On radioactive labeling of proliferating cells: The graph of labeled mitosis

J. Theoret. Biol. (1969) 22, 468-492

On Radioactive Labeling of Proliferating Cells: The Graph of Labeled Mitosis? B. V. BRONK Medical Department, Brookhaven National Laboratory, Upton, New York, U.S.A., and QueensCollege of the City University of New York, Flushing, New York, U.S.A. (Received 15 April 1968, and in revisedform 13 August 1968) The interpretation of the results of radioactive labeling during the state of DNA synthesisis discussedin terms of the model of proliferating cells in which the time spent in each state is determinedstochastically. Using renewal equationsfor the labeled and unlabeledpopulations, the birth rate of labeled cells, the labeling index, and various aspectsof limiting forms of the labeling index are discussed.It is shown that the index can be very well representedby its asymptotic value plus the product of a cosinefunction (with a period slightly lessthan the doubling time) and a decaying exponential for times long after labeling. Modifications due to cell death and differentiation are considered.It is shown that someof the resultsobtained are rather insensitiveto a uniform death rate or differentiation. This is in contrastto the behavior of the fractions-in-statewhich are found to depend strongly on the multiplication rate at the end of the cycle. It is found that the asymptotic labeling index is given by the fraction of time in S rather than the log-phasefraction in S as one might expect. This makesa significant differenceif the state labeledhasa transit time short comparedto the cycle time and occursat one end of the cycle. 1. Introduction The development of the methods of flash labeling of cells with radioactive materials has made the measurement of the average lengths of the various states of the cell cycle population a standard procedure (Hughes et al., 1958; Quastler & Sherman, 1959). Since the separation of a population into synchronous sub-populations by labeling is rather elegant in concept, a more precise theoretical treatment of the labeling curves resulting from this procedure is of interest in itself. More important, however, is the probability that a considerable amount of additional quantitative biological information can be extracted from experiments which make use of these techniques. This j Work supported under the auspices of the U.S. Atomic Energy 468

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information will become more accessible as automatic counting techniques are developed, since these will make the counting of larger statistical samples practicable. We shall discuss the labeling experiments in terms of a model which has been used elsewhere to explain effects of fractionation during radiation experiments (Paskin, Bronk & Dienes, 1968; Bronk, Dienes & Paskin, 1968). The main assumptions of the model follow. The cycle is divided into four states with respect to DNA synthesis. The states, Gr, S, Gz and M, are designated in this paper by the subscripts i = 1, 2, 3 and 4, respectively. Each of these states is assigned a mean transit time, li, a variance, of, of the transit times about that mean, and a probability, fi(t) dt, of the transit time being located in the interval around t. It is assumed in this model? that the time a cell spends in each state is independent of the time it spends in the others. It follows that the overall time spent in traversing the entire cycle has a probability densityf(t) which is given by the fourfold convolution of theyi( The mean transit time for the cycle is T,=

c

ti

i=l

(1)

and its variance is 4 o’, = 1 r7i’. i=

1

The model is applied in this article to cells growing freely in uncrowded conditions such that interactions and cell death can be ignored. The latter restriction is for convenience only, and the minor modifications needed to include a uniform and constant death rate will be discussed also. Using this model of the cell cycle we derive renewal equations for the average number of labeled cells born at any time after labeling. The birth rate for labeled cells is discussed in terms of the general birth rate for the model. An equation is given for fluctuations in the labeled birth rate which arise due to the stochastic nature of the transit times. However, other, probably more important sources of fluctuations are only qualitatively discussed. The expression for the fraction of mitotic cells labeled is derived. The behavior of this fraction immediately following labeling is discussed. Its oscillatory decay to an asymptotic fraction is exphcitly exhibited. In the limiting case in which the rri approach zero the expression for this fraction assumes a simplified form which shows the features which are due to the exponential age distribution and the transit times only. t This assumption should be useful as a first approximation relations exist.

even in cases where cor-

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2. The Idealized Model The label we have specifically in mind is tritiated thymidine which is incorporated into the cell’s DNA during synthesis (S state). A short time after exposure to the DNA, the unincorporated portion is completely removed from the culture. Since the time for this exposure is short compared to the transit times, we will consider it instantaneous. We assumethat every cell in S at the time of labeling is labeled, and that no cells in the other states are labeled. We shall make a further idealization which is rather severe but which makes the analysis much more tractable. This is the assumption that every cell initially labeled has only labeled descendents. It naturally follows that unlabeled cells are descendents only of cells which are not initially in S. We will now consider briefly and qualitatively the effect of departures from this behavior. The amount of labeled thymidine present per cell must halve on the average during each division. If the detection technique reveals a relatively small number of 3H decays (of the order of 15 or 20) per labeled cell shortly after labeling, one can expect that after three or four cycles, a detection technique allowing the same length of time for decays to register will show one or two decays per labeled descendent. Since the half life of tritium is very large in comparison with the cycle time, the total number of decays per unit time for the entire cohort of labeled cells remains essentially constant over the duration of an experiment. However, becauseof the presenceof background, it is a common laboratory practice to neglect cells with a grain count indicating one or two decays per cell.? Hence, after three or four cycles, about half the labeled cells will be hidden in the background. The detection time may be doubled, of course, but eventually fluctuations in the number of decays per cell due to the stochastic nature of radioactive decay will obscure the later peaks in the per cent labeled mitosis curves. This effect is more pronounced since the peaks in the curve decay rapidly (exponentially we seelater) to the asymptotic fraction of the mitotic cells which are labeled. Besidesthe fluctuations due to practical limits on the length of detection time, there will be fluctuations which are more interesting biologically due to the unequal distribution of label to the two daughters of a labeled cell.; Specifically one might ask, to what extent does a parent chromosome normally remain a material entity during cell division ? t D. G. Baker of the biology department at Brookhaven brought this practice to the author’s attention. $ Trucco (1965) has made a start on the analysis of this problem by considering distribution of label for a deterministic (definite cycle time) population with the labeled radicals divided according to a binomial probability distribution on each cell division.

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There is an additional perturbation on the initial distribution of label, due to the fact that cells in all stages of synthesis do not pick up the same amount of label.? All these variations must eventually be considered in detail in order to extract full information from experimental data, such as grain counts, which are available for analysis. For the present, however, we continue with our idealized model. 3. The Labeled Population

We will begin by deriving a renewal equation for the labeled population.: Let t = 0, be the time (instantaneous) of flash labeling. The probability that a cell labeled at t = 0 has exactly j descendents at a time t later is denoted by p;(t). Similarly, the probability that a cell born at t = 0 has j descendents at time t is denoted by p j(t). We define two generating functions G*(s, t) = $ py(t)s’ j=

G(s, t) = 5 j=l

1

(2) pj(t)s’

We let n*(t) (or n(t)) designate the number of descendents at time t of a cell which was labeled (or born) at time t = 0. A complete set of initial conditions for the starred (labeled) quantities (n*(t), PT, etc.) includes age-in-state distributions at the time of labeling. The corresponding condition for the unstarred quantities is simply a delta function at age zero in G,. We will use the following properties of generating functions, which are obtained by taking derivatives of equation (2). G*(l, t) = 1 aG*(l, t)/c?s = n*(t) (3) a’G*(l, t)/&’ = [n*(t)12 - II*(~), the bars denoting expected values. Let L(t) dt denote the probability that a cell which is in state S at time t = 0, will divide (finish mitosis) during the interval around t. Using the property of the generating functions, that tells us that the generating function for descendents of two independent cells born at time t = 0, is given by the square of the generating function for one such cell we write G*(s, t) = [ 1 - F,(t)]s + [ G2(s, t - t’)L(t’) dt’, h

(4)

t Alpen & Johnston (1966) found a greater variance in the grain count of erythropoietic cells in a given part of the S state than one would expect on the basis of a Poisson distribution of decays. This tends to indicate a variable rate of incorporating thymidine. $ The derivation follows closely that given by Harris (1959) for a population of cells without labeling.

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where

FL(Z) = 1 Jyt’) dt’ b

(5)

is the probability that a labeled cell divides less than t hours after labeling. This implies that the probability that only one labeled cell exists at time t arising from a cell labeled at t = 0 is given by [ 1 -FL(t)]. The second term in (4) gives the terms of the generating function arising from a labeled cell dividing at any time t’ earlier than t. Taking the first derivative of equation (4) with respect to s gives [G*b

j)]’ = [I -FL(j)]

+2 (‘G(s, j- t’)G’(s,

t- /‘)L(t’)

dt’.

16)

Setting s equal to one and using equation (3) we arrive at the renewal equation for the labeled population per initially labeled cell. H*(I) = [I -F,(t)]+2

(k(t-f’)L(r’) i,

dt’.

(7)

From equation (7) one may verify that for large t. .* grows exponentially at the same rate CI as is well known for n(t) (e.g. Harris. 1959). That is n”(t) = c ear[ + o(t)]

(-co)

where c is a random variable of order of magnitude ci = (ln 2/T,)[l

+(ln 2a2,/2Tc”)].

(8) one, and (9)

If we take the second derivative of equation (4) and set s equal to one we obtain ~ ___ (IO) [n*(t)12-n*(t) = 2 )‘~r~2(1--I’)--,(j-f’)+[n(t--1’)]2}L(t’)dt’. b By considering an equation for the unlabeled population which corresponds to equation (10) for the labeled, one can show that [T(t)-E’(t)]+ grows asymptotically with the same exponential growth rate as does n(t). This may be shown by assigning an exponential growth rate to z(t) and ii(t) and comparing both sides of the resulting equation. The same procedure may be followed with equation (10) and one then finds? that {[n*(t)]‘[n*(t)]‘}” - ear for t > > T,. (11) This standard

deviation

is per single initially labeled cell. If one starts with

t For times t, longer than T, but still during the first few cyclesfollowing labeling, oneexpectsa decayingoscillatorytermto contributeto the rate of changeof the standard deviation.This wouldappearin a mannersimilarto the oscillatoryterm of equation(21).

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N labeled cells initially, the standard deviation divided by the average number present is of course reduced by a factor (l/J%). The ratio of the standard deviation to the number actually present at long times, t, is a random variable, X, of order of magnitude, one. When t is large, the number of cells of a given age is also large for ages less than the cycle time. We can therefore conclude that later fluctuations are moderated by a factor like the (l&?) discussed above and x is determined by early fluctuations. Since for most labeling experiments one can start with a large initially labeled population, we are justified in limiting ourselves to this brief discussion of fluctuations and concentrating on average behavior. The discussion of per cent labeled mitosis curves is facilitated by the introduction of birth rates for both labeled and unlabeled cells. We let B*(t) represent the average birth rate (per initially fubeZed cell) of labeled cells t hours after labeling. Since two new cells emerge following every “birth” (division), B*(t) = 2 dn*(t)/dt. (12) Similarly, B(t) is defined as the average birth rate (per initial cell) at time t for descendents of cells which were at the beginning of G, at t = 0. The renewal equations for these quantities are obtained from equation (7) and the similar equation for n(t). They are B*(t) = 2 1 [B0 - 2’) +3(t - t’)]LJt’) b = 0

dt’

t>o

(13)

t I 0.

where L(t)dt is defined as the probability that a cell Iabeled at t = 0, completes the first mitosis following labeling during the interval around time t. We also have B(t) = 2 /-[B(r-I’)+&?-t’)]f(i’) ‘0 and by definition

dl’

t > 0,

(14)

B(O) = 1. Equation (14) and the properties of its solutions were discussed in some detail in Bronk, Dienes & Paskin (1968). The following properties will be of interest to us here. First we have an iterative solution B(t) = 2 2”S’“‘(t), “=I

474 where f’“‘(t) I”“(2) Iff(t)

z /ftn-0 ES(Z)

“(t’),f(t-

t’) dt’

(n = 2,. .),

(16)

is a Gaussian? with oc/Tc 5 .3, we have to a rather good approximation B(t) = ;;I 3”&(t) n= 1

117)

where g,(t) = [2nno2,]-+

exp [-(t-n~C)2/2nof].

(18) When f(t) has a small coefficient of variation (c.v. = g,/T,j, there is little overlap between terms of the series solution for several cycles following t = 0. Therefore, one or at most two terms of (15) or (17) is sufficient to approximate the birth rate very closely. That is, B(t) z 2f~~“f(“‘)(t)+2m+1~(m+1)(t)

(19)

for mT, < t I (m +l)T,. Equations (15) to (19) suggest a general approach to asymptotic properties of the renewal equation. The probability that the sum of the lifetimes of all the ancestors of a cell in the rrth generation has a value between t and t +dt is given byf’“‘(t)dt. According to the central limit theorem,f’“‘(t) approaches a Gaussian for large II, regardless of the form of f(t). For moderate C.V. and t of the order of several cycles, a very few terms of (15) contribute significantly to B(t). We may therefore use any convenient function with appropriate cc and T, to obtain asymptotic approximations to the birthrate resulting from the truef(t). For functions of the form j(t) = ~~~j ?/‘t”- ’ em”‘.

(20)

the exact solution of the renewal equation is well known (e.g. Feller, 1966, problem 1, chapter XIV). The asymptotic birthrate obtained in Bronk. Dienes & Paskin (1968) using such functions with C.V. < 0.707 is given by B(t) z (l/r,)

exp (zl)(l

$2 CC”’ cos w,t),

(21)

where w. = iYr(i $-CT’ in 2/7-:)/T,,

(22)

and A = 2n2aZ,lTc3. -f Since a cell can’t

have a negative transit time we mean .f(f) = [2m,2]- 5 exp [--(I ~T,)2,‘203 -0

(23) t 1 0, 1 < 0.

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According to the argument of the preceding paragraph we may use these equations to obtain general results for labeling experiments which do not depend on the exact form off(t). Before proceeding to determine the behaviour of B*(t) with the above results we will study the components which determine the detailed form of L(t). The time that a labeled cell spends before its first division is the sum of the additional time it spends in S following labeling, plus the time it spends in Gz and M. Let the probability that a cell initially in S waits between t and t+dt hours before leaving S be W,(t)dt. Then L(t) = f W2(t’)f3,4(t-

t’) dt’

where f3,4 is the density for the sum of the transit times in G, and M which is given by f

Since the population is usually labeled when in asynchronous growth, we will obtain the log-phase form of W, for use with equation (24). Suppose that ai(t is defined as the probability that a cell in state i has been in state i for a time in the neighborhood of t hours and that the functional form of ai is known. Then in terms of this age distribution the waiting time distribution? for the ith state is

~i(t> = 4 (ai(t’)fi(t + t’),[l -Fi(t’)]) dt’

i = 1, 2, 3,4.

(26)

‘0

If the population is in log-phase growth prior to labeling then the asymptotic age-in-state distribution is applicable. This is (e.g. Bronk, Dienes & Paskin, 1968), q(t) = ci e-“[l - Fi(t)] i = 1, 2, 3,4 t>o (27) where Fi(t) = J’fi(t’)

dt’.

(28)

0

ci is the normalization

constant, given by ci = cc(1-I,)-’ m Ii = ‘e-“Ifi J0

(29) dt

i = 1, 2, 3,4.

t This function is discussed in detail in Bronk, Dienes & Paskin (1968).

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Using the above asymptotic distribution phase population K(t) = cr(1-Ii)-l[ea’l,= 0

in equation (26), we obtain for a log1 e’(‘-“‘fi(t’)

dt’]

t 2 0,

(30)

t < 0.

Here CIis the growth rate given in equation (9). Now for a convolution of the form (13), it is well known that the Laplace transform of the left hand side is given by the product of the Laplace transforms of the two factors in the integrand (times two in this case). Now the Laplace transform of B(t) is well known in terms of the Laplace transform off(t). (It may be expressed as a sum of partial fractions when f is of the form (20). See for example Bronk, Dienes & Paskin (1968), Appendix III.) Using these results we obtain for the Laplace transforms of B*(t). 8(B*) Similarly,

= 2Y(L)/[1-2Y(j)].

(31a)

using (24) and (25) we obtain (31b)

g(L) = ~wzMf3wYf4)? and from (30) and (26)

If the fi are of the form (20). the inverse transform of the partial fraction expansion of the right-hand side of (31) is simply a sum of exponentials in time with fairly complicated coefficients. The exact form of B*(t) could be studied as was done for B(t) in Bronk, Dienes & Paskin (1968). Proceeding via the exact solution is unnecessary however, since all the properties of B*(t) necessary for the discussion of the labeled population can be obtained directly from the various forms already given for B(t). We therefore proceed using in the appropriate regions equations (15), (17). (19) and (21). We have again the iterative solution B*(t)

=

f

2”+

n= 1

’ /,(“)(t

-

t’)L(t’)

dt’,

(32)

where f(“)(t) is given by (16) and may be replaced by (18) when t is large. For small C.V. B*(t) may be expressed in a form similar to that of equation (19) except that in this case the convolution in (32) spreads the contributing terms over three generations. The most important approximation to B*(f) is obtained from equation (21).

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Then B*(t) = (2/T,) cut f dt’ +2Re”“O-“)’ f e-(a+iOo-“)t’L(t’) dt’ +o(e-At) e -q(f) (33) s I 0 0 1 for large t, with oo, and 1 as given in (22) and (23). Since L(t) approaches zero rapidly for t of the order of the cycle time, the two integrals in (33) are very nearly constants, which we define, starting from the left as C, and C,,. We will later determine values for C, and C,, for the limiting case when 02, 3, ,+ -+ 0 which should be good approximations when t3, 4 is short and &(t) has small variance. We return first to the discussion of L(f). We will show that the graphical form of L(t) can be described and visualized rather well in terms of departures from its limiting form for the deterministic case. Let us first suppose that cr3 and 64 both approach zero. In this case the limiting form off3, 4 is a Dirac delta function,

1

f3,4W = W-t,,,) The integral in (24) is then given by

r3.4 = t,i-t,

(34)

r;(t) = Wz(t - t,, 4), 03 = L74= 0.

(35) In many real cellular systems one finds that f3,4 is short compared with < 1, where 03,4 = (0: +oi)*. In that case f3,4 appears as f2 and c4/t3,, a spike compared to the broader distribution W, and equation (35) is a valid approximation. We will later see that even when t3,.+/Tc = 0.25, a fairly good approximation is obtained for the labeling curve when (35) is used to compute it (see Fig. 1). Since JVz(t) is zero for t < 0, L(t) is zero for I < f3,4. When the C.V. Off,(t) is small, there is very little contribution from the second term within the bracket of (30) when I is small. Let us compare the two terms of (30) for a particular case. This amounts to assigning a value to c in the inequality I

e-“‘x(f)

-< cl,.

(36)

0

If in our example fi is a Gaussian then both sides of (36) may be obtained from tables of the error function. Taking the case T, = 10-O hours and fi = 3ai = 6 hours, we find that for t < 1.86 hours that c 5 0.023, and according to the well known asymptotic properties of the error function, c decreases very rapidly as t -+ 0. This means that Wz(t) - eat for small t. For the purpose of obtaining the approximate shape of L(t) near t = f3,4 let us assume that tag, 4 < < l/a, and that f3,4 is approximately a Gaussian. In this T.A. 31

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I o-

0 9-

08-

er <+ 2

OG-

0,504.

0 3-’

0 2-

0 l-

!! 0

FIG. 1. Labeling index versus time calculated by direct numerical method (solid line); equation (60) for first peak (dashed line); asymptotic method of equation (79) [0.594 + 0.59 e-h’ cos (w,t - n)] (solid circles). tl = 1.5, tz = 6.0, t3 = 1.5, tr = 1 .O (hr); u1 = 0.5, u2 = 1.85, u3 = 0.5, u4 = 0.33, C.V. = 0.2; 1 = 0.079, a0 = 27~ (0.1028).

case ear changes slowly in the region of interest so that the convolution may be approximated by a constant, times an error function and we see that for =3,4 small but greater than zero L(t) is negligible for t < (t,,,- 2a,,,). As t increases beyond that point, L(t) increases with the sigmoid shape of the error function until t N” t3,4+2a3,4 at which time it joins into the form of the right hand side of (35). For 6 - (tz + t, ,4+a2), we can determine the appearance of L(r) by looking at the limiting form of equation (35) as a2 approaches zero. Then we easily obtain from (35) and (30) L(t) = a e”(‘-‘%4)/(eat2 - 1) t3.4 < t 5 b.3.4, (37) = 0 otherwise, for,

a2 =

a3 = a4+0.

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Using similar arguments to those in the paragraph above, we see that for cr’3,4 = 0, but c2 not zero, the function on the right-hand side of (37) is multiplied by a sigmoid curve which begins its decrease at about (tz, 3, 4 - 20,) and approaches zero rapidly for t > (tz, 3,4 + 2a,). For c3,4 non-negligible, a similar argument shows that the decreasing sigmoid takes over from the exponential increase of L(t) at (t2, 3, 4 - 2rr,, 3, ,J and L(t) becomes negligible for f > (b,3,4+2a2,3,4). Now let us apply the limiting forms (35) and (37) for L(t) to obtain approximate values for the constants C, and C,, in B*(t) = (2/T’) e”‘{C,+2Re[C,, e(iwo-a)t] +o(e-“‘)} t > T,. (38) Let us assume first that rr3,4 is small so that we can use (35) for L(t). Then using (30)

If gz were negligible, the second term inside the bracket of the integrand would jump abruptly from zero to Zz at (f’-t3,J = t,. For non-negligible CJ~,the integral gains a negative contribution from the region where (t’- t,,,) is to the left? of 1, which is made up by an approximately equal positive contribution for t’ to the right of t,. Therefore, even for non-negligible g‘2 we have C I z ctZ,(l-I,)-’ emat3+V2. (40) Now in obtaining (39) we assumed c3, 4 was negligible, but a similar argument to that made above for c’z shows that C, gains and loses equal contributions as 03,4 becomes non-negligible. We may therefore use the approximation (40) for C, with realistic gi. We may use (37) to obtain for C,, C,, z [~/(a-/I)] e-Bm~4[e(a-B)f*- l]/(eazz- 1) for o,,,,~/T~ < < 1 (41) where p 3 (a+iw,-A).

The integral defining C,, is a complex quantity. It would be difficult therefore to apply such arguments as were used for C, to see how nonnegligible ci affect C,,. In a numerical example considered later however we will see that C,, for realistic (TVdoes not change very markedly from its limiting value. Before leaving the birthrate we will show how the renewal equations give a birthrate consistent with expectations in the deterministic case. This is t The upper limit for which the second term in the bracket equals Z,/2 in magnitude is actually at a point tz = tz - aoi which is slightly to the left of tz. C, is therefore slightly less than the estimafe (40).

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the case? in which the cells spend a fixed time in each state so that the fi(t) become J(t--tJ. In this case, f(t) becomes s(t-T,) and (13) can be written B(t) = 2&t-T,)+2B(t-7”) t > 0, (4’) = 1 t = 0, = 0 t < 0. The right hand side of (42) may be iterated again and again to give B(t) = -f 2”%@--,T,) n=O

t20

(43)

0, = 0, which may be inserted into (13) to finally produce B*(t) = g 2”+9(t--nT,) n=O

t20

(44)

fJ, = 0.

4. Percentage Labeled Mitosis The quantity which is most conveniently counted during labeling experiments is the fraction of mitotic figures labeled. Let M*(t) denote the number of mitotic cells labeled, and M(t) denote the number of unlabeled mitotic cells a time t after labeling. Then the labeling index or fraction of mitotic cells labeled is M*/(M + M*). We will calculate the time development of the labeling index for the case in which the cells are in asynchronous log phase growth at the time of labeling. Asynchronous log phase growth is characterized by asymptotic age-in-state distributions, or equivalently asymptotic waiting time distributions (equation (30)) and by asymptotic fractions in each of the states which we denote by nilPI, for the ith state. We will first use the following argument to obtain the fraction-in-state in a form somewhat more precise than the approximation usually given. The population of a clone proliferating in log-phase for a time t is proportional to ecrr.If the cells which are at the beginning of the cycle have been proliferating in log-phase for a time t, then the cells which have spent a time _t in the cycle since their last division have been proliferating on the average for a time (t-i). Hence the number of cells of ages between t and t+d_t divided by the total number of cells is proportional to e-“!dt. Now we note that cells of age _tin our model may be in any of the four states. If we integrate our fractional age density, e-@ d_t multiplied by the probability that a cell has not reached state i by time _t over all t, then we shall have t Von Foerster (1959) has called such a population

“equivivant”.

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the mean value of the asymptotic fraction of all the cells which are in all the states preceding i (designated by the subscript bi) n,i/PI,

= C, 1 e-@[l -F&)]3

d_ts

(45) where C,, is the normalization constant for the age density and Fbi(t) is the probability that a cell takes a time longer than t to traverse all the states before i. Now if we similarly define the asymptotic fraction of cells that have not entered the states beyond i as n,(i+ r,/Pl, then nilPlm=

(46)

nb(i+l~IP~m-nbilP~m~

or (47) where fbi(f) is the density for the sum of the transit times for the states preceding i. It is given by a convolution over the appropriate densities. We have integrated by parts to obtain (47) from (45) and (46). Note that

We may then rewrite the second integral in (47) as - CnJ ] e-a(r-“)f,(t- t’) [ema”&( dt’ d_t (4% 0i 0 I Since fi(t) is zero for negative argument we may change the upper limit of the t’ integral to co. We then make the transformation of variables (_t- t’)--+t” and change the order of integration and we finally obtain from (49) and (47) ~JPI,

= C, 7 e-“tfbi(t) dt I- 7 e-“‘j-i(t) [ z C, in,/P (un-normaliled)]. The constant C, is given by

dt’ , I

Cl ’ = t {ni/P (un-normalized)}

(50)

(51)

i=l

and is very nearly equal to 2.0 which is the value obtained from (51) when (50) is evaluated for ci + 0. In this limit we have from (50) i-l

ni!J,

= 2eyatbf(l -es=‘*), where tbi = C j=l which is the usual value obtained without the probabilistic argument Puck & Steffan, 1963). In terms of (47), equation (40) becomes C I z a2I,I, e -af~42(n2/P14))- l. tj,

(52) (e.g. (53)

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It is now relatively simple to obtain usable expressions for the labeling index in terms of the various quantities already considered. We express the numerator of the labeling index as the sum of two terms M*(t) = M;(t)+M,*(r).

(j-1)

M:(t) is defined as the average number of labeled mitotic cells which have still not divided t hours after labeling. M,*(t) is then the average number of labeled mitotic cells at t hours after labeling which have experienced at least one division by that time. If a given cell is in synthesisat the time of labeling, then the probability that it will arrive at M a time t’ < t hours later, and not divide during the interval between t’ and t is

f’ I w2(r”)f3(r-t”) 0

dt”[l -F4(r-r’)]

(55)

We note thatfJt> for all i must be zero for negative argument, therefore the upper limit in (55) can be extended to (+ co). If we multiply (55) by the number of cells in S at the time of labeling and integrate over t’, we obtain

1

4 W2(f”)f3(f’- ,“) dt” [l -F4(r- t’)] dt’. (56) 0 [0 where PO is the total number of cells alive at the time of labeling, and n,/PIL is the fraction of cells in state 2 at that time. We next observe that M,*(t) consistsof those labeled mitotic cells for which the sum of the number of hours past labeling until the most recent division plus the time it took from that division to reach state M is a time t” which is lessthan t. Then M;(t) = n~,P~&j

t”

1

[l -F4(tt”)] dt”, i B*(t’)&(t”t’) dt’ (57) [b wheref,,(t) is the probability density for the total time spent in the first three states which is given by a threefold convolution as in (48). While the above expressions are true for any initial distribution of the population, we will obtain explicit formulae for the labeling index for a logphase population. In this case the denominator of the labeling index is M,*(t) = .#‘j&,j

0

M(t) + M*(t) = n4/PlmP0 e”‘.

(58)

It is instructive to consider briefly the form of the labeled mitotic wave for a log-phase deterministic popu1ation.t For this case all the ei are zero, t At the time of writing this paper I received a preprint from E. Trucco and P. J. Brockwellin whichsome aspects of this problem,andin particularthe deterministiccase arediscussed. I would like to thank G. J. Dienes for helpful discussions of the deterministic aswell asotheraspectsof the problem.

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and therefore the asymptotic waiting time distribution becomes Wi( t) = CIe”‘/(e”” - 1) ti > t 2 0, (59) = 0 i = 1,2,3,4, otherwise, where we have used equations (26), (27) and the fact that [ 1-F,(t)] is a step function which changes discontinuously from one to zero at t = ti for ai = 0. Before proceeding let us ask what biological thought experiment this population represents. We can imagine that we have a population arising from a single cell with rri small enough so that after de-synchronization the waiting time distributions are represented to sufficient accuracy by (59) and the percent in state by (52). Then at the time of labeling, the rri all go to zero and the population proceeds deterministically. For such a population we obtain from (56) and (57) M,* = n,/PI,P, 0+0

j- Wz(t’-tf3) dt’ f--14

a3 = aq = 0,

(60)

a1 = a2 = a3 = a4 = 0.

(61)

and M,* = n,/PI,P, 0+0

i B*(t’-tr,,,,)

dt’

rztq

Notice that equation (60) holds even when aI and aZ are non-negligible. The deviation of this expression from the actual value of Mb* for realistic a3 and a4 is indicated by the dashed line in Fig. 1. If we use equation (59) with some care on the end points, (60) becomes M:(t) = C,{exp @[(t-la n f2) u 0] -expcc[((t-t,,, u 0) n tJ} (62) with C, = n2jPI,Po/(ea’z1). OZ-+O

The symbol (X u y) means the greater of x and y, and (x n u) means the lesser of x and JJ. We use equation (44) to write M,* similarly M,* = n,/PI,

f 2”‘-n~+‘;(t’)

tl=1

dt’,

(63)

t-PIT,

which becomes after using equation (37) for L(t), M,*(t)=C,

f 2”{expcc[(t--nT,+t,nt,,,,,-t,,,)uO] “=I --expcdt - nT, u h,,- b,,)) (64) If we evaluate M,* for t = nT,+ T, z < T, we find that the only non-zero term of (64) is C,2n(exp~C(~+t4nt2,3,4-t3.4)uOJ-exp~[(tut3,4-t,,4)nt2l).

(65)

484

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BRONK

We note that because we have assumed a log-phase deterministic population distributed according to (59) and (52) that the denominator of the labeling n = 0,1,2,... index is given for t = nT, + z by n,/P ,P,2” ear. 0 I T i T,. (66) a.4-+b A careful inspection of (62), (64) and (65) shows that the labeling index is in the form of a wave with period T,. If we use (52) and (64) together with (62) or (65) we find in terms of our reduced time z that the labeling index can be written M*/(M +M*) = [L, -L2] (eaf3/1 -e-“‘$ (67) where ewar T < t, L, = e-“‘3 t, 5 T < t,,, (68) ea(r2- r) t,,, 5 T, and

e -[II e-“‘J,’

L, =

T -C t3.4 t,,,~T~tt,,, t, 13 * 4Ir.

eauz- 7) We exhibit this deterministic

2

(69)

I I

labeling index in Fig. 2 where the t, are chosen

4

6

- 8

IO

12

i4

16

I8

20

T~rrse ihr)

FIG. 2. Labeling index for a deterministic but log-phase population. t3 = 1.5, t4 = 1 .O (hr); ~7~= az = ~7~= u4 = 0.0.

fl = 1.5, t2 = 6.0,

to simulate typical Chinese hamster cell parameters (; = 1.5, 6.0, 0.5, I-0). To the scale of the figure, the deviation of the deterministic labeling index from a perfect trapezoid is not visible. Notice that the labeling index in our example is zero until T = t, at which time it increases until T = 1,. 4. lt

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then remains constant and equal to one until r = t,,,. It then decreases and is again zero at r = t2, a ,4 when all the labeled cells have completed mitosis. If t4 were longer than t,, we would have a different situation. The labeling index would increase from z = t, until it reached a maximum value of (nh4\ me--bf2~3)at z = t2,3. Then when all the cells were in mitosis it would decrease slowly until z = t,,, at which time it would decrease more rapidly until it reached zero at r = t,, 3,4. We now proceed to obtain approximations more useful for comparison with experiment. First we note that if g3 and rr4 are small compared with T, then we may use equation (60) with W2(t) now given by (26) since rr2 # 0. Dividing equation (60) by expression (58) then gives the first peak of the labeling index. Such a numerical calculation has been made for a population with C.V. = 0.2 and a realistic distribution of the LT~.It is represented by the dashed line in Fig. 1. The solid line in Fig. 1 represents a numerical calculation of the labeling index for the same parameters where the model considered is the same, but the method of calculation is the age transfer method of Kemmey (see Bronk, Dienes & Paskin, 1968). This method is independent of the renewal equations and therefore gives us a good check on the validity of our approximations. One can see that while the first peak is approximately the same shape for moderate as for negligible c3 and oh, the non-zero g’3 and cr4 gives a curvature to the initial labeling index and lowers the peak by a couple of percent. We now rewrite M,*(t) in a form suitable for exhibiting the oscillatory behavior M:(f) = nJPI,P,

i jPfb4(t’)B*(t” - t’) dt’[l - F4(t - t”)] dt”. (70) 00 The factor (1 - F4) restricts the effective domain of t’ to values comparable to t, that is, large with respect to the cycle time. Because of the presence of &(t) however, the contribution from the t’ integral is non-negligible only for t’ < T,. Therefore, the argument of B* is effectively large and we may use (38) to evaluate (70). Proceeding in this manner we obtain, Mf(O = nJPI,P$/T,{(e

‘,4GJr+2Re(GIJII

@*lb>>.

(71)

Here Jr1 = 7 e-B”&Jt’) 0

dt’ [

1 - 1 em8’f4(t) dt , 0 I

and P = cw+icuo-l.

6’2)

486

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J, is similarly

defined except that p in J,, is replaced by CIin J, and therefore according to (50), (73) J, = Wh/f’~,. For times long compared with T,, the labeling index will be Mz/(M + M*) obtained from (71) and (58). The first term of this ratio gives the asymptotic labeling index. We use (73) to obtain M*/(M+M*)j, = n2/Plm(2/Tca)C,. (74) Using (40) and (50) to evaluate (74) we have M*/(M+M*)l, = (21,1, e-“‘3*4)(l,/rJ. (75) If (4.2 cx2/2) < < 1 the first factor in (75) is very close to one, and the asymptotic labeling index very closely approximates the ratio of the time in synthesis to the cycle time. It is interesting that this ratio comes out equal to the value one would estimate for nz/Pjao if one were to naively neglect to take into account doubling. It is important to note that if the state labeled is early in the cycle, then (75) gives us an asymptotic labeling fraction which is slightly less than the logphase percent in the labeled state. The reason for this is that part of the unlabeled population begins to double immediately, while there is a delay in the onset of doubling of the labeled population. Conversely, if the labeled state is quite late in the cycle and t, is small compared to T,, (75) predicts that the asymptotic labeled fraction is substantially greater than the original log-phase per cent in the labeled state. This is of course due to the fact that in this case the labeled cells double before most of the others. Taking the first factor of (75) as equal to one and comparing (75) with equation (52), it is easy to estimate that the asymptotic labeling index becomes about 40”;, less (greater) than the log-phase per cent when the transit time for the labeled state is short compared to T,, and the labeled state occurs near the beginning (end) of the cycle. Another important feature of (75) is that it is not affected by the rate of increase of the population, nor by a death rate which is uniform over the cycle.? This will be discussed below. Before turning to these modifications, we will evaluate the correction term in the labeling index which arises from the second term of (71). This is

~~~/~~,>~~~/~~,>~1~4/T,~~~C~~~J~~~’i”o~”~’lP1.

(76)

If we evaluate the magnitude of (76) at t = 0, we will have the zero intercept of the straight line portion of the logarithmic graph of the peaks. This can be seen in Fig. 3 where we show results obtained via the age transfer t It seems clear that this should be so since we are assuming that all cells are affected uniformly. It is useful however to see how this comes about from the equations, since “In mathematics, what appears ‘obvious’ is not always true” (C. Z. Bronk, personal communication).

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80

Time Ihr)

FIG. 3. Logarithmic plot of the magnitude of the difference between labeling index peaks and asymptotic value. t1 = 1.5, tz = 6.0, tB= 1.5, t4 = 1.0 (hr); 0 represents ~1 = 0.5, u2 = 1.28, ~3 = 0.5, o4 = 0.33, C.V. = 0.15; l represents g1 = 0.5, az = 1.84, CT3= 0.5, u4 = 0.33, C.V. 0.20.

method for our Chinese hamster cycle parameters with two different c.v.‘s. It is actually the magnitude of the difference of the peak from the asymptotic index which is plotted against the time in hours. The phase of (76) at t = 0 will give the time lag between the labeling time and the first peak, or between subsequent peaks and nT,, where n is an integer (this is approximate, since the period is slightly less than T,). We can obtain a simplified expression for the constant coefficient of the exponential in (76) when or -+ 0. Let us call this limiting value R. The meaning of R can be inferred from a series of plots as in Fig. 3 for smaller and smaller C.V. The zero intercepts would approach a limit which is given by the magnitude of R. To obtain R we use C,, as given by equation (41), CI = In 2/T,, o0 = (2n/T,), and J,,

=

e-~rl.2,3(1-e-br4).

We find R = (n,/PI,)-‘[2

In 2/nT,(a+io,)]

ea’4 sin (0,,f2/2) (1 -e-“‘4-i”0’4)* .,-ioo(ta+f2/2)

(77)

488

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BRONK

For t, < < T,, the (a+&,) in the denominator cancels the complex factor involving t, in the numerator so in that casethe time lag before the peak in the labeling index is (t3+tz/2), a reasonable result. We evaluated R for the Chinese hamster ti (1.5, 6.0, 1.5, 1*O)and obtained R = 0.596 e-4.99-oi (78) The limiting zero intercept given by the magnitude in (78) is within plotting error of the value we obtained graphically in Fig. 3 for C.V. = 2. The limiting value of the phase shift is just one-half T, which meansthat we get the first minimum at t = T,, and the second maximum, one half cycle, or five hours later. Now let us take our approximate formula in the form M*(t)/[M(t)

+M*(t)]

= M*/(M +M*)I,

+a e-” cos (o,t-a),

(79) and seefor realistic parameters how well it fits the labeling index we have calculated numerically by the completely different age transfer method. The first term in (79) we estimate from (75) and the footnote on page 479. The decay parameter A is obtained from (23). The constant a, we obtain from the zero intercept of Fig. 3. [Note that a turns out to be quite close to the limiting value given in equation (78).] The phase angle 6, we choose arbitrarily to agree with (78). In Fig. 1, formula (79) is shown with the actual constants used. The results obtained using the approximation (79) are shown by the solid dots and are in rather good agreement with the results of the more precise age transfer calculation which is shown by the solid line. Our estimate for 6 is apparently slightly larger than it should have been. 5. Some Modifications to the Model Thus far, we have restricted ourselves to the ideal situation in which no cells are removed from the cycle, and each cell that completes a cycle gives rise to precisely two new cells at the beginning of G. We will now consider the simplest generalizations in which these restrictions are relaxed. In case I we relax the first restriction by allowing cells to be removed from the cycle (e.g. by death) at a uniform rate throughout the cycle. In case2 we concentrate the change at the end of mitosis by allowing the cell (or other biological entity) to give rise to either more or lessthan two individuals at the end of a cycle. In case 1 let us consider a probability of death per unit time interval equal to Kdf, where K is constant throughout the cycle. The probability that a cell which enters a given state (i.e. we identified it as alive when it entered the state) and has been in that state a time r, is still alive at time t is eeKt. More familiarly, the fraction of cells alive t hours after initially counting them and excluding division is eeKt. Now this probability is

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independent of the specific transit time for a cell, therefore the number of cells doubling in the neighborhood of c hours after division is equal to f(t)eVKtdt. We therefore modify equations (7), (lo), (13) and (14) by replacingf(t) by f(t)e-“’ and L(t) by L(t)e+. Let us temporarily restrict ourselves to K sufficiently small that the total number of cells is still increasing in time. We can then assume that the cells eventually attain exponential growth, with a new growth rate g. In this case if we study equation (14) for t large enough so that B(t) is exponential we obtain 1 f e -(.+K)r)f(tf) &‘. -= (80) 2 i0 But this is the same equation we would have for c( if K were equal to zero. We therefore conclude that & =

X-K.

(81)

For case 2 we consider a situation where the number of new individuals produced per cycle is given by a fixed? integer, m. The only modification made in equation (6) and in the similar equation for G is that G2 in the integral is replaced by G”. Then it follows that in the right-hand side of the renewal equations (7), (lo), (13) and (14) the integer 2 is replaced by M. A further modification that may easily be included is that of the m cells produced, some may be removed, say by differentiation, before entering the next cycle. Suppose that each new cell produced has a probability p of remaining in the pool under consideration. This is easily accounted for by further replacing m in the renewal equations by mp. The consequences of this replacement are equally simple. Everywhere the constant 51appears, it is replaced by a new constant g. The constant & is obtained from u by replacing (In 2) everywhere it appears in the definition of u [equation [9)] by In (mp). In both cases, one may proceed to the asymptotic situation by the same reasoning as is used for the unmodified model. That is, we reason by the central limit theorem (see equations (15) to (20)) that the birth rate after several generations will have the same functional form for any regular f(t) with a given c and T,. We therefore solve the renewal equation exactly for f(t) of the form given in equation (20) (see for example, Appendix III. Bronk, Dienes & Paskin, 1968), and use this exact solution to obtain a general asymptotic result. In this manner one finds that the only modification to the birth rate B(t), is that CIis replaced by cror & as appropriate. Proceeding via the same reasoning we obtain B(t) in the form of equation (21) and use this function to obtain the new B*(t) as was done in equation (33). This new B* is of the same form but with CI replaced again by g or @as appropriate. ‘1 This integer could itself be a random variable.

490

1). V. BRONK

For case 1, the constants C, and C,, as defined in equations (33) and (38) are unchanged since the positive exponents in it from the birth rate is cancelled inside the integrals by the negative exponent multiplying L(r). The labeling index is unaffected in case 1 since the denominator as well as both terms of the numerator are all multiplied by the same exponential factor. If we study the fraction-in-state using the same argument as that given in the paragraph preceding equation (49, we find that the number of cells of age _tat time t would be proportional to e(ZmX)‘t--!)if one did not kill cells during the present cycle in which they are found. To account for the cells lost during the present cycle, we multiply by an additional factor e-” which causesK to drop out of the age-in-state density and the fractions-instate are thus left identical to those given in equations (50) and (52) with the unmodified LY.Before turning to case 2, we remark that interesting changes are to be expected when the death or removal rates from the various states are not equal. For case 2, the same reasoning as given in the above paragraph leads to changed fractions-in-state. The form is still that of equations (50) and (52), but LXis replaced by cc.The constant C, now becomes c, = mp/(mp- 1). =.

(8.2)

The fact that C, becomesnegative? if (mp) is lessthan one is not crucial. This is becausethe other term in (52) also becomesnegative when CI= In (HZ~) is negative. The meaning of g (or cl) negative is that the population is decreasing. If g is negative and sufficiently large in magnitude the kinematics may be described as a rapid depletion of the early states to a lower level followed by a similar depletion of the late states. This would be followed by further such ordered depletions until all the cells were gone. In order to approximate a steady condition for the asymptotic fraction-in-state the characteristic time for depletion (w - (l/g) for negative c() should be long compared to the cycle time T,. If one starts far from the steady condition, then one needs (-c( < 1.) [see equation (23)) to give sufficient time to approach a steady condition. We emphasize that the result for the fraction-in-state for case 2 is quite different from that of case 1, with a pile-up of cells in M being a striking characteristic of case 2 for a decreasing population. In view of the above changes in fractions-in-state, it is remarkable that there is rather little effect of a case 2 modification on the labeling index. In ‘t The ideathat the formalismcould be pushedthrough in this casewassuggested to the author by G. J. Dienes.Although C, has a singularity when mp= I, the other factor has a zero so that the product is easilzhown to be continuous at this point.

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fact, we will first show that the asymptotic labeling index is invariant for a wide range of g = ln(mp). First, we note using equations (21) and the modified renewal equations that the factor 2 outside the bracket in equation (71) is now replaced by (mp). Dividing the modified first term of (71) by the modified (58) we find, again using the arguments that led to equations (39), (40), (53) and (73) that M*/(M+M*)l, = [(mp)l,l, e-E’3*4] (f2/Tr). (83) The factor Crz which replaced the 2 in equation (53) for C, was cancelled by = the occurrence of the same factor in the denominator of the modified Jr. Now for g202/2 < < ET,,

we find that Ii 25 e-2, and therefore M*/(M+M*)\,

% t,/T,

for In (mp) < < -2-

(84)

a(c.v.)’

Reviewing the origin of the oscillatory term in (71) it is seen that o0 (see equations (22) and (23)) changes very slowly as a function of g, and A not at all. The only change which may occur for the oscillatory term of the labeling index is that the constant C,, changes for case 2. Finally, examining equation (56) we see that a can only affect the initial labeled mitotic wave by the change in form of W=(t). This change is not very drastic and we find that in general, the labeling index does not depend very strongly on the multiplication rate. It is finally remarked that the invariance of the results to the modifications we have considered depends on the implicit assumption that the differentiated or dead cells are experimentally distinguished from those of the original populations. If this is not true, these same results imply that the graphs will be altered by a reduction of the labeling index in a way which depends on the number of altered cells which are counted.

ALPEN,

REFERENCES M. E. (1966). U.S. Naval Radiological San Francisco, California.

Defense Lab. Report,

E. L. & JOHNSTON,

USNRDL-TR-1104,

BRONK, B., DIENES, G. J. & PASKIN, FELLER, W. (1966). “An Introduction

A. (1968).

Biophys.

.7. 8, 1353.

to Probability Theory and Its Applications”,

Wiley. HARRIS, T. E. (1959). In “The Kinetics of Cellular Proliferation”, F., Jr., ed.). New York: Grune and Stratton, Inc.

Vol. II,

London:

HUGHES, W. C., BOND, & SHERMAN, F. G.

V. P., BRECHER,

G., CRONKITE,

pp. 368-381 (Stohlman,

E. P., PAINTER,

(1958). Proc. natn. Acad. Sci. U.S.A. 44,476.

E. P., QUASTLER,

H.

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A., BRONK, B. & DIENES, G. J. (1968). Brookhaven Symp. Biol., Number 20, pp. 169-178. PUCK, T. T. & STEFFEN, J. (1963). Biophys. J. 3, 379. QUASTLER, H. & SHERMAN, F. G. (1959). Expl. Cell Res. 17,420. TRUCCO, E. (1965). Bull. math. Biophys. 27. TRUCCO, E. & BROCKWELL, P. J. (1968). “Percent Labeled Mitosis Curves in Exponentially Growing Cell Populations” (Preprint). VON FOERSTER, H. (1959). In “The Kinetics of Cellular Proliferation”, pp. 382-407. (Stohlman, F., Jr., ed.). New York: Grune and Stratton, Inc. PASKIN,