cell divisions were synchronous. The CSD curves obtained with chick fibroblasts proved to be ‘S-shaped’. in contrast to the approximately linear curves obtained by Smith e al. (1 977) with human fibroblastic cells. f Similar results with chick cells have been reported both by Ryan (1979) and by Smith et al. (1980). Our attempts to simulate the S-shaped curves using our earlier computer programs proved to be unsatisfactory. We have therefore written two new computer programs which provide alternative implementations of the underlying model (Table 2). Both of the new programs allow us to start the simulations with an arbitrary distribution of cell types-i.e. arbitrary proportions of ‘stem’ cells, ‘committed’ cells and ‘terminally differentiated’ cells-rather than with a single cell. In this sense the current programs give a more realistic account of the experimental situation than did the earlier ones. In addition, each of the new programs allow us to vary the cell-cycle times of the different classes of dividing cells (stem cells and committed cells). The first, and simpler, new program, termed AVCLONE. is J . C . Angello and J . W. Prothero I . Newly. committed cell Committed cell N-1 N-2 h v d N= 7 probability of recycling: P,-the probability of commitment: A-the number of divisions remaining prior to terminal differentiation: TTren,-the stem-cell-cycle time ( h r ) : TLt,,,,,,-the cormvirred cell-cycle time (hr) (see text). Fig. 2. Pattern of cell proliferation postulated in the model: P,-the Table 2. Analytical features of the model and the computer implementations Variables in the computer implementations AVCLONE (Synchronous cell divisions: deterministic) CLONE (Asynchronous cell divisions: probabilistic) Parameters of the underlying model ,V is the maximum number of divisions undergone by newly committed cells. P, is the probability of commitment (in the program CLONE). F, is the prescribed fraction of cells and T,,,,,, hhich commit at each stem-cell division (in the program AVCLONE). Tc,,,,,,,, represent the duration of the committed- and stem-cell cycles, respectively. f,. and f, f, , represent the initial proportions ( i t . at the beginning of a computer run) of stem. committed and terminally differentiated cells. designed to simulate the average behaviour of an indefinitely large number of cells. In this program. as in the earlier program. the ‘cells’ all divide synchronously. Unlike the earlier program. however. the cell-cycle times can be varied from cell division to cell division. In AVCLONE a fixedfraction (FC. Table 2) of the ‘cells’ are ‘committed’ at each division, in see accordance with the probability of commitment. The behaviour of AVCLONE is completely deterministic-i.e. could be reduced to a set of algebraic formulae. Clonal attenuation We have found AVCLONE useful for making trial runs to simulate C S D curves (see below). Thus one can use AVCLONE to determine the number of committed cell compartments, cell-cycle times, and the way in which the probability of commitment is likely to vary. Since AVCLONE is a relatively small program, it is inexpensive t o run. However, in order to obtain some insight into the stochastic nature of commitment, we have written a more elaborate program, called CLONE, in which the ‘commitment events’ are in fact probabilistic-i.e. generated with a random number generator. In CLONE, a certain number of ‘cells’ (maximum of forty) can be ‘plated’ at random phases of the cell cycle. As with AVCLONE, the cell-cycle times can be varied arbitrarily from cell division to cell division. In CLONE. a complete history is maintained for each ‘cell’. For example, a complete genealogical record would be maintained for the following sequence of events: a cell starts out (i.e. is ‘plated’) as a ‘stem cell’, divides a few times and ‘commits’, the committed progeny then proceed through several ‘differentiative’ divisions and finally become terminally differentiated. In each run of CLONE. we assign a different ‘seed’ value to the random number generator. For each seed value, a different sequence of probability values is generated. Thus, slightly different proportions of stem cells will ‘commit’, even if all the parameters4.e. mean probability of commitment, cell-cycle times, number of committed cell compartments and proportions of cell types-are identical in each run. Thus, by using CLONE, we obtain a measure of the effects of simple stochastic variation. Because this program uses a good deal of memory and requires a moderate amount of execution time, it is fairly expensive to run. Computer simulations Once a model has been implemented in the form of a computer program, it is possible to carry out computer simulations. This is achieved by varying the parameters of the model so as to give a ‘best fit’ to the data. In our study we have used mean deviation between a simulation and the experimental data as a measure of best fit (see below). We elected to begin the simulations at PDL-6. As indicated above, our approach is to carry out exploratory simulations with AVCLONE. The initial proportions of cell types (stem, committed and terminally differentiated) were determined by simulation of the empirical data of thefirst C S D curve (PDL = 6). Then numerical values were assigned to N and T,. and the ‘probability of commitment’ (which specifies the proportion of stern cells committing at each successive division) was varied in order t o obtain the best fit possible to the next C S D curve (PDL = 12). Subsequently, new values were assigned to N and T,, and the process was repeated until the best fit to the PDL = 12 curve was achieved. Once a reasonably good fit to the CSD curve at P D L = 12 was achieved, the values of N and T , were held constant and the probability of commitment alone was further varied to generate the succeeding C S D curves-i.e. at PDLs 15, 18 and 22. Having roughly defined the parameters using AVCLONE, a detailed simulation was carried out using CLONE. Three to five simulations were made, each with a different seed value. for each C S D curve. The mean percentage of clones of each size, as well as the standard deviations. were then calculated. The variation associated with the simulated C S D curve for a particular PDL could then be compared with the variation in the empirical data. The comparisons show that the deviations within each simulation are smaller than those for the empirical data. (See Fig. 3 for an example comparing the simulated and empirical data obtained for cells at PDL-12.) Therefore factors other than P , may exist which cause scatter in the empirical data (see Discussion). J . C. Arigello arid J. W . Prothero Clone size Fig. 3. Comparison of the standard deviations associated with the experimental data and with the simulation of the P D L ~ CSD curies. The variation associated with the simulation is. in every instance. smaller than the variation 12 asociated irh the empirical data. Results and discussion of simulations The solid lines fitted to the data of Fig. I represent the results of the best-fit computer simulations. The mean fit at each P D L was calculated according to the formula: Percent deviation = -L 100 -y I Y,’ - Yi‘1 /.VF, where J,: = the percentage of clones >X for the ith simulated data point, Y: = the percentage of clones >Xfor the ith empirical data point. and ti = the total number of data points. For example, at PDL = 12, 77.9% of the clones were experimentally found to contain at least two cells (i.e. clone size 3 X = 2. first empirical data point). The ‘best fit’ to the PDL = 1 2 data was found when 83.4% of the clones contained at least two cells. Therefore, or 7. loo. At PDLs 12. 15. 18 and 22. the mean fit was 4.5. 9-5. 14.3 and 29.4%, respectively (Table 3). The computer simulations of the data are. on the whole, in excellent agreement with the data (see Fig. 1 and Table 3). It is possible that the greater ‘YO deviations’ at later PDLs are the consequences of some inadequacy in the current model. However, it is more likely that this progressi\e error is an artifact of using ‘% deviation’ as a measure of ‘goodness of fit’. Recall that the more negative slopes of the CSD curves from older cultures mean that these cultures yield very small numbers of large clones. Hence, even a small absolute difference between the empirical and stimulated data translates as a large ‘% deviation’. For example, at PDL = 18, 17.9% of the clones were empirically found t o contain at least sixty-four cells, while 21. 1% was determined by simulation. Hence. a difference of 3.2% (21.1 - 17.9) corresponds to an 1 8 O 6 deviation. Again. at P D L = 22. 1.2% of the clones were empirically Clonal attenuation Table 3. Sensitivity analysis: variations in N a n d T, ~~ Variations in N t $ : 'Best-fit' parameters*+$ PDL ~~~ ~ (%)§ ("/.I =6 N =8 ("6) Variations in TL*$: T,,,, = TL<,",,,, (?h) Mean %I deviation *N=l. + T,,,,, = 2/3 Tcomm. $ Values of P, as described by the 'best-fit' curve in Fig. 7. 5 Determined as the ' O h deviation' of the simulated clone-size distribution (CSD) curve from the corresponding empirically derived curve. The determinations at PDLs 12, 15 and 18 utilized seven data points/CSD curve, while the determinations at PDL-22 utilized six data points/CSD curve. .o 80* I I Fig. 4. The proportion of stem. committed and terminally differentiated cells as a function of lifespan of the culture. Curves are derived from computer simulations of the CSD curves. determined to contain at least sixty-four cells, whereas 0.3% was determined by simulation. In this case, a difference of 0.9% (1.2-0.3) represents a 75% deviation. In the course of simulating the data, we found that the best fit to the CSD curves was obtained when N (the number of 'differentiative' cell divisions) was set t o seven. This is the same number we (Prothero & Gallant, 1981) obtained in our earlier simulations of the human fibroblast data of Smith el al. (1977). We also found that a significant improvement in the fit to the CSD curves (of chick cells) was obtained when the stem-cell-cycle time (T,,,,,,)was shorter than that of committed cells (Tco,,,,,,). The simulations provide detailed information about the kinetics underlying clonal attenuation. In Fig. 4, the proportion of each cell type (stem, committed and terminally J. C. Aiigello arid J . W. Prothero Fig. 5. Tlir ribsolure fiumhcr. of stem. committed and terminally differentiated cells i n the total population. i.\prcsst.d :IS a funcrion of the llfespan of the culture. Curves were obtained hv computer simulation of the CSD c u r \ c \ . N o w especiall!. that the ahsolutc nurnher of stem cells increases throughout 80nh of the lifespan of the cult urc. differentiated) is illustrated at each stage of the lifespan. It can be seen that the relative number of stem cells a n d of committed cells declines at -40% and at -80% of lifespan, respectively. Of perhaps even more interest is the behaviour of the three cell populations in absolute ferrns (Fig. 5). The number of stem cells rises to a maximum at -8000 of lifespan and then falls dramatically to zero. The number of committed cells likewise increases. until -95% of lifespan and then declines precipitously as the strain senesces. The population of coininitred cells can be further analysed in terms of subpopulations. which are those cells having between one and seven remaining divisions. Figure 6 shows clearly that each subpopulation of comniitted cells reaches a maximum size and then declines to zero. The sequential order i n which this occurs is. of course. predetermined by the initial assumptions of >t'obligatory cell di\risions after commitment. However. the quantitative relationships among the subpopulations is solely dependent on the way in which P, changes. Sensitivity analysis I t is useful to k n o w how stringent the model parameters are. This can be assessed by investigating the effects on the 'cgoodness of fit' to the empirical data when simulations are carried out uith parameter \,dues different from the *best-fit' values. Using the program AVCLONE. we ha1.e varied each parameter individually (keeping the other parameters constant) and determined the mean '('6 deviation' of the resulting CSD curves from the empirically derived curves. The results are summarized in Tables 3 and 4. As may be expected. a different \.slue for any of the parameters will change the quantitative relationships among all of the cell types. Some changes will be more pronounced earlier in the lifespan of the culture. and some will be most noticeable in the later stages. I n the following section. we have given a short description of the inujor changes which occur (see also Table 5). Clonal attenuation I001 Fig. 6. The dynamic change in the relative sizes of the subpopulations of committed cells, expressed as a function of the lifespan of the culture. iV = 1-7 identifies each compartment of committed cells. For example. cells in compartment 1 have seben divisions remaining. Cells in compartment 2 have six divisions remaining. and so on. Cells in compartment 7 have one division remaining prior t o terminal differentiation. Table 4. Sensitivity analysis: variations in P,* P,” ‘Best-fit’ p, (Oh) P,’ (V”) PDL (%I)+ I2 15 18 22 Mean %deviation * ‘Best-fit’ values of N and TLare: N = 7: T,,,,,,, 213 T,,,,,,,,,,. = + See footnote (3) t o Table 3. P,: values of P , as described by the ‘best-fit’ curve in Fig. 7. P,”: values <: P, (see Fig. 7). P,’: values > Pc(see Fig. 7). Table 5. Summary: effects of using other parameter values N (obligatory divisions) TL PL N-8 Progeny from each newly committed cell Stem cells Maximum PDL N-6 T,,,,,, = T<<,,,,,,, Greater Lesser 256 High Too many High LOW Rate of accumulation not optimal Too few Low J. C. Angello and J. W. Prothero 1 0 20 Number of stem cell divisions Fig. 7. The change in Pi with successive stem-cell divisions: the 'best-fit' P , (-). The sensitivity analysis of P , (see rrxt) was performed with values of P, ivhich were either greater ( P c ' )or less (Pi') than the 'best-fit' values given by fL. Variations of N (see Table 3) I f N (the number of obligatory cell divisions after commitment) is either increased or decreased by I . the 'mean (yo deviation' of the set of simulated PDL curves is approximately 2-fold greater than if N = 7. When N is increased from 7 to 8. a larger number of obligatory divisions is required of newly committed cells. This reduces the proportion of terminally differentiated (post-mitotic) cells throughout most of the lifespan of the culture. However, the larger number of obligatory divisions eventually results in too many cells being produced (i.e. a maximum PDL which is too high). When N is decreased from 7 to 6, the smaller number of obligatory divisions by newly committed cells cannot produce a sufficient number of progeny to simulate the empirically determined maximum PDL. Variarions o T (see Table 3) f The relationship between T,,,,,, and Tconlm governs the relative number of stem cells in the mass population. If T,,,,, = Tc,,llInl. stem-cell population divides, on average, in parallel the with the committed-cell populations. However. if T,,,,, < Tcol,,ll,, the stem-cell population then divides more often than the committed-cell populations. Early in the lifespan of the culture, when P, is low. this difference in the two T,s results in an increased number of stem cells, and an increased proportion of stem cells. The 'best-fit' values of T, are T,,,,, = 2/3 T,.,,,,,,,,. In other nords. we have found that the best fit to the empirical data is achieved when stem cells cycle 30% faster than committed cells. If T,,,,, and T,.,,,,,,,, have the same value, the 'mean O h deviation' of the simulated CSD curves from the empirical curves is nearly 3 times greater, mainly because the number of stem cells does not increase rapidly enough. Varia(ions of P, (Table 4) The 'best-fit' of the simulated CSD curves to the empirically derived curves was obtained when P , increased progressively at successive stem-cell divisions. The shape of the resulting curve ( P , 1 ' . stem-cell division number) was approximately exponential (Fig. 7, solid line). In Clonal attenuation order to test the effect of different P,s at the same stem-cell division, parallel curves were constructed above and below the ‘best-fit’ P, curve (see Fig. 7, dashed lines). The resulting (greater and lesser) values of P, were read from the curves and used for the sensitivity analysis. When lesser values of P, (P,”) are employed, the fit to the empirical data is about 4 times worse through PDL = 18. More significantly, the fit to the last CSD curve (PDL = 22) is an order of magnitude worse (see Table 4). The primary reason for these discrepancies is that lower values of P, permit too many stem cells to recycle (recall, P , = 1 - P,). One consequence is that too many cells are generated-i.e. the maximum PDL is too high. When greater values of P, ( P , ’ ) are employed, the fit to the empirical data is reasonable through PDL = 15, but is 2-3 times worse at later PDLs (see Table 4). The main reason for this discrepancy is that too many stem cells commit early in the lifespan of the culture. As a result, the stem-cell population never achieves a size which can sustain the culture long enough-i.e. the maximum PDL achieved is too low. Stem-cell-derivedclones One of the advantages of the computer program, CLONE, is that it records the life history of each ‘cell’. As a result, the simulated clonal progeny of individual stem cells can be compared with each other and with experimental data. Because commitment is probabilistic in our model, there should be significant variation in the distributions of cell types which are generated by the initial stem cells, especially at the earlier PDLs when P, is relatively small. For example. if both daughters of the first clonal division commit, the maximum clone size would be 2 x 2’ = 256 cells, and 100% of the cells would be post-mitotic after eight cell divisions. At the other extreme, if none of the stem-cell progeny commit, 100% of the cells would be mitotically active after the time required for eight committed cell divisions and the clone size would be 256 cells (or greater if T,,,, is less than T,,,,). The characteristics of most of the clones naturally, will be intermediate between these two values. To document the predicted variability among progeny derived from a cohort of stem cells, the clonal progeny of stem cells from simuZated PDL-12 cultures were analysed for mitotically active and for post-mitotic ‘cells’ at a time when eight committed-cell divisions had occurred. An average of approximately 70% of the ‘cells’ were mitotically a c t i v e 4 . e . were either stem cells or committed cells-and the range among the individual clones was 44-100% (Table 6). Experimentally, 13-day-old clonal cultures of PDL-12 cells were labelled for 20 hr with [ -‘HITdR, and the largest clones (those > 256 cells) were autoradiographically assayed for ‘% labelled nuclei’. An average of about 60% of the cells within the clones were labelled, and the range was - 2 9 4 9 % (Table 6). The similarity of the experimental and simulated stem-cell data, especially the range of ‘% cycling cells/clone’, reinforces the hypothesis of a probabilistic basis for clonal attenuation. DISCUSSION We have shown that a simple model of clonal attenuation is capable of explaining the experimental results obtained with chick embryo fibroblasts. This model is essentially the same as that which we proposed to explain the human fibroblast clonal data of Smith et a/. (1 977). However, the computer program which we used to implement the model and simulate the Smith data was not as useful when we attempted to simulate the chick data. This was because the CSD curves obtained from the chick cells were more ‘S-shaped’ than were the CSD curves obtained from the human cells. At this time we presume that this difference is due J . C. Angello and J . W. Prothero Table 6. Stem-cell~derived clones Percent di\.iding cells Experimental data 0 1 Simiilation ( / I = 9 ) SD Range 5) 5p.2* 69.4: !I= 28.9-88.9 44-100 sample rire. * Labelled nuclei. t Stem cells plus committed cells. to a different distribution of cell t y p e s 4 . e . stem. committed and terminally differentiated-in the two tissues of origin. Therefore. we have written two new computer programs ( A V C L O N E and CLONE) which offer the flexibility of choosing the initial distribution of cell types. and with which we can simulate the chick data more satisfactorily. A major result of this study is the most detailed currently available characterization of the kinetics underlying clonal attenuation. Perhaps the most striking prediction of the model is that the n u t n b ~ r stem cells increases exponentially throughout 80% of the lifespan of the of strain (Fig. 5 ) . a feature which is not apparent when cell numbers are expressed on a percentage basis (as in Fig. 4). This prediction can be tested whenever a suitable marker for stem cells is developed. I t is scarcely necessary to emphasize that it is important to understand the dynamics of the stem-cell subpopulation in order to understand the behaviour of the total cell population. The model incorporates the idea that an orderly sequential process is involved in chick fibroblast clonal attenuation : committed cells which have N divisions remaining are the daughters of cell Lvhich previously had N + 1 divisions remaining (Fig. 6). This aspect of the model is consistent with the concept that the final steps of commitment are a deterministic process. genetically programmed and. to a large extent. irreversible. Similar ideas have been advanced for the myogenic (Kligman & Nameroff. 1980). the erythroid (Prothero. Starling & Rosse. 1978) and epidermal (Potten et 01.. 1952) cell lines. Computer simulation of our data leads to an estimate of seven obligatory cell divisions after the commitment ‘event‘. The simulations also suggest faster cell-cycle times for stem cells than for committed cells (T,,,,,, T,,, ,,“,). = 2/3 The concept of a few ‘differentiative’ divisions (seven) for fibroblastic cells has been proposed by us in an earlier report (Prothero & Gallant. 198 1). and utilized by Jones & Smith (1982) in the development of their model. The concept has also been proposed recently for a totally different proliferating system. the murine epidermis, by Potten et al. (1982). These authors. also drawing upon computer simulation. concluded that the most appropriate model for their system was one containing three ’transit proliferating subpopulations’. Furthermore, the proposed cycle times for stem cells and for ’transit‘ cells were different. In contrast to our results with fibroblasts. however. Potten ef al. (1982) deduced that the cycle time of epidermal stem cells u a s longer than that of ‘transit’ cells. It may be that the length of the stem-cell cycle relative to that of the committed daughters is cell type- or even tissue-specific. In our fibroblastic system. if N (the number of obligatory committed cell divisions) is either increased or decreased by 1. and the other parameters are held constant, the ‘goodness of fit‘ to the empirical data deteriorates significantly. If N = 8. the larger number of obligatory committed cell divisions reduces the proportion of terininalli*dflerenriafed cells in the early and midpassage cell populations. In addition. too many cells are ultimately produced (i.e. the Clonal atteiiuation maximum P D L is too great). If N = 6, the smaller number of obligatory divisions by committed cell produces too few cells (i.e. the maximum P D L is too low). We have also found that the best fit to the empirical data is achieved when stem cells cycle faster than do committed cells (T,,,, = 2/3 T,,,,,,). Unless this constraint is imposed, the size of the stem-cell population is too small to generate a sufficient number of newly committed cells at the earlier PDLs. As a result, the relative number of larger (64 and 128 cell) clones at the earlier PDLs is too low. When T,,,, < T,,,,, this also has a favourable consequence at later PDLs-it assures that the stem-cell population will begin to decline sooner in the lifespan of the culture than if T,,,, = T,,,,,,. This contributes to the dramatic fall in the CSD curve at PDL = 22. Once the values of N , T,,,, and T,,,, are established, variation in P , is the sole determinant of how rapidly the strain will senesce. We have made the simplest initial assumption-that the value of P, does not decrease-and have determined by computer simulation that the value of P, increases roughly exponentially with each successive stem-cell division (Fig. 7, solid line). A sensitivity analysis was designed to investigate how well determined the values of P, are. First, two curves parallel to the ‘best-fit’ P, curve were drawn (Fig. 7, dashed curves). One curve described values of P, greater than the best fit values (P,’) and the other curve described values which were less (P,”). Values of P, for each stern-cell division were then read from these curves and used to simulate the empirical CSD curves. Curves whose starting values of P, were only 20% different from the best-fit value of P, yielded simulations whose fit was 2-8 times worse (‘mean % deviation’, Table 4). Especially in the case of smaller values of P, (P,”), a difference from the best-fit simulations occurs early (by PDL = 12) and continues to get worse throughout the lifespan of the simulated culture. Quite simply. too many stem cells recycle when P, is too low and, consequently, too few cells become ‘newly committed’. The consequence of higher values of P, (P,’) is more subtle because simulations even up to PDL = 18 are reasonable. However, by this time P, is very near a value of 0.5. Further stem-cell divisions lead to such a dramatic decrease in the stem-cell population that, by P D L = 22 there are virtually no newly committed cells being produced, and the fit to the empirical data becomes quite unacceptable. For all of the perturbations tested. excepting the variations in T,, an important constraint which is not met is a simulation of the empirically determined maximum P D L of the cell strain. When N = 8, or when lower values of P, (P,”, Fig. 7) are used, the final simulated P D L is too large; when N = 6, or when higher values of P, (Pc’,Fig. 7) are used, the final P D L is too small. Both the value of N and the manner of variation of P, are crucial to obtaining the correct final PDL. While the ‘percent deviation’ from the average experimental data (Tables 3 and 4) is a reasonable method for evaluating the quality of the simulations, another aspect of the statistical analysis is the variation in both the experimental data and the simulations. We have determined that the deviations within each simulation are smaller than those for the experimental data (see Fig. 3 for comparison at PDL-12). Therefore. factors other than P, may exist which cause scatter in the experimental data. One possibility is the existence of a small ‘probability of terminal differentiation’ (PTD). According to the concept of PTD, committed cell is not obligated to proceed through all a seven differentiative divisions. Rather, the daughter of any committed cell may become terminally differentiated (post-mitotic). The probability of terminal differentiation may be very small. This idea would be consistent with data reported by Smith & Whitney (1980). We have tested the effect of varying P,, on the goodness of fit of the simulations and, so far, have 42 J. C. Ailgello and J . W. Prothero found no improvement. However. we do not dispute the results of Smith & Whitney (1980). We assume that the effects of varying P , are too subtle to be observed by our C S D analysis. , Another possible explanation of the scatter in the empirical data is that there are subpopulations of stem cells iir pica. each of which completes its first in uitro division with a different initial value of P,. With regard to this possibility. it should be noted that the current computer programs treat all stem cells as equivalent, which is, no doubt, an oversimplification. Identifying possible subsets of stem cells is another topic for future research. With the use of the program. CLONE. it was shown that the progeny of stem cells from midpassage cultures may form a variety of populations: the range among these populations is substantial (Table 6 .Experimental data to test this prediction were obtained by labelling ) clones with I'HITdR for the last 20 hr of a 2-week clonal growth period. The largest clones-i.e. those derived from stem cells-contained an average of -60% labelled nuclei with a range of about 3&90% (Table 6). Assuming that the cells which incorporate I 'HITdR are also capable of dividing. the experimental data are in good agreement with the predictions of the model. This is consistent with the hypothesis that commitment is a probabilistic event. In summary. a relatively simple model can simulate the experimental C S D curves obtained from chick embryo fibroblasts. The parameters of the model are: P,, the probability of commitment: N . the number of differentiative divisions: and T,. the cell-cycle times. The model offers the following explanation of the dynamics which govern clonal attenuation: each daughter of a stem-cell division is capable of being committed to a deterministic mitotic sequence. namely. seven divisions. culminating in terminal differentiation. Commitment is probabilistic and governed by the function P,. P, is relatively small during the first few in z,itro stem-cell divisions and thereafter increases roughly exponentially each time a stem-cell divides. The absolute number of stem cells increases throughout most of the lifespan of a fibroblastic culture. and then declines precipitously as P, becomes greater than 0.5. ACKNOWLEDGMENTS This work was supported in part by USPHS Grant No. 2 PO1 AG-01751 from the National Institutes of Health to Dr George M. Martin. We thank Drs Jonathan Gallant and George Martin for their comments on a draft of the manuscript. and Ms Doris Ringer for preparation of the manuscript.

Journal

Cell Proliferation – Wiley

Published: Jan 1, 1985

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