J. theor. Biol. (1977) 69, 429-445
Cell Kinetics: A Model of Stimulated
Populations
V. DE MAERTELAER Groupe de Physique, Institut de Recherche Interdisciplinaire en Biologie Humaine et Nucleaire (L.M.N.) et Laboratoire de Statistique MPdicale, Fact&P de MPdecine, U.L.B., Bd de Waterloo no 115, B- 1000 Brussels, Belgium AND P. GALAND Unite de Biologie, Institut de Recherche Interdisciplinaire en Biologie Humaine et Nucleaire (L.M.N.), Fact& de Medecine, U.L.B., Bd de Waterloo n* 115, B- 1000 Brussels, Belgium (ReceiLled 30 May 1977) In the course of developing models of the kinetics of cell proliferation and differentiation, we attempted to determine the mathematical relationship between the responsiveness of cells towards a differentiation stimulus and their age, i.e. the time elapsed since their last division. The experimental model was oestrogen induced cornification of the vaginal epithelium in castrated mice. Under hormonal stimulation, the migration process of basal cells is widely intensified, resulting in the multiplication of the epithelial cell layers and the cornification of the tissue. Migration of cells is the first easily detectable manifestation of the differentiation process. The responsiveness of a stimulated basal cell to a differentiation stimulus may thus be expressed by its susceptibility to migrate which we quantitatively express by the migration rate of that cell. Since this is not experimentally measurable, it is evaluated by an indirect approach involving the following steps : (I) An experiment gives access to a set of points { Yjob”(a),j == 1, . . . , n} of a function Y(u) which indirectly depends on the relationship between the migration rate and cell age. (2) Various kinds of functions {PLc-D(a), i = 1, . . , m} are successively assumed for describing the migration rate of “a”-aged G cells. (3) For each of those m hypothesized functions, Y(a) is calculated and then confronted with the experimental data { YJob’(u)). This procedure has led us to exclude all but one of the hypothesized migration rate functions. Specifically, a progressive increase of the cell susceptibility to migrate with cell age is not likely to occur. The only kind of function that 429
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gave rise to theoretical values compatible with the experimental results suggests that the cell susceptibility to migrate enhances very slightly with cell age until a “critical point” after which it increases abruptly. Biological implications are discussed. 1. Introduction A very exciting biological problem concerns the mechanisms by which cell population kinetics are controlled. An important step in the understanding of these mechanisms is to unravel the sequence of events that constitute the response of a cell population to a mitogenic agent. This approach could reveal the existence of critical stages, corresponding to rate-limiting steps that are influenced by control mechanisms. The proliferative activity of several cell populations increases when exposed to the proper stimulus. This is true for continuously dividing populations, or for populations that display a very low level of proliferating activity under ordinary conditions (Baserga, 1968). The application of the stimulus is followed by a time-interval which is about 6-9 h in the vaginal epithelium of castrated mice (a renewing system) stimulated by oestradiol (Biggers & Claringbold, 1955; Perrotta, 1962; Beato, Lederer, Boquot & Sandritter, 1968) and which varies from 12 h to 4 days in most mammalian populations that ordinarily proliferate slowly (Baserga, 1968). After that lag, a burst of DNA synthetic activity appears, which is followed by a wave of mitoses. In some cell populations, an increase in cell loss from the germinative compartment is observed following stimulation of their proliferative activity. This increase appears, in pluristratified layers tissues, as a rapid movement of cells from the basal layer (germinative population), toward the diffcrentiating compartment, which could result in a transient hyperplasia of the tissue. This phenomenon has been observed, among others, by Barker & Walker (1966) on the vaginal epithelium in mice stimulated by oestrogen, by Lessard, Wolff & Winkelmann (1968) in stripped-guinea-pig epidermis, by Potten & Allen (1975) in mouse dorsal epidermal cells stimulated by removal (plucking) of cornified cells. The experimental system considered was the oestrogen-induced cornification of the vaginal epithelium of castrated mice. When hormone deprived, this epithelium consists of a basal and a superficial cell layer (see Fig. I) (Biggers & Claringbold, 1955; Green, 1959; Husbands & Walker, 1963; Galand, Leroy & ChrCtien, 1971). The germinative compartment (from which any cell of the epithelium derives by mitotic division), is represented by the basal layer only. Under normal conditions, a steady-state number of basal cells is ensured by a more or less accurate balance between cell production by mitosis and cell loss by upwards migration into the superficial layer.
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FIG. 1. Schematic representation of (a) the vaginal epithelium in castrated mice, (b) the vaginal epithelium in castrated mice 12 h after oestradiol injection. S = superficial layer; B = Basal layer; Z = intermediate layer.
After ocstrogenic stimulation and after a lag, proliferative activity is increased, as manifested by a peak of mitotic activity (Perrotta, 1962: Galand et al., 1971). The increase in the number of cell layers observed 12 h after oestrogen injection, is attributable to a direct migration of pre-existing basalcells, i.e. cellsthat did not undergo a further mitosis sincethe stimulation (Galand & Vandenhende, 1973). These authors suggested that G,-cells preferentially leave the basal layer and that “the susceptibility of cells to respond to an oestrogen stimulation by moving to an intermediate position is greater for cells that are farther in G1”. The aim of this paper is to investigate the relationship between the age of a cell, i.e. the time elapsed since its last division, and its responsivenessto the differentiation stimulus (viewed as the “susceptibility, to migrate, which i?, quantitatively expressed by the migration rate). Experimentally, it is not possible to measurethis relationship directly. In other terms, the migration rate as a function of cell age is not directly accessible.Nevertheless, when a fraction of the basal cells is labelled with a pulse of [3H]thymidine prior to stimulation, it is possible to measure the proportion of labelled cells which contribute to the formation of the intermediate layer during the period between stimulation and observation. This proportion does not represent a migration rate for a given age value. Indeed, first it results from the contribution of a large cohort of cells with an age range equal to T, (width of the cohort of the previously labelled cells); secondly, it results from a migration process that takes place during a definite time-interval (the time z that elapsesfrom stimulation till observation). Our mathematical approach to the problem consisted in : (i) Stimulating the experimental results, hypothesizing different types of functions relating the cell migration rate to the cell age. (ii) Then comparing the values generated by this way to those observed experimentally. A two-phase model of the cell cycle was used (seeBurns & Tannock, 1970; Smith & Martin, 1973; de Maertelaer & Galand, 1975). (a) The intermitotic phaseconsistsof a G-phaseand a C-phase (seeFig. 2). The C-phaseincludes the usual S, G, and M phasesas defined by Howard &
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-1
FIG. 2. Diagram summarizing hypothetical sub-phase is noted
the mean features of the model used for steady-state. by *. D = differentiation compartment.
A
Pelt (1953). In some cell populations, each cell must pass through a necessary part of G, before entering S-phase and/or just after having completed mitosis. (b) The cells leave G-phase randomly (in accordance with Lajtha’s (1966) concept of G,) to enter either C-phase with a constant rate, k,, or the differentiation compartment (here the superficial layers) with a constant rate, k L. The biological bases for this kind of model have been widely considered by Burns & Tannock (1970) and Smith & Martin (1973). In Burns & Tannock’s original model, it was assumed, moreover, that C-phase duration is a constant. This restrictive hypothesis is not needed here. Unstimulated basal cells are in steady state (cell production being compensated by cell loss). The rates of cell exit (k, and kL) from G-phase are constant. The initial conditions of the system are thus described by k, = k,. We have investigated here how they must be changed in order to describe the kinetic behavior of a stimulated cell population. 2. Material
and Methods
Since the migration rate as a function of cell age cannot be measured directly, we referred to an experimental procedure which provides results that indirectly depend on that relationship. The following experimental scheme (see Fig. 3) was used (for further details, see Galand & Rognoni, 1975). Castrated mice were given a single intraperitoneal (i.p.) injection of [jH]thymidine. (This pulse labelling distinguishes the cells that were in S-phase at labelling time.) Then, after different times ranging between
CELL (a)
Time
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of lobelling
7T-l-~~~,,,, ..~~.--Is1 cycle
TCI --~--. 2nd cycle
(b)
Tune
of stlmulatlcn
(! ~0)
(c)
Time
of oSservatio1
Ii =Ti
-
nI.
Intermedlote
-.--.-i r--.. ...___ --.-‘i‘---CA
layer
LF .I
FIG. 3. Schematic representation of the labelled cells position in the cell cycle, (a) when tritiated thymidine is injected, (b) at the time of stimulation, (c) at the time of observation. Labelled cells are represented by stippling. The age-density function of cells is visualized b:y the height of the cell-cohorts.
(‘&,, + TM) and (TG2 + TM+ O-r), randomly selected animals were given 1 /lg-178 oestradiol administered i.p. (TG2 and TM are respectively the mean duration of G,-phase and of M-phase and 6’ is the minimum cell cycle duration.) Each animal was killed at time z (here 12 h) after oestradiol injection. The vagina was dissected and treated for autoradiography. z was greater than T,-mean duration of S-phase, and, for mathematical convlenience, was chosen less than w-the minimum time needed by G-stimulated ~31s to reach the mitotic phase-so animals are killed before any significant change in proliferative activity under oestrogen action is detectable (Biggers 6~ Claringbold, 1955; Perrotta, 1962). The ratio of the number of labelled cells in intermediate layers to the total number of labelled cells in both intermediate and basal layers was then scored on each autoradiograph. The experimental results are shown in Fig. 7. 3. Mathematical
Model
As a result of stimulation, the cell flow rates out of G-phase are modified. Consequently, the age-density function (Trucco, 1965) of cells in G-phase becomes time-dependent. This is mathematically described by 2k,(t) N(t)&(a, t) where 2k,(t) N(r) is the generalized birth rate (Trucco, 1965). N(t) is the number of basal cells at time f; k,(t) is the rate of basal cell
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production at time f. This means that 2k,Nf,(a, t)da is the expected number of G-cells at time t, with age within the infinitesimal interval (a, a+&‘). If the time of application of the stimulus is referred to as f = 0, the agcdensity function of G-cells before and till stimulation is represented by 2/c,(O) N(O)f,(a, 0). The function &(a, 0) is a decreasing exponential since. as assumed by the modei, unstimulated cells leave G-phase randomly (see Introduction). Then f,(a,
0) = f,(o,
0) e-(kG+kr)o
wheref,(o,
= e-j“
o) = 1
p = k,-+k,. A useful property, since it makes the calculations easier, is that the birth rate remains constant between t = 0 till t >, w, where w represents the first time labelled cells that were in G at t = 0, reach mitosis. The time interval (0, w) is the minimum time needed for the stimulus being actually “received” and translated by the cells, phs the C-phase duration. Since in this paper, the age-density function of cells in G-phase is only considered for time values between [0, t], with t equal to 12 h and o > 12 h, the birth rate is a constant (that we designate by 2k,N). Before stimulation, the cell flows from G-phase towards C and D are constant. Following application of the hormonal stimulus: the flow rate of upwards migration of G-cells (towards 0) becomes an increasing function of cell age “a” (Galand & Vandenhende, 1973) and is assumedto vary with time (there is no argument, apriuri, supporting the contrary). It is designated by P,,,(u, t) (see Fig. 4).
G-Phase
FIG. 4. Diagram summarizing the mean features of the model used for stimulated cell populations. The validity of this model covers, at least, the period between the application of the stimulus (here oestrogen injection) and the time when stimulated G-cells reach mitosis. D = differentiation compartment.
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So, we introduced a “generalized migration rate” depending on age and time, and that we designate by P +&, t). The quantity PG-rD(u, t)2k,Nf,(a, t) da represents the migration rate of cells of age within the infinitesimal interval (a, u+da), at time t. The flow of G-cells towards C-phase is also affected. Indeed, after a latent period of about 12 h, the mitotic rate increases markedly until about 30 h after application of the stimulus (Biggers & Claringbold, 1955; Perrotta, 1962; Beato et al., 1968). Similarly as for upwards migration, we introduced the function PC&u, t) that we call the “generalized flow rate from G to C”. P(u, t) is defined as the “generalized flow rate out of G-phase”. Since we are searching the dependency between the generalized migration rate and cell age, but not between the generalized migration rate and time, and since no experimental information is available about the time evolution of P,,,(u, t), we substituted that generalized rate by its mean over the time interval [0, T]: p,,,(u)
i P,-Au,
= o-rm___--.--&u,
where P c+D(u) is the mean migration P,,,,(u), defined as:
Of, df
- ,
(1)
t> dt
rate of “a” aged G-cells. Similarly.
is the mean flow rate towards C of “a”-aged G cells, and P(u), defined as:
(3)
is the mean global flow rate out of G-phase of “&-aged
4. Mathematical
cells.
Formulation
The experiment described under section 2 provides us with the value of the ratio Y(u) of labelled intermediate cells to all labelled cells, for some particular positions of the labelled cells cohort in the cycle. The analytical form of this ratio can be derived as follows.
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(A)
P.
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NUMERATOR
The density of “a’‘-aged cells leaving G towards D at time t, is P,,,(a) . 2k,Nf,(a, t). Then, if (a’-z) is the age of a infinitesimal cell density at t = 0, the contribution of that infinitesimal cell density to the intermediate layer during the time interval [0, z] is: 2k,N~P,,,(a’-Tft’)
.f,(a’-Tft’,
t’) dt’.
(4)
0
Considering the labelled cells exclusively, a’ belongs to the restricted interval (a-T,, a) where “a” is the age at observation time of the oldest labelled cells, and a > z. Thus, among all initially labelled cells the number that have reached the intermediate layer between t = 0 and t = z is 2k,N
i da’jdt’ a-Ts
P,,,(a’-zft’)
.f,(a’-T+t’,
t’)
0
(5)
Note that, if z < a < r + r,, equation (5) remains valid with the further condition that &(a - z + t’, t’j = 0 when a’ - T+ t’ < 0. (B)
DENOMINATOR
The denominator, i.e. the total number of labelled cells at time z, is calculated as follows :
In this case, all labelled cells have divided once since their labelling. Then, the number of labelled cells at t = z is equal to the number of labelled basal cells at t = 0, i.e. the number (2k,NTs) of labelled cells that have entered G before t = 0, minus half the number of labelled cells that left G before stimulation. The latter number represents the labelled cells that joined the superficial layer under unstimulated conditions. [They do not appear in the counting of the experimental ratios Y(a).] The number of labelled cells at t = z is thus: 2k,NTs-3
(
(I--c
2k,NT,-
j
ll-T$--r ePTs - 1
P
>*
2k,N empa’ da’ >
(6)
In this case, the number of labelled basal cells at t = 0 is equal to the number of labelled cells that entered G before that time, minus half the number of those cells that already left G at t = 0, plus the number of labelled
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cells that have not yet divided at t = 0, i.e. (1-r 2k,N(a-2)-s 2k,N(a-z)j 2k,N e-““’
da’
0
>
+k,N(T,-a
+T)
= k,N T,+1-e-“(‘-‘) . (7) P > The number of labelled cells at observation time, is the number of labelled basal cells at t = 0 (equation 7) plus the number of labelled cells that were in the first cycle at t = 0 but that divided during (0, T), i.e. forr
2Ts-a+z+1-e-“‘“-” P (C)
< a < T,+?.
> RATIO
Y(U)
Y(u) is equal to the ratio of equation (5) to equation 20-irS da’i
dt’PG+Ja’--r+t’)
Y(a) = T,+e -da-r)
2 =
i da’ j dt’P,,,(a’--z+t’) -.-__a-Ts O 2T s -a+z+lk!?“‘“-‘)
(6) i.e.
.&(a’--z+t’,t’) (9
epTS- 1 --.-. _ P
Y(a) is equal to the ratio of equation (5) to equation
y(a)
(8)
(S), i.e.
.fJu’--r+t’,t’) (l(J) P
withf,(a’--t+t’,t’) = 0 when a’--t+t’ < 0. Note that when the function P c+D(u) has a breakpoint, those equations are still valid for “a”, less or equal to the abscissa value (designated by “aC”) of that breakpoint. But for a > a,, the numerator of Y(a) is described by a much longer expression, unsuitable for reproduction.
5. Mean Migration
Rate Functions
We have envisaged some a priori possible biological mechanisms, each giiving rise to a different relationship between the mean migration rate and cell age. For each of these assumed PiG.+D(u), we simulated (see section 7) the
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corresponding Y(u) function that we confrontated with the experimental data. The following cases were considered : (1) The biochemical and morphological changes that the cell undergoes when sejourning in G-phase do not influence its susceptibility to respond to the stimulus. In this case, the P(u) function is (as illustrated in Fig. 5) an horizontal straight line, i.e. P&(u)
= PO
(PO > 0)
(2) Responsiveness increases with cell age (without any point of inflexion) consequently to continuous metabolic changes. These might be linked, for example to the continuous accumulation (destruction) during G-phase of some substance (s) promoting (inhibiting) the cell response to a stimulating agent, or to changes in cell membrane with age. Decreasing migration rates with cell age were not envisaged here since radically contradictory to experimental results (Galand & Vandenhende, 1973).
I IQ
/
I
30
I
I
1
50
a : Ccl I age (hours1 FIG. 5. The mean migration rate function P c..D(u) is assumed to be constant or to increase with cell age, without any breakpoint. Curve (a) (- - -): Pa-&z) = O-0333 h-l, curve (b) (- - -): PG.+&) = (0GO67+04003a) h-l, curve (c) (- - -): PG+&) = 0+067 exp (0.02~) h-l. Corresponding to each of those mean migration rate functions, curves (-) are the simulated proportions Y(u) of the labelled intermediate cells, among all labelled cells. They are functions of the age “a” of the oldest labelled cells at observation time,
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In this case, two possibilities were considered (see Fig. 5): (a) P,,,(a)
increaseslinearly with age: P&(a)
(b) P,,,(u)
= PO+va
(I’ > 0)
increasesexponentially with age P:,,(a)
:= POera
(3) If G cells, are either not sensitive to the stimulating agent or cannot respond to it, unless they have passeda specific control point (for example thle completion of some metabolic rate-limiting step), the Pc.+Ju) function has a breakpoint corresponding to a cell age value called “q”. Under this point. G-cells do not respond to the stimulating agent. Nevertheless, in order to take into account that those cells can migrate like unstimulated cells, P(u) can be different from 0 for a < u,. (In the present paper, the irltercellular variability on the ~1, value is neglected.) We assumed (see Fig. 6) (a) P”(a) = PO for a Q a, = 03 for a > a, (b) P5(n) = P" for a d a, = PO+v(u-a,) for u > u,
FIG. 6. The mean migration rate function P o+D(a) is assumed to have a breakpoint in a =ma,. Curve a (- - -): PC+&) = 0.0333 h-l, a < 43 h; PC.-,(a) = cc, a :- 43 h. Curve b (- - -): PG,D(a) = OGO67 h-l, a < 43 h; P G-&z) r= OX)O67+O~Ol(a-43) h-‘, a > 43 h. Corresponding to each of those mean migration rate functions, curves (--I are the simulated proportions Y(u) of the labelled intermediate cells, among ail labelled cells. They are functions of the age “a” of the oldest labelled cells at observation time. 28 T.B.
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(4) For reasons that will appear clear later on, a variant of P’(a) was also introduced, i.e. P(u) = PO +a for a > a, = PO + vu, + v2(u - a,) for a > a,(~, > 0). 6. Age Density Functions
In equation (S), the age density function 2k,N .fG(u, t) of G-cells needs still to be specified. The first derivative of &(a, t) with respect to cell age, taken along the pathway p = u-t = constant is
2k,N . df&, a-p) = -P(a) da
After integration
.2k,N
. f,(a, a - p)
(11)
of equation (11) from p to a:
- i P(a’) da’ (12) ( Cl--l > This equation is valid when a 3 t (p > 0). When p < 0, the knowledge of fc(u, t) is immediate since j&z, t) = j&z, a) for t > a and &(a, a) can be deduced from equation (12). No experimental clue suggests a particular relationship between the mean migration rate and the mean rate of cell flow from G -+ C. In this paper, we have assumed that the evolution of P c+c(a) with cell age is similar to that of P,,,(u), i.e. that: PC-*&) = ~.P,,,(u) (13) where I is a constant. The same simulations as those reported here, have been performed but with the inverse hypothesis, i.e. that if one of the functions P&a) or P,,,(u) is increasing with cell age, the other is decreasing. In that case, the same conclusions as those presented in this paper, have been reached. At this stage, equation (12) permits to derive the analytical forms of the &(a, t) corresponding to the assumed P c+D(u) functions. Those analytical forms are listed in the Appendix A. f,(a,
t) = f,(a-
t, 0) . exp
7. Results
The evolution of Y (the proportion of labelled intermediate cells among all labelled cells at observation time) with “a” (the age of the oldest labelled cells at observation time) has been simulated according to different
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441
hypotheses about P,,,(u) (th e migration rate of “a’‘-aged cells averaged on the time interval z between stimulus injection and observation). Y(a) depends on at least five parameters: T,, p, z, PO, ,I. Depending on the f’orm assumed for Pc_,D(a), parameters v, a,, and/or v2 may also intervene. T, is available from various kinds of experimental data including the classical fraction of labelled mitoses (FLM) curve (Quastler & Sherman, 1959) and the double labelling method (Wimbler & Quastler, 1963). p can be evaluated from the FLM curves (de Maertelaer & Galand, 1975). In the simulations presented, Ts has been taken as 9 h and p as 0.01 h. T (an experimental condition) was 12 h. A has been arbitrarily taken as 0.5. The value of this parameter does not affect the shape of the Y(u) curves. For convenience in drawing the figures, the values for I” and v have been selected as to giving raise to Y(a) functions that covered more or less the same range of values. The curves Y(a) shown in Fig. 5 have been simulated by assuming P,,,(a) = O-0333 h-’ for curve (a), P,++D(u) = (0~0067+0NMI3u) h-’ for curve (b), and P,,,(a) = 0.0067 e:0’02ah-’ for curve (c). The curves Y(u) shown in Fig. 6 have been simulated under the assumption that Pc+D(u) = P” up to a = a, [a, = 43 h; P” = 0.0333 h-’ for curve (a)
o= Cell
age
(hours1
The mean migration rate function P o+D(u) (- - -) is PG+D(u) = (84067 a) h-l, a < 43 h; P,,,(a) = [0~021O+OX067) (u-43)] h-l, a > 43 II. Curve (-) is the corresponding simulated proportion Y(u) of the labelled intermediate cells, among all labelled cells, as a function of the age “a” of the oldest labelled cells at observation time. 0 = experimental points. FIG. +04003
7.
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GALANI)
and P” = OW67 h-r for curve (b)]. After the breakpoint PG+D(u) = a~ [curve (a)] and Pc_,D(a) = [0~0067+0~01(u-43)] h-’ for curve (b). In Fig. 7, PC+&) = (0~0067+0~0003u) h-’ up to a, = 43 h, and P,,&) = [0.0210 +OM67(u-Us)] h-’ for a > a,.
8. Discussion and Conclusions In the mouse vaginal epithelium stimulated by oestrogen a rapid migratory movement of cells from the basal layer is observed (Galand et al., 1971). Consequently, a new cell layer can be observed (the intermediate layer). This process starts 12 hours after oestrogen administration before any noticeable change in proliferative activity. The simulated curves represented by solid lines in Figs 5, 6 and 7 are the theoretical proportions Y(u) of labelled intermediate cells among all labelled cells at time T after stimulation as a function of the age of the oldest labelled cells at observation time. These curves have been calculated by assuming different kinds of functions for the mean cell migration rate of “Z-aged G-cells. Those mean cell migration rate functions P,,n(u) are represented in Figs 5, 6, 7 by broken lines. The shape of the curves simulated under the assumption that PC,,(u) increases continuously with cell age (Fig. 5) is clearly incompatible with the experimental data represented by solid circles in Fig. 7. Scanning the space of the parameters of Y(u) reveals that this incompatibility is general, i.e. not linked to a particular set of values of the parameters. The rise in the Y(u) function always precedes by far that in the experimental data. Contrarily when P,,,(a) is a biphasic function, as represented in Figs 6 and 7, the Y(a) simulated curves rise after a lag, as in the experimental data. The Y(a) curve shown in Fig. 7 satisfactorily fits the experimental data. It has been obtained by assuming a very slight increase of P,,.(a) up to a = a,, followed by an abrupt increase of this function. It seems that no obvious biological implications can be drawn from the fact that the simulated Y(a) function is more convenient when P,,,(u) increases very slightly before a = a, than when PG_n(u) is a constant till c1 = a,. If this increase is real, it is, in any case, nearly imperceptible and could be due to the fact that we neglected the variability on the uE value. Our results suggest that the cells have to pass a certain time in the G,phase before being fully sensitive to the stimulating agent. The existence of such a cell-age delay before the cell responds to the differentiation stimulus (the response being here an upward migration) suggests that the cell must either perform biochemical process(es) or accumulate a critical amount of some substance(s) in order to be able to respond to the stimulus. This emphasizes the possible existence of a critical step in the G,-phase of the
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KINETICS
cell cycle with respect to the responsiveness to the induced differentiation process. This emphasizes also that G,-cells are non-equivalent with respect lo the extracellular signal. If the G, cells are really G-cells, and unless the mechanisms of the response to extracellular and intracellular mediators are different, the G-cells will also be non-equivalent with regard to the intracellular signals that govern the cell progression in the cycle, namely the signal(s) for entry into C-phase and the signal(s) for entry into Dcompartment. In that case, this non-equivalence of G-cells would indicate that the concept of “undeterministic” G-phase does not hold true, at least concerning the differentiation pathway. However the alternative hypothesis e>.ists that, before the critical step, G, cells are not in G-phase, but still in the postmitotic part of C-phase (see Fig. 2). In that case, the stimulus would act specifically on G-cells by increasing the migration rate. This would suggest an analogy between the response of stimulated cells towards their differentiative and proliferative activities. Indeed, there are indications (Brooks, 1975, 1976) that G-cells respond to a stimulus by increasing their transition probability (from G to C), i.e. by increasing the rate of cell flow from G to c’. We are greatly indebted to Dr J. Otten for its constructive criticism of the manuscript and to Dr E. &hell-Frederick for having corrected the English text. We would also like to thank Mrs D. Leemans for typing of the manuscript. This work was performed under Contract of the “Minis&e de la Politique Scientifique” within the framework of the Association Euratom-Universities of Pisa and Brussels. P.G. is a Maitre de Recherches of the Fonds National de la Recherche Scientique.
REFERENCES BASERGA, R. (1968). Cell tiss. Kinet. 1, 167. BARKER, T. E. & WALKER, B. E. (1966). Anat. Rec. 154, 149. BEATO, M., LEDERER, B., BOQUOT, E. & SANDRITTER, W. (1968). Expl. Cell BAGGERS, J. D. & CLARINGBOLD, P. J. (1955). J. Anat. 89, 124. BROOKS, R. F. (1975). J. Cell. Physiol. 86, 369. BROOKS, R. F. (1976). Nature 260, 248. BURNS, F. J. & TANNOCK, I. F. (1970). Cell. riss. Kinet. 3, 321. DIE MAERTELAER, V. & GALAND, P. (1975). Cell tiss. Kinet. 8, 11. GALAND, P., LEROY, F. & CHRETIEN, J. (1971). J. Endocr. 49, 243. GALAND. P. & VANDENHENDE. J. (1973). Anat. Rec. 176.455. GALAND; P. & ROGNONI, J. B: (la75). Am. J. Anat. 144; 533. GREEN, J. A. (1959). Anat. Rec. 135, 247. HOWARD, A. & PELC, S. R. (1953).Heredity 6, 261. HUSBANDS, M. E., JR. & WALKER, B. E. (1963). Anat. Rec. 147, 187. LAJTHA, L. G. (1966).J. Cell Camp. Phys. 67 Suppl. 1, 133. LESSARD, R. J., WOLFF, K. & WINKELMANN, R. K. (1968). J. Invest. Dermat. PERROTTA, C. A. (1962). Am. J. Anat. 111, 195. POTTEN, C. S. & ALLEN, T. D. (1975). J. Cell Sci. 17,413.
Res. 52, 173.
50, 171.
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AND
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GALAEU‘D
QUASTLER, H. & SHERMAN, F. G. (1959). ExpI. Cell. Res. 17, 420. SMITH, J. A. & MARTIN, L. (1973). Proc. Natn. Acad. Sci. USA 70, 1263. TRUCCO, E. (1965). Bull. math. Biophys. 27, 285. WIMBER, D. E. & QUASTLER, H. (1963). Expl. Ceil Res. 30, 8.
APPENDIX
A
Analytical forms of the age-density functions corresponding to the migration rate functions assumed under section 5. Since a11 the agedensity functions considered in this paper can be derived fromfz(a, t) and fz(u, t), only these two functions are written out: fi(a,
t) = exp [-~(n-i)+$evu(e-”
-11,
j,!(u, t) = exp [-p(o--~)--P”lvoi+~],
cl 6 CIC 2
= exp
[
-~(a-t)-P”t-vaf+v~+~-~~(o,-a)2
I
,
a, < a < a,+f, 2
= exp
[
-~(n--?)--POf+(v,-v)a,f-v,af+~~
1
) a > a,+t.
APPENDIX
B
List of Symbols and Definitions (A)
VARIABLES
OF THE MODEL
When the basal cells are in an equilibrium-state k, = flow rate constant of cells from G-phase into C-phase. k, = migration rate constant of cells from G-phase into the upper layers of the tissue. p = k,+k,. k, = rate of cell production. 8 = minimum cell cycle duration. T, = duration of S-phase. TG2 = duration of G,-phase. TM = duration of M-phase.
CELL
fier a perturbation
445
KINETICS
of the equilibrium-state
of the tissue
P,,c(a,
t) = generalized flow rate at time t of “a’‘-aged G-cells into the C-phase. Pc+.(a, t) = generalized migration rate at time t of “a’‘-aged Gcells into the differentiation compartment. P(a, t) = generalized global flow rate out of G-phase (flow tn C+ migration to D) at time t of “a’‘-aged cells. PC&a) = mean flow rate of “a’‘-aged G-cells into the C-phase, averaged on the time r between the application of the stimulus and the observation time. PGdD(a) = mean migration rate of “a’‘-aged G-cells into the differentiation compartment, averaged on the time t between the application of the stimulus and the observation time. P(a) = mean global flow rate out of G-phase (flow to CS migration to 0) of “a’‘-aged cells averaged on the time t between the application of the stimulus and the observation time. 2k,N = birth rate of basal cells. 2k,N. &(a, t) = age-density function of ceils in G-phase. (B) VARIABLE
LINKED
TO THE EXPERIENCE
t = 0: time at which the stimulus is injected. z = time of observation (= time at which the animal is killed). Y(a) = proportion of labelled intermediate cells among alI labelled cells, as a function of the age of the oldest labelled cells at observation time.
Cell Kinetics: A Model of Stimulated
Populations
V. DE MAERTELAER Groupe de Physique, Institut de Recherche Interdisciplinaire en Biologie Humaine et Nucleaire (L.M.N.) et Laboratoire de Statistique MPdicale, Fact&P de MPdecine, U.L.B., Bd de Waterloo no 115, B- 1000 Brussels, Belgium AND P. GALAND Unite de Biologie, Institut de Recherche Interdisciplinaire en Biologie Humaine et Nucleaire (L.M.N.), Fact& de Medecine, U.L.B., Bd de Waterloo n* 115, B- 1000 Brussels, Belgium (ReceiLled 30 May 1977) In the course of developing models of the kinetics of cell proliferation and differentiation, we attempted to determine the mathematical relationship between the responsiveness of cells towards a differentiation stimulus and their age, i.e. the time elapsed since their last division. The experimental model was oestrogen induced cornification of the vaginal epithelium in castrated mice. Under hormonal stimulation, the migration process of basal cells is widely intensified, resulting in the multiplication of the epithelial cell layers and the cornification of the tissue. Migration of cells is the first easily detectable manifestation of the differentiation process. The responsiveness of a stimulated basal cell to a differentiation stimulus may thus be expressed by its susceptibility to migrate which we quantitatively express by the migration rate of that cell. Since this is not experimentally measurable, it is evaluated by an indirect approach involving the following steps : (I) An experiment gives access to a set of points { Yjob”(a),j == 1, . . . , n} of a function Y(u) which indirectly depends on the relationship between the migration rate and cell age. (2) Various kinds of functions {PLc-D(a), i = 1, . . , m} are successively assumed for describing the migration rate of “a”-aged G cells. (3) For each of those m hypothesized functions, Y(a) is calculated and then confronted with the experimental data { YJob’(u)). This procedure has led us to exclude all but one of the hypothesized migration rate functions. Specifically, a progressive increase of the cell susceptibility to migrate with cell age is not likely to occur. The only kind of function that 429
430
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gave rise to theoretical values compatible with the experimental results suggests that the cell susceptibility to migrate enhances very slightly with cell age until a “critical point” after which it increases abruptly. Biological implications are discussed. 1. Introduction A very exciting biological problem concerns the mechanisms by which cell population kinetics are controlled. An important step in the understanding of these mechanisms is to unravel the sequence of events that constitute the response of a cell population to a mitogenic agent. This approach could reveal the existence of critical stages, corresponding to rate-limiting steps that are influenced by control mechanisms. The proliferative activity of several cell populations increases when exposed to the proper stimulus. This is true for continuously dividing populations, or for populations that display a very low level of proliferating activity under ordinary conditions (Baserga, 1968). The application of the stimulus is followed by a time-interval which is about 6-9 h in the vaginal epithelium of castrated mice (a renewing system) stimulated by oestradiol (Biggers & Claringbold, 1955; Perrotta, 1962; Beato, Lederer, Boquot & Sandritter, 1968) and which varies from 12 h to 4 days in most mammalian populations that ordinarily proliferate slowly (Baserga, 1968). After that lag, a burst of DNA synthetic activity appears, which is followed by a wave of mitoses. In some cell populations, an increase in cell loss from the germinative compartment is observed following stimulation of their proliferative activity. This increase appears, in pluristratified layers tissues, as a rapid movement of cells from the basal layer (germinative population), toward the diffcrentiating compartment, which could result in a transient hyperplasia of the tissue. This phenomenon has been observed, among others, by Barker & Walker (1966) on the vaginal epithelium in mice stimulated by oestrogen, by Lessard, Wolff & Winkelmann (1968) in stripped-guinea-pig epidermis, by Potten & Allen (1975) in mouse dorsal epidermal cells stimulated by removal (plucking) of cornified cells. The experimental system considered was the oestrogen-induced cornification of the vaginal epithelium of castrated mice. When hormone deprived, this epithelium consists of a basal and a superficial cell layer (see Fig. I) (Biggers & Claringbold, 1955; Green, 1959; Husbands & Walker, 1963; Galand, Leroy & ChrCtien, 1971). The germinative compartment (from which any cell of the epithelium derives by mitotic division), is represented by the basal layer only. Under normal conditions, a steady-state number of basal cells is ensured by a more or less accurate balance between cell production by mitosis and cell loss by upwards migration into the superficial layer.
CELL
KINETICS
431
FIG. 1. Schematic representation of (a) the vaginal epithelium in castrated mice, (b) the vaginal epithelium in castrated mice 12 h after oestradiol injection. S = superficial layer; B = Basal layer; Z = intermediate layer.
After ocstrogenic stimulation and after a lag, proliferative activity is increased, as manifested by a peak of mitotic activity (Perrotta, 1962: Galand et al., 1971). The increase in the number of cell layers observed 12 h after oestrogen injection, is attributable to a direct migration of pre-existing basalcells, i.e. cellsthat did not undergo a further mitosis sincethe stimulation (Galand & Vandenhende, 1973). These authors suggested that G,-cells preferentially leave the basal layer and that “the susceptibility of cells to respond to an oestrogen stimulation by moving to an intermediate position is greater for cells that are farther in G1”. The aim of this paper is to investigate the relationship between the age of a cell, i.e. the time elapsed since its last division, and its responsivenessto the differentiation stimulus (viewed as the “susceptibility, to migrate, which i?, quantitatively expressed by the migration rate). Experimentally, it is not possible to measurethis relationship directly. In other terms, the migration rate as a function of cell age is not directly accessible.Nevertheless, when a fraction of the basal cells is labelled with a pulse of [3H]thymidine prior to stimulation, it is possible to measure the proportion of labelled cells which contribute to the formation of the intermediate layer during the period between stimulation and observation. This proportion does not represent a migration rate for a given age value. Indeed, first it results from the contribution of a large cohort of cells with an age range equal to T, (width of the cohort of the previously labelled cells); secondly, it results from a migration process that takes place during a definite time-interval (the time z that elapsesfrom stimulation till observation). Our mathematical approach to the problem consisted in : (i) Stimulating the experimental results, hypothesizing different types of functions relating the cell migration rate to the cell age. (ii) Then comparing the values generated by this way to those observed experimentally. A two-phase model of the cell cycle was used (seeBurns & Tannock, 1970; Smith & Martin, 1973; de Maertelaer & Galand, 1975). (a) The intermitotic phaseconsistsof a G-phaseand a C-phase (seeFig. 2). The C-phaseincludes the usual S, G, and M phasesas defined by Howard &
432
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-1
FIG. 2. Diagram summarizing hypothetical sub-phase is noted
the mean features of the model used for steady-state. by *. D = differentiation compartment.
A
Pelt (1953). In some cell populations, each cell must pass through a necessary part of G, before entering S-phase and/or just after having completed mitosis. (b) The cells leave G-phase randomly (in accordance with Lajtha’s (1966) concept of G,) to enter either C-phase with a constant rate, k,, or the differentiation compartment (here the superficial layers) with a constant rate, k L. The biological bases for this kind of model have been widely considered by Burns & Tannock (1970) and Smith & Martin (1973). In Burns & Tannock’s original model, it was assumed, moreover, that C-phase duration is a constant. This restrictive hypothesis is not needed here. Unstimulated basal cells are in steady state (cell production being compensated by cell loss). The rates of cell exit (k, and kL) from G-phase are constant. The initial conditions of the system are thus described by k, = k,. We have investigated here how they must be changed in order to describe the kinetic behavior of a stimulated cell population. 2. Material
and Methods
Since the migration rate as a function of cell age cannot be measured directly, we referred to an experimental procedure which provides results that indirectly depend on that relationship. The following experimental scheme (see Fig. 3) was used (for further details, see Galand & Rognoni, 1975). Castrated mice were given a single intraperitoneal (i.p.) injection of [jH]thymidine. (This pulse labelling distinguishes the cells that were in S-phase at labelling time.) Then, after different times ranging between
CELL (a)
Time
433
KINETICS
of lobelling
7T-l-~~~,,,, ..~~.--Is1 cycle
TCI --~--. 2nd cycle
(b)
Tune
of stlmulatlcn
(! ~0)
(c)
Time
of oSservatio1
Ii =Ti
-
nI.
Intermedlote
-.--.-i r--.. ...___ --.-‘i‘---CA
layer
LF .I
FIG. 3. Schematic representation of the labelled cells position in the cell cycle, (a) when tritiated thymidine is injected, (b) at the time of stimulation, (c) at the time of observation. Labelled cells are represented by stippling. The age-density function of cells is visualized b:y the height of the cell-cohorts.
(‘&,, + TM) and (TG2 + TM+ O-r), randomly selected animals were given 1 /lg-178 oestradiol administered i.p. (TG2 and TM are respectively the mean duration of G,-phase and of M-phase and 6’ is the minimum cell cycle duration.) Each animal was killed at time z (here 12 h) after oestradiol injection. The vagina was dissected and treated for autoradiography. z was greater than T,-mean duration of S-phase, and, for mathematical convlenience, was chosen less than w-the minimum time needed by G-stimulated ~31s to reach the mitotic phase-so animals are killed before any significant change in proliferative activity under oestrogen action is detectable (Biggers 6~ Claringbold, 1955; Perrotta, 1962). The ratio of the number of labelled cells in intermediate layers to the total number of labelled cells in both intermediate and basal layers was then scored on each autoradiograph. The experimental results are shown in Fig. 7. 3. Mathematical
Model
As a result of stimulation, the cell flow rates out of G-phase are modified. Consequently, the age-density function (Trucco, 1965) of cells in G-phase becomes time-dependent. This is mathematically described by 2k,(t) N(t)&(a, t) where 2k,(t) N(r) is the generalized birth rate (Trucco, 1965). N(t) is the number of basal cells at time f; k,(t) is the rate of basal cell
434
V.
DE
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AND
P.
GALAND
production at time f. This means that 2k,Nf,(a, t)da is the expected number of G-cells at time t, with age within the infinitesimal interval (a, a+&‘). If the time of application of the stimulus is referred to as f = 0, the agcdensity function of G-cells before and till stimulation is represented by 2/c,(O) N(O)f,(a, 0). The function &(a, 0) is a decreasing exponential since. as assumed by the modei, unstimulated cells leave G-phase randomly (see Introduction). Then f,(a,
0) = f,(o,
0) e-(kG+kr)o
wheref,(o,
= e-j“
o) = 1
p = k,-+k,. A useful property, since it makes the calculations easier, is that the birth rate remains constant between t = 0 till t >, w, where w represents the first time labelled cells that were in G at t = 0, reach mitosis. The time interval (0, w) is the minimum time needed for the stimulus being actually “received” and translated by the cells, phs the C-phase duration. Since in this paper, the age-density function of cells in G-phase is only considered for time values between [0, t], with t equal to 12 h and o > 12 h, the birth rate is a constant (that we designate by 2k,N). Before stimulation, the cell flows from G-phase towards C and D are constant. Following application of the hormonal stimulus: the flow rate of upwards migration of G-cells (towards 0) becomes an increasing function of cell age “a” (Galand & Vandenhende, 1973) and is assumedto vary with time (there is no argument, apriuri, supporting the contrary). It is designated by P,,,(u, t) (see Fig. 4).
G-Phase
FIG. 4. Diagram summarizing the mean features of the model used for stimulated cell populations. The validity of this model covers, at least, the period between the application of the stimulus (here oestrogen injection) and the time when stimulated G-cells reach mitosis. D = differentiation compartment.
CELL
435
KINETICS
So, we introduced a “generalized migration rate” depending on age and time, and that we designate by P +&, t). The quantity PG-rD(u, t)2k,Nf,(a, t) da represents the migration rate of cells of age within the infinitesimal interval (a, u+da), at time t. The flow of G-cells towards C-phase is also affected. Indeed, after a latent period of about 12 h, the mitotic rate increases markedly until about 30 h after application of the stimulus (Biggers & Claringbold, 1955; Perrotta, 1962; Beato et al., 1968). Similarly as for upwards migration, we introduced the function PC&u, t) that we call the “generalized flow rate from G to C”. P(u, t) is defined as the “generalized flow rate out of G-phase”. Since we are searching the dependency between the generalized migration rate and cell age, but not between the generalized migration rate and time, and since no experimental information is available about the time evolution of P,,,(u, t), we substituted that generalized rate by its mean over the time interval [0, T]: p,,,(u)
i P,-Au,
= o-rm___--.--&u,
where P c+D(u) is the mean migration P,,,,(u), defined as:
Of, df
- ,
(1)
t> dt
rate of “a” aged G-cells. Similarly.
is the mean flow rate towards C of “a”-aged G cells, and P(u), defined as:
(3)
is the mean global flow rate out of G-phase of “&-aged
4. Mathematical
cells.
Formulation
The experiment described under section 2 provides us with the value of the ratio Y(u) of labelled intermediate cells to all labelled cells, for some particular positions of the labelled cells cohort in the cycle. The analytical form of this ratio can be derived as follows.
436
V.
DE
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AND
(A)
P.
GALAND
NUMERATOR
The density of “a’‘-aged cells leaving G towards D at time t, is P,,,(a) . 2k,Nf,(a, t). Then, if (a’-z) is the age of a infinitesimal cell density at t = 0, the contribution of that infinitesimal cell density to the intermediate layer during the time interval [0, z] is: 2k,N~P,,,(a’-Tft’)
.f,(a’-Tft’,
t’) dt’.
(4)
0
Considering the labelled cells exclusively, a’ belongs to the restricted interval (a-T,, a) where “a” is the age at observation time of the oldest labelled cells, and a > z. Thus, among all initially labelled cells the number that have reached the intermediate layer between t = 0 and t = z is 2k,N
i da’jdt’ a-Ts
P,,,(a’-zft’)
.f,(a’-T+t’,
t’)
0
(5)
Note that, if z < a < r + r,, equation (5) remains valid with the further condition that &(a - z + t’, t’j = 0 when a’ - T+ t’ < 0. (B)
DENOMINATOR
The denominator, i.e. the total number of labelled cells at time z, is calculated as follows :
In this case, all labelled cells have divided once since their labelling. Then, the number of labelled cells at t = z is equal to the number of labelled basal cells at t = 0, i.e. the number (2k,NTs) of labelled cells that have entered G before t = 0, minus half the number of labelled cells that left G before stimulation. The latter number represents the labelled cells that joined the superficial layer under unstimulated conditions. [They do not appear in the counting of the experimental ratios Y(a).] The number of labelled cells at t = z is thus: 2k,NTs-3
(
(I--c
2k,NT,-
j
ll-T$--r ePTs - 1
P
>*
2k,N empa’ da’ >
(6)
In this case, the number of labelled basal cells at t = 0 is equal to the number of labelled cells that entered G before that time, minus half the number of those cells that already left G at t = 0, plus the number of labelled
CELL
437
KINETICS
cells that have not yet divided at t = 0, i.e. (1-r 2k,N(a-2)-s 2k,N(a-z)j 2k,N e-““’
da’
0
>
+k,N(T,-a
+T)
= k,N T,+1-e-“(‘-‘) . (7) P > The number of labelled cells at observation time, is the number of labelled basal cells at t = 0 (equation 7) plus the number of labelled cells that were in the first cycle at t = 0 but that divided during (0, T), i.e. forr
2Ts-a+z+1-e-“‘“-” P (C)
< a < T,+?.
> RATIO
Y(U)
Y(u) is equal to the ratio of equation (5) to equation 20-irS da’i
dt’PG+Ja’--r+t’)
Y(a) = T,+e -da-r)
2 =
i da’ j dt’P,,,(a’--z+t’) -.-__a-Ts O 2T s -a+z+lk!?“‘“-‘)
(6) i.e.
.&(a’--z+t’,t’) (9
epTS- 1 --.-. _ P
Y(a) is equal to the ratio of equation (5) to equation
y(a)
(8)
(S), i.e.
.fJu’--r+t’,t’) (l(J) P
withf,(a’--t+t’,t’) = 0 when a’--t+t’ < 0. Note that when the function P c+D(u) has a breakpoint, those equations are still valid for “a”, less or equal to the abscissa value (designated by “aC”) of that breakpoint. But for a > a,, the numerator of Y(a) is described by a much longer expression, unsuitable for reproduction.
5. Mean Migration
Rate Functions
We have envisaged some a priori possible biological mechanisms, each giiving rise to a different relationship between the mean migration rate and cell age. For each of these assumed PiG.+D(u), we simulated (see section 7) the
438
V.
DE
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AND
P.
GALAND
corresponding Y(u) function that we confrontated with the experimental data. The following cases were considered : (1) The biochemical and morphological changes that the cell undergoes when sejourning in G-phase do not influence its susceptibility to respond to the stimulus. In this case, the P(u) function is (as illustrated in Fig. 5) an horizontal straight line, i.e. P&(u)
= PO
(PO > 0)
(2) Responsiveness increases with cell age (without any point of inflexion) consequently to continuous metabolic changes. These might be linked, for example to the continuous accumulation (destruction) during G-phase of some substance (s) promoting (inhibiting) the cell response to a stimulating agent, or to changes in cell membrane with age. Decreasing migration rates with cell age were not envisaged here since radically contradictory to experimental results (Galand & Vandenhende, 1973).
I IQ
/
I
30
I
I
1
50
a : Ccl I age (hours1 FIG. 5. The mean migration rate function P c..D(u) is assumed to be constant or to increase with cell age, without any breakpoint. Curve (a) (- - -): Pa-&z) = O-0333 h-l, curve (b) (- - -): PG.+&) = (0GO67+04003a) h-l, curve (c) (- - -): PG+&) = 0+067 exp (0.02~) h-l. Corresponding to each of those mean migration rate functions, curves (-) are the simulated proportions Y(u) of the labelled intermediate cells, among all labelled cells. They are functions of the age “a” of the oldest labelled cells at observation time,
CELL
439
KINETICS
In this case, two possibilities were considered (see Fig. 5): (a) P,,,(a)
increaseslinearly with age: P&(a)
(b) P,,,(u)
= PO+va
(I’ > 0)
increasesexponentially with age P:,,(a)
:= POera
(3) If G cells, are either not sensitive to the stimulating agent or cannot respond to it, unless they have passeda specific control point (for example thle completion of some metabolic rate-limiting step), the Pc.+Ju) function has a breakpoint corresponding to a cell age value called “q”. Under this point. G-cells do not respond to the stimulating agent. Nevertheless, in order to take into account that those cells can migrate like unstimulated cells, P(u) can be different from 0 for a < u,. (In the present paper, the irltercellular variability on the ~1, value is neglected.) We assumed (see Fig. 6) (a) P”(a) = PO for a Q a, = 03 for a > a, (b) P5(n) = P" for a d a, = PO+v(u-a,) for u > u,
FIG. 6. The mean migration rate function P o+D(a) is assumed to have a breakpoint in a =ma,. Curve a (- - -): PC+&) = 0.0333 h-l, a < 43 h; PC.-,(a) = cc, a :- 43 h. Curve b (- - -): PG,D(a) = OGO67 h-l, a < 43 h; P G-&z) r= OX)O67+O~Ol(a-43) h-‘, a > 43 h. Corresponding to each of those mean migration rate functions, curves (--I are the simulated proportions Y(u) of the labelled intermediate cells, among ail labelled cells. They are functions of the age “a” of the oldest labelled cells at observation time. 28 T.B.
440
V.
DE
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AND
P.
GALAND
(4) For reasons that will appear clear later on, a variant of P’(a) was also introduced, i.e. P(u) = PO +a for a > a, = PO + vu, + v2(u - a,) for a > a,(~, > 0). 6. Age Density Functions
In equation (S), the age density function 2k,N .fG(u, t) of G-cells needs still to be specified. The first derivative of &(a, t) with respect to cell age, taken along the pathway p = u-t = constant is
2k,N . df&, a-p) = -P(a) da
After integration
.2k,N
. f,(a, a - p)
(11)
of equation (11) from p to a:
- i P(a’) da’ (12) ( Cl--l > This equation is valid when a 3 t (p > 0). When p < 0, the knowledge of fc(u, t) is immediate since j&z, t) = j&z, a) for t > a and &(a, a) can be deduced from equation (12). No experimental clue suggests a particular relationship between the mean migration rate and the mean rate of cell flow from G -+ C. In this paper, we have assumed that the evolution of P c+c(a) with cell age is similar to that of P,,,(u), i.e. that: PC-*&) = ~.P,,,(u) (13) where I is a constant. The same simulations as those reported here, have been performed but with the inverse hypothesis, i.e. that if one of the functions P&a) or P,,,(u) is increasing with cell age, the other is decreasing. In that case, the same conclusions as those presented in this paper, have been reached. At this stage, equation (12) permits to derive the analytical forms of the &(a, t) corresponding to the assumed P c+D(u) functions. Those analytical forms are listed in the Appendix A. f,(a,
t) = f,(a-
t, 0) . exp
7. Results
The evolution of Y (the proportion of labelled intermediate cells among all labelled cells at observation time) with “a” (the age of the oldest labelled cells at observation time) has been simulated according to different
CELL
KINETICS
441
hypotheses about P,,,(u) (th e migration rate of “a’‘-aged cells averaged on the time interval z between stimulus injection and observation). Y(a) depends on at least five parameters: T,, p, z, PO, ,I. Depending on the f’orm assumed for Pc_,D(a), parameters v, a,, and/or v2 may also intervene. T, is available from various kinds of experimental data including the classical fraction of labelled mitoses (FLM) curve (Quastler & Sherman, 1959) and the double labelling method (Wimbler & Quastler, 1963). p can be evaluated from the FLM curves (de Maertelaer & Galand, 1975). In the simulations presented, Ts has been taken as 9 h and p as 0.01 h. T (an experimental condition) was 12 h. A has been arbitrarily taken as 0.5. The value of this parameter does not affect the shape of the Y(u) curves. For convenience in drawing the figures, the values for I” and v have been selected as to giving raise to Y(a) functions that covered more or less the same range of values. The curves Y(a) shown in Fig. 5 have been simulated by assuming P,,,(a) = O-0333 h-’ for curve (a), P,++D(u) = (0~0067+0NMI3u) h-’ for curve (b), and P,,,(a) = 0.0067 e:0’02ah-’ for curve (c). The curves Y(u) shown in Fig. 6 have been simulated under the assumption that Pc+D(u) = P” up to a = a, [a, = 43 h; P” = 0.0333 h-’ for curve (a)
o= Cell
age
(hours1
The mean migration rate function P o+D(u) (- - -) is PG+D(u) = (84067 a) h-l, a < 43 h; P,,,(a) = [0~021O+OX067) (u-43)] h-l, a > 43 II. Curve (-) is the corresponding simulated proportion Y(u) of the labelled intermediate cells, among all labelled cells, as a function of the age “a” of the oldest labelled cells at observation time. 0 = experimental points. FIG. +04003
7.
442
V.
DE
MAERTELAER
AND
P.
GALANI)
and P” = OW67 h-r for curve (b)]. After the breakpoint PG+D(u) = a~ [curve (a)] and Pc_,D(a) = [0~0067+0~01(u-43)] h-’ for curve (b). In Fig. 7, PC+&) = (0~0067+0~0003u) h-’ up to a, = 43 h, and P,,&) = [0.0210 +OM67(u-Us)] h-’ for a > a,.
8. Discussion and Conclusions In the mouse vaginal epithelium stimulated by oestrogen a rapid migratory movement of cells from the basal layer is observed (Galand et al., 1971). Consequently, a new cell layer can be observed (the intermediate layer). This process starts 12 hours after oestrogen administration before any noticeable change in proliferative activity. The simulated curves represented by solid lines in Figs 5, 6 and 7 are the theoretical proportions Y(u) of labelled intermediate cells among all labelled cells at time T after stimulation as a function of the age of the oldest labelled cells at observation time. These curves have been calculated by assuming different kinds of functions for the mean cell migration rate of “Z-aged G-cells. Those mean cell migration rate functions P,,n(u) are represented in Figs 5, 6, 7 by broken lines. The shape of the curves simulated under the assumption that PC,,(u) increases continuously with cell age (Fig. 5) is clearly incompatible with the experimental data represented by solid circles in Fig. 7. Scanning the space of the parameters of Y(u) reveals that this incompatibility is general, i.e. not linked to a particular set of values of the parameters. The rise in the Y(u) function always precedes by far that in the experimental data. Contrarily when P,,,(a) is a biphasic function, as represented in Figs 6 and 7, the Y(a) simulated curves rise after a lag, as in the experimental data. The Y(a) curve shown in Fig. 7 satisfactorily fits the experimental data. It has been obtained by assuming a very slight increase of P,,.(a) up to a = a,, followed by an abrupt increase of this function. It seems that no obvious biological implications can be drawn from the fact that the simulated Y(a) function is more convenient when P,,,(u) increases very slightly before a = a, than when PG_n(u) is a constant till c1 = a,. If this increase is real, it is, in any case, nearly imperceptible and could be due to the fact that we neglected the variability on the uE value. Our results suggest that the cells have to pass a certain time in the G,phase before being fully sensitive to the stimulating agent. The existence of such a cell-age delay before the cell responds to the differentiation stimulus (the response being here an upward migration) suggests that the cell must either perform biochemical process(es) or accumulate a critical amount of some substance(s) in order to be able to respond to the stimulus. This emphasizes the possible existence of a critical step in the G,-phase of the
CELL
443
KINETICS
cell cycle with respect to the responsiveness to the induced differentiation process. This emphasizes also that G,-cells are non-equivalent with respect lo the extracellular signal. If the G, cells are really G-cells, and unless the mechanisms of the response to extracellular and intracellular mediators are different, the G-cells will also be non-equivalent with regard to the intracellular signals that govern the cell progression in the cycle, namely the signal(s) for entry into C-phase and the signal(s) for entry into Dcompartment. In that case, this non-equivalence of G-cells would indicate that the concept of “undeterministic” G-phase does not hold true, at least concerning the differentiation pathway. However the alternative hypothesis e>.ists that, before the critical step, G, cells are not in G-phase, but still in the postmitotic part of C-phase (see Fig. 2). In that case, the stimulus would act specifically on G-cells by increasing the migration rate. This would suggest an analogy between the response of stimulated cells towards their differentiative and proliferative activities. Indeed, there are indications (Brooks, 1975, 1976) that G-cells respond to a stimulus by increasing their transition probability (from G to C), i.e. by increasing the rate of cell flow from G to c’. We are greatly indebted to Dr J. Otten for its constructive criticism of the manuscript and to Dr E. &hell-Frederick for having corrected the English text. We would also like to thank Mrs D. Leemans for typing of the manuscript. This work was performed under Contract of the “Minis&e de la Politique Scientifique” within the framework of the Association Euratom-Universities of Pisa and Brussels. P.G. is a Maitre de Recherches of the Fonds National de la Recherche Scientique.
REFERENCES BASERGA, R. (1968). Cell tiss. Kinet. 1, 167. BARKER, T. E. & WALKER, B. E. (1966). Anat. Rec. 154, 149. BEATO, M., LEDERER, B., BOQUOT, E. & SANDRITTER, W. (1968). Expl. Cell BAGGERS, J. D. & CLARINGBOLD, P. J. (1955). J. Anat. 89, 124. BROOKS, R. F. (1975). J. Cell. Physiol. 86, 369. BROOKS, R. F. (1976). Nature 260, 248. BURNS, F. J. & TANNOCK, I. F. (1970). Cell. riss. Kinet. 3, 321. DIE MAERTELAER, V. & GALAND, P. (1975). Cell tiss. Kinet. 8, 11. GALAND, P., LEROY, F. & CHRETIEN, J. (1971). J. Endocr. 49, 243. GALAND. P. & VANDENHENDE. J. (1973). Anat. Rec. 176.455. GALAND; P. & ROGNONI, J. B: (la75). Am. J. Anat. 144; 533. GREEN, J. A. (1959). Anat. Rec. 135, 247. HOWARD, A. & PELC, S. R. (1953).Heredity 6, 261. HUSBANDS, M. E., JR. & WALKER, B. E. (1963). Anat. Rec. 147, 187. LAJTHA, L. G. (1966).J. Cell Camp. Phys. 67 Suppl. 1, 133. LESSARD, R. J., WOLFF, K. & WINKELMANN, R. K. (1968). J. Invest. Dermat. PERROTTA, C. A. (1962). Am. J. Anat. 111, 195. POTTEN, C. S. & ALLEN, T. D. (1975). J. Cell Sci. 17,413.
Res. 52, 173.
50, 171.
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AND
P.
GALAEU‘D
QUASTLER, H. & SHERMAN, F. G. (1959). ExpI. Cell. Res. 17, 420. SMITH, J. A. & MARTIN, L. (1973). Proc. Natn. Acad. Sci. USA 70, 1263. TRUCCO, E. (1965). Bull. math. Biophys. 27, 285. WIMBER, D. E. & QUASTLER, H. (1963). Expl. Ceil Res. 30, 8.
APPENDIX
A
Analytical forms of the age-density functions corresponding to the migration rate functions assumed under section 5. Since a11 the agedensity functions considered in this paper can be derived fromfz(a, t) and fz(u, t), only these two functions are written out: fi(a,
t) = exp [-~(n-i)+$evu(e-”
-11,
j,!(u, t) = exp [-p(o--~)--P”lvoi+~],
cl 6 CIC 2
= exp
[
-~(a-t)-P”t-vaf+v~+~-~~(o,-a)2
I
,
a, < a < a,+f, 2
= exp
[
-~(n--?)--POf+(v,-v)a,f-v,af+~~
1
) a > a,+t.
APPENDIX
B
List of Symbols and Definitions (A)
VARIABLES
OF THE MODEL
When the basal cells are in an equilibrium-state k, = flow rate constant of cells from G-phase into C-phase. k, = migration rate constant of cells from G-phase into the upper layers of the tissue. p = k,+k,. k, = rate of cell production. 8 = minimum cell cycle duration. T, = duration of S-phase. TG2 = duration of G,-phase. TM = duration of M-phase.
CELL
fier a perturbation
445
KINETICS
of the equilibrium-state
of the tissue
P,,c(a,
t) = generalized flow rate at time t of “a’‘-aged G-cells into the C-phase. Pc+.(a, t) = generalized migration rate at time t of “a’‘-aged Gcells into the differentiation compartment. P(a, t) = generalized global flow rate out of G-phase (flow tn C+ migration to D) at time t of “a’‘-aged cells. PC&a) = mean flow rate of “a’‘-aged G-cells into the C-phase, averaged on the time r between the application of the stimulus and the observation time. PGdD(a) = mean migration rate of “a’‘-aged G-cells into the differentiation compartment, averaged on the time t between the application of the stimulus and the observation time. P(a) = mean global flow rate out of G-phase (flow to CS migration to 0) of “a’‘-aged cells averaged on the time t between the application of the stimulus and the observation time. 2k,N = birth rate of basal cells. 2k,N. &(a, t) = age-density function of ceils in G-phase. (B) VARIABLE
LINKED
TO THE EXPERIENCE
t = 0: time at which the stimulus is injected. z = time of observation (= time at which the animal is killed). Y(a) = proportion of labelled intermediate cells among alI labelled cells, as a function of the age of the oldest labelled cells at observation time.
