A threshold model of antigen-antibody dynamics

J. theor. Biol. (1977) 65,499-512

A Threshold Model of Antigen-Antibody

Dynamics

PAUL WALTMAN? The University

of Zowa, Iowa City, Zowa 52242, U.S.A.

AND E. BUTZ~ Okanagan College, Kelowna, British Coiumbia, Canada and The University

of Iowa, 52242, U.S.A.

(Received 28 October 1975, and in revisedform

1 June 1976)

A mathematical model of antigen stimulated, antibody production is derived. The model uses threshold criteria to initiate proliferation of lymphocytes and to initiate antibody production (i.e., to indicate termination of differentiation of the lymphocytes). These thresholds introduce time lags into the differential equations. The qualitative behavior of the model is illustrated by numerical computations. These computations show that the phenomena of low level tolerance and of the anamnestic response to a second challenge are a natural consequence of the model. A proof that the resulting mathematical problem is well posed is also included. 1. Introduction

The complex sequence of events which comprises the mammalian response to antigen challenge has undergone considerable investigation in recent years. In particular the mechanism of antibody production has been clarified considerably. Although there are still some fundamental, unanswered questions, enough is known to attempt to formulate qualitative models. Several such models (Bell, 1970, 1971a,b; Richter, 1975; Hoffman, 1975) exist. The present paper also formulates a model of the antigen stimulated, antibody response although the focus here is on the narrow question of the t Research supported by Public Health !bvice Grant IROlCA18639-01 from the National Cancer Institute and by National ScienceFoundation grant BMS74-18648 from Division of Regulatory Biology. 499

500

P.

WALTMAN

AND

E.

BUT2

technical and abstract difficulties of modeling the thresholds which are such an obvious part of the immune response. How one models the threshold events sets the tone for the rest of the theory. In his pioneering work, Bell (1970,197lu) uses the familiar S-shaped function kx/l +x to reflect thresholds. However, this only weakly reflects the known character of threshold events. Borrowing from the theory of epidemics (Cooke, 1967; Hoppensteadt & Waltman, 1970, 1971) we represent the threshold phenomena by functional integral constraints. While our representation of thresholds in the immune response represents a radical departure from traditional modeling techniques, our representation of the remainder of the immune system is conservative-all of the immune events we consider are part of the current dogma. Thus our work is intended to complement rather than compete with existing theories of Bell, Hoffman, Richter (lot. cit.). There are other types of immune models in the literature. For example, Bell (1973) considered a growing antigen modelled as a predator-prey system. This study has been continued by Pimbley (1974a,b,c) and by Hsu & Kazarinoff (1976). There is a statistically based model of Bruni, Giovenco, Koch & Strom (1976). The work of Cohen (1970, 1971) and Cohen & Milgrom (1971) contains the idea of a threshold, particularly with a view towards tolerance. Also focusing on a model to include tolerance is the work of Coutinho & Mijller (1974) and Cohn & Blomberg (1975). Although differential equations are not utilized in the last few papers cited above, perhaps some of the ideas there could be expressed in mathematical terms. Functionally it is apparent that the immune system is an adaptive control system. Our fundamental (long term) research objective is to explore this point of view, exploiting the very extensive developments in modern control theory. Our optimism that such an approach will contribute to a basic understanding of immunological events stems from the observation that the immunologists and the adaptive control theorists deal with the same fundamental problem: how to maintain the functional integrity of a system in the face of unpredictable changes in the system’s environment. Before control theory can be utilized, however, the points at which the system switches from one state to a qualitatively different state (these are the potential control points) must be identified. The point of this paper is to provide a kinetic description of the antibody production mechanism which identifies these points via thresholds. One of the thresholds used also provides the “immunological memory” necessary to explain the rapid response to a second challenge by the same antigen. The use of terms like “thresholds” and “memory”, suggest a similarity with neurophysiological phenomena. The viewpoint that .neurophysiology and immunology share a fundamental mechanism has been discussed by

A THRESHOLD

MODEL

OF

ANTIGEN-ANTIBODY

DYNAMICS

501

Jerne (1973, 1974, 1975). Although we do not specifically consider a network model, the proposed dynamics are consistent with this viewpoint and a network model, using the type of thresholds considered here, is certainly possible. 2. Immune Events and the Biological Model Since our primary interest is in the mathematical description of threshold events, we have chosen to imbed these phenomena in a relatively simple theory of the immune response to antigen challenge. As a minimal or skeletal immune model we consider three kinds of populations; antigen molecules, antibody molecules and cellular lymphocytes. Antigen is assumed to initially interact with molecular receptors on the cellular surface of a lymphocyte. After some threshold event on the lymphocyte surface membrane, the lymphocytes are assumed triggered into mitotic proliferation. The lymphocytes are assumed to continue dividing until some appropriate population level is attained; i.e. another threshold is exceeded, at which time the lymphocytes are assumed triggered to synthesize and secrete free antibody molecules. The latter then interact with the remaining free antigen, usually in sufficient quantity to eliminate the original antigen population. In actuality the immune response is considerably more complicated than what we have described and it is appropriate to comment on some of the omissions in the model. First of all, we consider the antigen to be “sufficiently processed”, meaning that whatever macrophage, T-cell interaction, cross linkage, etc., that is required has already been accomplished. The T-cell : B-cell interaction which is known to be required in many cases is the subject of much current research but is not fully understood. Secondly, we have treated the lymphocyte both as a “memory” cell and a “plasma” (antibody secreting) cell, marking the completion of differentiation by a threshold, but ignoring the fundamental distinction in roles. Also we have ignored, but could have included, equilibration time and initial catabolism of the antigen [Weigle, 1967 (chapter 2)]. These complicate the model but do not add to the point we wish to make. It may also be noted that a T-cell : B-cell interaction is emphasized in Hoffman’s (1975) mathematical model whereas in the model by Bell (1970,1971), a distinction between plasma and memory cells is maintained. Therefore it is possible to construct a mega-model, even now, which more accurately reflects the details of the immune mechanism. However, the analysis of such a mathematical model would appear intractable. An excellent account of the known properties of the immune response to antigen challenge can be found in the review article of Nossal (1973).

502

P.

WALTMAN

AND

3. The Mathematical The immune

E.

BUTZ

Model

model we consider is schematized in Fig. 1.

Antigen -

threshold

The dynamic model corresponding to Fig. 1 is constructed using the familiar concepts of chemical kinetics and birth and death processes. Antigen (Ag) is assumed to react with free receptor sites (R) on the surface of the B-cell lymphocyte resulting in an antigen-receptor bound complex (Ag - R). Therefore, Ag+R+Ag-R where k, and kd are the association and disassociation constants, respectively. The time of triggering of the B-cell into mitotic proliferation is assumed to be (for the moment) an arbitrary function of free antigen, free receptors, and antigen bound to receptors. Assuming that a lymphocyte which reaches the proliferation threshold at time t was initially stimulated at time z,(t) in the past, we have jf,(Ag,R,Ag-R)du

= 8,.

(2)

For the general model we shall put only mathematical restrictions on the functionf,. It clearly should involve the proportion of receptors which are bound with antigen, but one might also want to include some dependence on the total amount of antigen present or some sensitivity to the rate at which receptors are being occupied. Further discussion of this point will appear in section 5 where a specific function is chosen for a numerical illustration. Determining rl(t) is necessary since a realistic description of the proliferation kinetics at time t must be dependent on antigen and lymphocyte concentrations at a time in the past when the approach to threshold was initiated. That is, if Ag(r) and L(z) are the antigen and lymphocyte concentrations at 7 = z(t) in the past, then the proliferation “reaction” in the immune model may be written symbolically as, Ag(z) + L(z) -2 c.&(t) where c1 is simply a mitotic amplification threshold prerequisite for proliferation.

factor and /c

(3) indicates the

A THRESHOLD

MODEL

OF

ANTIGEN-ANTIBODY

DYNAMICS

503

Proliferation of lymphocytes is assumed to continue until some threshold number, &, of B-cells is reached at which time the lymphocytes (which at this stage are referred to as plasma cells) begin to secrete free antibody molecules (Ab). The duration of this proliferation phase may be determined from the time integral,

j+) fiCW1 du = ‘4

(4)

where againf, is (for the moment) an arbitrary function. The determination of z2(t) is necessary in order to describe the next stage in the model, the kinetics of antibody proliferation. Again we assume that the presence of a threshold mechanism means that proliferation rates at time t will depend on antigen and lymphocyte levels at a time in the past when the approach to threshold was initiated. Symbolically, this may be represented as Ag(z) + L(z) 3 FAb(t)

(5)

where /3 is an antibody amplification factor. The final phase in the model schematized in Fig. 1 is the interaction of free antibody with free antigen to form an antigen-antibody complex (Ag -MI) according to the kinetic relation, Ag+Ab

+ Ag-Ab

(61

where k: and ki are appropriate association and dissociation rates, respectively. We now translate the above kinetic relationships into mathematical relationships. Let x(t) = concentration y(t) = concentration z(t) = concentration

of free antigen molecules at time t of free receptor molecules at time t of free antibody molecules at time t,

i.e. z(t) = Ah(t). y(t) = NO, A reasonable assumption in equation (1) is that the reaction is essentially irreversible (& = 0), in which case we have that $0 = &(t),

dx Tit- - -4)Y(O,

0 I t I tz,

x(0) = xg where we have written r for k,, where t, is the time at which antibody is

504

P.

WALTMAN

AND

E. BUT2

first available, and where x,-, is the initial antigen concentration. If we let of antigen bound to receptor molecules, w = Ag -R in the above discussion, then equation (1) yields a similar relationship for w(t), dw - = rw(t)y(t), 0 I t < 03, (8) dt w(0) = 0. w(t) be the concentration

w(t), which gives the bound receptor concentration, together with y(t) determine the total concentration of lymphocytes, L(t). We write equation (2) as

1 hCx(~)~Y(S), WI ds = ml.

(9)

If t, is the instant of initial lymphocyte proliferation, we take Tl(t) z 0, t < tl. The equation for free receptors, y(t), from the kinetic relations (1) and (3), is given by dy -dt = - rx(Oy(t) + crrxC~:21(t>luCzl(t>lHCt - hl

(10)

Y(O) = Yo, where H(t - tl) formally expresses the fact that no receptor population growth is possible until initial proliferation at time tl, i.e. H(t) = 0, t c 0 and H(t) = 1, t 2 0. The time delays associated with the second threshold mechanism may be determined from equation (4) where we define T2(t) = 0, t 5 t2 and t2 is the initial instant of free antibody secretion. Specifically, one has &f2C~(~)+w(sll

ds = m2.

(11)

Finally, the population dynamics of free antibody follow from the kinetic relationships (5) and (6). In the latter relation we again assume an irreversible reaction (/CA= 0). Therefore, dz -dt = - =4)z(0 z(0) = 0

+ B~~C~z(t)lYC~z(t)l~Ct - tzl

(11’)

where k: = s and where H[t - t2] explicitly indicates that no antibody is freely secreted until t = t2. The rate of decay of free antigen, x(t), given by equation (7) must now be modified to incorporate elimination by interaction with free antibody.

A THRESHOLD

MODEL

OF

ANTIGEN-ANTIBODY

DYNAMICS

505

From equation (6), with ki = 0, we then have that dx - = - rx(t)y(t)-sx(t)z(t), 0 I t < co, dt x(0)

= 0.

Another modification which must be made to this system of equations is to include a term in equation (11’) which simulates the spontaneous decay of the free antibody which does not interact with antigen. A simple approximation to this situation may be achieved by assuming that the rate of antibody decay is proportional to the amount present. That is, the term, - yz(t), must be added to the right-hand side of equation (1 l’), where y can be determined from the half-life of a free antibody molecule. Thus equation (11’) becomes dz (13) -dt = - smm f BrxC~*(t)luC~z(t>l~(t- t2) - YZW, z(0) = 0. Note also that z(0) = 0 implies z(t) E 0, 0 I t I t2. The six equations (8)-(13) constitute our model. 4. Mathematical

Preliminaries

The system of equations x’(t) = - rx(t)y(t)

Y’(t) = - rdMO+

- sx(t)z(t),

x(0) = x0

~rxCzl(t>lu[~~(t>lH(t- tJ,

Y(O) = Yo

z’(t) = -sx(t)z(t)-yz(t)+Jh[~(t)]y[~~(t)]H(t-

tz),

z(0) = 0

w(0) = 0

w’(t) = rxW(O,

~10) = 0, &MY(~)+

(14)

t 5 t,

WI ds = m2, t 2 tz m

= 0,

t I t2

where tl and tz are given by ~.flEx(s), [DAY

Y(S), w(s)] ds = ml

+ WI ds = mz

or by tl = + 03, tz = f 00, if no such ti are defined by the above integrals,

506

P.

WALTMAN

AND

E.

BUT2

were developed in the preceding section. By a solution of such a system we mean six continuous functions, x(t), y(t), z(t), w(t), zr(t), r2(t) defined on [0, oo), x(t) and w(t) differentiable there, y(t) differentiable on (0, tl), (f,, t2), (t2, co) and z(t) differentiable on (0, tJ, (tz, oo), such that if these functions are put into equation (15), an identity results. That such a problem is well posed mathematically, is not immediate because of the functional nature of the delays and forms the content of the following theorem. The proof, which is deferred to the end of the paper, depends on an early paper of Driver (1963). TJ~E~RI~M

1

Let r, s, 6 B, ml, m2, y be non-negative constants. Let R’ = [0, a~), fi : Rf x R’ x Rf --P R’ be continuous, locally Lipschitzian, and fi(x, y, w) >Oifx>0,y>0.Letf2:Rf-+Rf be continuous, locally Lipschitzian, and f2(5) > 0 if 5 > 0. Then there exists a unique solution of the system (14) which depends continuously on the initial conditions and parameters.

5. Qualitative

Properties of the Model

Very little is known about the process by which the target cells are triggered to proliferate (although “patch” formation in the work of De Lisi & Perelson (1975) suggests a way to provide a threshold mechanism). Thus the choice of a function fi(x, y, w) in the system (14) is somewhat arbitrary at this time. For present purposes we take fitx, Y, w) = k,xy+kzw

(15) where k, > 0, k, 2 0 (k, = 0 produces mathematical difficulties in the proof in section 6). Equation (15) states that the triggering mechanism depends on the amount of antigen bound to receptors [w(t)] and on the rate of such antigen binding (w’/r = xy). ki and k2 reflect the weight assigned to each, and for numerical computation we choose kl = k2 = 1.0. The choice of fi seems much more straightforward. We take f2(q) = q, i.e. define z2(t) by

y(s)+ w(s) is proportional to the total (bound and unbound) lymphocyte concentration. (It would also be possible, and reasonable, to let the threshold parameters, m, and m2, depend on the initial antigen concentration, x,,.) Having chosen fi and f2, the system (14) can be solved numerically. On [O, tJ, it is merely a system of ordinary differential equations; thereafter

A THRESHOLD

MODEL

OF

ANTIGEN-ANTIBODY

DYNAMICS

507

the system can be solved numerically by ordinary differential equations methods (we used a packaged fourth order Runge-Kutta) for sufficiently small stepsize, since both zl(t) and zz(t) are less than t. [See ahead, inequality (18). A similar inequality exists for am.] For the parameter values indicated in the caption of Fig. 2 (which were chosen for computational convenience and were not intended to be realistic), the system (14) was solved numerically on [0, 50.01. At t = 50.0, the program was stopped and antigen added (an amount equal to the antigen at time zero). The results are shown in Figs 2 and 3, depicting free antigen and free antibody, respectively. Antigen present at time zero decreases slowly until antibody first appears -a delay of 8.375 units-whereupon antigen-antibody complexes form, accounting for the rapid decrease of free antigen shown in Fig. 2 and the rapid increase in free antibody shortly thereafter. Free antibody reaches a peak at about 15 time units and begins exponential decay. When the second challenge is presented at 50 time units, the response is immediate, free antibody reaches its maximum concentration in less than three time units, and this maximum concentration is considerably higher than that of the first challenge, even though the same amount of antigen is involved. This follows, qualitatively, the known properties of the antigen driven, antibody response. The fact that rr(SO+O) # 0 provides all of the “immunological memory” necessary to initiate immediate proliferation when the antigen is presented for the second time.

I

10-33 10-a-

, 0

8

16

I

I

I

I

I

I

I

24

32

40 Time

4%

56

64

72

I

FIG. 2. A plot of the log of free antigen V~XE.I.IS time. A second chdenge of antibody Occurs at t = 50.0. Parameters are x0 = 5.0, ya = 1.0, ml = 15, ma = 15, r 4.02, s = o-05, dl = 10, p = 20, y = o-2.

P. WALTMAN

508

AND E. BUT2

1 IO.0

0 2E 0 s -I

\

I.0

10-l

6

10-2

~

0

I I6

I 24

I I 32 40 Time

I 48

I 56

I 64

FIG. 3. A plot of the log of free antibody versus time corresponding to tigure 2. Note the immediate response to the second challenge in contrast to the delay in the response to the first challenge.

6. Proof of the Basic Existence Theorem In this section a proof of Theorem 1 is given. The trigger times, t,, t2, are assumed to satisfy tI c &-this is the normal case but the proof can be modified to cover the other cases. Let x(t), y(t), w(t) be the unique solution of the initial value problem x’ = -rxy y’ = -my

(161

w’ = rxy x(0) = x0 > 0,

Y(O) = Yo ’ 0,

w(0) = 0,

valid for t =Z 0. x(t), y(t) satisfy x(t) = x0 exp

y(t)=yoexp

[

-riy(s)ds

[-r%x(s)drl/

1

>O >O

and since monotone decreasing, are bounded. Thus w(t) > 0, t > 0, and w’(t) ( M, some M, and solutions of equation (16) do extend to R’ as claimed. Since for such functions fi[x(t), y(t) w(t)] > 0,

is strictly monotone

increasing,

the integral

exists or properly

diverges

A

THRESHOLD

MODEL

OF

ANTIGEN-ANTIBODY

DYNAMICS

509

to sco. If

then the above functions and r,(t) = rz(t) = z(t) = 0, form a solution of the system (14) on [0, co). This would be a manifestation of low zone unresponsiveness. If there exists a number t, such that

Ifib(

Y(S), WI

ds = ml>

then the above functions are a soiution of the system on [0, tJ. First of all it will be shown that the solution can be extended to a larger interval [0, t, +h]. Let y*(y*) denote a lower (upper) bound for y(t) over [0, tJ. [In this case y, = y(tl) and y* = y(0) but in later arguments y may not be monotone.] Let I = Kw> Nl~W2 < x < 2% Y*/2 < Y < 2y*, 0 < w < Zw(t,)} and apriori restrict h < ml/y” where

Y* = SUP,f(X, Y, w) > 0. Consider the system x’ =-my y’ = - rxy + crrx(zl)y(rl) w’ = rxy

G = fib,

(17)

YTWM-liI4~1>,Y(d,

WWI,

with initial data Q(t) = 0, 0 5 t I tl 40 y(t)

= solution of equation (16),

0 < t _< t,.

w(t) 1 The right-hand side of equation (17), (x, y, w) E I, and the initial data satisfy all of the hypotheses of Driver (1963, Theorem 3) [the requisite Lipschitz condition on the initial data is satisfied since x(t), y(t), w(t) are all solutions of an ordinary differential equation]. Note that infr fi(x, y, 2) = y* > 0. Thus there exists a solution of equation (17) defined on [0, t, + h] for some h > 0. For this solution, [x(t), y(t), w(t)] E I, 0 I t I t, +h and hence z,(t) I t-ml/y*

< tl.

(18)

It is the inequality (18) that permits the numerical solution of the system of functional differential equations by ordinary differential equation techniques.

510

P.

WALTMAN

AND

E. BUT2

Let t2 = t, +h. Since x(tJ and y(tJ are positive, the above procedure may be repeated with tl replaced by t2 to generate a solution on [0, t3]. Continuing the argument generates a sequence (tn} and a solution of equation (15) valid on [0, tJ. One of the following three alternatives must hold.

(a) A point

t*

is reached such that ~szC~(0+~(01

dt =

m2.

(b) The extension procedure covers the real line and

~hCw(t)+~(Ol dt 5 mz. (c) There is a point T which cannot be reached i.e. supn tn < T

and

fn sup, sfi[y(t)+

w(t)] dt c nr2.

0

If alternative (b) occurs, the problem is uniquely solved, and no antibody production occurs [z(t) z O]. If alternative (a) occurs, the problem is solved on [0, t *], and the equations must be reformulated to now take into account the variables z(t) and the and the additional delay z2(t). This is discussed below. We now show that (c) is not possible. If(c) occurs then there exists a maximal right open interval, which without loss of generality we take to be [0, T), on which a solution x(t), y(t), zl(t) exists. Since z,(t) is non-decreasing, (r; > 0) and bounded above, lim t-.r zdt> = N--3 exists and from equation (18) z,(T-)

I T-ml/y*

< T.

We next establish that lim sup,,,- y(t) is finite. Suppose lim sup,,,- y(t) = + 03. If y(t) is eventually strictly monotone increasing (so that y(t) > y[zl(t)], t E [T,, T)), then a simple Gronwall-type argument provides boundedness. Ify(t) is not eventually monotone increasing there exists a sequence Y,, --P T such that y(y,) is a local maximum. Delete from the sequence {q,} those values where y(nJ I y(~i), i = 1, . . . , n- 1. Thus at q,, y(q,J is a local maximum, and y(t) takes on the value y(qJ for the first time. Let r: denote the largest value less than Q, where y($,) = y(~,-,). Thus on [& q,], y(t) is monotone increasing and

A THRESHOLD

MODEL

OF

ANTIGEN-ANTIBODY

y(t) > y(x), x E [0, t). In particular, y(t) satisfies

y(t) > y[rl(t)]

DYNAMICS

511

on this interval. Here

y’(t) 5 r%Yo)

or

ykb) s

YM)

ew trmh

- dJ1

= vh- A exp Crmh - 601 22YCr,,- A exp Credfl, - fib- Al. Hence by an inductive argument it follows that

Y(4-31. ukL-~>exp Lraxdr,- 1-4kllCexP r~xdm-r,- Al = Y(L2) exp Pmb---rln-211

Thus (y(~,)> is bounded above, and by the choice of qn, y(t) is bounded above. Since x(t) is monotone decreasing, Em,,,- x(t) exists. Further, since y(t) is bounded, x(T-) > 0. Since z,(T-) exists, x[r,(T-)]y[z,(T-)] exists and is finite. From this it follows that y’(t) is bounded on [O, T], and hence by a simple application of the mean value theorem that lim sup,,,.. y(t) = lim inl& y(t). The extension procedure can now be applied, contradicting the maximality of T. Thus the solution can be continued to the right until t * where

if2

[Y(S) + WI

ds = m2

or to [0, co) if no such t * exists. When the second threshold triggers, as noted above, two additional functions z2(t) and z(t) must be considered. However, the technical arguments are essentially the same as those above except that in the extension procedure only choices (b) and (c) (without the integral) are considered. We omit the details. Continuity in the parameters and initial conditions follows from standard arguments of ordinary differential equations (see, for example, Hoppensteadt & Waltman, 1970) or by using the theorems of Driver (1963). REFERENCES BELL, G. I. (1970). J. theor. Biol. 29, 191. BELL, G. I. (1971a). J. theor. Biol. 33, 339. BELL, G. I. (1971b). J. theor. Biol. 33, 379. T.B.

34

512

P.

WALTMAN

AND

E.

BUTZ

BELL, G. I. (1973). Math. Biosci. 16, 291. BRUM,C., GIOVENCO, M. A., KOCH,G. & STROM, R. (1975).Math. Biosci. 27, 191. COHEN, S. (1970).J. theor.Biol. 27, 19. COHEN, S. (1971).In Cellular Interaction and the Immune Response (eds. Cohen, S., Cudkowicz,G. & McCluskey,R. T.). Basel:S.Karger. COHEN, S. & MILGROM,M. (1971).J. Zmmunoi. 107, 115. COHN,M. & BLOMBERG, B. (1975).Stand. J. Zmmunol. 4, 1. COUTINHO, A. & MILLER,G. (1974). &and. J. Immunol. 3, 133, COOKE, K. L. (1967).In Diff”erentia1 Equations and Dynamical Systems (eds.Hale,J. K. & LaSalle,J. P.). NewYork: AcademicPress. DELISI, C. & PERELSON, A. S. (1975).J. theor. Biol. 62, 159. DRIVE& R. (1963).Contrib. Di$ E&s. 1, 317. HOFFMAN. G. W. (1975).Eur. J. Immunoi. 5. 638. HOPPENS~DT, HOPPENSTEADT, Hsu, IN-DING,

F.’ & WALTMAN, P. (1970). F. & WALTMAN, P. (1971). & KAZARINOFF, N. P. (1977).

Math. Biosci. 9, 71. Math. Biosci. 12, 133. J. Roy. Sot. Edin. (in press).

JERNE, N. K. (1973).Ski. Am. 229,52. JERK, N. K. (1974). Ann. Zmmunol. (Inst. Pasteur),125,373. JERNE, N. K. (1975).The Immune System, The HarveyLectures,preprint. NOSSAL, G. S. V. (1973). In Essays in Fundamental Immunology, (ed. Roitt, I.). Oxford: BlackwellScientificPublishingCo. PIMBLEY, G. H. JR.(1974a). Math. Biosci. 20, 27. G. H., JR. (1974b). Math. Biosci. 21, 251. G. H.. JR. (1974c).Arch. Rational Mech. Anal. 55.93. RICHTE;, P. H.‘(1975).Eur. J. Zmmunol. 5, 350. WEIGLE, W. 0. (1967). Natural and Acquired Immunological Unresponsiveness. Cleveland: PIMBLEY~ PIMBLEY.

World PublishingCo.,