Kinetics of cell-mediated cytotoxicity: Stochastic and deterministic multistage models

Kinetics of Cell-Mediated Cytotoxicity: Stochastic and Deterministic Multistage Models*

ALAN S. PERELSON AND CATHERINE A. MACKEN+ Theoretical Dioision, L>osAlumos Natronal Luhorutoq~, Los Alumos, New Mexico 87545 Receiced 30 March I984

ABSTRACT One of the body’s major defenses against viral diseases and tumors is the killing of abnormal cells by host defense cells, such as T lymphocytes. The mechanism by which killing is accomplished is unknown. Here we develop both stochastic and deterministic models for the kinetics of killing in aggregates which contain a single lymphocyte and multiple target cells (~57;~conjugates), as might be seen early in an immune response, and in aggregates containing multiple lymphocytes and a single target cell ( L,,T conjugates), which is characteristic of the late phase of a successful immune response. Comparing our models with data, we rule out the possibility of certain classes of lytic mechanisms and draw attention to the characteristics of likely mechanisms. Our stochastic model can be viewed as a specialized application of queueing theory to cell biology. For certain choices of arrival-time and service-time distributions, stochastic and deterministic models.

1.

we find an exact correspondence

between

our

INTRODUCTION

As part of the body’s immune response to disease-causing agents that infect cells and then reside intracellularly, mechanisms have been devised to contain the infection by killing the diseased cell and preventing the replication of the pathogen. The most familiar example of this phenomenon occurs in viral or parasitic infection, The virus takes command of the biochemical mechanisms of a cell in order to reproduce itself. As part of this process, viral proteins are made, some of which may be inserted into the cell membrane. If this occurs, cells in the immune system, such as cytotoxic T lymphocytes

*Work performed under the auspices of the United States Department +Permanent address: Centre for Computing and Biometrics, Lincoln bury, New Zealand. MATHEMATICAL

BIOSCIENCES

70:161-194

(1984)

of Energy. College, Canter-

161

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ALAN S. PERELSON AND CATHERINE

A. MACKEN

(CTL), may recognize the infected cell as abnormal and kill it. As part of the process of cell death, intracellular nucleases are activated which almost assuredly destroy the viral DNA. Besides CTL, other host defense cells, such as natural killer (NK) cells and macrophages, may also kill abnormal cells. Host defense cells have been shown to play a role in killing tumor cells and may be involved in protecting us against more frequent occurrences of cancer. Much has been learned about how cytotoxic cells kill other cells from in vitro studies. In such experimental systems, target cells, the virally infected cells or tumor cells that are supposed to be killed, are mixed with host defense cells, generally called effector or killer cells, and the kinetics of their interaction monitored. Although many of the details that we discuss below are true for all host defense cells, we shall limit our remarks to the best-studied example-cytotoxic T lymphocytes. The CTL is a white blood cell that resides in the blood and lymph nodes and on occasion leaves the peripheral circulation to crawl among the body’s tissues looking for abnormal cells. The CTL returns to the circulation via the lymph system. Once the CTL encounters an appropriate target cell, it binds to the target cell, remaining in contact with it for a period of between a few minutes and a few hours. If the CTL detaches from the target cell after a very short period of time, nothing happens to the target cell. However, if the CTL remains in contact with the target cell for some critical period, generally of order 3-10 min, before detachment, then the target cell will eventually die. At the time of detachment the target cell may look normal or may show unusual membrane activity, but is definitely alive. After an elapsed time, generally of order 10 min-2 hr, the target cell spontaneously disintegrates-with both nuclear and plasma membrane lesions being observed. The exact mechanism of death is unknown, but in some aspects it resembles osmotically induced cell lysis. Further details can be found in a number of recent reviews [l, 2, 4, 15, 31, 321. The events that occur during the critical contact period are not yet understood at the molecular level. This mysterious process has been colloquially referred to as the CTL giving the target cell a “kiss of death.” More accepted terminology is “programming the target cell for lysis” or “delivering a lethal hit,” reflecting two differing popular views-the killer cell may activate a process within the target cell causing it to self-destruct (i.e., programming it), or the killer cell may deliver a toxic agent (i.e., a lethal hit) to the target cell. Many investigators believe that the former view is incorrect. For example, experiments have shown that target cells whose metabolism has been irreversibly poisoned by glutaraldehyde are killed efficiently [3]. Thus we shall adopt the prevailing view and refer to this crucial event as lethal hitting. Two biochemical facts known about lethal hitting are that the process

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is dependent upon the presence of Ca ++ in the medium [lo, 11, 231 and that the process is substantially slowed or stopped if the CTL is at room temperature (e.g., 20°C) rather than 37°C. The act of killing a target cell does not damage the CTL, and the CTL is able to kill repeatedly [22]. Once differentiated, the CTL does not need to synthesize proteins to kill, and this is probably true even for repeated killing 1331. One can describe the cytotoxic process as occurring in a number of stages: (1) The CTL must find the target cell. The CTL may search randomly or use a more directed hunting process. (2) The CTL must recognize and bind to the target. (3) The CTL then delivers a lethal hit to the target cell, a temperaturedependent step requiring Ca++. (4) The target cell disintegrates. Enzymatic destruction of nuclear material, loss of integrity of the plasma membrane, and abnormal ion and water flows are all observed during the process. Disintegration does not require the presence of the CTL, although it may occur faster if the CTL remains bound to the target cell [32]. (5) The CTL recycles to begin the process anew [22]. It may to this with a target still bound to its surface, or it may first disengage from the target cell [301. In vitro, it is possible to study the lethal hitting and target cell disintegration stages by forming CTL-target-cell conjugates at room temperature or in the absence of Ca++ -that is, under conditions which prevent the infliction of a lethal hit. Warming the CTL-target-cell conjugate to 37°C in the presence of Ca+ + allows the events of lethal hitting and target disintegration to proceed. If the conjugate is incubated in a semisolid medium such as agar, then the CTL is prevented from recycling and can only kill the target cell or cells to which it is initially conjugated. By incorporating into the medium a dye, such as trypan blue, which is excluded from living cells but which permeates the membrane of dead cells, one can monitor the kinetics of cell lysis. In Section 2 we discuss a general stochastic model of lethal hitting due to Perelson and Bell [26]. By comparison of the model with data on the kinetics of lethal hitting, we will be able to eliminate some obvious choices for the state transition rates in the model. A biologically interesting model will be shown to be consistent with available data. In Section 3 we propose a stochastic model for target cell disintegration and then, in Section 4, show how the models for lethal hitting and target cell disintegration can be combined into a two-stage model for the kinetics of target cell lysis. Predictions of the model are compared with data. Section 5 contrasts the two-stage

ALAN

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S. PERELSON

AND CATHERINE

A. MACKEN

stochastic model with deterministic models that yield comparable results. In Section 6 we discuss the biological relevance of our results and directions for future research. The models we develop are a special class of queueing-theory models [5, 17, 181. Typically, queueing theory deals with the stochastic arrival of customers, say at a bank, who line up at teller windows and then are served one at a time. The service time is also stochastic. The analogy which underlies our model is that the CTL serves death. For a cell to be served, it must be lethally hit. Thus the probability of generating a lethal hit corresponds to the probability of a customer arriving. The probability of a cell disintegrating d time units after being lethally hit corresponds to the probability of a customer being served in time d. This analogy should become clearer after our models are introduced, but it is a useful one for readers with a background in stochastic processes to keep in mind. 2.

MODELS

OF LETHAL

HITTING

In an experiment by Zagury et al. [35], the kinetics of lethal hitting were measured in conjugates containing one lymphocyte and one, two or three target cells. We shall refer to such conjugates by the abbreviation LT,, , where n =l, 2, or 3. The experiment utilized the fact that lethal hits are not delivered if Ca+ ’ is absent or if the medium is at room temperature. Conjugates were formed at room temperature, then placed in a Ca++-conmining medium at 37°C. After a predetermined time interval (3, 5, 10, or 15 min), a Ca++ -chelating agent, EDTA, was added to the medium. Once EDTA was added, the Ca++ concentration dropped sufficiently low that lethal hits could no longer be generated. Target cells that were hit before EDTA was added went on to lyse. At the end of a 3-hour incubation period, which is sufficiently long that any lethally hit cell would have lysed, the number of dead target cells in each conjugate was measured by phase-contrast microscopy. Thus the experiment determined the fraction of LT, conjugates in which the CTL delivered a lethal hit by time t, where t was the time of EDTA addition. For LT, and LT, conjugates, Zagury et al. reported from the experiment the fraction of conjugates in which the CTL delivered lethal hits to at least m < n of the target cells by time t. Thus we shall also be interested in predicting the fraction of LT,, conjugates in which at least nt < II target cells are hit by time t. A.

MODEL

FOR THE RANDOM

DELIVERY

OF LETHAL

HITS

IN LT,,

CONJUGATES

Consider a population of LT,, conjugates. Because a CTL may behave differentily in conjugates of different sizes, we shall keep track of the conjugate size by attaching the subscript II to the parameters and variables in

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our theory. In experiments, the time at which a target cell receives a lethal hit seems to vary. Perelson and Bell [26] modeled that variation by assuming lethal hits are delivered at random times. In the remainder of this section we describe their model and then go on in the next section to build upon it and develop a model for the complete lytic process in multicellular conjugates. Let the number of target cells hit within a conjugate to be the state of the conjugate. Then a continuous-time Markov chain, with states {O,l,. , n}, for which transitions can occur only from state m to state m + 1, will describe the lethal hitting process. Such a process is known as a pure birth process, since the state of the system always increases by one. The rate of the process is determined by the state transition rate h,,( m, t), where X,,(m, t) At + o( A t) is the probability that an LT,z conjugate with m cells hit at time t has m + 1 cells hit at time t + At. By standard methods (cf. Feller [S]) one finds that P,,( m, t), the probability that m target cells in an LT,, conjugate have been lethally hit by time t, satisfies the following forward Kolmogorov differential equations:

dp,,(0,t> dt dP,,(m, dt

t>

(14

=-&,(O,t)P,,(O,t),

L,t)P,,(m-l,t),

=-x,,(m,t>P,,(m,t)+k,(m

lgm
dP,,(n>t)=x,,(n-l,t>P,,(n-l,t)

(lc)

dt

At the beginning

(lb)

of the process no cells are lethally hit, so


and

l
P,,(m,O)=O,

(2)

To solve these equations we make explicit assumptions about h,,( m, t). Different choices of h,,( m, t) correspond to different biological models. (a) Assume h,,(m, t) = A,, = constant. With h,,(m, t) = A,,, the transition probability does not vary with t, the time cells remain in culture at 37°C nor with m, the number of lethal hits already delivered. Under this assumption Equation (l), with initial conditions (2), can be solved sequentially, first for m = 0, then for m = 1, etc., yielding

P,,(m,t)=

P,,(n,t)=l-

(X,,f)n:eph.“,

O
,z (X,it)n’e-h,,r c m! n*= 0

(3b)

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A. MACKEN

Equation (3) describes a Poisson distribution truncated at n. This modified Poisson distribution has been discussed by Newell [24]. The mean of the distribution, i.e., the expected value of m, is given by ‘I-’ (n - m)(A,,t)“’ C m! m=0

E,,(m,t)=n-e-‘u’

As t + co, E,( m, t) + n, i.e., all II cells in the conjugate will eventually be hit. (b) Assume X,,(m, t) = (l- m/n)h,,, A,, = constunt. In this model we again assume that the time cells remain in culture at 37°C is unimportant in determining the transition rate, and that the CTL delivers hits at a constant mean rate X,,. However, in addition we assume that the CTL cannot distinguish between hit and unhit target cells, and thus it randomly delivers lethal hits, one at a time, to all attached target cells. Only hits to previously unhit cells will change the state of the conjugate. Thus, the probability of a state change in a small time At, when m of the n cells have already been hit, is the product of the probability h,, At that a hit to some cell occurs and the probability 1 - m/n that the hit is to a previously unhit cell. The solution of Eq. (l), with h,, (m, t) given as above, is

In the Appendix this is proven by induction. From Equation (4) we see that P,,(m, t) is given by a binomial tion where the probability of a success is p(t)

=l_

em%,‘/,r,

distribu-

(5)

This is not surprising, since hits to any one of the n cells occur as a Poisson process with the probability of a hit in a small time At given approximately by (h,, A t)(l/n). Then for each cell, the probability of at least one hit in time t is 1 - emhJ/“. Hits to distinct cells are independent (since all hits occur independently). Thus, the number of cells hit in time t is a binomial random variable, i.e., the number of “successes” in n independent trials, each with probability p(t) of success. From the binomial form of P,,( m, t) we deduce that the expected number of hits by time t is

Again, as t + co, E,,( m, t) -+ n, i.e., all cells are eventually

hit.

Many other choices of h,,( m, t) are possible. For example, one might assume the CTL depletes biochemical resources in delivering a lethal hit, and

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thus the rate at which subsequent hits are generated progressively diminishes. This would be reflected in choosing X,( m, t) as a decreasing function of m. Conversely, on might assume that critical biochemical pathways need to be turned on within the CTL before it can generate the first lethal hit, but that once turned on, subsequent hits can be generated at a progressively faster rate. This would be reflected in choosing h,( m, t) as an increasing function of m. Although these and other hypotheses can be formulated and converted into predictions of P,, (m, t), to test them would require substantially more data than are currently available. Before expending the effort to collect additional data, it seems worthwhile to test models (a) and (b), which contain only a single parameter, A,, , and which describe feasible scenarios. Models (a) and (b) describe different biological situations. In model (a) we view the CTL as behaving in an intelligent manner, attacking only those cells that it has not previously hit. One cell at a time is hit, and thus the state transition rate equals the hitting rate, A,,. In model (b) we view the CTL as being indiscriminate and attacking any cell to which it is bound. As in model (a), the CTL can only hit one target cell at a time. Even though all target cells are at risk, the probability of generating two hits at the same time is vanishingly small, i.e., o(A t) as At + 0. However, now a previously hit cell may be hit again. For future reference, we shall call this model the RIH or random indiscriminate hitting model. There is an alternative biological scenario of the lethal hitting process which gives rise to the same mathematical formulation for P,, (m , t) as does model (b). A current, popular view about lethal hitting is that the interaction between the CTL and the target cell is a local membrane event that occurs at the site of attachment between the two cells. In an LT,, conjugate there are n sites of attachment or “membrane attack areas.” If one assumes that these areas are identical in the sense that they generate lethal hits at times picked from the same probability distribution, then the sequential activation of these areas with no delay between the delivery of a hit and the activation of the next area leads to model (a), whereas the simultaneous activation of the membrane attack areas leads to model (b). If the activation is sequential, then the rate of hitting A,, would not depend on m, the number of targets already hit. If the activation is simultaneous and the CTL generates hits at a net rate h,,, then each of the n membrane attack areas must generate hits at rate h,,/n. Further, the probability that m out of the n areas have generated a hit by time t is given by a binomial distribution, i.e.

(6) where p(t) is the probability that a membrane attack area has generated a hit by time t. For hits generated with a constant mean rate h,,/n, the probabil-

168

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AND CATHERINE

A. MACKEN

ity of no hit being given by time t is exp(- X,,t/n). Thus, p(t)=lexp( - A,,t/n) and Equation (6) is equivalent to Equation (4). When we wish to emphasize the membrane-attack view of CTL action, model (a) will be called the sequential-attack model and model (b) the simultaneous-attack model. Although the RIH and simultaneous-attack models give rise to identical predictions, they derive from different assumptions about CTL behavior. To emphasize the difference, we describe each model in terms of the random activation of membrane attack areas. We stress, however, that the RIH model is by derivation not tied to the notion of membrane attack areas. In the RIH model at time 0, one membrane attack area picked at random is activated. It generates a lethal hit a random time H later, where H is picked from an exponential distribution. The clock is then set back to zero, and the process begins anew. In the repetitions of the process the same membrane attack area can be activated repeatedly, and thus a hit target can be hit again. On the other hand, in the simultaneous attack model all n areas are activated at time 0, each area is fired once at a random time picked for that area from an exponential distribution, and the process is not repeated. Thus in the RIH model the same cell can be hit many times, whereas in the simultaneousattack model a cell is hit only once. B.

COMPARISON

OF MODELS

WITH

EXPERIMENTAL

DATA

A comparison of theory with experiment is given in Tables 1 and 2. The experimental data of Zagury et al. [35] on the fraction of conjugates with one TABLE

1

Comparison of Data with the Sequential-Attack (Poisson) Model in Which Each CTL Delivers Lethal Hits at a Constant Mean Rate Only to Unhit Target Cells Fraction of conjugates with lysis of ut leasr 2

target cells

LT,

3 target cells LT,

LT,

Time (min) Predicted” Measuredb Predicted’ Measuredb Predicted’ Measured’ 3 5 10 15

26 .49 .84 .96

26 .42 ND .v2

.08 .19 .4x .69

ND* .30 ND .80

.Ol .05 .22 .43

ND .04 ND .45

“Using Equation (3) with h, = 0.33, the value estimated by Perelson and Bell [26]. bMeasured by Zagury et al. [35]. ‘Using Equation (3) with A, = 0.16, the value estimated by Perelson and Bell. *Not determined in experiment by Zagury et al.

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or more target cells hit by time t was used to estimate h,,. Our estimator of h,, has a simple form which is independent of the model. We simply set ~~(1, t), the observed proportion of LT, with at least one target cell hit by time t, equal to the model prediction for this proportion, 1- P,,(O, t), and solve for A,. Note, from Equations (3a) and (4), that for either the simultaneous or the sequential model, the probability that at least one target cell is hit by time t is given by 1 - P, (0, t) = 1 - e-ill’.

Hence, one can estimate

(7)

X,, from

or equivalently X,, = - + ln[lFurther

details of the estimation

y,,(l, t)].

procedure

are given in Perelson

and Bell

WA. With A,, determined, the two models can then be tested for fit by comparing predictions with observations of the fraction of conjugates with at least two and with three target cells hit by time t, data not used to estimate h,,. As shown in Table 2, the simultaneous (RIH) model gives a poor fit to the data and hence can be rejected. The sequential model gives a good fit to the data (Table 1). TABLE 2

Comparison of Data with the Simultaneous-Attack (Binomial) Model in Which Each CTL Delivers Lethal Hits at a Constant Mean Rate to Unhit and Previously Hit Conjugated Target Cells Fraction of conjugates with lysis of at least 3

2 target cells

LT,

target cells

LT,

LT,

Time (min) predicteda measuredb predictedC measuredb predictedC measurcdb 3 5

.15 .32

.26 .42

.06 .14

NDd .30

,003 .Ol

ND .04

10 15

.65 .84

ND .92

.37 .58

ND .80

.07 .17

ND .45

%sing Equation (4), h, = 0.33. bMeasured by Zagury et al. [35]. ‘Using Equation (4), h, = 0.16. dNot determined

in experiment

by Zagury

et al

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ALAN S. PERELSON AND CATHERINE

A. h4ACKEN

Although this one experiment is clearly not sufficient to prove the sequential model is correct, it does indicate that a modified Poisson distribution is accurate enough to make quantitative predictions. Further, it presents an intriguing biological question. If the sequential model is correct, the CTL must be able to sense which cells have been hit and which have not. When micromanipulating LT, conjugates, Zagury et al. [35] observed that two target cells separated easily from the CTL, whereas the last one remained attached to the lymphocyte even after vigorous treatment. The one cell that remains firmly attached might be the one currently under attack. Sanderson [36] has noted by microcinematography that after a cell has been lethally hit it undergoes zeiosis, i.e., its membrane undulates rapidly, so that the cell appears as if its membrance were boiling. Perelson and Bell [26] proposed that zeiosis could break the firm attachments between the CTL and the hit target, thus allowing the CTL to redistribute its surface receptors and make a firm attachment to an unhit target cell. Recent experiments by Bongrand, Pierres, and Golstein [37] in which graded shear flows were used to disrupt lymphocyte-target-cell conjugates also suggest that a threshold level of binding strength is necessary for T-cell-mediated cytolysis. Although the Perelson-Bell scenario is plausible, other mechanisms might be employed in distinguishing hit from unhit cells. The agreement of the sequential model with data suggests that determining this mechanism is an important endeavor worthy of further experimental investigations. In what follows we shall assume that lethal hits are given sequentially and that the distribution function for lethal hitting times is given by Equation (3). A consequence of the Poisson nature of Equation (3) (or more generally the memoryless property of continuous-time Markov chains) is that the time intervals between lethal hits are distributed exponentially [17, 181. To make this statement more precise, let Hr , Hz,.. ,H,,be random variables denoting the elapsed time between 0 and the first hit, the first hit and the second hit, etc. (H,can be interpreted as the time it takes a CTL to focus its attention on an unhit target cell and deliver a lethal hit for the i th time.) The probability distribution function, F,,(h), describing the probability that a lethal hit is generated in a time H less than h, is exponential, i.e.

F,,(h) =lwith corresponding

probability

ephnh,

(ga)

density function

ff,( h) = h,,e-“,lh.

(8b)

The expected value of H, is

E(H,) =1/X,,,

i =1,2 ,...,n,

(8c)

KINETICS OF CELL-MEDIATED

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and the variance of H, is

vdff,) =W:,,

i =1,2 ,..., n.

These facts will play a role in our subsequent C.

MODEL FOR THE DELIVERY

OF LETIfAL

(Ed)

modeling. HITS IN L,,T CONJUGATES

Under the appropriate conditions, multiple cytotoxic T lymphocytes may bind to a single target cell, producing conjugates of the form L,,T. In experiments reported in Perelson et al. [27] conjugates of this form with n = 1, 2, 3, and 4 were studied. Each of the CTL in a conjugate may generate a lethal hit. Let H,‘, i = 1,2,.. ., n, be random variables denoting the time at which the i th CTL first lethally hits the target cell. The probability that a target cell has not been lethally hit by time t is simply the probability that none of the n conjugated CTL has delivered a lethal hit by time t. Thus H;>t,...,H,:>t).

Pr(targetnothitbytimet)=Pr(Hi’>t, If each CTL in the conjugates

acts independently

Pr( H{ > t, H;> 2,...,

H,f>t)=

in generating

lethal hits,

fiPr(H,‘>t). i=l

Under the assumption of either model (a) or (b) that A,,(O, t) = h,,, Equation (3a) or Equation (5) gives the probability that a single CTL does not hit the target cell by time t as Pr( H,‘> t) = e-‘*,

i =1,2 ,...,n,

where h is the rate at which a single CTL in an L,,T conjugate lethal hits. Considering all n CTL in the conjugate, Pr(target

not hit by time t) = Pr( H > t) = e-““,

(9) generates

(10)

where H is a random variable denoting the time at which the target cell is first hit, i.e., H = min( H{, Hi,. . . , H,:}. Although we have assumed the n CTL bound to a target cell all act independently, it is possible that interactions among the CTL will occur, especially as n becomes large (say n > 3). For example, if a tight adhesion needs to be made between a CTL and a target cell before a lethal hit can be generated, we might expect crowding of the CTL to decrease the rate at which lethal hits could be generated. Thus one might expect h, > h, > . . . > h ,), where A, denotes the rate at which a CTL delivers hits when in an L, T

ALAN S. PERELSON AND CATHERINE

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A. MACKEN

conjugate. A similar effect was found by Perelson and Bell in their analysis of LT,conjugates. When three target cells were conjugated to a single lymphocyte, the rate at which the CTL delivered lethal hits was 0.16 mini i, whereas with two target cells in the conjugate, the rate was 0.33 mini’. If the rate at which a CTL delivers a lethal hit depends on the conjugate size, nh in Equation (10) should be replaced by nh,,. Because h,, is unknown and needs to be estimated from data, it will simplify matters to write Equation (10) as Pr( H > t) = Kx:J,

(II)

where h’,, = nh,. If by fitting data on the kinetics of hitting in conjugates containing different numbers of lymphocytes, one finds X,, = nX, where h is some constant, then one can conclude that the CTL in a conjugate all act independently. Conversely, if h’,, f nX, then one can conclude that there is some interaction between the CTL.

3.

MODELS

OF TARGET-CELL

DISINTEGRATION

The molecular basis of target-cell disintegration is not yet understood. The process is though to involve multiple biochemical events such as the activation of nucleases [31], phospholipases [9], and proteases [28]; decreased integrity of the nuclear [31] and plasma membranes [14]; and abnormal ion and water flows. To the extent that such events occur sequentially, we can consider the disintegration process as being composed of c substages, each substage representing a single event or set of events occurring simultaneously. By this modeling procedure we can decompose a process containing both parallel and series events into a c-stage model. If we assume that the time spent in each stage is random and has an exponential distribution with mean l/p, then an Erlang or gamma distribution with integer shape parameter c > 0 and scale parameter p is the appropriate descriptor of the disintegration process [5, 181. As we show in Section 4, more complex models of the disintegration process can be formulated in which the residence-time distribution for each substage differs. However, the currently available data are insufficient for identifying the many parameters characteristic of complex models. Until the gamma distribution is shown by experiment to be inadequate we shall model disintegration as a sequence of c identical substages. Using a gamma distribution for the probability distribution of target cell disintegration times, Pr(D
where

‘-’ C

x=o

(pd)’

__

k!

’

(12)

D is a random variable describing the time after being lethally hit at

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which a target cell disintegrates. The mean of a gamma distribution is c/p, and thus E(D), the expected value of D, is given by E(D) = c/p. In an LT, conjugate, let D,, i =1,2,. , n, be the disintegration times of the n target cells. We assume the D, are independent random variables, identically distributed according to the gamma distribution given in Equation (12). 4.

KINETICS

OF TARGET-CELL

LYSIS

By combining the sequential model for lethal hitting given in Section 2 with the c-stage model for target-cell disintegration in Section 3, we obtain a model for the kinetics of target-cell lysis in multicellular conjugates. We first consider L,,T conjugates and then LT,, conjugates. Our analysis of L,,T conjugates can be considered a special case (n = 1) of the later analysis of L7;, conjugates. When n CTL are attached to a single target cell, the rate of lethal hitting is X,,; whereas when only one CTL is bound, the hitting rate is Xi. Thus, to analyze an L,T conjugate we could simply proceed as for an LT, conjugate with X, replaced by X,,. However, for pedagogic reasons we shall first treat L,,T conjugates and then proceed to the more complex analysis of L7j, conjugates with n > 1. A.

LYSIS

IN L,,T CONJUGATES

Let T be a random variable representing the time at which the target cell in an L,T conjugate lyses. For the target cell not to lyse by time t (i.e., for T > t), one of two mutually exclusive events must have occurred: either (1) it has not been lethally hit by time t, or (2) it has been hit by time h < t but it has not disintegrated in the remaining t - h time units. The probability that the target cell in an L,T conjugate has not lysed by time t is the sum of the probabilities of these two events. Thus from Equation (11) and the law of total probability, Pr(T>t)=KXL’+

s

OfPr(D>~-h)f,,(h)&,

where f,,( h ) is the probability density function One finds from Equation (11) that

of the random

fH( h) = A’ne-x~h.

Combining

pr(

Equations

T>

1) =

(13) variable

H.

(14)

(12), (13), and (14) gives

eph:z’ +

A’,,

‘-’

J‘epa(rph)c 0

k=O

[d--h)lke-X',h~~, k!

(15)

ALAN

174 Finally,

evaluating

S. PERELSON

A. MACKEN

the integral for c = 1 to 4, we find ,&+-”

p(t)

AND CATHERINE

= Pr(Td

+ (-

p)ce-h:,r

t) =l-

(16) ’

(0)’ where

A=l,

c-1.

A=&‘,,-&+h:,-2~,

c= 2;

A=f~*(X:,-~)*~*+~(h:,--)(h:,-2~)t+~~-33h:,~+3p2, c=3; and

A=B~3(A:,-~)3t3+IPz(A:,-p)2(h:,-2~)t2+~(~~-p)(h:;l-3X:,~')c +(X:,-2~)(~~-2h,l.L+2~2), c= 4. In an L,,Tconjugate, T,the time of target-cell lysis, is the sum of the two random variables H and D. Thus, E(T),the expected value of T,is simply

E(T)=E(H)+E(D)=++;. II The predictions of this multistage model were compared with experimental data collected by B. Bonavida’s group at UCLA [27]. As shown in Figure 1, substantial agreement was found between theory and experiment when target-cell disintegration was modeled as a c-stage process with c = 2 or 3. B.

LYSIS

IN L7;, CONJUGATES

In conjugates which contain one CTL and n target cells, the theory is much more complicated because it must predict n separate functions describing the probability that by time t one, two,. . . , n, target cells have lysed. Further, the order in which target cells are hit is not necessarily the same as the order in which they lyse. Thus many sequences of random events will need to be examined in order to determine the kinetics of lysis. As in Section 2, let H,,i= 1,2,. , n, be the random variables describing the elapsed time between successive lethal hits. We assume each lethal hit is delivered to a different target cell and that the hits are generated sequentially. Once a target cell is hit, it begins disintegrating. As defined in Section 3, D, is a random variable describing the time that elapses between the delivery of the i th lethal hit and the disintegration of the i th lethally hit target cell (see Figure 2). The time taken for a target cell to lyse is the sum of the time taken for the CTL to deliver a lethal hit and the time taken for the target cell to disintegrate. Thus the first target cell to be hit lyses at time H, + D,.When

KINETICS

OF CELL-MEDIATED

-

I

0

30

I

I

60 Time

I

I

90

175

CYTOTOXICITY

I

I

120

1

.8-

I

. 150 ‘0

30

150

30

0

(min)

FIG. 1. The kinetics of target-cell lysis in L,,T conjugates tally as described in [27]. The solid line is the best-fitting

I 60

1

I 90

60

90

Time

(mln)

I

I I 120 150

120

150

were measured experimentheoretical curve found by

nonlinear least-squares regression [c = 3 in (a) and c = 2 in (b), (c), and (d)]. The data points indicate the fraction of conjugates lysed in a total of ten separate experiments, See [27] for further details. Reproduced with permission from Perelson et al, [27].

multiple target cells are conjugated to the CTL, the second target cell lyses at time Hi + h; + D,. If Hz + D2 c D,, then the second cell to be hit will lyse before the first cell to be hit. One can easily observe the number of target cells in an L7;, conjugate that have been lysed by time t. In what follows we shall compute P,, (k, II, t), the probability that by time t, k out of n target cells in an LT,, conjugate have lysed. Our derivation will be general. Once it is complete, we shall impose the particular modeling assumptions that we have discussed in Sections 2 and 3 to characterize the distribution of hitting and disintegration times. We begin by noting that from the law of total probability

P,,(k,n,r)=~~p,,(k,n,IIH,=h)/,,,(ll)dh(17)

ALAN S. PERELSON AND CATHERINE A. MACKEN

176

o*

I

I t

HI

*

time

FIG. 2. A possible sequence of hitting and disintegration times for an LT, conjugate. Incubation at 37°C begins at t = 0. At a random time Hl one of the target cells is lethally hit. A random time H, later the other target cell is lethally hit. The disintegration process is assumed to begin once a target cell is hit. The first target cell to be hit takes a random time D, to disintegrate, whereas the disintegration of the second target cell takes a random time Dz. As shown here, it is possible for the cell hit last to disintegrate first. By time t, both target cells in the conjugate have been lethally damaged, but only one cell has lysed.

where P,,(k, n, tlf?, = h) is the conditional probability that k out of n target cells have lysed by time t, given that the first lethal hit is delivered at time h. For h > t, no cells can be hit or lyse by time t and thus P,(k,n,tlH,=h,h>t)=6,,,

where a,,

is the Kronecker

(18)

delta function

defined by

Using this fact, the integral from 0 to cc in Equation two integrals. Hence we find

f’,,(k,w)=l:P,(kn,tlH,=h)f,l(h)dh+6,,Pr(H,>t).

(17) can be split into

(19)

In order to evaluate P,( k, n, t[ HI = h) recall that the lysis time of the first cell to be hit is HI + D,. If D, 6 r - h, then the first cell to be hit lyses by time t; whereas if D, > t - h, then this cell lyses after time t. In order that k cells lyse by time t, k - 1 of the remaining n - 1 cells must lyse during (h, t] if D, < t - h, whereas k of the remaining n - 1 cell must lyse during (h , t] if D,>t-h.Thus, P,(k,n,tJH,=h)=P,,(k-l,n-l,t-h)Pr(D,t-h).

Substituting

Equation

(20) into Equation

(20)

(19) we find

P,(k,n,r)=jo’P,(k-l,n-1,1-h)Pr(DI~r-h)f,,(h)dh +

‘P~(k,n-l,t-h)Pr(D,>f_h)f,,(h)dh+6,,,Pr(H,>t). /0 (21a)

KINETICS OF CELL-MEDIATED

By defining

CYTOTOXICITY

177

for all values of n and t P,(O,O,t)

=l

(2lb)

and for i > j and for i< 0,

p,,(i,j,l)=O Equation

(21) becomes

a recursion

relation

which

can

(21c) be solved

for

P,,(k n, f). We now impose particular assumptions on the distribution of hitting and disintegration times. As discussed in Section 2, hitting appears to occur sequentially with hitting times H,, i = 1,2,. . , n, identically distributed according to an exponential distribution, with density function fH(h) given by Equation (8). Because the random variables H, all have the same distribution, we drop the subscript i from H,.Similarly, we use the same gamma distribution to characterize c-stage disintegration of each of the n target cells and hence drop the subscript on D,.For notational simplicity we also drop the subscript n, denoting the conjugate size, from the parameter X,, and from P,,( k, n, t). Combining Equations (8) (12), and (21), we find the recursion relation P(k,n,t)=XJorp(k-l,n-l,r-h)

[,dr k,- h)lk k=O

+

.

e-A,,

dh

i

fp(k,n-l,t-h)e-““~h’ (22)

SkoepA*.

We now illustrate the use of Equation (22) for c =l. To begin the recursion we need to evaluate P(O,l, Z) and P(l,l, t). Because P( - l,O, t) = 0 and P(O,O, t) = 1, we find from Equation (22) P(O,l,

1) = Ab-““P”‘e-“h

P(O,l,

t) =

d/r + ePhr,

which evaluates to hemp’ - pe-“,

Af Pt

A-P i

(l+At)e-“‘,

h=p.

(23)

178

ALAN

S. PERELSON

AND CATHERINE

A. MACKEN

The case h = p can be obtained from the expression for X # p by 1’Hbpital’s rule and will be dropped from further consideration. The probability of one cell being lysed by time t in an LT, conjugate can be derived from Eq. (22) or from the fact that P(l,l, t)+ P(O,l, t) =l. Either way, one obtains he-” -pe+r P(l,l,t)=lA_ . (24) P Continuing

the recursion,

p(o 2 t)= > 3 P(1,2,

one finds

X*e-2”‘+X(h-2CL)e-‘“+X”+(~--)(211+X)e~”’ (P - X)(&J - A)

? (25)

h

t) =

(P - Q2(2P

- A)

x [2h(h-&-*“-(~-2A)(2~-h)eP” +2( /&- h)(2/.l-

A) e-“+x”

+{(~-h)(2~-X)~t-2/~*+3h~-22X~}e~~’],

(26)

P(2,2,t)=l-P(O,2,t)-P(1,2,t).

(27)

and Further, 1

P(O,3, t) =

2P(P - X)(2P - X)(3P - A) x [(~-AA)(~~-AX)(~/.L*+~A~+A*)~-~’ -2X(p

- X)(3p - A)(2p + A)eP(P+x)’

+ X2(211- X)(3/L - h)e~‘*‘+X’r-2A3EL,-3pr], P(1,3,

(28)

x

f) = 2P(P - U2(2P

- V2(3P - A)

x[{2P(P--h2(2P--)(3P-Q(2P+W -(~-X)2(24~3-8Xp2-6A2~+3A3)}e~X’ -~{A&L-A)(~~-X)~(~~~.-X)I -(2~-A)(3~-A)(2~3-A~z-33h2~+3A3)}e~(P+A)’ +2~h(3~-h)(2~3-66X~+3A2)e-2P’ -3X( p - A)(Zp - X)2(3p - A) e-(2P+h)r +6pA*(p-A)(2~-h)e-~~‘],

(29)

KINETICS OF CELL-MEDIATED P(2,3,t)=

CYTOTOXICITY

179

h ~P(P--)~(~cL-~~(~cL-~ x[{p'(p-X)2(2p-A)2(3p-h)t2 -2p(p-A)(2p-h)(3p-A)(2pz-3hp+222)t +12p5-40Ap4+73X2p3-59h3p2+21h4p-3A')AeF"' -2p(2p-A)2(3p-X)(p2-3iip+332)e-pr

and P(3,3,t)=l-P(0,3,t)-P(1,3,t)-P(2,3,t).

(31)

When the disintegration process contains more than one stage, i.e., c > 1, or when the conjugate contains three or more target cells, i.e., n > 3, the computations become very tedious. Consequently, we have used the symbolic-manipulation computer program MACSYMA[21] implemented on a VAX computer to carry out the computations for c = 2 and produce a FORTRAN subroutine containing the expressions for P( k, n, t). Our MACSYMA program is listed in Table 3. For c = 3 we could evaluate the recursion relation using MIACSYMA,but found the resulting expressions too complex to utilize for subsequent model fitting via nonlinear least squares. We thus recommend the use of purely numerical methods for evaluating the recursion relation when c>, 3. A comparison of theory with experimental data obtained by Zagury et al. [35] on the kinetics of lysis in LT, and LT, conjugates is given in Figure 3. Details of the parameter estimation and least-squares fitting procedures are given in [19]. Here we simply point out that reasonable agreement to data can be obtained with our stochastic model. As discussed in [19], we believe that much of the discrepancy between the theoretical curves and the data points in Figure 3 is due to the design of Zagury’s experiment, which was not intended for collecting data to test a quantitative theory. Based on the data shown in Figure 3, Zagury et al. [35] conclude that approximately 30-min time intervals separate the lytic events in LT, con-

180

ALAN

S. PERELSON

AND CATHERINE

A. MACKEN

TABLE 3 MACSYMA

Program

for Solving

the Recursion

Relation

(22)”

HIT(T):=L*EXP(-L*T)S POISCOl(T):=l$ POISCIl(T):=POISCI-1l~T)*MU*T/I$ LYSIS(T):=EXP(-MU*T)*SUM(POISCIl~T),I,O,C-11s INTEGlCI,Kl(T):=IF ELSE

I>(K-I)

THEN

0

(INT:RISCH(LYSIS(T-H)*SPROBCI,K-ll(T-H) *HIT(H),H),

UPPER:LIMIT(INT,H,T), LOWER:LIMIT(INT,H,O), RATSIMP(UPPER-LOWER))$ INTEGZCI,Kl(T):=IF ELSE

I<1

THEN

EXP(-L*T)

(INT:RISCH((~-LYSIS(T-H))*~PR~B[I-I,K-~I(T-H)*HIT(H),H),

UPPER:LIMIT(INT,H,T), LOWER:LIMIT(INT,H,O), RATSIMP(UPPER-LOWER))$ PROBCO,Ol(T):=l$ PROBCI,Kl(T):=INTEG1~I,Kl(T)+INTEGZCI,Kl~T~$ SPROBCI,Kl(T):=RATSIMP(PROBCI,Kl~T))$ CUMPEI,Kl(T):=RATSIMP(SUM(SPROBCJ,Kl(T),J,I,K))$ NUMER:TRUE$ ASSUME(T>O)$ DECLARE(PROB,INCREASING)$ DECLARE(SPROB,INCREASING)$

“To run

the program

PROBCI,Kl(T),

one

i.e.,

assigns

a value

to c and

then

asks

the program

to evaluate

P(i, k,t).

jugates. From our theory we can derive general expressions for the expected time of lysis of the target cells in an LT,, conjugate. In [19] we show that by using our best estimates of the parameters X and p we predict approximately 30-min intervals between lytic events. In the next section we illustrate the computation of these expected values. C.

EXPECTED

TIME OF TARGET-CELL

LYSIS

Let T, be a random variable representing the time at which the i th lethally hit target cell in an LT, conjugate lyses. According to the two-stage model, the expected value of T, is

E(T,)=E(H,+H*+

---H,+n,)=++;.

(32)

It is difficult to determine the order in which the target cells of an LT,, conjugate are hit. One usually observes only the time at which the i th lytic

KINETICS

OF CELL-MEDIATED 1.0

181

CYTOTOXICITY

I

_ (a)

,

I

I

I

I

I

t

LT2

I

I

I

I

60

40

I

I

00

I

I

00 Time

FIG. 3. experiments,

,

100

I

I loo

120

I

120

(min)

The results of a least-squares fit to the data of Zagury et al. [35]. In their 200 LT, and 200 LT, conjugates were incubated at 37°C. At regular intervals,

each conjugate was examined by phase-contrast microscopy, and the presence of at least one (i = 1). at least two (i = 2). or three (I = 3) lysed target cells noted. In the experiment not all conjugates contained lymphocytes which were cytotoxic in the assay. We therefore introduced an additional parameter, f,,, the fraction of conjugates that contain a viable CTL. The predicted fraction of L7;, conjugates with k target cells lysed by time t therefore becomes i,P(k, n,r). Lethal hitting was assumed to occur at random at rate X,,, as determined by Perelson and Bell [26]. Disintegration was assumed to occur at random with the distribution of disintegration times given by a gamma distribution with c = 2. (a) LT2 conjugates, A, = 0.33 min i, pLz= 0.046 min-‘, and fz = 0.91; (b) LT, conjugates, h, = with permission from Macken 0.16 min ‘, p,, = 0.045 min i, and f3 = 0.94. Reproduced and Perelson [19].

ALAN S. PERELSON AND CATHERINE

182

A. MACKEN

event occurs, i = 1,2,. . , n. Let ql,, i = 1,. . . , n, be the ordered times of target cell lysis, i.e., 7;,, is a random variable representing the time at which the i th lytic event is observed. Thus the sequence qi,, T&, . ,7; ,,) is simply the set of lysis times ( Ti, T2,. . , T, } arranged in increasing order. The probability that the i th lytic event has occurred by time t is easily obtained from the results of the previous section. If the i th lytic event is observed by time t, then at least i target cells have lysed by time t. Thus 2 P(k,n,t). X=1

Pr(7;,,Gt)=

(33)

To obtain the expected value of T,,, we note that 1251 (34)

E(7;,,)=JoDPr(7;,,>t)dr. provided

the integral converges. Thus,

where P( k, n, t) is given by the recursion of Equation (22). We need to compute only n - 1 expected lysis times from Equation (35). The n th expected lysis time can be determined as follows. The set { T,,, , T(,, , . , T(,,, } contains the same elements as the set of unordered lysis times, {T,,T, ,..., T,,}. Thus,

2 E(T,,,)=E

2

T~~~)=~~~lT)=,S~(T)~

[,=l

(36)

r=l

Substituting

Equation

(32) for E(T,), one finds

k E(T,,,)= “;^+l) +y, r=l

and thus n(n+l) E(h)=

2x

12+;-

c

1

E(q,,j.

(37)

r=l

To illustrate the usefulness of Equation (37) consider an LT, conjugate with Tl and T, being the times of lysis of the first-hit and second-hit target cells. The ordered lysis times are TcI, = min(T,, T,) and T(,, = max(T,, T,).

KINETICS

OF CELL-MEDIATED

CYTOTOXICITY

183

One would expect I?(?,,) < E(T,) = l/A + c/p, because the mean of the minimum of the lysis times should be less than the mean of T, (Tl not always being less than T,). Similarly, one expects E(T,,,) > E(T2) = 2/h + c/p. Thus we write E(7;,,)=;+;-&(c). where E(C) is a function implies

(38)

to be determined.

Once F(C) is found, Equation

(37)

(39) Now, by direct application

of Equation

(35) we compute h

i+‘EL

E(?;,,)=f%(0,2,t)df=

2cL(II+X)’

‘+‘_V4p+3V

0

/

h

/J

c=l

’ (40)

c= 2.

4&+#’

Evidently, for c = 1 and 2, e(c) > 0, which implies E(T,,,) < E( Tl) and E( q2)) > E( T2) as expected. For LT, conjugates, similar results hold. Again it is convenient to express E(T,,,) in terms of E(T), i.e., let

and then by Equation

(37) E(7;,,)=;+$+e,(c)+e,(c).

By integrating find

P(O,3, t) and P(1,3, t) and substituting

(41) into Equation

h(3/~ +2A) Et(l) =

~P(P+W~P+V’ X (216~~ + 549Aj~~ + 504A2$ + 197X3~ + 28h4)

El (2)

=

27/.~(/.~+ A)2(2p + h)3 P(4/.L + X) E2 (1)

= ~P(cL+~)~(~P++

(35) we

184

ALAN

S. PERELSON

AND CATHERINE

A. MACKEN

and &z(2)=

h2(3p+A)(192p3+260Xp2+114h2p+19A3) 108p(p+h)3(2p+A)3

(42)

In [19] these formulas are used to show that with the parameter values characterizing Zagury’s experiment, E(7;,,), E( ?&), and E(T,,,) are approximately 30 min, 60 min, and 90 min. These results were quite surprising, because we found that in LT, conjugates the average time between lethal hits was of order 5 min. One might expect that cells all hit within a short time of each other would lyse within a short time of each other. The reason our model predicts such long intervals between lysis of target cells is that disintegration times exhibit high variability due to the long-tailed exponential distributions of each of the c events in the disintegration process. 5.

DETERMINISTIC

MODELS

If large populations of aggregates are studied, the variability in the proportion of conjugates with k out of n targets lysed by time t is very low. The mean behavior is therefore an adequate description of the process. In this case, deterministic methods, which generally only model the mean, are commonly employed. One should, however, be cautioned that there are cases in which deterministic methods do not even predict the mean correctly (cf. [7]). However, here we show that for certain classes of multistage models, deterministic methods predict not only the mean, but also the complete probability distribution P(k, n, t). As we shall see, the difficulty with the deterministic approach is that large numbers of differential equations have to be solved. However, once these equations have been solved, we will have gained detailed dynamic information about the changes in the state of LT,, conjugates during cytolysis. The deterministic models that we develop are of a quite different character than our previous stochastic models. The deterministic models might be described as being developed from the “bottom up.” For an L7;, conjugate we define a set of basic states so that a transition between states consists of either a hit or a disintegration, but not both. We then construct models for the transition rates between states and write a differential equation for the fraction of conjugates in each state at time t. From the solution of these state equations we determine quantities of interest, such as the fraction of L7;, conjugates with exactly k target cells lysed at time t. Because the evolution of the system only depends on its current state, the models that we develop are equivalent to continuous-time discrete-state Markov processes (i.e., continuous-time Markov chains). Further, because our variables are the fraction of conjugates out of an infinite population that are in a given state, we are in essence computing the probability of being in a given state. Hence it is not

KINETICS OF CELL-MEDIATED

185

CYTOTOXICITY

surprising that our deterministic models give the same results as the stochastic models developed in earlier sections. To contrast this “bottom up” approach with the stochastic model of Section 4, we note that in developing the stochastic model we were only interested in predicting the experimentally observed quantity P( k, n, t). Thus, we took a “top down” approach and by the use of conditional probabilities introduced the minimum amount of detailed information about the state of a conjugate during the lytic process. As will become obvious from the treatment to follow, we could have adopted a “bottom up” Markov-chain approach to derive our stochastic results. To illustrate our method, we begin by analyzing LT, conjugates. Later we consider more complex conjugates. A.

LT, CONJUGA

TES

Consider a population of LT, conjugates. Let xi, x2, and x3 be the fraction of conjugates in which the target cell is unhit, lethally hit, and lysed, respectively. If X is the rate of lethal hitting and p the rate of disintegration of lethally hit target cells, then

CL --Ax,,

(434

dx,_-Ax,-w,,

(43b)

dt

dt

and x3 is determined

by the condition x,+x,+x,=1.

Initially,

all conjugates

are unhit, so

Xl(O)=l, Solving Equation

(43c)

x2

(0)

=

0,

and

x3 (0) = 0.

(43) with the above initial conditions,

x1( 1) = e-X’,

(44)

we find

(454

p(eFp’-epA’), x*(t)=&

and he-“’

+(t)=l-

-

pe-At

A_

P

(45c)

186

ALAN

S. PERELSON

AND CATHERINE

A. MACKEN

To compare this result with our previous stochastic model, notice that x3(t), the fraction or relative frequency of lysed conjugates in an infinitely large population, is just P(l,l, t), the probability that one target cell in an LT, conjugate has lysed by time t. Likewise x,(t)+ x2(t) is just P(O,l, t). Comparing Equations (45) and (24), we find that the deterministic and stochastic models give the same formulas for P(l,l, t) and P(O,l, t). As in the case of our stochastic models, we can allow target-cell disintegration to be a multistage process. We only need to replace x3 by the variables where c is the number of stages in the disintegration X~,X~,...,XC+~, process, x, + 2 is the fraction of conjugates with target cells having undergone i stages in the disintegration process, and x,+~ is the fraction of cells lysed. Moreover, now we can easily allow each step in the disintegration process to occur at a distinct rate, p,, i = 1,2,. . . , c. Thus,

dx, -=_ dt

XX,,

dx2_ - xx,dt

PlX,,

J&J_ -P,-2xi-1dt

(46) Pi-lx,,

i=3 ,...,c+l,

-dx,.+z=/.&,x(.+1. dt

With x1 (0) = 1 and x, (0) = 0, 1~ j < c + 2, the Laplace transform tion (46) is

of Equa-

x, = - 1 S+X’ %=

(s+x)(s+~;;~+d~~~~,(s+a,~l i=2 ,...,c+l,

(47)

x

hw2 c+2=

. . . PC

S(S+h)(s+111)(S+t2)...(S+~L,)’

where X(S) is the Laplace transform of x(t). When all the disintegration rates are the same (i.e., p, = p for all i), Equation (47) is easy to inverse-Laplace-transform [29]. For this case we find

(48)

KINETICS OF CELL-MEDIATED

CYTOTOXICITY

187

Comparing Equation (48) with the value of P(l,l, t) given by Equation (22), we see they are precisely the same. One advantage of the differential equation method is that it is simple to generalize the disintegration model into one in which each of the i steps, c, occurs with a potentially different rate CL,. In fitting our i =1,2,..., stochastic multistage disintegration model to data, we found that with p, = p for all i, c = 2 generally gave the best agreement of theory with data [19, 271. For the case c = 2, with pi # p2, x4( z), the fraction of LT, conjugates lysed by time t, is given by

(49) In future analyses of experimental data, it will be of interest to see if this formula gives better agreement than the limiting form of this equation with k1= P2 = CL,viz.,

x4(t)

B.

=l-

2 -Xr+{Xll(h-~))t--2h~+$2}e-” P e (A - PI2

’ (50)

L7;, CONJUGATES

When more than one target cell is present in a conjugate, deterministic models become complex. Each target cell can be in any of c + 2 states. The number of states of an LT, conjugate is the number of ways of choosing n objects from c + 2 types with repetition, of

’ +(c+2)-1 . Thus a total n 1 ( equations would be needed to dei.e.,

n ‘i ” -1 ordinary differential ( ) scribe the fraction of conjugates in each state. In order to illustrate our method of modeling LT, conjugates we shall only consider LT, conjugates with c = 1, the generalization to larger conjugates and higher values of c being obvious. Let x,, be the fraction of LT, conjugates with one target cell in state i and the other target cell in state j, where, as before, state 1 = unhit, state 2 = hit, state 3 = lysed. Experimentalists generally do not distinguish the target cells of a conjugate. (The matrix with elements x,, is therefore symmetric, and we need only consider the “upper triangular” portion with i < j.) The six states of an LT, conjugate and the transition rates between them are illustrated in Figure 4. Because lethal hits are assumed to be delivered sequentially to the target cells in an LT, conjugate, only one target cell at a time is at risk, and hence the transition rate between states (11) and

188

ALAN

S. PERELSON

x XII

AND CATHERINE

A. MACKEN

/J

-x12

-

Xl3 I whit I lysed

I unhlt I hit

2 unhit

L

x22 2 hit

‘23 I hit I lyscd

2 lysed FIG. 4. State transition diagram for an LT, conjugate. The fraction of conjugates in state v is represented as x,,, where state 1 = unhit. state 2 = hit, and state 3 = lysed. Hits are assumed to be given sequentially at rate X. With only one cell at risk at a given time, the is assumed to occur at rate transition rate between xtt and xtz is h, not 2X. Disintegration p per hit target cell. Thus sz2 decays

at rate 2~~ whereas

.xzi decays

at rate p.

(12) is A. Once hit, target cells all undergo the disintegration process simultaneously. Hence, both target cells in state (22) are disintegrating at rate II, and the net transition rate to state (23) is 2~. From Figure 4 we construct the following differential-equation model:

dx,,

hx

-=_

11,

dt dx,, --=h-(A-tCL)x12,

dt

63 dx,, _ dt dx,3 ~

dt

=

(51) h2 -2PXn3 2px22

+

Ax,,

-

PX23

9

-dx,, =px*3. dt Solving

these equations

by Laplace-transform

methods,

with

the initial

KINETICS

OF CELL-MEDIATED

condition

x,,(O) = 1, x,,(O) = 0 for all i, j with ij > 2, we find

xll(

CYTOTOXICITY

189

t) = epX’,

X12(t)

(524

= :em”‘(l-

e-p’),

(52b)

x,3(t)=$ee”‘(pt-l+ep”),

(524

e-w

e-hr

x*2(t) = x2

e-(“+h)t

P(2P-A) + (2P-h)(P---X) - FL(PL-A)I ’

A

x23(t)= ~ P-X

P-A

P

i

2hem2@

2p-x

2$ - 2h/J + A2

1

+ t-

(p-2A)epp’

(p+h)ep(p+X”_

- /L(p-A)(2p-A)

1 (524

1 XePX’,

X33(t)=l_h~~~)‘+h(p-2h):-“‘+

(5-W

(p_h;l;;;;_A)

(P--h)

[

1

+3x*/l - A3 l ~/.&At e-hr P-h (P4A2h-h?

2p3 -

_

5h$

(52f)

Notice that Equation (52f) and Equation (27) for P(2,2, t) when c = 1 are identical. Once again the deterministic and stochastic methods give the same results. However, as formulated here, the differential-equation model contains more information about the system. Thus, for example, while our stochastic model predicts P(1,2, t), the fraction of conjugates with one cell lysed, the differential-equation model tells us what proportion of those conjugates with one cell lysed also contain an unhit target or a hit target. In an infinite-sized population, a proportion may be interpreted as a probability. Therefore

and

190

ALAN S. PERELSON AND CATHERINE

A. MACKEN

Thus, given the solution to our deterministic model, we can recover the solution to the stochastic model, but not vice versa. As noted earlier, one could generate a Markov-chain model in which the state of each target cell in a conjugate is followed. Such a model would lead to a set of forward Chapman-Kolmogorov equations which are analogous to the differential equations of our deterministic model (cf. [18]). In comparing our deterministic and stochastic models one should keep in mind one additional feature of the stochastic model. As developed in Section 4, the stochastic model allows us the freedom to choose arbitrary probability distributions for the hitting and disintegration times, H and D. Thus, in some sense it is fortuitous that we chose for D a gamma (p, c) distribution, with c an integer, the distribution which results from c sequential exponential (p) processes. With this choice, disintegration is easily represented in a deterministic model by c sequential linear state transitions. If we had chosen for D a gamma distribution with a noninteger shape parameter, then one could not construct in an uncontrived manner a deterministic model involving parameters h and p which would give precisely the same P( k, n, t) as the stochastic model. Our top-down stochastic approach allows one to test very general distributions for the hitting and disintegration times by fitting models to data. When distributions are found that adequately describe the kinetics of the hitting and disintegration processes, one can attempt to open the black box that these distributions represent and interpret these processes in a mechanistic manner.

6.

CONCLUDING

REMARKS

We have developed multistage stochastic and deterministic models of the processes by which lymphocytes kill virally infected or otherwise altered target cells to which they are attached. Our models are based upon the observation that there is a critical period of time during which contact between a CTL and a target cell is absolutely necessary for eventual cell lysis. In a population of CTL-target-cell conjugates the length of this critical period is seen to be variable. During the critical period we suppose that the CTL delivers a lethal hit to the target cell, with the exact timing of the event being random. By definition, once a lethal hit has been received by the target cell, the target cell will eventually go on to lyse, even if separated from the CTL. Operationally, the lethal hit may be a single discrete event or the result of an accumulative process, which when it reaches a threshold value leads to eventual death of the cell. By defining the state of a conjugate containing n target cells to be the number of target cells hit by time t, we developed a pure birth-process model of lethal hitting. Assuming either (a) all target cells in a conjugate were simultaneously at risk of being attacked or (b) only one target cell at a time

KINETICS OF CELL-MEDIATED

CYTOTOXICITY

191

was at risk, and then comparing the model’s predictions with data, we were able to rule out the simultaneous-risk model. There is an increasing amount of recent data suggesting that lethal hitting involves the Ca++-dependent secretion of a mediator by the CTL [6, 13, 16, 381. If such a putative mediator were secreted into the medium surrounding the CTL, all target cells in the vicinity, whether bound to the CTL or not, would be affected-contrary to observation [34] and to our analysis. Thus it would seem that any putative mediator would necessarily have to be secreted only at the contact region or in the intercellular space between the CTL and a single target cell. Building upon the assumptions that (1) hitting occurs sequentially, and (2) a target cell begins disintegrating immediately after being hit, we were able to develop a two-stage stochastic model that mimics the observed kinetics of target-cell lysis. We discovered that a number of choices could be made for the probability distribution of disintegration times. Given currently available data, we do not believe that one can uniquely assign a distribution to these times, although we found that a gamma distribution, which derives from sequential exponential processes, fits experimental data much more closely than an exponential distribution. Hence, we deduce that the disintegration process is a complex series of biochemical events, whose kinetics can be roughly described by a distribution such as the gamma. Although stochastic models allow one to calculate variability in experimental results, for many purposes it is simplier to deal with deterministic models. In Section 5 we showed how one could formulate ordinary-differential-equation (0.d.e.) models of a process of sequential hitting followed by simultaneous disintegration of all lethally hit target cells. Our o.d.e. models implicitly assume that an infinite population of conjugates are being studied so that one can justifiably use the continuous variable x, representing the fraction of conjugates in state i. For the class of two-stage models in which hitting times are distributed exponentially and disintegration times are distributed as a sum of exponent& we were able to show an exact correspondence between the stochastic and deterministic models. For other choices of the disintegration-time distribution this might not be the case. Much remains to be done. The two-stage model we have proposed is sufficiently general that is should apply to killing by other types of host defense cells, for example NK cells and macrophages. At present we are engaged in comparing the model to data on NK-mediated killing. Some assumptions in the model may be overly restrictive and need to be relaxed. For example, we have assumed the CTL delivers a single lethal hit and then the target cell disintegrates in a killer-cell-independent process. However, some recent data [27,32] suggest that the continued presence of the CTL may speed up the kinetics of lysis, perhaps as a result of further hits to the target cell. Extending our theory to incorporate multiple hits leads to interesting and challenging stochastic models [20].

192

ALAN S. PERELSON AND CATHERINE

A. MACKEN

APPENDIX We prove that the solution to Equation (1) with A,,(m, t) = (1- m/n)A, is given by Equation (4). When m = 0, we have A,(O, t) = A,, and Equation (la) has solution P,(O, t) = emXn’. Thus Equation (4) is correct for m = 0. Now assume P,, (m, t) is given by Equation (4) for 1~ m < n - 1. Then from Equation (lb) p, ( m,

t) = e-(1 -m/nW

x(1_

le(l ~m/nh,T~ J0

to yield

hm

P,(m,t)=+-

n-m+1 n

e-Lqmp*exp

which can be rearranged

n( n-~+l)(mTl)

- yA,t)jde-““‘/“(l_

e&T/!1)“‘~1d7,

Because I

e-h,7/fl(l_

e-“.‘/“)nl-ldr=&(l_

/0

e-h+‘)m,

(A.11

n

one recovers Equation (4). All that remains is to show that Equation correct when m = n. In this case Equation (lc) yields

which upon use of Equation

(4) is

(A.l) becomes P,,(n,t)=l-e-“n’.

Thus Equation

(4) is valid for m = n, completing

the proof.

A.S.P. is the recipient of N.I.H. Research Career Development Award 5 KO4AIOO4.50-5. We thank C. Wofsy for critically reading and commenting on the manuscript. C.A.M. would like to thank G. Bell and the Theoretical Biology and Biophysics Group at Los Alamos for their hospitality.

KINETICS

OF CELL-MEDIATED

193

CYTOTOXICITY

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