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Expansion of Mutant Stem Cell Populations in the Human Colon
J. theor. Biol. (1996) 178, 381–385
Expansion of Mutant Stem Cell Populations in the Human Colon M B† Department of Anatomy and Cell Biology, Medical Sciences Building, University of Toronto, Toronto, Ontario M5S 1A8, Canada (Received on 14 July 1995, Accepted in revised form on 18 September 1995)
In general, it is presumed that colonic epithelial stem cells are the principal cell type at risk of incurring the series of somatic mutations leading to carcinoma, since all other epithelial cell types are short-lived. Mutant stem cell clonal expansion increases the risk for subsequent mutations and is therefore a potentially important step in carcinogenesis. The stem cells reside in colonic crypts, simple tubular foldings of the epithelium, and thus counting crypts provides an indirect means to determine stem cell numbers. The normal crypt population is known to expand through a process of crypt replication and this is thought to result in a corresponding expansion of the epithelial stem cell population. A simple mathematical model of the population dynamics of normal and mutant crypts (crypts containing mutant stem cells) is developed and used to estimate a lower bound on the relative rate of expansion of the mutant stem cell population. The model predicts that if mutant and normal crypt populations expand at the same rate, and if the mutation rate is small relative to the rate of growth, then the fraction of clusters of mutant crypts composed of only a single mutant crypt should steadily decrease with age towards one-half. Aberrant crypts are easily recognizable lesions in human colon which have frequently been shown to contain cells with K-ras and occasionally APC gene mutations. Application of the model to recent counts of aberrant crypt cluster sizes indicate that the aberrant crypt population, and the contained mutant stem cell population, is expanding substantially faster than normal. 7 1996 Academic Press Limited
increased mutation risk. Simple methods capable of demonstrating abnormalities in the relative rate of expansion of clones of mutant stem cells in man are therefore of interest, and will require techniques both for the identification of mutant clones and also for the extraction of useful information from identified clones. It is the latter that will be dealt with here. A careful scan of the colonic epithelium will occasionally reveal crypts of abnormal morphology and histochemistry; these are the aberrant crypts (Bird, 1987; McLellan & Bird, 1988a,b; Roncucci et al., 1991a, b; Pretlow et al., 1994). They are often found in clusters thought to represent clones derived from a mutant stem cell. In fact, many aberrant crypts contain cells with K-ras or APC mutations (Stopera et al., 1992; Pretlow et al., 1993; Vivona et al., 1993; Smith et al., 1994; Jen et al., 1994; Yamashita et al.,
Introduction There is strong evidence that many colorectal carcinomas result from accumulated somatic mutations, presumably starting with the mutation of an epithelial progenitor or stem cell (Fearon & Vogelstein, 1990; Nishisho et al., 1991; Gordon et al., 1991; Powell et al., 1991; Vogelstein & Kinzler, 1993). Expansion of the clones derived from such mutant stem cells is thought to be an important stage in carcinogenesis for at least two reasons: (i) the increasing numbers of mutant cells directly increases the probability that at least one of these cells will receive the additional mutations leading to malignant transformation; and (ii) clonal expansion entails DNA replication with its associated † E-mail: bjerknes.crypt.med.utoronto.ca. 0022–5193/96/040381 + 05 $18.00/0
381
7 1996 Academic Press Limited
382
.
1995). Although their role in human carcinogenesis remains contentious (Jen et al., 1994; Yamashita et al., 1995; Pretlow, 1995), aberrant crypts are nonetheless a good test case for the study of the expansion of a population of readily identified mutant clones. When first formed, aberrant crypts are single isolated structures, but with time they branch producing clusters of related aberrant crypts (McLellan et al., 1991). This description of the growth of clusters of aberrant crypts is reminiscent of the crypt cycle, a continuous crypt replication process which has been most thoroughly studied in the normal mouse small intestinal epithelium (Bjerknes, 1986, 1994, 1995; Cheng & Bjerknes, 1985; Totafurno et al., 1987), but which also seems to occur in human colon (Cheng et al., 1986). In normal intestine, the crypt cycle results in the slow expansion of the crypt population through a process of crypt budding and fission. It appears that newly formed aberrant crypts expand their numbers through a similar process. Thus, aberrant crypts arise from two sources; from spontaneous somatic mutation of a stem cell in a normal crypt, or from a crypt-cycle-like replication of existing aberrant crypts. The latter mechanism has the potential to expand the mutant stem cell pool rapidly and thus may warrant serious concern. It is a simple matter to write equations describing the expansion of normal and mutant crypt populations and the conversion of normal into mutant crypts. In this work I demonstrate that if mutant and normal crypt populations expand at the same rate, then these equations predict that with age, about one-half of mutant crypt clusters should contain only a single crypt. If fewer than one-half are actually found to be unicryptal, then this is evidence that the mutant crypt population is expanding faster than normal. These ideas are applied to the analysis of recently published data on the distribution of the size of human aberrant crypt clones, where it was found that fewer than one-half of clusters of aberrant crypts were unicryptal; most were polycryptal (Roncucci et al., 1991a,b; Pretlow et al., 1994). It is concluded that the aberrant crypt population, and hence the mutant stem cell population contained in them, is expanding substantially faster than normal. A Mathematical Model of Aberrant Crypt Dynamics Our goal is to determine if the population of mutant stem cells found in aberrant crypts is growing faster than the stem cell population found in normal crypts. Since the rate of growth of a stem cell pool is reflected in the rate of growth of the population of
crypts containing those stem cell (Bjerknes, 1986, 1994, 1995; Cheng & Bjerknes, 1985; Totafurno et al., 1987), we will compare the expansion rates of the populations of aberrant vs. normal crypts in order to explore the relative expansion rates of the populations of mutant vs. normal stem cells. We will assume that the normal crypt population expands at rate k1 proportional to the current population size, and that l is their rate of conversion into aberrant crypts (presumably through spontaneous somatic mutations in a contained stem cell; Bjerknes, 1995). We also assume that single isolated aberrant crypts give rise to larger clusters of aberrant crypts at a constant rate k2 , again in proportion to current population size. Note that k1 and k2 may be interpreted either as the respective rates of replication of normal and mutant crypts, or in the presence of significant crypt loss as the respective net population expansion rates. Also note that we will ignore the presumably negligible probability that stem cells in adjacent normal crypts would undergo the same mutation thus producing a mutant cluster without crypt replication. Under these assumptions, the growth of the populations of normal and aberrant crypts at time t may be approximated by the system of differential equations N '(t) = (k1 − l)N(t)
(1)
S '(t) = lN(t) − k2 S(t)
(2)
C '(t) = k2 S(t),
(3)
where N(t) is the number of normal crypts, S(t) is the number of single aberrant crypts, and C(t) is the number of clusters containing two or more aberrant crypts. These equations have solutions [assuming N(0) = N0 and S(0) = C(0) = 0] N(t) = N0 eat
(4)
S(t) =
lN0 at (e − e−k2 t ) b
(5)
C(t) =
lN0 (k eat + a e−k2 t − b), ab 2
(6)
where a = k1 − l and b = k1 + k2 − l. Here, all aberrant crypts were treated as though they behave identically. This should be an adequate first approximation, but as markers are developed which allow subpopulations of aberrant crypts to be identified (for example those containing different mutations), this system of equations could easily be expanded by addition of variables representing the singleton and larger clusters of each of the subtypes, each with their own mutation and replication rates (Bjerknes, 1995).
The total number of aberrant crypt clusters will grow as T(t) = S(t) + C(t) =
lN0 at (e − 1), a
(7) (8)
and, therefore, the fraction of aberrant crypt clusters which should be found as singletons is s(t) 0
S(t) a eat − e−k2 t = . T(t) b eat − 1
(9)
There are two observations about this equation which will lead us to our final result. First, with time, the fraction of single aberrant crypts tends to the limit s 0 lim s(t) t4a
(10)
383
Discussion Since fewer than one-half of aberrant crypt clusters are single (Roncucci et al., 1991a,b; Pretlow et al., 1994), we concluded that the aberrant crypt population is expanding at an increased rate in comparison with normal crypts. We may estimate a lower bound on the magnitude of the increase from recent data. Pretlow et al. (1994; their fig. 4) found in a study of 15 patients that only one of 42 clusters of aberrant crypts were single. Thus, an estimate of an upper bound on the asymptotic fraction of clusters of single aberrant crypts is s E 1/42. This is an inequality because s(t) is a decreasing function of time, and we do not know whether the data were collected from individuals whose crypt dynamics had reached the asymptotic state. Rearranging eqn (14) we obtain 1 r= −1 s
= a/b
(11)
= (k1 − l)/(k1 + k2 − l)
(12)
e
1 k1 /(k1 + k2 ),
(13)
= 41.
where eqn (13) follows from the additional, and probably reasonable, assumption that the spontaneous mutation rate, l, is small relative to the crypt population expansion rates k1 and k2 (Bjerknes, 1995). Second, is the observation that the fraction of single aberrant crypts, s(t) is a decreasing function of time (see Appendix). Therefore, the fraction of aberrant crypt clusters containing a single crypt starts at 1 (at t = 0) and decreases steadily towards the asymptotic value k1 /(k1 + k2 ). Equation (13) may be rewritten as s = 1/(1 + r),
(14)
where r = k2 /k1 . This tells us that if the rates of expansion of normal and aberrant crypts are equal (r = 1) then asymptotically one-half of aberrant crypts should be single. Since measurements of aberrant crypt clusters in human colon revealed that far fewer than one-half of the clusters were composed of single crypts (Roncucci et al., 1991a,b; Pretlow et al., 1994), we may safely conclude that the aberrant crypt population is growing faster than is the population of normal crypts, and that in turn the mutant stem cell population contained in the aberrant crypts expands at an increased rate. This conclusion holds even though the samples may not have been obtained at asymptotic ages because s(t) is steadily decreasing in time.
1 −1 1/42
(15) (16) (17)
Thus, given this admittedly rough estimate of s, the mutant stem cell pool contained in the aberrant crypt population is growing q40 times faster than normal. This may be an overestimate if single aberrant crypts are more easily missed during scoring than are large clusters, and thus, the reader should be cautioned that the data analysed here was not collected with our purpose in mind. Vagaries of the experimental methods used may have biased the distribution of cluster-sizes obtained; an issue which future measurements of cluster-size distributions must address. It is important to stress that the conclusion that the mutant stem cell pool is growing q40 times faster than the normal stem cell pool does not necessarily mean that the mutant stem cells are cycling 40 times faster than normal stem cells, a feat which might not even be possible. Rather, the magnitude of the increase in the rate of growth of the mutant stem cell pool suggests that increased retention of stem cell offspring, which normally would be lost through cell differentiation or death, could be a major factor in the increased rate of expansion. Recruitment into the cell cycle of non-cycling mutant stem cells, if such exist, offers an additional source. However, increased proliferation probably does make a contribution because aberrant crypts are known to have proliferative abnormalities (Roncucci et al., 1993; Pretlow et al., 1994; Yamashita et al., 1994), but
384
.
it is unclear from these previous studies whether the stem cells are affected. Crypt loss is also a potential concern, but it should be recalled that the model parameters k1 and k2 may be interpreted as representing the net expansion rates of normal and mutant crypts, respectively. Furthermore, if significant loss of aberrant crypts occurs, say as a result of immune surveillance, then the mutant stem cells must be replicating at an even higher rate than is indicated above, in order to counter the loss. Another issue, about which little is known, is the actual stem cell content of normal and mutant crypts. There are strong theoretical reasons for suspecting that the stem cell content of normal crypts is variable (Bjerknes, 1986, 1994, 1995; Totafurno et al., 1987), but nothing is known about possible effects of mutations on either the variability or the average stem cell content. We have largely avoided this issue here by focusing on the relative rates of expansion, rather than the absolute sizes of the mutant and normal stem cell populations. It is also important to stress that the somatic mutations afflicting aberrant crypts need not necessarily affect the stem cells directly in order to yield an increased growth rate. This could result, for example, if the mutation affected cell adhesion in a way which stimulated crypt replication (Augenlicht, 1994). Stimulation of the replication process would, in turn, stimulate growth of the stem cell population. Regardless of whether the mutation impacts directly or indirectly on stem cell growth rates, the mutant stem cell pool in aberrant crypts does appear to be expanding at an abnormal rate, a potentially dangerous situation. Much effort is being spent in the search for dietary factors which might alter the rate of progression towards tumors (Bruce et al., 1993; Magnuson et al., 1993; Kawamori et al., 1994; Sutherland & Bird, 1994; Takahashi et al., 1994; Thorup et al., 1994) and thus, there is considerable interest in the design of simple methods of analysis of the results from these trials. The application of variations of the analysis introduced here may offer another useful approach. The approach introduced above is not limited only to the study of the rate of expansion of aberrant crypts. The ideas may be applicable to the study of any mutation for which markers are developed allowing identification of the afflicted crypts.
This work was supported by a grant from the MRC of Canada.
REFERENCES A, L. H. (1994). Adhesion molecules, cellular differentiation, and colonic crypt architecture. Gastroenterology 107, 1894–1898. B, R. P. (1987). Observation and quantification of aberrant crypts in the murine colon treated with a colon carcinogen: preliminary findings. Cancer Lett. 37, 147–151. B, M. (1986). A test of the stochastic theory of stem cell differentiation. Biophys. J. 49, 1223–1227. B, M. (1994). Simple stochastic theory of stem-cell differentiation is not simultaneously consistent with crypt extinction probability and the expansion of mutated clones. J. theor. Biol. 168, 349–365. B, M. (1995). The crypt cycle and the asymptotic dynamics of clusters of mutant crypts in the intestinal epithelium. Proc. Roy. Soc. Lond. B 260, 1–16. B, W. R., A, M. C., C, D. E., M, A., M, S., S, D., et al. (1993). Diet, aberrant crypt foci and colorectal cancer. Mutat. Res. 290, 111–118. C, H. & B, M. (1985). Whole population cell kinetics and postnatal development of the mouse intestinal epithelium. Anat. Rec. 211, 420–426. C, H., B, M., A, J. & G, G. (1986). Crypt production in normal and diseased human colonic epithelium. Anat. Rec. 216, 44–48. F, E. R. & V, B. (1990). A genetic model for colorectal tumorigenesis. Cell 61, 759–767. G, J., T, A., S, W., C, M., G, L., A, H. et al. (1991). Identification and characterization of the familial adenomatous polyposis coli gene. Cell 66, 589–600. J, J., P, S. M., P, N., S, K. J., H, S. R., V, B. & K, K. W. (1994). Molecular determinants of dysplasia in colorectal lesions. Cancer Res. 54, 5523–5526. K, T., T, T., K, T., S, M., O, M. & M, H. (1994). Suppression of azoxymethane-induced rat colon aberrant crypt foci by dietary protocatechuic acid. Jpn. J. Cancer Res. 85, 686–691. M, B. A., C, I. & B, R. P. (1993). Ability of aberrant crypt foci characteristics to predict colonic tumor incidence in rats fed cholic acid. Cancer Res. 53, 4499–4504. ML, E. A. & B, R. P. (1988). Specificity study to evaluate induction of aberrant crypts in murine colons. Cancer Res. 48, 6183–6186. ML, E. A., & B, R. P. (1988). Aberrant crypts: potential preneoplastic lesions in the murine colon. Cancer Res. 48, 6187–6192. ML, E. A., M, A. & B, R. P. (1991). Sequential analysis of the growth and morphological characteristics of aberrant crypt foci: putative preneoplastic lesions. Cancer Res. 51, 5270–5274. N, I., N, Y., M, Y., M, Y., A, H., H, A. et al. (1991). Mutations of chromosome 5q21 genes in FAP and colorectal cancer patients. Science 253, 665–669. P, S. M., Z, N., B-B, Y., B, T. M., H, S. R., T, S. N., V, B. & K, K. W. (1991). APC mutations occur early during colorectal tumorigenesis. Nature 359, 235–237. P, T. P. (1995). Aberrant crypt foci and K-ras mutations: earliest recognized players or innocent bystanders in colon carcinogenesis? Gastroenterology 108, 600–603. P, T. P., B, T. A., F, C., C, C. & K, E. L. (1993). K-ras mutations in putative preneoplastic lesions in human colon. J. Natl. Cancer Inst. 85, 2004–2007. P, T. P., C, C. & O’R, M. A. (1994). Aberrant crypt foci and colon tumors in F344 rats have similar increases in proliferative activity. Int. J. Cancer 56, 599–602. P, T. P., R, E. V., O’R, M. A., C, J. C., A, S. B. & S, T. A. (1994). Carcinoembryonic antigen
in human colonic aberrant crypt foci. Gastroenterology 107, 1719–1725. R, L., M, A. & B, W. R. (1991a). Classification of aberrant crypt foci and microadenomas in human colon. Cancer Epidemiol. Biomarkers Prev. 1, 57–60. R, L., P, M., F, R., D G, C. & P L, M. (1993). Cell kinetic evaluation of human colonic aberrant crypts. Cancer Res. 53, 3726–3729. R, L., S, D., M, A., C, J. B. & B, W. R. (1991b). Identification and quantification of aberrant crypt foci and microadenomas in the human colon. Hum. Path. 22, 287–294. S, A. J., S, H. S., P, M., H, K., M, A., B, B. V. & G, S. (1994). Somatic APC and K-ras codon 12 mutations in aberrant crypt foci from human colons. Cancer Res. 54, 5527–5530. S, S. A., M, L. C. & B, R. P. (1992). Evidence for a ras gene mutation in azoxymethane-induced colonic aberrant crypts in Sprague-Dawley rats: earliest recognizable precursor lesions of experimental colon cancer. Carcinogenesis 13, 2081–2085. S, L. A. M., & B, R. P. (1994). The effect of chenodeoxycholic acid on the development of aberrant crypt foci in the rat colon. Cancer Lett. 76, 101–107. T, M., M, T., Y, M., K, T., Y, K. & E, H. (1994). Effect of docosahexaenoic acid on azoxymethane-induced colon carcinogenesis in rats. Cancer Lett. 83, 177–184. T, I., M, O. & K, E. (1994). Influence of a dietary fiber on development of dimethylhydrazine-induced aberrant crypt foci and colon tumor incidence in Wistar rats. Nutr. Cancer 21, 177–182. T, J., B, M. & C, H. (1987). The crypt cycle. Evidence for crypt and villus production in the adult mouse small intestinal epithelium. Biophys. J. 52, 279–294. V, A. A., S, B., M, A., B, W. R., H, K., W, M. A., S, H. S. & G, S. (1993). K-ras mutations in aberrant crypt foci, adenomas and adenocarcinomas during azoxymethane-induced colon carcinogenesis. Carcinogenesis 14, 1777–1781. V, B. & K, K. W. (1993). The multistep nature of cancer. Trends Genet. 9, 138–141. Y, N., M, T., O, A., O, M. & E, H. (1995). Frequent and characteristic K-ras activation and absence
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APPENDIX In order to show that s(t) = S(t)/T(t) is a decreasing function of t is suffices to show that its derivative is negative for t q 0. But s'(t) = [S '(t)T(t) − S(t)T '(t)]/T(t)2
(A.1)
= a(b eat e−k2 t − a eat − k2 e−k2 t )/b(eat − 1)2 . (A.2) Then, since 0 Q a Q b, it suffices to show that b eat e−k2 t − a eat − k2 e−k2 t Q 0, which in turn is equivalent to showing that b eat − a ebt − k2 Q 0 (multiply through by ek2 t q 0. Expanding the exponentials and combining terms yields b eat − a ebt − k2 = ab[(a − b)t 2/2! + (a 2 − b 2 )t 3/3! (A.3) + ···+(a m − 1 − b m − 1 )t m/m!+···]. (A.4) But this is negative since 0 Q a Q b implies that (a m − b m ) Q 0 for all m e 1. To summarize, we have shown that s'(t) Q 0 which in turn implies that s(t) is a decreasing function of time.
Expansion of Mutant Stem Cell Populations in the Human Colon M B† Department of Anatomy and Cell Biology, Medical Sciences Building, University of Toronto, Toronto, Ontario M5S 1A8, Canada (Received on 14 July 1995, Accepted in revised form on 18 September 1995)
In general, it is presumed that colonic epithelial stem cells are the principal cell type at risk of incurring the series of somatic mutations leading to carcinoma, since all other epithelial cell types are short-lived. Mutant stem cell clonal expansion increases the risk for subsequent mutations and is therefore a potentially important step in carcinogenesis. The stem cells reside in colonic crypts, simple tubular foldings of the epithelium, and thus counting crypts provides an indirect means to determine stem cell numbers. The normal crypt population is known to expand through a process of crypt replication and this is thought to result in a corresponding expansion of the epithelial stem cell population. A simple mathematical model of the population dynamics of normal and mutant crypts (crypts containing mutant stem cells) is developed and used to estimate a lower bound on the relative rate of expansion of the mutant stem cell population. The model predicts that if mutant and normal crypt populations expand at the same rate, and if the mutation rate is small relative to the rate of growth, then the fraction of clusters of mutant crypts composed of only a single mutant crypt should steadily decrease with age towards one-half. Aberrant crypts are easily recognizable lesions in human colon which have frequently been shown to contain cells with K-ras and occasionally APC gene mutations. Application of the model to recent counts of aberrant crypt cluster sizes indicate that the aberrant crypt population, and the contained mutant stem cell population, is expanding substantially faster than normal. 7 1996 Academic Press Limited
increased mutation risk. Simple methods capable of demonstrating abnormalities in the relative rate of expansion of clones of mutant stem cells in man are therefore of interest, and will require techniques both for the identification of mutant clones and also for the extraction of useful information from identified clones. It is the latter that will be dealt with here. A careful scan of the colonic epithelium will occasionally reveal crypts of abnormal morphology and histochemistry; these are the aberrant crypts (Bird, 1987; McLellan & Bird, 1988a,b; Roncucci et al., 1991a, b; Pretlow et al., 1994). They are often found in clusters thought to represent clones derived from a mutant stem cell. In fact, many aberrant crypts contain cells with K-ras or APC mutations (Stopera et al., 1992; Pretlow et al., 1993; Vivona et al., 1993; Smith et al., 1994; Jen et al., 1994; Yamashita et al.,
Introduction There is strong evidence that many colorectal carcinomas result from accumulated somatic mutations, presumably starting with the mutation of an epithelial progenitor or stem cell (Fearon & Vogelstein, 1990; Nishisho et al., 1991; Gordon et al., 1991; Powell et al., 1991; Vogelstein & Kinzler, 1993). Expansion of the clones derived from such mutant stem cells is thought to be an important stage in carcinogenesis for at least two reasons: (i) the increasing numbers of mutant cells directly increases the probability that at least one of these cells will receive the additional mutations leading to malignant transformation; and (ii) clonal expansion entails DNA replication with its associated † E-mail: bjerknes.crypt.med.utoronto.ca. 0022–5193/96/040381 + 05 $18.00/0
381
7 1996 Academic Press Limited
382
.
1995). Although their role in human carcinogenesis remains contentious (Jen et al., 1994; Yamashita et al., 1995; Pretlow, 1995), aberrant crypts are nonetheless a good test case for the study of the expansion of a population of readily identified mutant clones. When first formed, aberrant crypts are single isolated structures, but with time they branch producing clusters of related aberrant crypts (McLellan et al., 1991). This description of the growth of clusters of aberrant crypts is reminiscent of the crypt cycle, a continuous crypt replication process which has been most thoroughly studied in the normal mouse small intestinal epithelium (Bjerknes, 1986, 1994, 1995; Cheng & Bjerknes, 1985; Totafurno et al., 1987), but which also seems to occur in human colon (Cheng et al., 1986). In normal intestine, the crypt cycle results in the slow expansion of the crypt population through a process of crypt budding and fission. It appears that newly formed aberrant crypts expand their numbers through a similar process. Thus, aberrant crypts arise from two sources; from spontaneous somatic mutation of a stem cell in a normal crypt, or from a crypt-cycle-like replication of existing aberrant crypts. The latter mechanism has the potential to expand the mutant stem cell pool rapidly and thus may warrant serious concern. It is a simple matter to write equations describing the expansion of normal and mutant crypt populations and the conversion of normal into mutant crypts. In this work I demonstrate that if mutant and normal crypt populations expand at the same rate, then these equations predict that with age, about one-half of mutant crypt clusters should contain only a single crypt. If fewer than one-half are actually found to be unicryptal, then this is evidence that the mutant crypt population is expanding faster than normal. These ideas are applied to the analysis of recently published data on the distribution of the size of human aberrant crypt clones, where it was found that fewer than one-half of clusters of aberrant crypts were unicryptal; most were polycryptal (Roncucci et al., 1991a,b; Pretlow et al., 1994). It is concluded that the aberrant crypt population, and hence the mutant stem cell population contained in them, is expanding substantially faster than normal. A Mathematical Model of Aberrant Crypt Dynamics Our goal is to determine if the population of mutant stem cells found in aberrant crypts is growing faster than the stem cell population found in normal crypts. Since the rate of growth of a stem cell pool is reflected in the rate of growth of the population of
crypts containing those stem cell (Bjerknes, 1986, 1994, 1995; Cheng & Bjerknes, 1985; Totafurno et al., 1987), we will compare the expansion rates of the populations of aberrant vs. normal crypts in order to explore the relative expansion rates of the populations of mutant vs. normal stem cells. We will assume that the normal crypt population expands at rate k1 proportional to the current population size, and that l is their rate of conversion into aberrant crypts (presumably through spontaneous somatic mutations in a contained stem cell; Bjerknes, 1995). We also assume that single isolated aberrant crypts give rise to larger clusters of aberrant crypts at a constant rate k2 , again in proportion to current population size. Note that k1 and k2 may be interpreted either as the respective rates of replication of normal and mutant crypts, or in the presence of significant crypt loss as the respective net population expansion rates. Also note that we will ignore the presumably negligible probability that stem cells in adjacent normal crypts would undergo the same mutation thus producing a mutant cluster without crypt replication. Under these assumptions, the growth of the populations of normal and aberrant crypts at time t may be approximated by the system of differential equations N '(t) = (k1 − l)N(t)
(1)
S '(t) = lN(t) − k2 S(t)
(2)
C '(t) = k2 S(t),
(3)
where N(t) is the number of normal crypts, S(t) is the number of single aberrant crypts, and C(t) is the number of clusters containing two or more aberrant crypts. These equations have solutions [assuming N(0) = N0 and S(0) = C(0) = 0] N(t) = N0 eat
(4)
S(t) =
lN0 at (e − e−k2 t ) b
(5)
C(t) =
lN0 (k eat + a e−k2 t − b), ab 2
(6)
where a = k1 − l and b = k1 + k2 − l. Here, all aberrant crypts were treated as though they behave identically. This should be an adequate first approximation, but as markers are developed which allow subpopulations of aberrant crypts to be identified (for example those containing different mutations), this system of equations could easily be expanded by addition of variables representing the singleton and larger clusters of each of the subtypes, each with their own mutation and replication rates (Bjerknes, 1995).
The total number of aberrant crypt clusters will grow as T(t) = S(t) + C(t) =
lN0 at (e − 1), a
(7) (8)
and, therefore, the fraction of aberrant crypt clusters which should be found as singletons is s(t) 0
S(t) a eat − e−k2 t = . T(t) b eat − 1
(9)
There are two observations about this equation which will lead us to our final result. First, with time, the fraction of single aberrant crypts tends to the limit s 0 lim s(t) t4a
(10)
383
Discussion Since fewer than one-half of aberrant crypt clusters are single (Roncucci et al., 1991a,b; Pretlow et al., 1994), we concluded that the aberrant crypt population is expanding at an increased rate in comparison with normal crypts. We may estimate a lower bound on the magnitude of the increase from recent data. Pretlow et al. (1994; their fig. 4) found in a study of 15 patients that only one of 42 clusters of aberrant crypts were single. Thus, an estimate of an upper bound on the asymptotic fraction of clusters of single aberrant crypts is s E 1/42. This is an inequality because s(t) is a decreasing function of time, and we do not know whether the data were collected from individuals whose crypt dynamics had reached the asymptotic state. Rearranging eqn (14) we obtain 1 r= −1 s
= a/b
(11)
= (k1 − l)/(k1 + k2 − l)
(12)
e
1 k1 /(k1 + k2 ),
(13)
= 41.
where eqn (13) follows from the additional, and probably reasonable, assumption that the spontaneous mutation rate, l, is small relative to the crypt population expansion rates k1 and k2 (Bjerknes, 1995). Second, is the observation that the fraction of single aberrant crypts, s(t) is a decreasing function of time (see Appendix). Therefore, the fraction of aberrant crypt clusters containing a single crypt starts at 1 (at t = 0) and decreases steadily towards the asymptotic value k1 /(k1 + k2 ). Equation (13) may be rewritten as s = 1/(1 + r),
(14)
where r = k2 /k1 . This tells us that if the rates of expansion of normal and aberrant crypts are equal (r = 1) then asymptotically one-half of aberrant crypts should be single. Since measurements of aberrant crypt clusters in human colon revealed that far fewer than one-half of the clusters were composed of single crypts (Roncucci et al., 1991a,b; Pretlow et al., 1994), we may safely conclude that the aberrant crypt population is growing faster than is the population of normal crypts, and that in turn the mutant stem cell population contained in the aberrant crypts expands at an increased rate. This conclusion holds even though the samples may not have been obtained at asymptotic ages because s(t) is steadily decreasing in time.
1 −1 1/42
(15) (16) (17)
Thus, given this admittedly rough estimate of s, the mutant stem cell pool contained in the aberrant crypt population is growing q40 times faster than normal. This may be an overestimate if single aberrant crypts are more easily missed during scoring than are large clusters, and thus, the reader should be cautioned that the data analysed here was not collected with our purpose in mind. Vagaries of the experimental methods used may have biased the distribution of cluster-sizes obtained; an issue which future measurements of cluster-size distributions must address. It is important to stress that the conclusion that the mutant stem cell pool is growing q40 times faster than the normal stem cell pool does not necessarily mean that the mutant stem cells are cycling 40 times faster than normal stem cells, a feat which might not even be possible. Rather, the magnitude of the increase in the rate of growth of the mutant stem cell pool suggests that increased retention of stem cell offspring, which normally would be lost through cell differentiation or death, could be a major factor in the increased rate of expansion. Recruitment into the cell cycle of non-cycling mutant stem cells, if such exist, offers an additional source. However, increased proliferation probably does make a contribution because aberrant crypts are known to have proliferative abnormalities (Roncucci et al., 1993; Pretlow et al., 1994; Yamashita et al., 1994), but
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.
it is unclear from these previous studies whether the stem cells are affected. Crypt loss is also a potential concern, but it should be recalled that the model parameters k1 and k2 may be interpreted as representing the net expansion rates of normal and mutant crypts, respectively. Furthermore, if significant loss of aberrant crypts occurs, say as a result of immune surveillance, then the mutant stem cells must be replicating at an even higher rate than is indicated above, in order to counter the loss. Another issue, about which little is known, is the actual stem cell content of normal and mutant crypts. There are strong theoretical reasons for suspecting that the stem cell content of normal crypts is variable (Bjerknes, 1986, 1994, 1995; Totafurno et al., 1987), but nothing is known about possible effects of mutations on either the variability or the average stem cell content. We have largely avoided this issue here by focusing on the relative rates of expansion, rather than the absolute sizes of the mutant and normal stem cell populations. It is also important to stress that the somatic mutations afflicting aberrant crypts need not necessarily affect the stem cells directly in order to yield an increased growth rate. This could result, for example, if the mutation affected cell adhesion in a way which stimulated crypt replication (Augenlicht, 1994). Stimulation of the replication process would, in turn, stimulate growth of the stem cell population. Regardless of whether the mutation impacts directly or indirectly on stem cell growth rates, the mutant stem cell pool in aberrant crypts does appear to be expanding at an abnormal rate, a potentially dangerous situation. Much effort is being spent in the search for dietary factors which might alter the rate of progression towards tumors (Bruce et al., 1993; Magnuson et al., 1993; Kawamori et al., 1994; Sutherland & Bird, 1994; Takahashi et al., 1994; Thorup et al., 1994) and thus, there is considerable interest in the design of simple methods of analysis of the results from these trials. The application of variations of the analysis introduced here may offer another useful approach. The approach introduced above is not limited only to the study of the rate of expansion of aberrant crypts. The ideas may be applicable to the study of any mutation for which markers are developed allowing identification of the afflicted crypts.
This work was supported by a grant from the MRC of Canada.
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APPENDIX In order to show that s(t) = S(t)/T(t) is a decreasing function of t is suffices to show that its derivative is negative for t q 0. But s'(t) = [S '(t)T(t) − S(t)T '(t)]/T(t)2
(A.1)
= a(b eat e−k2 t − a eat − k2 e−k2 t )/b(eat − 1)2 . (A.2) Then, since 0 Q a Q b, it suffices to show that b eat e−k2 t − a eat − k2 e−k2 t Q 0, which in turn is equivalent to showing that b eat − a ebt − k2 Q 0 (multiply through by ek2 t q 0. Expanding the exponentials and combining terms yields b eat − a ebt − k2 = ab[(a − b)t 2/2! + (a 2 − b 2 )t 3/3! (A.3) + ···+(a m − 1 − b m − 1 )t m/m!+···]. (A.4) But this is negative since 0 Q a Q b implies that (a m − b m ) Q 0 for all m e 1. To summarize, we have shown that s'(t) Q 0 which in turn implies that s(t) is a decreasing function of time.