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The dynamics of repetitive asymmetric cell division
Mechanisms of Ageing and Development, 6 (1977) 319 - 332 ©Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
THE
DYNAMICS
OF
REPETITIVE
ASYMMETRIC
319
CELL
DIVISION
HENRY R. HIRSCH Department of Physiology and Biophysics, University of Kentucky, Lexington, KY 40506 (U.S.A.)
(Received November 14, 1975;in revised form September 26, 1976)
SUMMARY
Asymmetric cell division results in one daughter cell that is of the same type as the mother and one daughter of a type that is significantly different. In the repeated asym. metric division process which is analyzed here, the latter daughter itself divides asymmetrically, and gives rise to a third cell type. This process may continue through many reproductive cycles, generating many different cell types and forming a pattern of repeated asymmetric division: However, it is assumed that the number of cell types is finite and that cells of the final (fully differentiated) type do not divide within the confines of a particular asymmetric division pattern. A system of linear equations describes cellular proliferation by repeated asymmetric division. Exact solutions of the equations have the form of binomial coefficients if cell death is ignored and all cell divisions are synchronous. If these restrictions are relaxed, approximate solutions can be obtained which are valid in the limit of large values of time. If the lifetime of each type of cell is either much smaller or much larger than the values of time for which the solutions are of interest, i.e. if each type of cell is either "short-lived" or "long-lived", the number of cells of the final type is proportional to a non-negative integral power of time. If one cell type is long-lived, the power of time is zero, and the number of cells of the final type is constant; if more than one cell type is long-lived, the number of cells of the final type increases as a positive power of time. These results are applied to prenatal growth in the human, to the population of neurons in the human brain, to the full population of cells in the adult human, and to the maintenance of a constant supply of blood cells. In all cases, the calculated number of cell divisions is consistent with observed limits on the number of population doublings in human diploid cell cultures. The power-law time dependence of growth produced by asymmetric division is compatible with data on prenatal weight gain. The repeated asymmetric division process provides a simple alternative to the clonal succession hypothesis in explaining the production of very large numbers of blood cells without invoking the necessity for many thousands of cell divisions.
320 INTRODUCTION The process of cellular reproduction in which a mother cell divides symmetrically into two daughters of the same type is so commonplace in single-celled organisms and in cell cultures that we are persuaded to think of it as being, in some sense, normal. The process of asymmetric division in which one of the daughters differs significantly from the mother is held to be an exception. Here the opposite position will be adopted. The proliferation of a cell population which undergoes repeated asymmetric division will be analyzed mathematically, and the results of the analysis will be applied to populations which are growing or which have achieved constant size. Restrictions imposed by a limit on the number of cell divisions will be considered. The process of repeated asymmetric cell division was used as a model of growth and development by Lillie [1], who, in 1929, observed that embryonic differentiation patterns form continuously branching dichotomies. He pointed out that in certain nematodes and rotifers few non-differential cell divisions occur throughout the life history of the organism. Differentiation usually coincides with asymmetric cell division in these organisms and, in selected instances, in many others. The view that asymmetric division is important in ontogenesis remains current. Holtzer [2] states that symmetric division is rare during the embryogenesis of metazoans and proposes a cell lineage of the myoblast which depends upon repeated asymmetric division. L~vtrup [3] presents a hypothetical filocyte differentiation pattern which is similar in form to the repeated asymmetric division pattern that is treated here. Differences between the daughter cells and the mother cell at each bifurcation in the differentiation pattern may be attributable to asymmetric division or to differences in the environments of the cells. In the present research, prenatal human growth is divided into four time intervals. Symmetric division during the first interval is followed by asymmetric division during the remaining intervals. Calculations based upon these division patterns indicate that the total number of cell divisions required to yield the number of cells in a newborn human is substantially less than the maximum number of divisions observed in fibroblast cell cultures. The number of nerve and skeletal muscle cells in a healthy human adult is approximately constant because the average longevity of the cells is greater than that of the organism as a whole. In other tissues, such as the blood, constant size is maintained by continual replacement of cells that die. The relative importance of symmetric and asymmetric division in this process is uncertain. Osgood's model [4] of erythrocyte differentiation employs both types of division. Kay's calculations [5] indicate that the enormous numbers of erythrocytes and leukocytes required during a human lifetime can be descended from a reasonable number of stem cells by tangential asymmetric division only if the stem cells each divide many thousands of times. It will be shown here that the required number of divisions is much smaller if the stem cells result from a process of repeated asymmetric division. Sheldrake [6] has suggested that the rate of aging may depend upon the degree to which wastes are eliminated from cells by asymmetric division. The results of the present research are useful in calculating the waste content of a population of cells which
321 undergoes tangential or repeated asymmetric division. A preliminary report of this work has recently appeared [7].
PROPERTIES OF A REPEATED ASYMMETRIC CELL DIVISION PATTERN Growth of a cell population by symmetric division can be represented by a branching "tree" pattern as shown in Fig. l(a). All cells are of the same type and are designated by the numeral "1". The number of cells increases exponentially with time. The simplest asymmetric division pattern, sometimes called a "tangential" pattern [5], is shown in Fig. l(b). Each time a cell of type 1 divides, it produces another cell of type 1 and a differentiated cell of type 2. The number of type-2 cells increases linearly with time if cell death is negligible, while the number of type 1 cells remains constant.
4
(a)
%
(b)
Fig. 1. Cell division patterns. (a) A symmetric pattern. All cells are of the same type. (b) The simplest asymmetric pattern, sometimes called "tangential". Cells of type 1 divide; cells of type 2 do not. See text. Fig. 2. A repeated asymmetric cell division pattern in which cell types 1, 2, and 3 divide, but type 4 does not Q"= 3). See text.
Figure 2 presents a more involved asymmetric division pattern. Each time a cell divides, it reproduces itself and gives rise to a new cell type. Differentiation of the earher type coincides with or follows cell division. The division of a type-2 cell, for example, produces daughters of types 2 and 3. In general, it will be assumed that there are / cell types which undergo asymmetric division in addition to a j + l'st cell type which does not differentiate or divide further within the pattern under consideration. In Fig. 2 , / = 3 and the fourth cell type does not divide. The lengths of the arrows shown in Fig. 2 differ to indicate that the cell types may differ in the average length of time they spend between divisions. They may also differ in lifespan. Cell types need not differ morpholog-, ically, but they must, at least in principle, be experimentally distinguishable. Perfectly symmetric division is thus excluded In a living organism, it is reasonable to suppose that sequences of symmetric and asymmetric division are often interspersed. Nevertheless, attention is here concentrated upon the repeated asymmetric division pattern (Fig. 2) in order to explore its capacity
322 to explain growth, and the maintenance of constant-sized cell populations. General division patterns consisting of symmetric as well as asymmetric subpatterns are most easily analyzed by treating the symmetric and asymmetric portions separately. The mathematical description of the symmetric growth pattern is simple and well known; it remains to analyze the asymmetric pattern. The calculations upon which this analysis is based are presented in the next section. They may be omitted on first reading by those who wish to proceed directly to the applications of the results of the analysis which are contained in the following section.
SYNCHRONOUS DIVISION OF LONG-LIVED CELL TYPES An asymmetric cell division pattern such as that shown in Fig. 2 is especially simple to analyze if cell death can be ignored, i.e. if all cell types are "long-lived". Let ni(t) be the number of cells of type i (/ + I ~> i ~> 1) which exist at time t, and let n l(t) be a known function of time. The equation which governs the increase in n i (t) is: ni(t + Ti-1) - hi(t) = ni_l (t), i/> 2
(1)
where Ti-i is the average interdivision time of cells of type i - 1. The number of cells of type i is greater at time t + Ti-~ than at time t because each cell of type i - 1 has, on the average, produced one daughter of type i. Cells of type i may also divide during this same time interval, but such division does not change their number because they replace themselves, one for one, in asymmetric division. Thus eqn. (1) indicates that the increase in the number of type i cells during an interval of Ti-~ time units following time t is equal to the number of type i - 1 cells which exist at time t. Equation (1) is a compact representation of a set of i - 1 simultaneous linear equations. Exact solutions can be obtained if n ~ is constant and the values of Ti are equal for all i,j>~i>~ 1. Let X = tiTs,
(2)
where Tg is the value of the interdivision time characteristic of all cell types. X in effect measures time in units of Tg. In order to avoid ambiguity, it will be assumed that the number of cells of each type is counted at integer values of X, that cell division takes place synchronously ~ ( time units later, and that ~ < < 1. A celt of type i makes its first appearance at time X = (i - 1) + zSX and is first counted at time X = i. I f / 4 : j + 1, X - 1 is equal to the exact number of divisions experienced by a cell of type i. If i = / + 1, X - 1 is equal to the maximum number of divisions experienced by a cell of type j + 1. , Equation (1) can be rewritten as a function of X: ni(X + 1 ) - n i ( X ) = n i - l ( X ) , i > ~
2.
(3)
It can be verified by direct substitution that eqn. (4) is a solution of eqn. (3) for which n is constant: ni(X) = nl 6"~i--11, i ~> 2. (4)
323 C~/_-I1 represents the number of combinations of X - 1 items taken i - 1 at a time. The value of n i ( X ) is maximum if i = (X + 1)/2. In the event that X is even and (X + 1)/2 is half-integral, either the next higher or the next lower integer may be assigned to i without effect upon the calculated maximum value of ni(X). Once the assumption of equal interdivision times is introduced, the asymmetric cell division pattern can be regarded as a simple example of a deterministic, zero-sided, informationless DOL Lindenmayer system [8]. In lindenmayer's notation [8], the starting symbol or axiom is 1, and the production rule is i ~ i, i + 1. The growth function, eqn. (4), could therefore have been obtained with the use of the formalism presented by Herman and Vitanyi [9]. ASYNCHRONOUS DIVISION; FINITE CELL LIFESPANS
Equations (3) and (4) describe populations in which divisions of cells of all types are synchronized to occur at intervals of fixed duration. It is useful to relax this restriction and to introduce the possibility that cells of some or all types may have finite lifespans. Suppose that cells of type i have an average lifespan of Ti time units. It follows from eqn. (1) that ni_~(t) is the number of type-/ cells created between times t and t + T i - 1 . T h e number of cells dying during this interval is equal to the number that were created during an interval of equal duration which occurred T~ time units earlier. Thus the last term of eqn. (5) represents the modification of eqn. (1) required to take finite cell lifespans into account: ni(t + T i - l ) - ni(t) = n i _ l ( t ) - n i _ l ( t - Ti)
(5)
Equation (5) can be solved most easily with the use of Laplace transforms. If s denotes the complex frequency variable and N i the Laplace transform of ni(t). [lf
e-sTi ]
gi=Ni-'[ -sTi-'~-'l'e II
if> 2.
(6)
Let n~ be constant if t ~< T~ and zero if t > T I . Then I'/1
NI = - - (1 - e-sT1) $
(7)
Iterative application of eqn. (6) to eqn. (7)yields: i
l'I
(1 - e-STq)
n 1 q=l
M ---
S
(8)
i-1
II
(e'rq-1)
q=l
It is not difficult in principle to solve eqn. (5) by inverting the Laplace transform shown in eqn. (8). However, the presence in the transform of i - 1 infinite sets of poles
324 renders the inverse cumbersome. Useful results can be obtained by calculating the limiting form of ni(t) which is valid at large values of time. Large values of time in the function ni(t) correspond to small values of the magnitude of the complex frequency in its transform, Ni(s ). Thus, in eqn. (8), the expressions of the form eSTq -- 1 approach sTq as the value of time becomes large and the value of s approaches zero. The expressions of the form 1 - e-STq approach unity if the death of cells of type q can be ignored. These cell types are long-lived in the sense used in the preceding section. All other cell types are termed short-lived. For short-lived cell types, the expressions 1 - eSTq approach sTq as s approaches zero. With the help of these approximations, the complexity of eqn. (8) is substantially reduced: nl
i-I
i
Ni=~-ff - l'X (1/Tq) n q=l
Tq,
(9)
q=l qtF
where b is the number of long-lived cell types, and F is the set of short-lived cell types. The second product in eqn. (9) extends, as indicated, only over the latter types. The inverse of eqn. (9) is: t b-~
i-i
ni(t)=nl ~
i
n (1/Tq) I I Tq. a=l q=l
( b - 1)!
(10)
qeF
A number of results follow directly from eqn. (10). (1) The number of cells of the ith type increases as a power function of time at large values of time. (2) At least one cell type must be long-lived (b 1> 1) if cells of type i are to be present at large values of time. The lifespan of the long-lived type must exceed the largest value of time for which the function ni(t) will be calculated. (3) If all cell types are long-lived (b = i), eqn. (10) reduces to
t i-~ i-~ n (1/Tq). ni(t)=nl ( i - 1)! a=,
(11)
(4) If all cell types are long-lived (b = i), and, in addition, their generation times are equal (Tq = Tg for all q, i - 1/> q t> 1), eqn. (11) can be further simplified:
xi-1 ni(X)= nl ~
( i - 1)!
,
(12)
where X = t/Tq as in eqn. (2). If eqn. (4) is expanded as a power series in )(, its leading term, i.e. the term which dominates the series at large values of time, is identical to eqn. (12). (5) If the total number of long-lived cells is calculated by summing terms of any of the forms shown in eqns. (4), (11), or (12) over all cell types, 1 through j + 1, the j + l'st
325 term dominates the resulting sum at large values of time. Consequently, it is often an acceptable approximation to set the total number of cells equal to ni+ 1. (6) If the number of cells of type i is to be constant with respect to time, exactly one cell type must be long-lived (b = 1). The long-lived type will be assigned the subscript h. (7) If the single long-lived cell type is the i'th (h = i), the cell number ni(t) remains constant because the cells of all types ancestral to i are dead at large values of time. Equation (10) assumes the form: i--1
ni= nl I I
(13)
xo,
q=l
where xq = Tq/Tq is the maximum number of divisions experienced by a cell of type q. Consider the problem of maximizing ni under the constraint: i--I
X m ---- ~_~
(14)
Xq
q=l
= constant, where Xm is the maximum number of divisions performed by cells of all types ancestral to type i. The product over Xq in eqn. (13) can be regarded as the volume of an i - 1 dimensional parallelepiped in which xq is the edge length in the qth dimension. If the sum of the sides of this figure is constrained to a constant value, as in eqn. (14), the volume is maximized by choosing the values of Xq equal. Thus ni is maximized if Xq = x for all q, i - 1 >I q ~> 1, and eqns. (13) and (14) become: ni = n i x i-1
(15)
X.,, = (i - 1)x
(16)
and
respectively. Equation (15) is easily solved for x: (17)
x = (ni/nO 1/(i-0.
(8) If the single long-lived cell type is ancestral to the ith type (h < i), the cell number ni(t ) remains constant because type-/ cells that die are replaced by asymmetric cell division. The form of eqn. (10) which applies to this case is: i--I
ni=nl(Ti/th) II
xq,
(18)
q=l
where th is the maximum value of time at which cells of type h divide. Often this value will coincide with the lifespan of the organism in which the cell population is situated. In eqn. (18), as in eqns. (13) and (14), Xq represents the maximum number of divisions experienced by a cell of type q. I f q ~ h, Xq = Tq[Tq. I f q =h, Xq = th/Th.
326 Apart from the dimensionless factor Ti/th, eqn. (18) is identical to eqn. (13) If the constraint upon the maximum number of divisions shown in eqn. (14)is applied to eqn. (18), it follows that r/i will be maximized by the choice xq = x for all q, i 1 1> q >~ 1. Then Xm can be obtained from eqn. (16), and eqn. (18) becomes:
rl i = tl I (Tilth) Xi- 1
(19)
Solution of eqn. (19) for x yields:
= I ni/Ti J1/(i-l). x [ nl/thJ
(20)
APPLICATIONSAND DISCUSSION Examples of applications of the preceding results will be given below. In each instance, the number of cell divisions required to produce a cell population of a certain size will be calculated. Experiments reviewed by Hayflick [10] indicate that the number of divisions which can be performed by a strain of somatic cells is limited. Hayflick's own work [11] demonstrated a limit of approximately fifty population doublings in normal fibroblasts cultivated in vitro, and a number of the studies he reviewed [10] support the concept that a similar limit applies in vivo. However, contrary evidence exists. For example, in vivo studies of epithelial tissue by Strehler [12] and Cameron [ 13] show that the limit, if any, on the number of population doublings must be much greater than fifty. The distinction between the number of cell divisions and the number of population doublings should be noted. Merz and Ross [14] observed that many living cells in latepassage fibroblast cultures do not divide. As a result, the cells which retain the capacity to divide must do so more than once, on the average, each time the population doubles. The maximum number of cell divisions must exceed the maximum number of population doublings. Calculations by Good [15] and by Hirsch and Curtis [ 16] indicate that, in the forty-fifth passage, a small fraction of the dividing cells will have divided as many as 150 times. This figure will be adopted as an approximate estimate of the maximum number of divisions which a somatic cell can undergo.
Prenatal growth If asymmetric cell division patterns are responsible for embryonic and fetal growth, the number of cells in the developing organism should follow a combinatorial function of time, as in eqn. (4) or a power function of time, as in eqns. (10), (11), and (12). Available data on prenatal growth appear to be of insufficient accuracy to distinguish between these two cases. Zotin [17] reviewed evidence supporting a number of different "laws of growth", including the power law. Timiras [18] stated that over two hundred different mathematical functions of time have been applied to growth data, so it is evident that agreement, even close agreement, between theory and data provides no conclusive proof that the assumptions underlying the theory are correct. Nevertheless it is significant that
327 a number of studies, commencing with early work cited by Zotin [17], are consistent with the results obtained here. The cubic function, which is a special case of the general power function, was fitted to the fetal growth of a wide variety of species by Huggett and Widdas [19] and by Spencer and Coulombe [20]. Sikov and Thomas [21] fitted data on the prenatal growth of the rat with several mathematical functions. A cubic function o f time yielded reasonable agreement with the data, but other more complicated functions fit more closely. Osgood's studies [22] showed that increases in the weight of the human and in the weights of various o f the hematopoietic organs can be represented in fully logarithmic coordinates by a reasonably small number of straight-line segments. Thus, over substantial periods of prenatal and postnatal growth, the weight of the whole body, as well as the weights of the organs, are described by power functions. If the average cell mass is assumed constant, the same power functions can also be applied to the numbers of cells in the body or organs. TABLE I PRENATAL HUMAN GROWTH* Time interval (days) 0-9
9-42
42-115
115-266
250
13.6
Ratio of final weight to initial weight, r
1
106
Exponent, j" Number of dividing cell types,/"
0 1
9.1 9
Number of cell divisions, X-1
9
24
10
6
512
1,307,504
252
20
Cell number ratio,
5.4 5
3.25 3
ni+:/nl
Interdivision time, Tg (days)
1.00
1.37
7.30
25.2
*The first three rows represent Osgood's data [22]. Numbers in the remaining rows are calculated by methods described in the text.
Osgood's data on body weight are summarized in the first three rows of Table I. The prenatal growth period is divided into four time intervals, as indicated in the first row. The ratio of final weight to initial weight, r, and the exponent of time,/', with which the weight increases are given in the second and third rows, respectively. During the second, third, and fourth time intervals, it is assumed that the increase in cell number results from asymmetric division, that the total number of cells is approx-
328 imately equal to the number of cells of type / + 1, and that the number of cells of type / + 1 is proportional to the weight of the embryo or fetus. With the use of these assumptions, eqn. (10) can be fitted to the data which apply to each of the time intervals. Inspection of eqn. (10) shows that the number of long-lived cell types, b, must be equal to j" + 1. The number of cell types, / + 1, is minimized if it is assumed that all cell types are long-lived, i.e. if the number of dividing types,j, is equal to the experimental value of the weight-increase exponent, j" Accordingly, the values o f / w h i c h appear in the second, third, and fourth columns of the fourth row of Table I were obtained by rounding the corresponding values o f / ' t o the nearest integer. The assumption that all cell types are long-lived makes it possible to replace eqn. (10) with eqn. (11). The further assumption that all cell types within a particular time interval have the same interdivision time allows the use of eqn. (12), which is the limiting form of eqn. (4). The data are equally consistent with either of these two equations. The values of nj+l/nl shown in the sixth row of Table I were calculated with the help of eqn. (4) rather than eqn. (12) because it yields more conservative, i.e. larger, estimates of the number of cell divisions required during prenatal growth. The number of cell divisions, which appears in the fifth row, was obtained by choosing the smallest integer, X, which yielded a cell-number ratio, ni+~/n~, in excess of the weight ratio, r. The interdivision time, given in the seventh row, represents the ratio of the duration of the time interval to the number of cell divisions. The first time interval shown in Table I demands special consideration. Although there is no detectable weight gain [22], division is actively taking place during this time. The early cell divisions are synchronous and the cell number increases exponentially [18]. Some differentiation occurs by the 9th day, perhaps signifying the beginning of asymmetric division. Nevertheless, in the absence of precise information, it will be assumed that all divisions during the first 9-day period are symmetric. In order to calculate the number of divisions and the interdivision time, it is necessary to know the number of cells at the end of the 9th day. This can be estimated from the data shown in Table I as follows: between the 9th and the 266th days, weight increases by the ratio 106 × 250 X 13.6 = 3.4 × 1 0 9 . The newborn is constituted of 1.25 × 1012 cells [22]. If the average cell mass remains constant, between the 9th and 266th days, there are therefore 1.25 × 1012/3.4 X 109 = 368 cells on the 9th day. To grow these from a single cell requires no more than nine symmetric divisions which take place at a rate of one per day. Thus a total of 40 asymmetric divisions and 9 symmetric divisions, or 49 cell divisions in all, are sufficient to account for the full growth of a newborn human. This value is well within the limit of approximately 150 cell divisions which was assumed above. The calculated number of cells, given by the product of the numbers in the sixth row of Table I, exceeds the required total of 1.25 × 1012 by a factor of approximately three. The number of cell types required for this growth is 18, since the final type which appears during each time interval also serves as the initial type during the following interval. The average interdivision time increases monotonically throughout prenatal growth.
329
Mature populations Equations (13)-(17) describe asymmetric cell division patterns in which only the final cell type, / + 1, is long-lived. Examples of such long-lived cell types include neurons and skeletal muscle cells in the adult human. The maximum number of cell divisions, Xm, required to produce the 101° neurons in the human brain can be calculated from eqns. (16) and (17) if it is assumed that the maximum number of divisions, x, performed by each of the dividing cell types is the same. An estimate of the number of dividing cell types, j, can be obtained by adopting the value, 17, obtained above for prenatal human growth. With / = 17, nx = 1, and nj+ 1 = 10 l° , eqn. (17) yields x = 4 divisions, rounded to the next higher integer, and eqn. (16) yields Xm = 68 divisions. Most cell populations in the adult human divide rather slowly if at all. If popula. tions which divide rapidly are regarded as special cases which can be ignored for the present*, the maximum number of asymmetric divisions needed to produce an adult can be calculated as above with the help of eqns. (16) and (17). On the assumption that cell mass remains constant throughout postnatal life, the number of cells in a 68 kg adult is taken as 20 times the number of cells in a 3.4 kg newborn infant. Thus ni+l = 20 X 1.25 X 1012 cells = 2.5 X 10 ~3 cells. Ifn~ = 1 and/' = 17,x = 7 divisions, rounded to the next higher integer, and Xm = 119 divisions. The estimate / = 17 is very likely to be low, since only cell types involved in prenatal growth were taken into account. Fortunately, the calculated values o f x and Xm are rather insensitive to the value of/'. The use of a higher estimate of/' would have led to lower calculated values of Xm. However, the present values, 64 divisions for the growth of the brain and 119 divisions for the growth of the adult human, are consistent with the assumed maximum of 150 cell divisions. In a celt population of constant size, the average number of divisions performed by cells of each type is half the maximum number. In the example of the total adult cell population discussed above, the average number of cell divisions per cell type, calculated by dividing the result obtained from eqn. (17) by two, is 3.07. The average number of divisions performed by cells of all types ancestral to the/' + l'st is 17 X 3.07 or 52.2. This value is much smaller than the maximum number of divisions, 119, because, in a population of constant size, it is only the very rare cell that follows a path of maximum length through the asymmetric division pattern. The situation is quite different in expanding populations having many long-lived cell types. If all cell types are long-lived and cell divisions take place synchronously, it can be shown that the average number of divisions preceding the birth of a cell of type/' + 1 is X-= [j/(j' + 1)] X; as the number of cell types increases, the average number of divisions approaches the maximum number. Application of this result to the data in Table I shows that cells of type/' + 1 in the newborn human infant have divided an average of 45.9 times. The maximum number of divisions, 49, is not very much greater. Thus the large
*It will be shown in the next section that a cell population in which turnover is comparatively rapid need not be associated with a very high division rate.
330 difference in the maximum number of divisions between the adult and the newborn is not reflected by a corresponding difference in the average number of divisions. The average number of divisions, 45.9, is 6.3 divisions less than the average number obtained for the full growth of the adult, a difference which is reasonable in view of the mass ratio of 20 which was assumed between the adult and the newborn and in view of the drastic simplification involved in representing the development of the adult by a single asymmetric growth pattern. A constant-sized population with cell turnover Equations (16), (18), (19), and (20) are applicable to asymmetric division patterns in which the number of cells of the final type,/" + 1 is maintained constant by replacement of the cells that die. Consider the simple tangential pattern (Fig. lb) which was used by Kay [5] to calculate the number of divisions needed to produce a lifetime's supply of human erythrocytes and leukocytes. Kay proposed the existence of an initial pool of 4 X 101° cells which divide at a rate sufficient to yield 2.4 X 101° stem cells per day. Each stem cell gives rise to a clone containing an average of 20 blood cells. The length of a human lifetime was set equal to 2 × 104 days (approximately 56 years). In the notation used here, j = 1, nl = 4 × 10 l° cells, n:+l/T/+l = ne/T2 = 2.4 × 10 l° cells/day, and th = 2 × 104 days. With the use of these values, Kay calculated that cells in the initial pool would have to divide 12,000 times during the course of a lifetime. The same result can be obtained with the help of eqns. (16) and (20). A pattern which requires 12,000 divisions is clearly inconsistent with the limits which are held to apply to most diploid celt populations [10, 11]. Kay's suggestion [5], later expanded by Reincke etal. [23] is that blood cells are produced by a process called "clonal succession". This means, very briefly, that division follows a symmetric tree pattern (Fig. la), but the rate at which stern cells mature is controlled by a regulatory feedback mechanism. Although the blood cells appear at a constant rather than an exponentially increasing rate, only 54 divisions are sufficient to produce enough cells to last a lifetime [5, 23]. The phenomena of recovery from hemorrhage and radiation make it clear that regulatory mechanisms exist which serve to maintain the sizes of the blood-cell populations approximately constant. The kinetic properties of these mechanisms have been modelled by Lajtha et aL [24]. Their proposed regulatory model, like all feedback systems, is subject to exhaustion of its dynamic range, to overshoot, and even to oscillatory instability if the disturbances it is called upon to control are too extreme or if the required degree of regulation is too, great. These problems can be reduced in severity if the normal operating point of the system can be maintained in the absence of feedback. The full capacity of the feedback system is then available to oppose disturbances that tend to drive the system away from its operating point. In clonal succession, the normal operating point of the regulatory system, 2.4 X l010 stem cells per day, is set by feedback, while, in asymmetric division, the same operating point is maintained by the intrinsic division rate of the cells. With the latter mechanism feedback regulation would function only in response to an external disturbance, e.g. hemorrhage, disease, or radiation injury.
331 Arguing as Kay [5] did, "teleologically and unashamedly," an asymmetric division pattern would be advantageous with respect to the regulation of blood-cell populations if the number of divisions required to produce a lifetime supply of cells were not unreasonably large. As is shown in Table II a dramatic reduction in the number of divisions TABLE II PRODUCTION OF STEM CELLS BY ASYMMETRICDIVISION* Number o f dividing cell types, /
Number o f cell divisions per dividing type, x
Maximum number o f asym. metric divisions, X m
1 2 3 4 5 6
12,000 110 23 11 7 5
12,000 220 69 44 35 30
*Based on data by Kay [5]. See text. results if the simple tangential pattern (Fig. lb) is replaced by the more general asymmetric pattern (Fig. 2). The values of x and X m in Table II were calculated from eqns. (20) and (16), respectively, with the use of the same numerical values that were given by Kay [5] and cited above. Values of x were rounded to the next higher integer. With just two dividing cell types rather than one, the maximum number of asymmetric divisions decreases from 12,000 to 220. If there are six dividing cell types, the maximum number of asymmetric divisions decreases to 30. Beyond six, increases in the number of cell types offer little further reduction in the maximum number of divisions. In order to obtain the total number of divisions required to produce the full population of blood cells from a single ancestor, it is necessary to take into account the number of divisions required to yield the initial pool of 4 × 10 l° cells and the number of divisions needed to produce a clone of 20 blood cells from a stem cell. If it is assumed, in the interest of simplicity, that these divisions are symmetric, the growth of the stem-cell pool requires 36 divisions, and the growth of the clone requires five divisions, both values having been rounded to the next higher integer. Thus the numbers listed in the third column of Table II must be increased by 41 to obtain the total number of divisions needed to produce a lifetime supply of blood cells from one ancestor cell. It is clear that no conflict exists between the postulated limit of 150 cell divisions and the calculated number of divisions shown in Table II if the asymmetric division pattern contains three or more dividing cell types. None of the foregoing should be construed as an attempt to prove that asymmetric division is, in general, of greater importance than symmetric division. The calculations which have been presented simply purport to show that asymmetric division is a plausible alternative to symmetric division in several typical cases of cell-population growth and that the number of divisions required in each case is consistent with observed limitations on the number of population doublings in diploid cells.
332 REFERENCES 1 F. R. Lillie, Embryonic segregation and its role in the life history, Arch. Entw. Mech., I 18 (1929) 499. 2 H. Holtzer, Myogenesis, in O. A. Schjeide and J. de Vellis, (eds.), Cell Differentiation, Van Nostrand/Reinhold, New York, 1970, pp. 476-503. 3 S. L~vtrup, Epigenetics, Wiley, New York, 1974. 4 E. E. Osgood, Regulation of cell proliferation, in F. Stohlman, (ed.), The Kinetics of Cellular Proliferation, Grune and Stratton, New York, 1959, pp. 282-289. 5 H. E. M. Kay, How many cell generations?, Lancet, 2 (1965) 418. 6 A. R. Sheldrake, The ageing growth and death of cells, Nature, 250 (1974) 381. 7 H. R. Hirsch, Waste elimination in symmetric and tangential cell division, Gerontologist, 15 (5) (ll) (1975) 26. 8 A. Lindenmayer, Developmental algorithms for multicellular organisms: a survey of L-systems, J. Theor. Biol., 54 (1975) 3. 9 G. T. Herman and P. M. B. Vitanyi, Growth functions associated with biological development, Am. Math. Mon., 83 (1976) 1. 10 L. Hayflick, Cell senescence and cell differentiation in vitro, in H. Bredt and J. W. Rowen, (eds.), Aging and Development, Vol. 4, Schattauer, Stuttgart, 1972, pp. 1-37. 11 L. Hayflick, The limited in vitro lifetime of human diploid cell strains, Exp. Cell Res., 37 (1965) 614. 12 B. L. Strehler, The nature of cellular age changes, Syrup. Soc. Exp. Biol., 21 (1967) 149. 13 I. L. Cameron, Minimum number of cell doublings in an epithelial cell population during the life-span of the mouse, J. Gerontol., 27 (1972) 157. 14 G. S. Merz and J. D. Ross, Viability of human diploid cells as function of in vitro age, J. Cell Physiol., 74 (1969) 219. 15 P. I. Good, Subcultivations, splits, doublings, and generations in cultures of human diploid fibroblasts, Cell Tissue Kinet., 5 (1972) 319. 16 H. R. Hirsch and H. J. Curtis, Dynamics of growth in mammalian diploid tissue cultures, J. Theor. Biol., 42 (1973) 227. 17 A. I. Zotin, Thermodynamic Aspects of Developmental Biology, S. Karger, Basle, 1972. 18 P. S. Timiras, Developmental Biology and Aging, Macmillan, New York, 1972. 19 A. St. G. Huggett and W. F. Widdas, The relationship between mammalian foetal weight and conception age, J. Physiol., 114 (1951) 306. 20 R. P. Spencer and M. J. Coulombe, Fetal weight-gestational age relationship in several species, Growth, 29 (1965) 165. 21 M. R. Sikov and J. M. Thomas, Prenatal growth of the rat, Growth, 34 (1970) 1. 22 E. E. Osgood, Development and growth of hematopoietic tissues, Pediatrics, 15 (1955) 733. 23 U. Reincke, H. Burlington, E. P. Cronkite and J. Laissue, Hayflick's hypothesis: an approach to in vivo testing, Fed. Proc. Fed. Am. Soc. Exp. Biol., 34 (1975) 71. 24 L. G. Lajtha, R. Oliver and C. W. Gurney, Kinetic model of a bone marrow stem-cell population, Br. J. Haernatol., 8 (1962) 442.
THE
DYNAMICS
OF
REPETITIVE
ASYMMETRIC
319
CELL
DIVISION
HENRY R. HIRSCH Department of Physiology and Biophysics, University of Kentucky, Lexington, KY 40506 (U.S.A.)
(Received November 14, 1975;in revised form September 26, 1976)
SUMMARY
Asymmetric cell division results in one daughter cell that is of the same type as the mother and one daughter of a type that is significantly different. In the repeated asym. metric division process which is analyzed here, the latter daughter itself divides asymmetrically, and gives rise to a third cell type. This process may continue through many reproductive cycles, generating many different cell types and forming a pattern of repeated asymmetric division: However, it is assumed that the number of cell types is finite and that cells of the final (fully differentiated) type do not divide within the confines of a particular asymmetric division pattern. A system of linear equations describes cellular proliferation by repeated asymmetric division. Exact solutions of the equations have the form of binomial coefficients if cell death is ignored and all cell divisions are synchronous. If these restrictions are relaxed, approximate solutions can be obtained which are valid in the limit of large values of time. If the lifetime of each type of cell is either much smaller or much larger than the values of time for which the solutions are of interest, i.e. if each type of cell is either "short-lived" or "long-lived", the number of cells of the final type is proportional to a non-negative integral power of time. If one cell type is long-lived, the power of time is zero, and the number of cells of the final type is constant; if more than one cell type is long-lived, the number of cells of the final type increases as a positive power of time. These results are applied to prenatal growth in the human, to the population of neurons in the human brain, to the full population of cells in the adult human, and to the maintenance of a constant supply of blood cells. In all cases, the calculated number of cell divisions is consistent with observed limits on the number of population doublings in human diploid cell cultures. The power-law time dependence of growth produced by asymmetric division is compatible with data on prenatal weight gain. The repeated asymmetric division process provides a simple alternative to the clonal succession hypothesis in explaining the production of very large numbers of blood cells without invoking the necessity for many thousands of cell divisions.
320 INTRODUCTION The process of cellular reproduction in which a mother cell divides symmetrically into two daughters of the same type is so commonplace in single-celled organisms and in cell cultures that we are persuaded to think of it as being, in some sense, normal. The process of asymmetric division in which one of the daughters differs significantly from the mother is held to be an exception. Here the opposite position will be adopted. The proliferation of a cell population which undergoes repeated asymmetric division will be analyzed mathematically, and the results of the analysis will be applied to populations which are growing or which have achieved constant size. Restrictions imposed by a limit on the number of cell divisions will be considered. The process of repeated asymmetric cell division was used as a model of growth and development by Lillie [1], who, in 1929, observed that embryonic differentiation patterns form continuously branching dichotomies. He pointed out that in certain nematodes and rotifers few non-differential cell divisions occur throughout the life history of the organism. Differentiation usually coincides with asymmetric cell division in these organisms and, in selected instances, in many others. The view that asymmetric division is important in ontogenesis remains current. Holtzer [2] states that symmetric division is rare during the embryogenesis of metazoans and proposes a cell lineage of the myoblast which depends upon repeated asymmetric division. L~vtrup [3] presents a hypothetical filocyte differentiation pattern which is similar in form to the repeated asymmetric division pattern that is treated here. Differences between the daughter cells and the mother cell at each bifurcation in the differentiation pattern may be attributable to asymmetric division or to differences in the environments of the cells. In the present research, prenatal human growth is divided into four time intervals. Symmetric division during the first interval is followed by asymmetric division during the remaining intervals. Calculations based upon these division patterns indicate that the total number of cell divisions required to yield the number of cells in a newborn human is substantially less than the maximum number of divisions observed in fibroblast cell cultures. The number of nerve and skeletal muscle cells in a healthy human adult is approximately constant because the average longevity of the cells is greater than that of the organism as a whole. In other tissues, such as the blood, constant size is maintained by continual replacement of cells that die. The relative importance of symmetric and asymmetric division in this process is uncertain. Osgood's model [4] of erythrocyte differentiation employs both types of division. Kay's calculations [5] indicate that the enormous numbers of erythrocytes and leukocytes required during a human lifetime can be descended from a reasonable number of stem cells by tangential asymmetric division only if the stem cells each divide many thousands of times. It will be shown here that the required number of divisions is much smaller if the stem cells result from a process of repeated asymmetric division. Sheldrake [6] has suggested that the rate of aging may depend upon the degree to which wastes are eliminated from cells by asymmetric division. The results of the present research are useful in calculating the waste content of a population of cells which
321 undergoes tangential or repeated asymmetric division. A preliminary report of this work has recently appeared [7].
PROPERTIES OF A REPEATED ASYMMETRIC CELL DIVISION PATTERN Growth of a cell population by symmetric division can be represented by a branching "tree" pattern as shown in Fig. l(a). All cells are of the same type and are designated by the numeral "1". The number of cells increases exponentially with time. The simplest asymmetric division pattern, sometimes called a "tangential" pattern [5], is shown in Fig. l(b). Each time a cell of type 1 divides, it produces another cell of type 1 and a differentiated cell of type 2. The number of type-2 cells increases linearly with time if cell death is negligible, while the number of type 1 cells remains constant.
4
(a)
%
(b)
Fig. 1. Cell division patterns. (a) A symmetric pattern. All cells are of the same type. (b) The simplest asymmetric pattern, sometimes called "tangential". Cells of type 1 divide; cells of type 2 do not. See text. Fig. 2. A repeated asymmetric cell division pattern in which cell types 1, 2, and 3 divide, but type 4 does not Q"= 3). See text.
Figure 2 presents a more involved asymmetric division pattern. Each time a cell divides, it reproduces itself and gives rise to a new cell type. Differentiation of the earher type coincides with or follows cell division. The division of a type-2 cell, for example, produces daughters of types 2 and 3. In general, it will be assumed that there are / cell types which undergo asymmetric division in addition to a j + l'st cell type which does not differentiate or divide further within the pattern under consideration. In Fig. 2 , / = 3 and the fourth cell type does not divide. The lengths of the arrows shown in Fig. 2 differ to indicate that the cell types may differ in the average length of time they spend between divisions. They may also differ in lifespan. Cell types need not differ morpholog-, ically, but they must, at least in principle, be experimentally distinguishable. Perfectly symmetric division is thus excluded In a living organism, it is reasonable to suppose that sequences of symmetric and asymmetric division are often interspersed. Nevertheless, attention is here concentrated upon the repeated asymmetric division pattern (Fig. 2) in order to explore its capacity
322 to explain growth, and the maintenance of constant-sized cell populations. General division patterns consisting of symmetric as well as asymmetric subpatterns are most easily analyzed by treating the symmetric and asymmetric portions separately. The mathematical description of the symmetric growth pattern is simple and well known; it remains to analyze the asymmetric pattern. The calculations upon which this analysis is based are presented in the next section. They may be omitted on first reading by those who wish to proceed directly to the applications of the results of the analysis which are contained in the following section.
SYNCHRONOUS DIVISION OF LONG-LIVED CELL TYPES An asymmetric cell division pattern such as that shown in Fig. 2 is especially simple to analyze if cell death can be ignored, i.e. if all cell types are "long-lived". Let ni(t) be the number of cells of type i (/ + I ~> i ~> 1) which exist at time t, and let n l(t) be a known function of time. The equation which governs the increase in n i (t) is: ni(t + Ti-1) - hi(t) = ni_l (t), i/> 2
(1)
where Ti-i is the average interdivision time of cells of type i - 1. The number of cells of type i is greater at time t + Ti-~ than at time t because each cell of type i - 1 has, on the average, produced one daughter of type i. Cells of type i may also divide during this same time interval, but such division does not change their number because they replace themselves, one for one, in asymmetric division. Thus eqn. (1) indicates that the increase in the number of type i cells during an interval of Ti-~ time units following time t is equal to the number of type i - 1 cells which exist at time t. Equation (1) is a compact representation of a set of i - 1 simultaneous linear equations. Exact solutions can be obtained if n ~ is constant and the values of Ti are equal for all i,j>~i>~ 1. Let X = tiTs,
(2)
where Tg is the value of the interdivision time characteristic of all cell types. X in effect measures time in units of Tg. In order to avoid ambiguity, it will be assumed that the number of cells of each type is counted at integer values of X, that cell division takes place synchronously ~ ( time units later, and that ~ < < 1. A celt of type i makes its first appearance at time X = (i - 1) + zSX and is first counted at time X = i. I f / 4 : j + 1, X - 1 is equal to the exact number of divisions experienced by a cell of type i. If i = / + 1, X - 1 is equal to the maximum number of divisions experienced by a cell of type j + 1. , Equation (1) can be rewritten as a function of X: ni(X + 1 ) - n i ( X ) = n i - l ( X ) , i > ~
2.
(3)
It can be verified by direct substitution that eqn. (4) is a solution of eqn. (3) for which n is constant: ni(X) = nl 6"~i--11, i ~> 2. (4)
323 C~/_-I1 represents the number of combinations of X - 1 items taken i - 1 at a time. The value of n i ( X ) is maximum if i = (X + 1)/2. In the event that X is even and (X + 1)/2 is half-integral, either the next higher or the next lower integer may be assigned to i without effect upon the calculated maximum value of ni(X). Once the assumption of equal interdivision times is introduced, the asymmetric cell division pattern can be regarded as a simple example of a deterministic, zero-sided, informationless DOL Lindenmayer system [8]. In lindenmayer's notation [8], the starting symbol or axiom is 1, and the production rule is i ~ i, i + 1. The growth function, eqn. (4), could therefore have been obtained with the use of the formalism presented by Herman and Vitanyi [9]. ASYNCHRONOUS DIVISION; FINITE CELL LIFESPANS
Equations (3) and (4) describe populations in which divisions of cells of all types are synchronized to occur at intervals of fixed duration. It is useful to relax this restriction and to introduce the possibility that cells of some or all types may have finite lifespans. Suppose that cells of type i have an average lifespan of Ti time units. It follows from eqn. (1) that ni_~(t) is the number of type-/ cells created between times t and t + T i - 1 . T h e number of cells dying during this interval is equal to the number that were created during an interval of equal duration which occurred T~ time units earlier. Thus the last term of eqn. (5) represents the modification of eqn. (1) required to take finite cell lifespans into account: ni(t + T i - l ) - ni(t) = n i _ l ( t ) - n i _ l ( t - Ti)
(5)
Equation (5) can be solved most easily with the use of Laplace transforms. If s denotes the complex frequency variable and N i the Laplace transform of ni(t). [lf
e-sTi ]
gi=Ni-'[ -sTi-'~-'l'e II
if> 2.
(6)
Let n~ be constant if t ~< T~ and zero if t > T I . Then I'/1
NI = - - (1 - e-sT1) $
(7)
Iterative application of eqn. (6) to eqn. (7)yields: i
l'I
(1 - e-STq)
n 1 q=l
M ---
S
(8)
i-1
II
(e'rq-1)
q=l
It is not difficult in principle to solve eqn. (5) by inverting the Laplace transform shown in eqn. (8). However, the presence in the transform of i - 1 infinite sets of poles
324 renders the inverse cumbersome. Useful results can be obtained by calculating the limiting form of ni(t) which is valid at large values of time. Large values of time in the function ni(t) correspond to small values of the magnitude of the complex frequency in its transform, Ni(s ). Thus, in eqn. (8), the expressions of the form eSTq -- 1 approach sTq as the value of time becomes large and the value of s approaches zero. The expressions of the form 1 - e-STq approach unity if the death of cells of type q can be ignored. These cell types are long-lived in the sense used in the preceding section. All other cell types are termed short-lived. For short-lived cell types, the expressions 1 - eSTq approach sTq as s approaches zero. With the help of these approximations, the complexity of eqn. (8) is substantially reduced: nl
i-I
i
Ni=~-ff - l'X (1/Tq) n q=l
Tq,
(9)
q=l qtF
where b is the number of long-lived cell types, and F is the set of short-lived cell types. The second product in eqn. (9) extends, as indicated, only over the latter types. The inverse of eqn. (9) is: t b-~
i-i
ni(t)=nl ~
i
n (1/Tq) I I Tq. a=l q=l
( b - 1)!
(10)
qeF
A number of results follow directly from eqn. (10). (1) The number of cells of the ith type increases as a power function of time at large values of time. (2) At least one cell type must be long-lived (b 1> 1) if cells of type i are to be present at large values of time. The lifespan of the long-lived type must exceed the largest value of time for which the function ni(t) will be calculated. (3) If all cell types are long-lived (b = i), eqn. (10) reduces to
t i-~ i-~ n (1/Tq). ni(t)=nl ( i - 1)! a=,
(11)
(4) If all cell types are long-lived (b = i), and, in addition, their generation times are equal (Tq = Tg for all q, i - 1/> q t> 1), eqn. (11) can be further simplified:
xi-1 ni(X)= nl ~
( i - 1)!
,
(12)
where X = t/Tq as in eqn. (2). If eqn. (4) is expanded as a power series in )(, its leading term, i.e. the term which dominates the series at large values of time, is identical to eqn. (12). (5) If the total number of long-lived cells is calculated by summing terms of any of the forms shown in eqns. (4), (11), or (12) over all cell types, 1 through j + 1, the j + l'st
325 term dominates the resulting sum at large values of time. Consequently, it is often an acceptable approximation to set the total number of cells equal to ni+ 1. (6) If the number of cells of type i is to be constant with respect to time, exactly one cell type must be long-lived (b = 1). The long-lived type will be assigned the subscript h. (7) If the single long-lived cell type is the i'th (h = i), the cell number ni(t) remains constant because the cells of all types ancestral to i are dead at large values of time. Equation (10) assumes the form: i--1
ni= nl I I
(13)
xo,
q=l
where xq = Tq/Tq is the maximum number of divisions experienced by a cell of type q. Consider the problem of maximizing ni under the constraint: i--I
X m ---- ~_~
(14)
Xq
q=l
= constant, where Xm is the maximum number of divisions performed by cells of all types ancestral to type i. The product over Xq in eqn. (13) can be regarded as the volume of an i - 1 dimensional parallelepiped in which xq is the edge length in the qth dimension. If the sum of the sides of this figure is constrained to a constant value, as in eqn. (14), the volume is maximized by choosing the values of Xq equal. Thus ni is maximized if Xq = x for all q, i - 1 >I q ~> 1, and eqns. (13) and (14) become: ni = n i x i-1
(15)
X.,, = (i - 1)x
(16)
and
respectively. Equation (15) is easily solved for x: (17)
x = (ni/nO 1/(i-0.
(8) If the single long-lived cell type is ancestral to the ith type (h < i), the cell number ni(t ) remains constant because type-/ cells that die are replaced by asymmetric cell division. The form of eqn. (10) which applies to this case is: i--I
ni=nl(Ti/th) II
xq,
(18)
q=l
where th is the maximum value of time at which cells of type h divide. Often this value will coincide with the lifespan of the organism in which the cell population is situated. In eqn. (18), as in eqns. (13) and (14), Xq represents the maximum number of divisions experienced by a cell of type q. I f q ~ h, Xq = Tq[Tq. I f q =h, Xq = th/Th.
326 Apart from the dimensionless factor Ti/th, eqn. (18) is identical to eqn. (13) If the constraint upon the maximum number of divisions shown in eqn. (14)is applied to eqn. (18), it follows that r/i will be maximized by the choice xq = x for all q, i 1 1> q >~ 1. Then Xm can be obtained from eqn. (16), and eqn. (18) becomes:
rl i = tl I (Tilth) Xi- 1
(19)
Solution of eqn. (19) for x yields:
= I ni/Ti J1/(i-l). x [ nl/thJ
(20)
APPLICATIONSAND DISCUSSION Examples of applications of the preceding results will be given below. In each instance, the number of cell divisions required to produce a cell population of a certain size will be calculated. Experiments reviewed by Hayflick [10] indicate that the number of divisions which can be performed by a strain of somatic cells is limited. Hayflick's own work [11] demonstrated a limit of approximately fifty population doublings in normal fibroblasts cultivated in vitro, and a number of the studies he reviewed [10] support the concept that a similar limit applies in vivo. However, contrary evidence exists. For example, in vivo studies of epithelial tissue by Strehler [12] and Cameron [ 13] show that the limit, if any, on the number of population doublings must be much greater than fifty. The distinction between the number of cell divisions and the number of population doublings should be noted. Merz and Ross [14] observed that many living cells in latepassage fibroblast cultures do not divide. As a result, the cells which retain the capacity to divide must do so more than once, on the average, each time the population doubles. The maximum number of cell divisions must exceed the maximum number of population doublings. Calculations by Good [15] and by Hirsch and Curtis [ 16] indicate that, in the forty-fifth passage, a small fraction of the dividing cells will have divided as many as 150 times. This figure will be adopted as an approximate estimate of the maximum number of divisions which a somatic cell can undergo.
Prenatal growth If asymmetric cell division patterns are responsible for embryonic and fetal growth, the number of cells in the developing organism should follow a combinatorial function of time, as in eqn. (4) or a power function of time, as in eqns. (10), (11), and (12). Available data on prenatal growth appear to be of insufficient accuracy to distinguish between these two cases. Zotin [17] reviewed evidence supporting a number of different "laws of growth", including the power law. Timiras [18] stated that over two hundred different mathematical functions of time have been applied to growth data, so it is evident that agreement, even close agreement, between theory and data provides no conclusive proof that the assumptions underlying the theory are correct. Nevertheless it is significant that
327 a number of studies, commencing with early work cited by Zotin [17], are consistent with the results obtained here. The cubic function, which is a special case of the general power function, was fitted to the fetal growth of a wide variety of species by Huggett and Widdas [19] and by Spencer and Coulombe [20]. Sikov and Thomas [21] fitted data on the prenatal growth of the rat with several mathematical functions. A cubic function o f time yielded reasonable agreement with the data, but other more complicated functions fit more closely. Osgood's studies [22] showed that increases in the weight of the human and in the weights of various o f the hematopoietic organs can be represented in fully logarithmic coordinates by a reasonably small number of straight-line segments. Thus, over substantial periods of prenatal and postnatal growth, the weight of the whole body, as well as the weights of the organs, are described by power functions. If the average cell mass is assumed constant, the same power functions can also be applied to the numbers of cells in the body or organs. TABLE I PRENATAL HUMAN GROWTH* Time interval (days) 0-9
9-42
42-115
115-266
250
13.6
Ratio of final weight to initial weight, r
1
106
Exponent, j" Number of dividing cell types,/"
0 1
9.1 9
Number of cell divisions, X-1
9
24
10
6
512
1,307,504
252
20
Cell number ratio,
5.4 5
3.25 3
ni+:/nl
Interdivision time, Tg (days)
1.00
1.37
7.30
25.2
*The first three rows represent Osgood's data [22]. Numbers in the remaining rows are calculated by methods described in the text.
Osgood's data on body weight are summarized in the first three rows of Table I. The prenatal growth period is divided into four time intervals, as indicated in the first row. The ratio of final weight to initial weight, r, and the exponent of time,/', with which the weight increases are given in the second and third rows, respectively. During the second, third, and fourth time intervals, it is assumed that the increase in cell number results from asymmetric division, that the total number of cells is approx-
328 imately equal to the number of cells of type / + 1, and that the number of cells of type / + 1 is proportional to the weight of the embryo or fetus. With the use of these assumptions, eqn. (10) can be fitted to the data which apply to each of the time intervals. Inspection of eqn. (10) shows that the number of long-lived cell types, b, must be equal to j" + 1. The number of cell types, / + 1, is minimized if it is assumed that all cell types are long-lived, i.e. if the number of dividing types,j, is equal to the experimental value of the weight-increase exponent, j" Accordingly, the values o f / w h i c h appear in the second, third, and fourth columns of the fourth row of Table I were obtained by rounding the corresponding values o f / ' t o the nearest integer. The assumption that all cell types are long-lived makes it possible to replace eqn. (10) with eqn. (11). The further assumption that all cell types within a particular time interval have the same interdivision time allows the use of eqn. (12), which is the limiting form of eqn. (4). The data are equally consistent with either of these two equations. The values of nj+l/nl shown in the sixth row of Table I were calculated with the help of eqn. (4) rather than eqn. (12) because it yields more conservative, i.e. larger, estimates of the number of cell divisions required during prenatal growth. The number of cell divisions, which appears in the fifth row, was obtained by choosing the smallest integer, X, which yielded a cell-number ratio, ni+~/n~, in excess of the weight ratio, r. The interdivision time, given in the seventh row, represents the ratio of the duration of the time interval to the number of cell divisions. The first time interval shown in Table I demands special consideration. Although there is no detectable weight gain [22], division is actively taking place during this time. The early cell divisions are synchronous and the cell number increases exponentially [18]. Some differentiation occurs by the 9th day, perhaps signifying the beginning of asymmetric division. Nevertheless, in the absence of precise information, it will be assumed that all divisions during the first 9-day period are symmetric. In order to calculate the number of divisions and the interdivision time, it is necessary to know the number of cells at the end of the 9th day. This can be estimated from the data shown in Table I as follows: between the 9th and the 266th days, weight increases by the ratio 106 × 250 X 13.6 = 3.4 × 1 0 9 . The newborn is constituted of 1.25 × 1012 cells [22]. If the average cell mass remains constant, between the 9th and 266th days, there are therefore 1.25 × 1012/3.4 X 109 = 368 cells on the 9th day. To grow these from a single cell requires no more than nine symmetric divisions which take place at a rate of one per day. Thus a total of 40 asymmetric divisions and 9 symmetric divisions, or 49 cell divisions in all, are sufficient to account for the full growth of a newborn human. This value is well within the limit of approximately 150 cell divisions which was assumed above. The calculated number of cells, given by the product of the numbers in the sixth row of Table I, exceeds the required total of 1.25 × 1012 by a factor of approximately three. The number of cell types required for this growth is 18, since the final type which appears during each time interval also serves as the initial type during the following interval. The average interdivision time increases monotonically throughout prenatal growth.
329
Mature populations Equations (13)-(17) describe asymmetric cell division patterns in which only the final cell type, / + 1, is long-lived. Examples of such long-lived cell types include neurons and skeletal muscle cells in the adult human. The maximum number of cell divisions, Xm, required to produce the 101° neurons in the human brain can be calculated from eqns. (16) and (17) if it is assumed that the maximum number of divisions, x, performed by each of the dividing cell types is the same. An estimate of the number of dividing cell types, j, can be obtained by adopting the value, 17, obtained above for prenatal human growth. With / = 17, nx = 1, and nj+ 1 = 10 l° , eqn. (17) yields x = 4 divisions, rounded to the next higher integer, and eqn. (16) yields Xm = 68 divisions. Most cell populations in the adult human divide rather slowly if at all. If popula. tions which divide rapidly are regarded as special cases which can be ignored for the present*, the maximum number of asymmetric divisions needed to produce an adult can be calculated as above with the help of eqns. (16) and (17). On the assumption that cell mass remains constant throughout postnatal life, the number of cells in a 68 kg adult is taken as 20 times the number of cells in a 3.4 kg newborn infant. Thus ni+l = 20 X 1.25 X 1012 cells = 2.5 X 10 ~3 cells. Ifn~ = 1 and/' = 17,x = 7 divisions, rounded to the next higher integer, and Xm = 119 divisions. The estimate / = 17 is very likely to be low, since only cell types involved in prenatal growth were taken into account. Fortunately, the calculated values o f x and Xm are rather insensitive to the value of/'. The use of a higher estimate of/' would have led to lower calculated values of Xm. However, the present values, 64 divisions for the growth of the brain and 119 divisions for the growth of the adult human, are consistent with the assumed maximum of 150 cell divisions. In a celt population of constant size, the average number of divisions performed by cells of each type is half the maximum number. In the example of the total adult cell population discussed above, the average number of cell divisions per cell type, calculated by dividing the result obtained from eqn. (17) by two, is 3.07. The average number of divisions performed by cells of all types ancestral to the/' + l'st is 17 X 3.07 or 52.2. This value is much smaller than the maximum number of divisions, 119, because, in a population of constant size, it is only the very rare cell that follows a path of maximum length through the asymmetric division pattern. The situation is quite different in expanding populations having many long-lived cell types. If all cell types are long-lived and cell divisions take place synchronously, it can be shown that the average number of divisions preceding the birth of a cell of type/' + 1 is X-= [j/(j' + 1)] X; as the number of cell types increases, the average number of divisions approaches the maximum number. Application of this result to the data in Table I shows that cells of type/' + 1 in the newborn human infant have divided an average of 45.9 times. The maximum number of divisions, 49, is not very much greater. Thus the large
*It will be shown in the next section that a cell population in which turnover is comparatively rapid need not be associated with a very high division rate.
330 difference in the maximum number of divisions between the adult and the newborn is not reflected by a corresponding difference in the average number of divisions. The average number of divisions, 45.9, is 6.3 divisions less than the average number obtained for the full growth of the adult, a difference which is reasonable in view of the mass ratio of 20 which was assumed between the adult and the newborn and in view of the drastic simplification involved in representing the development of the adult by a single asymmetric growth pattern. A constant-sized population with cell turnover Equations (16), (18), (19), and (20) are applicable to asymmetric division patterns in which the number of cells of the final type,/" + 1 is maintained constant by replacement of the cells that die. Consider the simple tangential pattern (Fig. lb) which was used by Kay [5] to calculate the number of divisions needed to produce a lifetime's supply of human erythrocytes and leukocytes. Kay proposed the existence of an initial pool of 4 X 101° cells which divide at a rate sufficient to yield 2.4 X 101° stem cells per day. Each stem cell gives rise to a clone containing an average of 20 blood cells. The length of a human lifetime was set equal to 2 × 104 days (approximately 56 years). In the notation used here, j = 1, nl = 4 × 10 l° cells, n:+l/T/+l = ne/T2 = 2.4 × 10 l° cells/day, and th = 2 × 104 days. With the use of these values, Kay calculated that cells in the initial pool would have to divide 12,000 times during the course of a lifetime. The same result can be obtained with the help of eqns. (16) and (20). A pattern which requires 12,000 divisions is clearly inconsistent with the limits which are held to apply to most diploid celt populations [10, 11]. Kay's suggestion [5], later expanded by Reincke etal. [23] is that blood cells are produced by a process called "clonal succession". This means, very briefly, that division follows a symmetric tree pattern (Fig. la), but the rate at which stern cells mature is controlled by a regulatory feedback mechanism. Although the blood cells appear at a constant rather than an exponentially increasing rate, only 54 divisions are sufficient to produce enough cells to last a lifetime [5, 23]. The phenomena of recovery from hemorrhage and radiation make it clear that regulatory mechanisms exist which serve to maintain the sizes of the blood-cell populations approximately constant. The kinetic properties of these mechanisms have been modelled by Lajtha et aL [24]. Their proposed regulatory model, like all feedback systems, is subject to exhaustion of its dynamic range, to overshoot, and even to oscillatory instability if the disturbances it is called upon to control are too extreme or if the required degree of regulation is too, great. These problems can be reduced in severity if the normal operating point of the system can be maintained in the absence of feedback. The full capacity of the feedback system is then available to oppose disturbances that tend to drive the system away from its operating point. In clonal succession, the normal operating point of the regulatory system, 2.4 X l010 stem cells per day, is set by feedback, while, in asymmetric division, the same operating point is maintained by the intrinsic division rate of the cells. With the latter mechanism feedback regulation would function only in response to an external disturbance, e.g. hemorrhage, disease, or radiation injury.
331 Arguing as Kay [5] did, "teleologically and unashamedly," an asymmetric division pattern would be advantageous with respect to the regulation of blood-cell populations if the number of divisions required to produce a lifetime supply of cells were not unreasonably large. As is shown in Table II a dramatic reduction in the number of divisions TABLE II PRODUCTION OF STEM CELLS BY ASYMMETRICDIVISION* Number o f dividing cell types, /
Number o f cell divisions per dividing type, x
Maximum number o f asym. metric divisions, X m
1 2 3 4 5 6
12,000 110 23 11 7 5
12,000 220 69 44 35 30
*Based on data by Kay [5]. See text. results if the simple tangential pattern (Fig. lb) is replaced by the more general asymmetric pattern (Fig. 2). The values of x and X m in Table II were calculated from eqns. (20) and (16), respectively, with the use of the same numerical values that were given by Kay [5] and cited above. Values of x were rounded to the next higher integer. With just two dividing cell types rather than one, the maximum number of asymmetric divisions decreases from 12,000 to 220. If there are six dividing cell types, the maximum number of asymmetric divisions decreases to 30. Beyond six, increases in the number of cell types offer little further reduction in the maximum number of divisions. In order to obtain the total number of divisions required to produce the full population of blood cells from a single ancestor, it is necessary to take into account the number of divisions required to yield the initial pool of 4 × 10 l° cells and the number of divisions needed to produce a clone of 20 blood cells from a stem cell. If it is assumed, in the interest of simplicity, that these divisions are symmetric, the growth of the stem-cell pool requires 36 divisions, and the growth of the clone requires five divisions, both values having been rounded to the next higher integer. Thus the numbers listed in the third column of Table II must be increased by 41 to obtain the total number of divisions needed to produce a lifetime supply of blood cells from one ancestor cell. It is clear that no conflict exists between the postulated limit of 150 cell divisions and the calculated number of divisions shown in Table II if the asymmetric division pattern contains three or more dividing cell types. None of the foregoing should be construed as an attempt to prove that asymmetric division is, in general, of greater importance than symmetric division. The calculations which have been presented simply purport to show that asymmetric division is a plausible alternative to symmetric division in several typical cases of cell-population growth and that the number of divisions required in each case is consistent with observed limitations on the number of population doublings in diploid cells.
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