The dynamics of antibody secreting cell production: Regulation of growth and oscillations in the response to T-independent antigens

J. theor. BioI. (1980) 84,49-92

The Dynamics of Antibody Secreting Cell Production: Regulation of Growth and Oscillations in the Response to T-Independent Antigens ZVI GROSSMANt

Laboratory of Theoretical Biology, National Cancer Institute, National Institutes of Health, Bethesda, Maryland, 20014, U.S.A. RICHARD ASOFSKY

Laboratory ofMicrobial Immunity, National Institute ofAllergy and Infectious Diseases, National Institutes of Health, Bethesda, Maryland, 20014, U.S.A. AND CHARLES DELISI

Laboratory of Theoretical Biology, National Cancer Institute, National Institutes of Health, Bethesda, Maryland, 20014, U.S.A. (Received 13 November 1978, and in revised form 17 September 1979) We present an interpretation and analysis of the dynamic structure of the response of antibody secreting celIs to T-cel1 independent antigens. The observations of interest are sustained synchronous cycles in the number of such celIs, and the discontinuous, "staircase" like increase in cell numbers during the first few days of the response. The main features of the model from which these results can be deduced are the existence of time delays which separate different celIular events, a particular distinction between proliferative and differentative events among the celIs which will ultimately secrete antibody, and inhibition of the stimulation of the precursors of these celIs by antibody already formed. In addition to describing various characteristics of cyclical antibody production, the model makes a variety of predictions related to its principal hypotheses, including the effects of antibody passively given or withdrawn on the kinetics and magnitude of the subsequent antibody response. The model also predicts, consistent with observation, the existence of an optimal dose of antigen, high and low zone unresponsiveness, and high zone tolerance. It is consistent with the t Present address: Department of Mathematics, Weizmann Institute for Science Rehovot, Israel. 49

0022-5193/80/090049 +44 $02.00/0

© 1980 Academic Press Inc. (London) Ltd.

50

Z. GROSSMAN ET AL.

phenomenon of maturation of affinity of antibody, and also predicts fluctuations in affinity. A study of the mathematical properties of the model indicates that for a wide range of parameters, the system possesses multiple steady states and limit cycle behaviour. Variations in concentrations of antigen and antibody are found to have a small influence on the frequency of oscillations, but a large influence on their amplitude.

I. Introduction

The dynamics of an antibody mediated response to antigenic challenge had for many years been believed to consist of an exponential rise and fall in the number of antibodies (McMaster, 1953): the sort of behaviour which one expects intuitively on the basis of a simple negative feedback control in which antigen elicits, through cellular stimulation, secretion of the agent (antibody) which is responsible for its elimination. Data obtained at a cellular level over the past ten or so years, however, indicate that this view is considerably oversimplified. The dynamic patterns in the appearance of antibody forming cells (AFe) are varied and complex, showing fine structural features some of which are not observed serologically. Thus Jones et al.

601

(b)

50 c:


i5. "-

'"

~ a.

40

'2

'" 0

..J

10

Hours after immunization

FIG. 1. Antibody secreting cells are detected by their ability to form clear plaques on a red blood cell lawn. The number of plaque forming cells (PFC) denoted by solid circles, is therefore adopted as an operational measure of the number of antibody secreting cells. Staircase pattern arises in response to a single injection SSS III. Data from Jones et al., 1976. "Shelf times" correspond to the flat segments.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

51

(1976) found that the number of IgM secreting cells produced in response to a single injection of pneumococcal polysaccharide III (SSS III) does not rise smoothly, but exhibits step like changes, with "shelf times" of about 8 h (Fig. 1). Moreover, sustained fixed period cyclic responses to single injections of antigen (Fig. 2) appear to be characteristic of T -independent antigens (Britton & Moller, 1968). The step pattern, can also occur in response to

(0)

~ 104

a. '"
'"a.

~ a.

....0 ~

'"E

.0 ::J

:2

103

1::;::: ";: "

8

0>

4

(b)

~

'" 0

OJ

'"

0

...J

2 0

10

20 Doys

FIG. 2. LPS is typical of T cell independent antigens which elicit sustained cycles in the number of 10M secreting cells. Data (al from Britton & Moller (1968) and (b) Nielsen & White (1974).

T -dependent antigens, as was shown by Perkins et al. (1968) with red blood cells (Fig. 3). Oscillations in the dynamics of IgG secreting cell production have also been observed (Romball & Weigle, 1973); an example being the synchronous, in phase cycling produced in response to aggregated human immunoglobulin G(AHuIgG) (Fig. 4). Speculations on the origin of oscillations as well as a review of the experimental literature appear in reference (Weigle, 1975). There are several mechanisms by which oscillatory behaviour may be explained including various types of cell-cell interactions. In fact, an

52

Z. GROSSMAN ET AL. 10

6

r---------------........,

Hours FlO.

3. Staircase pattern in response to sheep red blood cells, from Perkins el al., 1968.

important aspect of many antibody responses is the presence of three interacting cell types: T cells, B cells and macrophages (Mosler & Coppelson, 1968). T cells consist of several distinct subpopulations which participate, with the aid of macrophages, in the initiation and regulation of antibody production by B cells. However since oscillations are observed even in animals depleted of T cells and for T-cell independent antigens (Briton & Moller, 1968) cellular collaboration is apparently not a requirement. We thus omit, in this initial effort, consideration of cell-eell interactions, about which little is known in any case, but do not exclude the possibility that they may modulate cyclic behavior in those responses in which they participate.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

53

260r-----------------, 240

.JI1

200

., ~

160

0; u

(-lindirect PFC (-- )Direct PFC

c

lD

Q "-

120

t2 Cl.

70

40

20

24

Days ofter immunization

FIG. 4. The number of direCt (IGM secreting cells) and indirect (lgG secreting cells) plaques as a function of time after a single injection of AHulgG. Data from Romball & Weigle, (1973).

The objective of the present study was to present a minimal model of the T -cell independent antibody response, which incorporates a particular and clearly defined picture of the stimulation and control of cell proliferation and differentiation. Basic features of the cellular and antibody response, such as the dependence on dose of antigen, affinity maturation, and the effects of manipulation of circulating antiboldy are accounted for. The model manifests fine structural details in the growth of the number of AFe's as well as oscillations. These served as important guidelines in the construction of this model.

2. The Model CA) ASSUMPTIONS CONCERNING GROWTH AND DIFFERENTIATION

There have been many efforts, at various levels of mathematical and biological sophistication, to model the humoral immune response (Bell, 1970, 1971a; 1971b; Bruni et a/., 1975; Waltman & Butz, 1977; Merrill, 1977; Mohler, Barton & Hsu, 1977; Dibrov, Livshits & Volkenstein, 1977; Freedman & Gutica, 1978; for reviews see, DeLisi, 1977 and Bell et at., 1977). In the formulation to be presented, the interplay of time delays with feedback inhibition of cell division and differentiation by antibody is of major importance, The inclusion of the former is dictated by physiological

54

Z. GROSSMAN ET AL.

facts; viz, the anatomic compartmentalization in the structure of the immune·system (Kallos, 1975), as well as by the delays, sometimes substantial, involved in cellular division and transformation (Dutton & Mishell, 1967). We shall adopt the nomenclature first used by Sercarz & Coons (1963) X ~ Y ~ Z, in which X is a resting small lymphocyte, Y is a stimulated lymphocyte, and Z is an antibody producing plasma cell. Two interactions with antigen (X ~ Y) and (Y ~ Z) are required to generate antibody forming cells. Perkins et al. (1968) defined "recruitment" as enlistment to differentiation and division from progenitor cells to functional (ultimately antibody-forming) cells; "nonrandom recruitment" such as enlistment spaced at specified time intervals; and progenitor cells as the ancestors of antibody forming cells (B cells (7)). The model, shown in Fig. 5, differs from

r.,-T,

-I

b ----~ 1/0 ~

1;>----

o

~

FIG. 5. Schematic diagram of the model.

those of Sercarz and Coons, and from that of Makimodan's group, in the existance of an X ~ y ~ X line of differentiation. It is based on the following view of the immune response. (1) The recruitment, differentiation and proliferation processes take place in localized areas (germinal centers) within the body and -areas immediately adjacent to them, mainly in the spleen and in the lymph nodes (Langevoort etal., 1963; Cohen, Jacobsen & Thorbecke, 1966 and Thorbecke, Jacobsen & Hochwald, 1965). Thorbecke et at. have shown that antibodies are synthesized in areas adjacent to germinal centers, but not by the centers themselves where the most active cell division occurs (Langevoort et al., 1963; Cohen et at., 1966). Germinal centers isolated

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

55

in vitro are stimulated to make antibody by addition to cultures of antigen. Adjacent areas are not stimulated to make antibody (Thorbeck et al., 1965).

(2) There is a constant, low-level supply of progenitors (small B lymphocytes or X cells) originating from stem cells in the bone-marrow, which are removed continuously with a rate proportional to their concentration. (3) When antigen is injected intravenously into the body, a small fraction of it is captured in the germinal centers within a short time. Such "capture" was first demonstrated by a group working with Nossal (Nossal, Ada & Austin, 1964) and seems always to occur when looked for (Weigle, 1975). The amount captured is a function of the total input which will not be specified in the model, and the antigen variable will refer to the localized fraction. The localizing time is assumed to be short compared to the characteristic times of the processes which follow, so that the initial condition for the antigen variable may be approximated by a step function when the input function has this form. (4) The antigen units (molecules, or molecule complexes) interact specifically with progenitor cells, probably by forming a complex, at the surface of the B cell, according to a mass-action law. The cells, so triggered (Y cells) become functional, possibly after a latent period. The interaction of antigen with macrophages prior to the interaction with the lymphocytes is not described explicitly. It is assumed to affect only weakly the relationship among the other system variables, except through the determination of the constant parameters. This assumption may need to be revised in a more elaborate model. (5) Triggered Y cells may further differentiate in two different ways. First, they differentiate spontaneously into new immunogenic, or memory cells, with a rate proportional to their number (ie. each triggered cell has a constant probability for differentiating at any given time). In so doing, the cells undergo some structural changes (not specified here) and become Y cells which undergo one or more divisions, and finally reappear in the germinal centers as new immunogenic X cells. The time delay between the beginning of differentiation of a cell and its reappearance as a new progenitor will be given by the differentiation-proliferation time. Y cells are "excited" lymphocytes, and will either relapse to the X (resting) state or will differentiate to the Z (plasma cell) state, in which antibody is synthesized. Note again that the present model provides for an X ~ Y ~ X line of division. (6) Alternatively, triggered cells may further interact with antigen and become, after a series of divisions and a maturation process,

56

Z. GROSSMAN ET AL.

antibody-forming cells, also called "plasma cells". A mass-action interaction is again assumed. (7) Mature plasma cells ("2 cells") secrete antibody molecules at a fixed rate (per cell) during their life time (Scharff & Laskov, 1970). The life -time of a plasma cell is a fixed period, after which it dies or becomes nonfunctional in a terminal way (Langevoort et ai., 1963; Dutton & Mishell, 1967). (Bl THE TIME DELAYED THREE POPULATION SYSTEM

Let X, Y and Z denote the concentrations of the progenitor cells (including memory cells), triggered cells and plasma cells, respectively (Fig. 5) and let Ag and Ab denote the concentrations of the antigen and antibody. Then the equations of the model are:

~~=s-elx-aAgx+eY(t-1'1)'

(1)

~~=aAg(t-1"2)X(t-72)-dY-a'AgY,

(2)

~~ = a"Ag(t-73) Y(t -73) -

a"Ag(t - T4) Y(t - 1"4)'

(3)

By assumption (Jones et ai., 1976), the X population increases at a rate s, from a constant source, and is removed at a rate e'X (lie' is the characteristic life time). X cells are triggered by antigen at a rate proportional to both Ag and X [assumption (4)J, with a proportionality constant a, and thus are removed from the X population at this rate. The cells triggered at (t - T2) are added to the Y population at the time t, where 7, is the latent period [assumption (4)], and hence the first contribution to the right hand side of equation (2). Y cells are removed at a rate d Y by spontaneous differentiation [assumption (5)]. The proportionality constant d may include also the effect of the natural decay rate of Y cells. Cells which started differentiating at the time (t -, 1), where T1 is the delay due to differentiation, increase the X population at time t, but there is a multiplication factor (el d) > 1 due to proliferation which has taken place during this time-interval. Lastly, by assumptions (6) and (7) Y cells may be further triggered by antigen yielding the term (-a'AgY) in equation (2). The cells removed from Y (per unit time) at time (t-T3), namely a'Ag(t-73)Y(t-73), multiplied by a factor (a"/ a') > 1 due to proliferation appear as a positive contribution to the Z equation. Those plasma cells which were added to the Z population at t - T4 are removed by terminal differentiation (74 - T3) units of time later, where T3

DYN AMICS OF ANTIBODY SECRETING CELL PRODUCTION

57

is the maturation~proliferationtime of plasma cells and f 55 74 -7"3 is their (fixed) life time. All the parameters defined above are positive numbers. Since there are time delays in the equations, initial conditions have to be specified over a finite time-interval rather than at one point in order to define a solution. This leads in a natural way, to a system response which depends upon the route of antigen administration. Since antigen (Ag) is a free parameter, equations (1) and (2) are independent of (3), and may be solved separately. Z is then found by direct integration of equation (3): Z

= aliI

3 '-T

Ag(t') Y(t') dt'.

(4)

t-7"4

This form, with vanishing integration constant, is valid since Y and Z are assumed zero for t =: O. The steady-state solution (Ko, Yo) to equations (1)-(2) for a given constant antigen concentration, Ag, is easily shown to be: K = o

Yo =

s(a'Ag+d) , [(aAg+ e')(a'Ag+ d) -eaAg] saAg [(aAg+ e')(a'Ag+ d) -eaAg]

.

(Sa)

Zo is obtained from (4): (5b)

If Ag =: 0, X o reduces to s/ e', which is the background equilibrium level of progenitors in the virgin system, and Yo = Zo = O. A necessary condition for the existence of a biologically acceptable steady state is that the denominator in equation (Sa) should be positive, since, by definition, the corresponding variables acquire only positive values. This condition leads to the inequality: -2 (6a) R =aa'Ag +[e'a'-(e-d)a]Ag+e'd.

It is evident that if B

55

e'a'- (e -d)a

(6b)

is positive, R is positive and a steady state always exists. If, on the other hand, B < then equation (6) is satisfied if Ag obeys one of the following conditions: (7a) either Ag>Ag;,

°

or Ag
(7b)

58

Z. GROSSMAN ET AL.

where critical values Ag~ are defined by

Ag~=-21,[ -B ±.JB2 -4aa'e'd]. aa

(8a)

If

AgrfO

(8b)

a positive steady state does not exist. The stability of the steady state is analyzed in Appendix A. (C) FEEDBACK SUPPRESSION (INHIBITION) BY ANTIBODY

As an initial attempt at describing a cyclic response, we consider the case in which antigen is long lived compared to antibody, the response is T independent, only one class of antibody, IgM, is produced and all B cells are identical with respect to their affinity for the antigen. Lipopolysaccharide (LPS) (Britton & Moller, 1968) and sal 0 antigen (Nielsen & White, 1954) are,the closest candidates for an antigen with such characteristics. Following Britton and Moller (1968), we shall adopt the interpretation that antibody produced during the response may block the antigen from stimulating competent cells. If the complex thus formed dissociates prior to catabolism, the free antigen can again stimulate cells. We will formulate these ideas mathematically, with emphasis on the role of the presence of time delays in the system, and illustrate the arising synchrony (i.e. the equality of the durations between subsequent peaks) for a range of conditions and situations. To the seven assumptions stated in section 2.B we add the following (Fig. 5): (8) The antibody secreted by Z cells can form a complex with the antigen, following a simple mass-action law. The complex thus formed does not interact with X and Y cells. (9) The antibody-antigen reaction is reversible. (10) Free antibody, bound antibody and bound antigen in lymphoid tissues are eliminated at fixed rates. Circulating complexes are removed to non-lymphoid tissues almost instantly (Dixon, 1954). We can now write the equation for the time evolution of antibody and antigen concentrations at the site of interaction. If 'Y is the rate of antibody secretion by mature plasma cells [assumption (7)], f3 the coefficient of association with antigen, C the complex concentration, p its dissociation rate, and () and K the rates of elimination of free antibody and complex,

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

59

respectively, we have the following equations: dAb dt = -yZ dAg

(3 AbAg - 0 Ab + pC,

.

(9)

d't= -(3AbAg+pC-aAgX-a'AgY;

(10)

dC ili= (3AbAg-(p +K)C.

(11)

The interaction term AbAg represents a prominent simplification. In general, blocking may involve a sequential combining of several antibody molecules to the determinants of. a multivalent antigen, and the release of free antigen by dissociation can similiarly involve several steps. However, the simple form assumed here is sufficient to account for the phenomena which are the interest of the present study. The last two terms of the right hand side of equation (10) represent loss of free antigen by complexation with lymphocytes. However, due to the small ratio of cells to antigen molecules, these contributions may be neglected. For example, we can write for the contribution of X cells to the fraction of antigen removed per unit time, i.e. to

dAgj 1 (X(O))( X ) dt Ag: Ag (aAgX) = aAg(O) Ag(O) X(O) =10. (2.10-

11 )

X~) =2.10- 10 ~)

(see Appendix C). Thus even if at the end of the response the number of lymphocytes has increased by a factor of 104 with respect to the initial number, the fraction of antigen removed per unit time is only about 2 x 10-6 and the total amount of Ag removed during the response is also estimated to be negligible. Next, we neglect antigen destruction altogether, as an approximation for the case of long-lived antigens like LPS. Inserting K =0 and adding equations (10) and (11), d

dt(Ag+C) = o.

(12)

We note that Ag+ C == Ag T is actually the total amount of antigen in the system, free and bound (blocked), and so Ag r = constant. Using this

60

Z. GROSSMAN ET AL.

notation, and dropping the small terms, equations (9) and (10) become: dAb dt"= yZ -{3AbAg-OAb+p(AgT-Ag), dAg

.

dt"= -j3AbAg+p(Ag T -Ag).

(13) (14)

We turn now to study the system (1}-(3), (13) and (14). Equating the left hand side of (13) and (14) to zero, and using (Sa) and (Sb) (where Ag= Ago), we obtain the following equation which must be satisfied by the steady-state value of Ag, Ago: saa/lfAg~

pO (AgT-Ag o) (aAgo+e'}(a'Ago+d)-eaAg o {3y Ago

(15)

As discussed in the second part of Appendix A, there is always one positive solution Ago < Ag;. This steady-state was found to be (locally) stable for the range of parameters used in the numerical simulation. If Ag T < Ag~, this is the only positive steady-state which can exist. If Ag T > Ag;, two additional steady-state solutions may exist (Appendix A), one of which was found numerically to be stable (the larger of the two). The stability of the steady-states was not studied analytically for the full model with regulated antigen.

3. Results (A) DOSE DEPENDENCE PATTERNS OF GROWTH

The solutions to equations (1)-(3) were studied analytically and numerically. They represent an approximation of the first phase of the antibody response. Their analysis helps to sort out the characteristics of the appearance of AFe and the dependence of the response on dose of antigen. As one might expect of a model in which antigen stimulates persistently, and in which no mechanism is present to regulate cell proliferation, one mode of behaviour is exponential growth. This occurs when the concentration of antigen lies in the interval Ag; :s Ag:s Ag;. The other possibilty is approach to a stable steady-state and this happens at either high (Ag ~ Ag;) or low (Ag:s Ag;) doses of antigen. [See equations (4)-(8) above and Appendix A.] A procedure for solving the equations is presented in Appendix B. These features of dependence on dose of antigen are preserved in the full model (see below). The short-time behavior of the system depends strongly on the initial conditions, the "loop-delay", 7" == 7"1 + 7"2, and the rates of the antigenic

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

61

interactions with the cells. If the initial conditions are specified by the assumption that the antigen accumulates "instantaneously", and if the interaction rates are fast compared to the loop delay, then fast changes in X and Y will occur once per time-interval of length 7" after antigen introduction. If condition (8b) holds, these changes may have the form of narrow peaks, or spikes, of increasing maxima (as discussed at the end of Appendix B).

The Z population is given in equation (4) by the difference of two integrals (maturation and death). The first transforms the peaks into ascending steps, and the second transforms earlier-formed peaks into smaller descending steps (which appear as dips in the staircase pattern). The "shelf-time" of the steps is equal to the loop delay T. The theory predicts that as time evolves, the peaks become less pronounced and the steps smoothen (Figs 6-8). This must happen because of the continuous nature of the dynamic system, which tends to distribute events which were initially simultaneous. In more general terms, the integration process causes an increase in continuity of the solution function, or "loss of memory of the initial conditions". The staircase pattern is thus a transient phenomenon, but it may exist throughout the growth stage if feedback suppression by antibody starts early enough, as is the case in the

1487

~ 0'1433

.3'"

o

47

904

Days

FIG. 6. Staircase-type pattern obtained by integrating equations. (1)-(3) and (13)-(14):

'T1

{3

== 0,8, 'T3 = 0,8, 'T4= 2, a = 50, a' '" 25, a" = 50, e =50, d = 10, e' =2,5, S =0·5, Ag(O) = 0,5, = 10, r = 4, ({3y = 20), (} = 5, p '" '" = 20, (K = 0). Z' "" Z +1 where Z is the concentration of

antibody-forming cells. All units are in inverse days, except for concentrations, which have been made dimensionless (see Appendix C). Switch-on of blocking after four days (see p. 91).

62

Z. GROSSMAN BT AL. 2·063 (0)

1-143 (b)

-",

0> 0·5717 j

1·032

14

14 Days

Days

FIG. 7. Staircase pattern of Z cell growth (a), and the corresponding curve for X cells (b), a""25, a ' =20, d=20, 8=0·2, 13""5, y=12·5, 8=2'5; other parameters as in Fig. 5. X' = X + 1. Switch-on of blocking after eight days.

800,....--------------------,

~

g

400

...J

o

8

10

Days

FIG. 8. Staircase pattern: '7'1=0,8, '7'3=1,5, '7'4=2, a""8, a' ''''4, a"=20, e=40, d=10, e ' = 8 = 2, Ag(O) = 5,13 = 200, 'Y = 0·05.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

63

examples. The actual magnitude of the steps will also be influenced by the necessary depletion of free antigen as the peak of the curve is approached. The choice of the numerical values for the parameters was based on the initial set presented in Appendix C, which was somewhat modified in order to demonstrate a variety of behaviors. In comparing the theoretical curves with those of Fig. 1, note that there is qualitative agreement with respect to a number of features: the general exponential increase, the steps, damping of the last steps (when suppression of the response is taking place; see the next section), and even the apparent manifestation of dips conjugate to the steps, to which no statistical significance has been previously attributed. Even the irregularity in the kinetics-which experimentally can of course arise for several reasons, related to the complexity of both the system and the techniques used-is reproduced to a certain extent by our relatively simple model (Fig. 8). It follows in the model from the existence of a few time delays which are not simply related in magnitude. Recruitment events and synchronized death events of Z cells interfere in a non-regular way. A staircase pattern exists for a range of parameters. However, it is enough to vary some of the parameters within a factor 2 or less to obtain patterns for which the approximate discreteness of the recruitment events and of Z cell death is masked and the growth is essentially smooth (as in some of the examples in the following section). Also, the pattern depends on the initial conditions, e.g. the manner of immunization. Hence this phenomenon, though not exceptional, may not be observed in every experiment in which measurements of APC are performed at short intervals. (B) FEEDBACK SUPPRESSION AND CYCLES

According to the results in Appendix A, a steady state number of AFC will be reached if the amount of antigen remaining uncatabolized is above Ag~ or below Ag~. Numerical simulations with the administration of a large amount of antigen lead to an equilibrium after a short phase of low-level antibody production, with the antigen mostly free (unblocked). The reason is that large doses preferentially transform Y to Z rather than to X. The X population is therefore not reseeded and the only new X cells which enter the system come from bone marrow precursors-a meager supply. Thus the formation of Z cells, and hence antibody production, is severly limited. Responses to smaller antigen doses end with the amount of free antigen at equilibrium below the threshold Ag~, and this may happen after a phase of exponential growth and antibody production (the immune response), or if the dose is very small-with only moderate and slow increase in antibody production (low zone non-responsiveness). The simulations thus show the

64

Z. GROSSMAN ET AL.

existence of an optimal antigen dose, leading to a peak of maximal magni~ tude, with diminishing peaks reached in response to decreased or increased doses. The model is therefore consistent with the classical pattern of dose-response relationship. The dependence of the peak in Z on antigen dose is not linear; this is due to the nonlinearity of the blocking process and to the combination of different growth rates with the time lags in the feedback control mechanism which enhance the tendency to "overshoot" (see below). For a large range of parameters, antigen dose in the immune response domain leads to oscillations rather than to a constant steady state. Oscillations in the number of antibody secreting cells (Z) (and therefore in the local antibody concentration) are driven by the concentration of free antigen. X cells are transformed to Y and Z cells with the consequent production of antibody and the blocking of antigen. As antibody production continues, free antigen is reduced to a level which is below its equilibrium value-practically all the antigen is blocked. At this point there is no longer enough free antigen to stimulate further Z cells production, but there are of course still a large number of Z cells; in part because they have a finite life, i = 7"4 - 7"3, and in part (due to 7"3) because previously stimulated Y cells are 05 r - 1 - , r - - - - - - - , - - - - - - - - - - - - , 6 8

/I II II

II II II II I, II 'I

en ~

34

II

I I , r I I

,I /

.0

I

\

25 Days

FIG. 9. Oscillations in the concentration of free antigen (Ag) and PFCs (Z) simulated numerically. TI == 0,3, T3'" 1·5, T4 == 2· 5, a'" 32, a' '" 5, a" '" 32, e' == 2 = s, Ag(O) '" 0.5, f3 '" 5, 'Y '" 0,5, 8 = 2, p = l/J == 0,1, (Ag;;- = 0,021, Ag~ = 2·92).

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

65

still maturing. In fact the free antigen concentration and the number of Z cells can be out of phase by nearly 180°, the latter peaking when theformeris near its minimum (Fig. 9). As the number of AFC begins to decline, and hence the local antibody level begins to decline, the equilibrium between free and bound antigen will begin to shift in favor of the former. Thus, as the number of AFC decreases, antigen increases until it peaks, at which point the cycle continues. Numerical simulations indicate that for a wide range of parameters, plots of Ag against Z approach closed curves, representing periodic solutions (Fig. 10). The forms of these curves on a logarithmic scale resemble

0151 H'''''''''~------_

0>

S

8'

00755

...J

o

05

10

Log(l)

FIG. 10. Sustained oscillations in the Ag-Z plane. Parameter values similar to those of Fig. 9.

triangles, with sides corresponding to three different phases during one cycle. Exponential increase of Z with exponential decrease in Ag (the diagonal), decay of Z with a slow variation in Ag (horizontal side), and increase of Ag with slow variation in Z (vertical side). These three events are identical to those suggested by Britton & Moller (1968): stimulation and blocking, shutoff of the response, release of free antigen from the complex. However, we have seen that it is the time delays which are responsible for

66

Z. GROSSMAN BT AL.

the chronologie separation of these events; in their absence all -three processes could have come to dynamic equilibrium, with the three-phase curve of Fig. 11 reducing to a point. Notice also the difference in the figure between the first response and the subsequent ones. It arises from the fact that only a small fraction of the antigen drives the continued oscillation. Several numerical simulations were carried out. For some parameter sets, the oscillations were damped and the solution converged to the steady state within a few cycles. But for many sets, integration over many dozens of oscillations demonstrate what seems to be sustained simple oscillations, approaching fixed period and amplitude. The frequency of the oscillations, unlike the amplitude, is only weakly affected by changing the antigen dose (Ag T ) and some of the parameters (notably with the exception of time delays). As the dose is reduced from the optimal value (for the first peak), the amplitude is decreased, and the oscillations disappear at some threshold dose (still above the non-responsiveness limit Ag~). Time delays of the order of one day can support oscillations of several days. Oscillations were demonstrated with and without the staircase-structure. If steps are manifest in the first wave, they tend to disappear in subsequent waves. This phenomenon is explained by two factors: first, the steps are transient in nature and are therefore masked as time evolves. Second, the secondary peaks usually involve only one wave of stimulated memory cells. The pool of X cells after the first peak is much larger than the pool in the virgin system, and the amount of exposed antigen is smaller than the initial dose, so that one wave of antibody can be sufficient for blocking. From the mathematical point of view, the sustained oscillations exhibited by systems responding to non-catabolized antigen are of a very interesting nature. Feedback systems with time delays, including biological systems, have been shown in the past to possess the potential of manifesting oscillatory behaviour. Usually sustained oscillations arise as a bifurcation from a stable steady state, when its stability is reversed as a delay is increased. However, numerical experiments indicate that this is not the case with the present model. Starting from initial conditions where the variables are near their steady-state value, but with the same parameters (including Ag T ) which previously led to oscillations, results in fast convergence to the steady state. Especially interesting from the mathematical point of view are the numerical experiments with addition of passive antibody (see below) which not only suppresses the first peak but also prevents the oscillations from occurring. Note that the initial amount of antibody is not a parameter in our equations. Hence it appears that the steady state is in fact locally stable, although sustained oscillations about it may arise for certain initial

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

67

.

I

1f

I

1·5

0·015 ..L N

I I I / f

I I

I

2·0 1·6 \·2

0·06

x O.B

0·04

I

~

>:

0-4

0

10

20

30

40

50 Days

3'0 (e)

o 13 1'5

S

10

20

30

40

50

Days

FIG. 11. Predicted dynamics of PFC (---) and free antigen concentration (-), (a), X (- -) and Y cells (-0-) (b), and free (~-) and bound h::>-) antigen, (c); as a function of time after immunization. Antigen is assumed to be eliminated according to equation 16 7"1 = 0,5,7"3 = 2·5, 7"4= 3,5, a = 1, a' = 4, a" = 12, fJ = 2, p = 30, i/J = 30·1(k = 0,1), Ag(O) =3. (Ag;;- = 0'17,Ag; = 3'683) s = 2, {3 = 5000, 'Y = 0·025.

68

Z. GROSSMAN ET AL.

perturbations. Thus the domain of attraction of the steady state in phase space is restricted. In summary, two important features of the model are the time delays and the combined action of negative and positive feedback (Fig. 6). Mathematically, these features bring about, for certain ranges of parameters, multiple steady states, partition of the phase space and associated threshold phenomena (see also Cohen et al., 1966). eCl ANTIGENIC DECAY

We now turn to the more general case of antigen which is eliminated at a rate proportional to the concentration of antibody-antigen complex. The governing equations, together with (1)-(3) for the lymphocytes, are equations (9)-(11) [without the terms aAgX and a'AgY in equation (10)J. The oscillations are now damped since Ag T is decreasing at the rate given by dAg T --=-K(AgT-Ag) dt

(16)

(16) is obtained by adding (10) and (11) and writing AgT=Ag+C. An example is shown in Fig. 11. In Fig. 11a the oscillations in AFCs are seen to be driven by damped oscillations in the free antigen. The Z oscillations become damped eventually, but may show an initial increase, as in the ratio of the third to second peak in this example. This is in spite of the reduction in Ag, since X is still increasing [Fig. 11(b)J. Fig. 11 (c) shows that the amount of bound antigen is decreasing during the first few weeks after immunization faster than the peak of free antigen. This tends to reduce the effect of antigen elimination on the amplitude of the oscillations. Thus, damped oscillations are possible if in the limit of fixed Ag T sustained oscillations exist. The latter are more easily analyzed, in principle (although this is not attempted here). A relatively small variation of the parameters in the example of Fig. 12 leads to the response which is plotted in Fig. 12a. Comparison with the data of Jones et at. (1976) on antibody and AFC response to sheep red blood cells (Fig. 12b) shows good qualitative agreement in the shapes of the curves and in their relationship to each other. (No attempt was made to obtain a better fit). Note that the lowest portion of the curves in this logarithmic scale corresponds to undetectable amounts of antibody and AFC. (There is also a difference of factor 3 between the time scales.)

0·33 ,----,.----,------,.--------,-----, 00405

1

0·21

::: 0'165

~

.3'"

o

10

20

30

40

8

10

50

Days

5·0

2·6

5·0

iii

JII

~

u 4·2

~

o

c;

.9

304

2'6

o

2

4

6

12

Time after immunization (days)

FIG. 12. (a) The antibody and antibody secreting cell response simulated numerically. 1·5, a=' o· 5, p =' 20, K = 0·1. The other parameters same as in Fig. 11. Ab' ;: Ab + 1. (b) Experimental data from Jones et al. Immunization with sheep erythrocytes (_0_); immunization with SSS III (-0-). 7'3 ='

70

Z. GROSSMAN ET AL.

4. Discussion (A) REGULARITIES IN THE DYNAMICS OF AFC.

The immune response is mediated by many different types of cells and molecules and probably posseses several modes of regulation. In some cases it seems that only a limited number of components is relevant. Complexity then arises from variability in the functional behaviour of the system due to interactions among its components. The present study is an attempt to push as far as possible the limits of a simple functional approach in describing a range of experimental phenomena. The ideas underlying the present model follow from suggestions and indications made by immunologists in the past; they have here been selected, modified and given a quantitative formulation. The notation used in the model for progenitor cells (precursors + memory), triggered cells and plasma cells, is somewhat similar to that of the XYZ model of Sercarz & Coons (1963). [Makinodan et at., (1969) also suggested the sequential production of two types of precursors, PC1 and PC2.] The main difference in the present model is the multiple increasing recruitment of Y cells; i.e. the repopulation and growth of the B cell pools as well as the simultaneous existance of two pathways of proliferation and differentiation, X...,» Y ~ X and X ~ Y ~ Z. Multiple recruitment is a term borrowed from Perkins et at., (1968), in the context of staircase structures. These authors, however, do not indicate the mechanism underlying this phenomenon. Our analysis suggests that this type of regulation in the growth pattern of the antibody response is important as a probe to the general mechanism of stimulation. Even when steps are not manifested, it is likely that the same mechanism is operating; doubling times are generally shorter than the cell cycle period, indicating the role of continuing recruitment of uncommitted cells. In reality cells may be distributed among several subpopulations at different differentiation phases which are more numerous and less distinct than in the functional classification of our model. [Proliferating cells can produce antibody (Smithies, 1965) at reduced rates before becoming mature plasma cells.] This, however, is expected to change mainly the quantitative details of the response and not the main qualitative features. An important aspect of the simulations is the sorting out of factors which affect the frequency of oscillations, from those which affect their amplitude. The former can be influenced by the various delays in the system, antigenantibody reaction rate constants and the rate of antibody clearance from the germinal centers. The cyclic rise and fall if the number of AFC is associated with the cellular events described above and their timing, most conspicu-

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

71

ously time delays in proliferation and differentiation; and by blocking of antigen by antibody (Briton & Moller, 1968; Pierce, 1968). Blocking was also suggested by Uhr and his colleagues (Urh & Bauman, 1961; Urh & Moller, 1968). It is worth remarking here that Weigle (1975), who has reviewed the subject of cyclical variations in antibody synthesis and has studied such variations in the antibody response to aggregated human gamma globulin has found it difficult to understand why cyclical rise and fall in AFC could occur with nearly identical frequency, although antibodies formed in the first cycle varied tenfold or more. The argument that frequency of oscillations depends on the concentration of antibody at some previous time should be reexamined in the light of our results. The frequency of oscillations, in contrast to their amplitude, is affected only slightly by the peak level of antibody or by the average amount of antibody during the oscillations. The frequency of the peaks remains essentially the same when oscillations are damped and levels of circulating antibody decrease. In contrast to the weak dependence of frequency on antibody concentration, the magnitude of the oscillations depends markedly on antibody concentration. In particular, this means that the antibody clearance and metabolism rates, and therefore also class, insofar as these rates are class dependent properties, will influence the amplitude of oscillation. The dependence of amplitude on antibody clearance rate from the germinal centers can be understood by recalling that in order for the number of AFC to rise again after falling to a minimum, free antigen must become available. This happens when the free antigen concentration is near its minimum and the rate of antibody production is therefore decreasing. The extent to which the clearance of free antibody shifts the equilibrium between bound and free antigen will, however, depend upon the rate at which antibody leaves its local environment. In the extreme case in which there is no clearance, antigen will not dissociate, there will be no further cellular stimulation and hence no oscillations. As the clearance rate increases, oscillations are expected to become more pronounced (after some threshold is passed). It should be noted that the level of circulating antibody may be quite weakly correlated with that of antibody in the germinal centers (Weigle, 1975). If equilibration of antibody concentration is not instantaneous and if antibody production is fast at the peak of the response (compared to the equilibration rate) then high concentrations can be formed locally and lead to blocking, while the external level is little affected. Alternatively, blocking may be accomplished by a small amount of circulating high affinity antibody, the oscillations of which are masked by the coexistence of a much larger amount of lower affinity antibody.

72

Z. GROSSMAN ET AL. (B) SOME GENERALIZATIONS AND PREDICTIONS

(i) The influence of antibody feedback on affinity maturation

Although the model has been presented primarily with T-independent antigens in mind, the mechanisms proposed to account for oscillations are fundamental and are therefore expected to playa role even in T~dependent responses. The importance of this is that T-dependent antigens, unlike those which are T -independent, usually elicit antibodies with a wide distribution of affinities, and our model makes some clear predictions about the regulation of this distribution. The predictions about affinity changes can be most readily discussed by first redefining some of the parameters in equations (1)-(3). It will be recalled that a and a', are composite parameters, reflecting not just affinity but proliferation rate as well. It is evident that the likelihood of stimulation depends upon the affinity of cellular receptors for antigen, 'but if all cells which bind the same amount of antigen proliferate at the same rate (Bell, 1970) the distribution of parameter values will reflect primarily affinity. We thus define a = Kk 1 and a' = Kk 2 , where K is a measure of affinity. Affinity maturation, Le. the observed increase in affinity with time after immunization, follows as a consequence of: (1) 'the mass action assumptions which have K and Ag entering the equations as a product, K Ag, (the argument can be generalized for different forms of interaction) and (2) the prediction of an optimum value for KAg; i.e. a value which leads to the fastest rate of increase of Z cells. To be more specific, the lower the antigen concentration, the higher the value of K which will be required to attain the optimum. Thus a decrease in antigen concentration selects high affinity cells preferentially, and the average affinity therefore varies inversely as antigen concentration. Evidently, if the free antigen concentration cycles as the result of being alternately bound and released by antibody, then affinity will also cycle. The effect will undoubtedly be somewhat more complicated than can be indicated by this model since changes in antibody affinity will also change the effectiveness with which they can bind and block antigen, and hence modulate stimulation. Rise and fall in affinity has in fact been reported by different groups, but the data are not extensive enough to determine the extent to which the considerations presented here contribute. As we noted earlier, the most sustained oscillations have been found with T-independent antigens which generally elicit monoclonal responses. In the most notable example of sustained oscillations with T-dependent antigens (Perkins et at., 1968; Makinodan et at., 1969), simultaneous affinity measurements were not performed. The closest candidates for comparison are the experiments of

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

73

Doria et at. (1972, in press) and Urbain et at. (1970). The former group showed that IgM affinity variations closely follow the number of direct plaques, while the latter showed that serum antibody affinity rises and falls in accordance with the level of serum antibody. These results are essentially in accord with the predictions of the model [Fig. 11(a)] which indicate that antigen concentration is low (and hence affinity is high) when the number of AFe is high, provided that the adjustment of affinity to antigen is rapid. In both experiments, however, only a single peak was observed and this is not sufficient information to rule out mechanisms other than those proposed here. Data on affinity variations toward antigens which elicit sustained oscillations, of which AHu IgG would be an excellent candidate, might be more revealing. Finally the model suggests the possibility that the structure of affinity variations may be more complex than simple oscillations which follow the rise and fall of APe. For example, when the concentration of free antigen is relatively high, the high affinity Y cells will be preferentially channeled into terminal antibody secreting cells (Z cells) rather than X cells. This would result in a relatively large number of high affinity antibody producing cells, but it will also tend to exhaust them because of lack of reseeding via the Y-X channel. Thus the next wave of differentiation, about twenty hours later, will necessarily be characterized by lower affinity. Consequently, there will be a drop in affinity, even when the local concentration of free antigen is not changing. In fact, Doria et at. (1972) observed a rise and fall in affinity during the first twenty hours after immunization preceding the long term rise and fall. There are however several other mechanisms which can account for this, including the presence of antibodies which cross react with the TNP determinants used in these experiments. Definitive statements will require additional simulations as well as detailed experimental studies of the dynamics of affinity regulation. ee)

INFLUENCE OF ANTIBODY PASSIVELY GIVEN OR REMOVED ON NUMBER OF AFC.

Several studies have demonstrated response suppression by passively administered antibody [reviewed by Uhr & Moller (1968)], i.e. by administration of antibody which is not produced by the system during the response. The suppression is specific and is believed to be caused by the blocking of antigenic determinants. Figure 13 illustrates the suppression which is predicted to result from antibody, administered at various times after immunization. Evidently, suppression depends upon the time interval separating antigen and antibody

74

Z. GROSSMAN ET AL. 0'176 ~---'r-I..----r----r-----r----, (a)

(b)

o

10

20

30

40

50

Days

0·176 r----,r--....,-----r-----r----, (c)

o

10

30

20

40

50

Days

Fig. 13. (a) Addition of passive antibody numerically, 13 time units after immunization. Parameters as in Fig. 12(a). Note the change in scales for Z' and for Ab withrespeet to Fig. 12(a). (b) Addition of small amount of passive antibody at time 15. (e) Addition of passive antibody at time 6. Ag' = Ag+l.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

75

administration. There is suppression of the second (small) peak if the antibody is added just after the first peak in Z, depending on the amount added [Figs 13(a) and (b)]. But even if the response in Z is initially suppressed completely, there is a later rise in the production rate [Fig. 13 (a)]. If antibody is added earlier, during the growth phase, there is suppression of the whole response [Fig. 13(c)]. Note, however, the delays in the suppression of Z and of Ab (i.e. the times between the sudden increase in antibody concentration and the subsequent peaks). These are measures of 7'3 and 7'4, respectively. (It would be interesting to test the theory and the constancy of the delays by measuring these quantities by adding passive antibody at different times during the rise phase.) Figure 14 illustrates the effect of antibody removal at various times, simulated numerically (by a fixed negative constant in the equation for antibody during a short time). The result is enhancement, which is stronger later in the response just after the shut-off of antibody production [Fig. 14(a)]. The antibody concentration increases considerably above the level it would have reached without antibody removal (note the logarithmic scale). Compare with the experimental curve, Fig. 15. The enhancement results from shifting the instantaneous equilibrium between free and bound antigen towards free antigen by removing antibody. This results in a free antigen concentration considerably above what it would have been at this point', and may lead to an overshoot in antibody production at a later time. That is, antibody is removed from Ag+Ab~AbAg leading to excess free antigen and an overshoot in feedback controls due to the delays. Indeed, the kinetics of the antibody forming cells manifests large secondary peak when the removal is made just after the end of the decline in Z. The change in the amount of antibody strengthened the tendency for cycling which the system indicates normally. If removal of antibody is performed earlier, smaller enhancement is manifested [Fig. 14(b) and 14(c)]. (D) MEMORY

Memory is another property exhibited primarily by T-dependent antigens and as such is somewhat peripheral to the main focus of this paper. However, the model, as formulated, does in fact manifest some of the main features of IgM memory. Short period memory follows from the repeated reseeding of the precursor population by differentiation of Y cells into X cells. Therefore, within the framework of our model T cells would act to regulate the balance between the Y ~ X and Y ~ Z conversion, and the parameters d, and at and possibility also the delays, 7'1 and 7'3 should properly be viewed as T-cell

76

Z. GROSSMAN BT AL. 0·336

r---r-:::;~-r----r---r-----,

0·\68

0·20

I ~

0'

0 ..J

0·336

0'405

Days

0·336.---.,-::rr---.---,---,-----,0·405

0·20

0·168

10

20

30

40

50

Days

FIG. 14. (a) Removal of antibody, numerically, at time 20. (b) Removal of antibody at time 14. (c) Removal of antibody at time 8. Parameters as in Fig. 13a.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

77

200 ,.---------:An:-t~i-':=::BSO:-::A-re-m-ov-e~d ------,

/00

0----0 Anli-BSA

Days

FIG. 15. Removal of circulating antibody followed by enhancement of serum antibody level (Data from Pierce, 1969).

dependent. Long period memory is possible within the model as a steady state phenomenon (or quasi steady state, if antigen is gradually catabolized), since X o can be much larger than its initial value, [equations (5a) and 15]. This type of memory thus depends on the degradation of the relevant antigen in the course of the immune response and on the parameters. Of course, it is possible that quite different mechanisms involving T cells and other factors regulate the enhancement or suppression of memory. For instance, the suppressive effects of antigen-antibody complex and nonimmunogenic soluble antigen may be considered. This will be done elsewhere. When the system is rechallenged with antigen after the primary response, a secondary response takes place. Several characteristiCs of the secondary IgM response are consistent with the numerical simulations (see Fig. 16). The amplification in the secondary response in the example was about 20 and its growth was faster, the peak being reached in half the time required in the primary response. When the priming dose was increased towards the

78

Z. GROSSMAN ET AL.

5xlQ3

I

I"I

,I (a)

z/X(O)

(b)/1

i'

I I I I

o

o

f

I

J

I

I ~. J

............ 10

30

20

36 40

,

60

Time

FIG. 16. Primary response, and secondary response to the same dose of antigen inserted at time 36(a), or to a doubled dose inserted at time 60(b). Peak time of primary response is 7 units, and for a secondary response - approximately 4 time units after rechallenging. a "" 1, a' "" 4, aU = 50, e "" 30, ,B = 5000, 'Y = 0·01 (131' = 50), () = 0·5, p = l/J = 5.

high-zone threshold, the magnitude of the secondary peak (in Z) became rather insensitive to the secondary dose, in agreement with the experimental evidence (Uhr & Finkelstein, 1967). This is explained by the fact that with an enhanced X cell pool multiple reseeding of the pool is not necessary before the antigen is blocked. Thus the magnitude of the secondary response depends mainly on the initial pool size rather than on its Ag-dependent amplification. It is predicted. that in cases where steps are exhibited by the primary response, there will not be many of them in the secondary response. In Fig. 16, the secondary response to one unit of Ag apparently involves one step, and the response to two units at a later time, with a reduced memory population and more antigen to block, shows two distinct steps. The model is consistent with the evidence (Uhr & Finkelstein, 1967) that it is considerably more difficult to inhibit priming by passive antibody than to inhibit the primary antibody response. This is so since only the total amount Ag T matters in determining the steady state level of the precursor pool, and the passive antibody does not change this quantity substantially, while it does affect the way by which the steady state is reached (Ld. the magnitude of the primary response). Finally, the model also predicts smaller changes in affinity durjng a secondary response, again because clone selection is associated with the proces~ of multiple recruitment which is less relevant to a secondary response.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

79

(El HIGH ZONE TOLERANCE

If the amount of antigen localized initially exceeds the threshold Ag;, a stable equilibrium can be maintained with very low level continuous production after a short, small magnitude burst of antibodies. The possibility of the existence of a steady state in this region is discussed in Appendix A.2, and its stability was demonstrated numerically. The lack of exponential growth, characteristic of an immune response, arises from the strong stimulation of available Y cells towards terminal differentiation into Z cells, at the expense of the non-antigen dependent differentiation into X cells, thus interfereing with the positive feedback loop (Fig. 5) and preventing the build-up of an immune response. The small amount of antibody produced blocks only a small fraction of the antigen. Repeated administration of antigen just increases the overdosage and strengthens the paralysis, with respect to stability and level of activity; hence there is memory of paralysis, or tolerance. On the other hand, degradation of the antigen is predicted, for this type of tolerance, to have the potential of causing a delayed response. Similarly blocking of antigen by passive antibody or by any other method of reducing the amount bound to lymphocytes, may break the tolerance. If partial stimulation of Y cells by antibody-antigen complex is allowed in the model, then repeated administration of low doses of antigen-even with gradually increased amounts-could lead to formation of high zone nonresponsiveness and tolerance, since it is now Ag T which determines the ratio between terminal differentiation and proliferative recruitment. Accumulation of antigen up to a paralyzing level can be done in this way without eliciting an appreciable response, by adiabatic shifting of the steady state. A natural situation in which a state of tolerance of this type can be envisioned is tolerance to self. The coexistence of the body's lymphocytes with self antigens may be related, partially, to the great abundance of the latter ones. The break down of tolerance by non-self or atlered-self antigens could of course involve complex mechanisms. An example of a gradual acquisition of tolerance without eliciting a response might be seen, conceptually, in some aspects of tumor growth. We would like to thank Dr Alan Perelson for reading the manuscript and for his comments. We thank DrG. Knott and Dr R. Shrager of the Division of Computer Research at NIH for their advice concerning the use of MLAB-simulation system. This study was supported in part by a grant from the United States-IsraelBilateral Science Foundation (BSF), Jerusalem, Israel. REFERENCES BELL, BELL, BELL,

G. 1. (1970). J. theor. Bioi. 29, 191. G. 1. (197la), J. theor. Bioi. 33,339. G. 1. (197lb). J. theor. Bioi. 33,379.

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BELL, G. I., PERELSON, A. S. & PIMBLEY, G. H. (eds) (1977). Theoretical Immunology. New York: Marcel Dekker Inc. BELLMAN, R. & COOKE, K. L. (1963). Differential Difference Equations, p. 449. New York: Academic Press. BRITTON, S. & MOLLER, G. H. (1968). J. Jmmunol. 100, 1326. BRUNI, C., BJOVENCO, M. A., KOCH, G. & STROM, R. (1975). Math. Biosci. 27, 191. CELADA, F. J. expo Med., 125, 199. COHEN, M. W., JACOBSEN, E. B. & THORBECKE, G. J. (1966). J. Immunol. 96, 944. DAVIE, J. M. & PAUL, W. E. (1971). J. expo Med. 134,495. DELISI, C. (1977). Math. Biosci. 35,1 (1977). DIBROV, B. F., LIVSHITS, M. A. & VOLKENSTEIN, M. V. (1977). J. theor. Bioi. 65, 609. DIXON, F. J. (1954). J. Allergy 25, 487. DORIA, G., SCHIAFFINI, M., GARAVINI, M. & MANCINI, C. (1972). J. Immunol. 109, 1245. DORIA, G., MANCINI, C. & DELISI, C. (1978) J. Immunol., 121,2030. DUTTON, R. W. & MISHELL, R. I. (1967). In Cold Spring Harbor Symp. Quant. Bioi. XXXII, 407. DUTTON, R. W. & MISHELL, R. I. (1967). J. expo Med. 126,443. FREEDMAN, H. I. & GUTICA, J. A. (1978). Math. Biosci. 37, 113. HUMPHREY, J. H. & KELLER, H. U. (1969). In Developmental Aspects ofAntibody Formation and Structure. (J. Ster2:e & I. Riha, eds), p. 485. C2:echslavacian Academy of Sciences. JERNE, N. K., NORDIN, A. A. & HENRY, C. (1963). In Cell-bound Antibodies (B. Amos & H. Koprowski, eds), p. 109. Wistar Institute Press. JONES,J. M., AMSBAUGH,D. F., STASHAK,P. W.,PRESCOTT, B.,BAKER,P.J. &ALUNG, D. (1976). J. Immunol. 116, 647. KALLOS, P. (1975). Progress in Allergy 1, 19. LANGEVOORT, H. L., ASOFSKY, R., JACOBSEN, E. B., DEVRIES, T. & THORBECKE, G. J. (1963). J. Immunol. 90,60. MAKELA, O. & NOSSAL, G. J. V. (1962). J. expo Med., 115, 231. MAKINODAN, T. & ALBRIGHT, J. F. (1966). Progr. in Allergy 10, 1. MAKINODAN, T., SADO, T., GROVES, D. L. & PRICE, G. (1969). Current Topics in Microbiol and Immunol 49, 80. McMASTER, O. D. (1953). In The Significance of the Antibody Response (0. Pappenheimer, ed.), ch. 2, pp. 13-45. New York: Columbia University Press. MERRILL, S. (1977). In Proceedings of the International Conference on Nonlinear Systems and Applications (V. Lakshmikantham, ed.), New York: Academic Press. MOHLER, R. R, BARTON, C. F. & Hsu, C. S. (1977). T and B cell Models in the Immune System in Theoretical Immunology. (G. I. Bell, A. S. Perelson & G. H. Pimbley, eds), New York: Marcel Dekker Inc. MOSIER, D. E. & COPPELSON, L. W. (1968). Proc. natl. Acad. Sci. U.S.A. 61, 542. NIELSEN, K. H. & WHITE, R. G. (1974). Nature 250,234. NOSSAL, G. J. V., ADA, G. L. & AUSTIN, C. U. (1964). J. expo BioI. Med. Sci. 52, 311. NOSSAL, G. J. V., SUSTIN, C. M. & ADA, G. L. (1965). Analysis of immunologic memory, Immunology (London) 9, 333. PERKINS, E., SADO, T. & MAKINODAN, T. (1968). J. Reticuloendothelical Soc. 5,11. PIERCE, C. W. (1969). J. expo Med. 130,365. ROMBALL, C. G. & WEIGLE, W. O. (1973). J. expo Med. 138, 1426. SCHARFF, M. D. & LASKOV, R. (1970). Progr. Allergy. 14,37. SERCARZ, E. E. & COONS, A. H. (1963). J. Immunol. 90, 478. SMITHIES, O. (1965). Science 149, 151. THORBECKE, G. J., ASOFSKY, R, HOCHWALD, G. M. & SISKIND, G. W. (1962). J. expo Med. 116,295. THORBECKE, G. J., JACOBSEN, E. B. & HOCHWALD, G. N. (1965). In Molecular and Cellular Basics of Antibody Formation, (J. Ster2:e ed.), p. 587 C2:echeslavacian Academy Sciences. UHR, J. W. & FINKELSTEIN, M. S. (1967). Prgr. Allergy. 10, 37.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION UHR, J. W. & BAUMANN, J. B. (1961). J. expo Med. 113, 935. URBAIN, J., VANAcKER, A., DE VOS-CLOETENS, C. & URBAIN-VASANTEN,

81

G. (1972).

Immunochem, 9, 121. WALTMAN, P. E. & BUTZ, E. (1977). J. theor. Bioi. 65,499. WEIGLE, W. O. (1975). Adv. in Immunol. 87, 1.

APPENDIX A

(1) Steady States and Stability

We study first the stability of equations (1)-(2) for a fixed Ag. We define new variables x, y by X=Xo+x, Y= Yo+y (AI) where X o and Yo are given in equation (5a). Inserting in (1) and (2) we obtain for x and y the equations dx dt =-(aAg+e')x+ey(t-71)'

(A2)

dy dt =: aAgx(t- 72) - (a'Ag+ d)y.

(A3)

Inserting x = X eAt, y = y eAt we have: -aAg e- A'T2 x+(A +a'Ag+d)y = O. (A4)

The solution for x and y will be non trivial only if the determinant of coefficients vanishes. Thus, the characteristic equation for A is: !(A) =:2+ [(aAg+e') + (a'Ag+d)lA

+ [(aAg+ e')(a'Ag+ d) - eaAg e- AT ] 7

=

1"1

=:

0,

(AS)

+ 72'

The roots (eigenvalues) A = Ai of (A5) are generally complex numbers, which we will suppose to be simple (the case of multiple or otherwise degenerate eigenvalues being of little biological interest). It is known that the Ai are enumerable and that the (infinite) sequence {Ai} can be ordered according to lAd. For any (integrable) initial functions XI(t), YI(t) defined for -1" ~ 7 ~ 0 a solution of the system (A2), (A3) can be written in the form x(t)

=

00

L i=-oo

ai eA,t,

Y(t)

= L

(3; eAt',

(A6)

i=-oo

where the sequences of complex numbers {ail, {(3i} are so ordered that the above series are real-valued.

82

Z. GROSSMAN ET AL.

Consider now equations (1) and (2) of section 2. If the initial functions Xr(t) and Yr(t) are chosen non-negative (as they should) for -T1::; t::; 0 and

-"2 < t ::; 0, respectively, then it is clear from inspection of the equations that X(t) and Y(t) cannot become negative for 0 < t < 00. To see that, note that if this is not the case, i.e. if X or Y or both become negative at a certain time,

then by continuity of the solution the zero axis must be crossed. Suppose that t1 is the first time at which this occurs and X, say, reaches zero, then from (1) dx dt

- = S+eY(t1 -Tl) >0

at this time and therefore X cannot become negative. Similar argument applies if Y is supposed to vanish first. Consider first the case in which Ag lies in the interval between Ag;;- and Ag~, as in (8b), so that the denominator of (5) is negative, equation (6) is not satisfied and no positive steady state solution exists. We shall show that a positive real root of (A5) exists in this case. f(A) is a continuous, rea. function of A. For A = 0, f(O) is identical to the denominator of (5) and so f(O) < 0. Also, a value Ao > 0 can be found such that

(aAg+ e')(a'Ag+d) -eaAg e-A.oT = 0,

(A6)

so that f(A o) > O. Thus, as A is continuously increased from 0 to 11. 0 , f(A) changes sign and therefore a positive value of Amust exist for whichf(A) = O. In fact, f(A) is monotonic in this region, so that only one such root exists. Since the asymptotic solution will be dominated by this real positive eigenvalue X and Y must grow without limit when the quantity of antigen is fixed between the critical bounds. Z, the concentration of the antibody forming cell population, which is obtained by integration with the help of (4), also possesses the same asymptotic behaviour. Note that dominance by a complex eigenvalue with (larger) positive real part is excluded, since a growing oscillatory solution necessarily requires the variables to alternate between negative and positive values, and this is not allowed, as we have seen. Next, consider the case where AgAg~, so that the denominator of (5) is positive and a positive steady state solution does exist. We shall show that this steady state is a stable equilibrium. To do so we have to show that all eigenvalues which contribute to the sums in (A6) have negative real parts. Assume first that a A exists which is real and positive. Then by the requirement on Ag, and since e -AT < 1,

(aAg + e')(a' Ag +d) - eaAg e-A.T > (aAg + e')(a' Ag +d) - eaAg > 0.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

83

Hence [(A) > 0 and A cannot be a root of (AS). The possibility of a complex eigenvalue with positive real part is again excluded since the asymptotic solution would be dominated by such roots and would become negative. Thus the only eigenvalues which do contribute must have negative real parts, and the solution of equations (1), (2) converges asymptotically to the steady state given by (Sa) and (Sb). In fact, the results concerning the asymptotic behaviour of (A2, A3) can be derived by application of a theorem by Bellman & Cooke (1964). For completeness, we shall now bring the formal, though less instructive, derivation. The characteristic equation (Ag) can be rewritten in the form

H(u)=(u 2 +pu+q)e"+1'=0

(A7)

where we have defined U

= AT, P = [(aAg+ e') + (a'Ag+d)}T, q = (aAg+ e')(a'Ag+d)r 2 (A8) 2

and l' = -eaAgr • p, q > 0, and l' is real, so that theorem 13.9 of ref. (40) applies. The theorem states, that a necessary and sufficient condition that all the roots of H(u) = 0 lie to the left of the imaginary axis is that (a) l' ~ 0 and (1' sin aw)/pa w< 1, or (b) -q < l' < 0 and (1' sin aw)/pa w< 1. Since l' < 0 here, condition (b) is the relevant one. ak (k ~ 0) denotes the sole root of the equation cot a

= (a 2 -q)/p,

(A9)

which lies on the interval (hr, k7T + 71'). The number w is defined separately for various cases. One of those is l' < 0 and p2;?: 2q, in which case w = 2. It can easily be shown from (A8) that p2 > 2q, so that w = 2 here. Then aw lies in the interval (271',371') and sin a w> 0, so that (1' sin aw)/pa w< 1. The necessary and sufficient condition for stability is thus reduced to -q < 1', or

q+1'>O.

(AIO)

But q + r is identical to ".2[(0) [see (AS)], the sign of which determines the existence of a positive steady state [equation (6)]. Thus, again, existence implies stability. (2) The Complete Model with Antibody Feedback

Consider equation (15) for the steady state value Ago, of the complete system with antibody regulation. When Ago is.varied continuously from 0 to

84

Z. GROSSMAN ET AL.

Ag;;-, the left hand side of (15) describes a trajectory from 0 to infinity, and the right hand side from infinity to a positive value, and hence these trajectories must intersect at some positive value of Ago < Ag;;- which is the steady state. Furthermore, both sides of (15) change monotonically in this region, so there is only one intersection. For Ago between Ag;;- and Ag~ the left-hand side of (15) is negative while the rignt-hand side is positive (cP - p =K 1> 0 and Ag T > Ago) and no solution exists. If Ag(O) > Ag~ we look for a solution Ago between Ag; and Ag(O). Rewrite (15) in the form

a (Ago) = RAg6, a (Ago) == [(aAg o + e')(a'Ag o + d) -eaAgo](Ag T -Ago)(1 + PAg o),

R =saa"i{3'Y/pcr.

(All)

P=K1{3/pcr.

Equation (15) can also be written as a4(Ag~ + a3Ag~ - a2Ag~ + alAgo - ao == 0,

a4 = aa'P, a3 = aa'+R +P(aa'AgT-B), a2 = aa'AgT-B+P(e'd-BAgT), al =ed -BAgT+Pe'd,

(A12)

ao= e'd. B is given in equation (6b). As before, we assume B < 0, so that all the coefficients in (A12) are positive. By Descarte's rule, the number of positive roots cannot exceed the number of variations of sign in the coefficients of the polynomial, so there can be at most three steady states, of which one has been identified. Consider equation (All), between Ag~ and Ag T • The left hand side is zero at both limits and positive between them, and the right hand side

Ag

FIG. 17. A representation of equation (All) showing the existence of multiple real roots.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

85

increases monotonically. In order to have multiple roots, the trajectories corresponding to (All) might look as in Fig. 17. First note that if there is one intersection in the region from Ag; to Agy, there must be two of them. It is clear from the figure that conditions which ensure the existence of three roots are that R is not too large (Le. sufficiently small initial slope of RAg 3 ), and given R, Ag T should not be too small (Ag!j.). A sufficient condition is that the maximum of a, a(Ag m ), should be greater than RAg;". (This is not a necessary condition; see the curve ending at Ag'~.) For simplicity, neglect the term proportional to (<{J - p) in (15). (Negecting this term amounts to assuming that antibody is removed from the site of interaction mainly while not in complex with antigen, and does not affect any of the qualitative results.) Equating the derivative of a (Ag) to zero and solving for Ag we find: Ag m = 3~a' [ aa'AgT -B +{(aa'Ag T -B)2-3aa'(e'd -BAgT)}1 /2] (A13) and the requirement becomes (2aa'Ag m

+ B)(Ag T - Ag m )2> RAg;".

(A14)

For large AgT, Ag T ~ 00, Ag m ~ 1Ag T and three positive steady states will exist if (but not only if) the parameters satisfy the condition (A15)

aa'>2R. APPENDIX B

Integration of Equations (1)-(3) We shall derive a recursion formula for the solution of the equations of growth with given initial conditions. For specifity a step function dependence will be assumed for the antigen, Ag(t) == 0, t
Ag(t) = Ag,

t:2:

(B1)

0,

so that dAg/ dt = 0 for t> 0, nd the initial conditions for the variables &re . Xr(t)=s/e',

Yr(t)

= 0,

2 r (t) = 0;

t
(B2)

Let us denote by X 1 (t), Y 1 (t) and Zl(t) the solutions of the equations on the first time interval of length 7' after injection of antigen where 7' = 7'1 + 7'2 is the delay "along the memory loop" (See Fig. 5): Zl(t)=Z(t);

0< t:s; 7'.

(B3)

86

Z. GROSSMAN BT AL.

Equations (1) and (2) can be rewritten as one second order differentialdifference equation [by solving equation (1) for X and inserting in (2)J: d2)(

d)(

dt 2 +(a +8) d't+a8X -l3y(t-r)X(t- r),

(B4)

with the new notations

a=aAg(t)+e',

l3=e,

y=aAg(t),

8 s a'Ag(t)+d.

(B5)

(f3 and yare not the same as those defined in Section 3.) The characteristic equation corresponding to (B4) was given in (AS). For simplicity we consider in the following the special case in which 72 = 0,7 = rl. For 0$ t < 7', by (B 1) and (B3),

a =aAg+e',

(3

= e,

y

= aAg,

8 = a'Ag+d,

«B5')

and

X=)(I,

Y=Yl.

(B4) becomes: d 2X

d?"""1 + (a + 8)X1 + a8X1 = s8,

O
(B6)

This is now an ordinary differential equation, and the initial conditions are the values of Xl and its derivative at the beginning of the time interval, namely at t = 0. dX1 dt (t = 0) = s -

aX1(0) = s(1- al e').

(B6')

Note the discontinuity in the derivative of X at t = 0, due 50 the discontinuity in a. The second condition is obtained from equation (1). The characteristic equation corresponding to (B6) is A2 +(a+5)+a8=0,

(B7)

and it has two roots, A = -a and A = -8. The solution is X 1 = X 10 + A 1 e

-a/

-lit

+B t e ,

X 10 =s/a.

(B8)

Using (B6') yields for the coefficients

B 1 =O,

At=SC,-;)·

(B9)

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

87

From (2), the equation for Y1 is: dY1

d(= yX1 -<5Y1 or

(B10)

and the solution is (Bll)

Inserting this in (Bl0) we obtain: ys

y

(B12a)

Y:O=-X10 = /3 8e"

and from the condition Y 1 (0):=: 0,

1) . 1 1 (1 B- 1 =-YlO -A 1 =-ys-+---8e' <5 -y e' ex

(B12b)

If ex, /3» liT, ale» 1 and 8 -y > 0, then fL > 0 and 8 1 < O. Y 1 [equation (Bll)] increases rapidly at first (during a time much shorter than T) due to

the decrease in the negative term proportional to B1 , and then decreases rapidly with the (positive) A 1 term, and remains approximately constant at a low level (YlO) for the rest of the time interval. These changes correspond to a rapid triggering of precursors into Y cells followed by a rapid differentiation of these. The same procedure may now be readily continued for the next time intervals. For T < t < 21', the parameters are as before, and T
(B13)

The left hand side of (B13) is the same as in (B6), but the non-homogeneous part includes a term proportional to e -etC, which is also a solution of the corresponding homogeneous equation. Hence the solution to the full equation will include now a term ("secular term") proportional to t e -at (:814)

We shall not calculate the constants in (B 14), but instead turn to the general case. The equation for the nth time-interval is 2

d X n (a +8)-+a8X dX -2-+ n :=: s8+{3yXn dt dt n

(

1

)

t-1' ,

(n -1)7":5 t:5 n7".

(B15)

88

Z. GROSSMAN BY AL.

The initial (or continuity) conditions are

Xn[t = (n -1}r] = X n- 1 [t = (n -1)1"],

(B16)

dXn [t = (n -1)1"] = dXn - 1 [t = (n -1)r]. dt dt The corresponding equation for Yn is 2

d Yn dYn ( -d 2 +(a+o)-+a8Yn ='Ys +,BYYn - 1 t-r) t dt

(B17)

with the conditions

Yn[t = (n -1)r] = Yn-1[t = (n -1)r],

(B18)

d Y-n --1 [ t=(n-1 ) r ] -d Yn [ t=(n-1 )7] =

dt

dt

and Xn can be obtained from Yn using the relationship Xn

=

1:.y (ddtY n + 8Y. ) n

•

(B19)

Note that (B 17) and (B 18) are valid only for n > 1 in the more general case with 72'" O. We shall use (B 17) rather than (B 15), since knowledge of Y will enable us to find the measurable quantity Z by direct integration [see equation (4)]. The solution is

n-1

-- y.nO + "i(A -"I L, t nj e + B ni e -BI. i=O

Inserting the free term (B21)

the terms proportional to A no and E no give no contribution (they satisfy the related homogeneous equations), and by collecting terms of the same time-dependence we obtain, after some algebraic manipulations which are omitted, the following system of algebraic equations for the other coefficients (B22)

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

where j = 1, ... , n -1, and Ann =Bnn =0. These equations are readily solved. Starting with j

89

= n -1 we have

f3y e"'T An,n-l = (n -1)(8 _ a) A n- 1 ,n-2, -f3ye

(B24)

ST

1

A n,n-2 = (n -1)(8 _ a) -(n -1)(n - 2)A n,n-l +{3y e"'T(A n_1,n_3 - (n - 2)TA n- 1 •n- 2

(B25)

and similiar expression for B n,n-2' Going with j down to 1, the coefficients are expressed in terms of previously found ones and those belonging to the previous time interval. Finally, A no and B no are calculated from the conditions (BI8). Consider the asymptotic behaviour of the solution given in (B20) when t and n become large. Reiterating (B21) the free term can be written as the series Y: nO

= [YS

as

1 + yf3 + (Yf3)2 +

a8

as

(Yf3) . .. as

n-l]

-1]

[(Yf3/ aB )n as y{3/ as-l

= ys

(B26)

If

(Yf3/ as) < 1, n"OO

YnO

~

ys/(aS -yf3).

(B27a) (B27b)

But using the definitions (B5) (with Ag(t) = Ag (B27a) is just the condition for the existence of a positive steady state, and the right hand side of (B27b) is equal to Yo [see equation (5)]. This result is consistent with Appendix A, where we have shown that the steady state is stable in this case-the time dependent part in (B20) approaches zero asymptotically and Y converges to the steady state. If (y{3/aS) > 1,

YnO

~ ( {3 ys ~ ) (Yf3)n 'Y -au au~,

(B28)

90

Z. GROSSMAN ET AL.

and since n is the integer part of (tIT), the limit can be approximated by an exponential growth function, in consistency with Appendix A again. The Z-equation is [by equation (3)] dZ dt = al/Ag[ yet - 7'3))- [Y(t- 1'4)].

(B29)

The contribution of Y nO to the right hand side of (B29), if 7'::::: 7'4 -7'3 > 7', is a positive constant, except for the times in which t - 7'3 or t - 7'4 coincides with the beginning of a new 7'- interval, when there are discontinuities in Y nO • Thus the contribution to the Z-trajectory is made out of a connected series of straight segments with positive changing slopes. The increments in slope are greater than the decrements, and the overall evolution is that of exponential growth. The short time behavior, however, is dominated by the time-dependent terms in (B20). A typical behavior for the early time interval for a range of the parameters is the one described above [see the discussion following (B12)], in which there is a fast increase in Y followed by a fast decrease and then a low, slowly changing segment. This "spike" pattern is repeated in the following time intervals with increasing amplitude (Fig. 7). The terms of the form t i e -at and t i e -lit, with j > 1, tend to smoothen the spikes and to make the behavior more regular. Referring to the Z population, a fast increase in yet -7'3) causes Z to increase rapidly to [equation (B29)], and as yet - 1'3) falls down, the increase in Z stops and its value remains at the level reached. This brings about the step-wise increase pattern (see Figs 6 and 7). In contrast, a spike increase in the Y (t - 1'4) term causes, in a similar fashion, a stepwise decrease in Z. However, since the decrease is due to an earlier spike its size will be smaller (by a factor which depends, roughly, exponentially on the time difference) than the adjacent ascending step. The pattern will thus be essentially that of growing steps with smaller corresponding "dips". A more extensive study could be done on the basis of the present analysis in order to specify the conditions in parameter space which favor the appearance of stepwise patterns of various kinds. Better kinetic studies of the growth patterns of antibody-forming cells might motivate further analysis. APPENDIX C

Estimate of the Parameters

We shall present here a first, rough estimate of the model parameters. The actual parameters used in the simulations are specified in the text or in the captions to the figures.

DYNAMICS OF ANTIBODY SECRETING CELL PRODUCTION

91

Consider first equations (1)-(3). We estimate the antigen concentration injected to the body to be the order of 5 x 10 17 , mol/I. The relevant amount localized in the germinal centers (spleen, nodes) is estimated to be about 1/100 (Nossal et aT., 1968) of this so that Ag(D) = 1 X 10 15 mol/l. The concentration of specific precursor cells in a non-immunized animal is estimated as X(O) = 105 cells/l (Humphrey & Keller, 1969; Davie & Paul, 1971). The life time of memory cells (and precursor cells) is assumed to be around 20 days (Celada, 1967); long-time memory in our model is due mainly to active reproduction of cells by residual antigen rather than due to long-living cells remaining from the original response to antigenic stimulation. A considerably longer life time for the majority of memory cells would favor oscillations with amplitudes which increase with time. Hence e' = 0·05 (days)-l. The background supply of new precursors from stem cells is thus S = e'X(O) = 5 x 103 (cells/day-lite~. It is quite likely that e' is not constant locally, at the site of interaction, but increases with the concentration of lymphocytes (because of non-uniform concentration throughout the physiological system and activity in other organs). Then e' should be interpreted as an effective average decay rate and its value might be considerably larger than 0·05. The differentiation rate of Y-cells into the memory cell channel is estimated as a one-fifth of a day, so that d == 5(days-~) and e is chosen to be around 20 (day)-l, allowing for an aberage of 2 divisions per cell during the time delay 7"1 (see below). a and a' are equal approximately to 1O-16l/(mol h), or a = a' = 2 x 1O-15 1/(mol day). a" is then chosen in the region between 2a' to 16a', allowing for 1 to 4 effective divisions during the maturation time 7"3 to plasma cells. 7"1 + 7"2 should be equal to the average shelf-time observed in staircase patterns and hence it is approximately 10 hours. Usually we assume for simplicity 7"2 = 0, so that 7" = 7"1 = 0·35 days. 7"3 and if = 7"4 -7"3 are of the order of one day each (27). There is an additional uncertainty in the estimate of these times, since in reality proliferating cells which are in the process of maturation do produce antibody at varying rates, while in the model we assume that only mature plasma cells do that. Although plasma cells (Thorbeeke et aT., 1962; Jerne, Nordin & Henny, 1963; Nossal, Sustin & Ada, 1965) are indeed the major source of antibody, we might choose "effective" times and rates in order to improve the approximation. We now turn to the equations for Ag and Ab. (3 is chosen, to be around 107 1/(mol s), or {3 = 1·5 x 10- 12 1(mol day). The arbitrariness is related to the fact that our model assumes a simple one to one interaction between Ag and Ab, while the process of blocking may be considerably more complicated. Hence parameters related to antibody production and interactions are actually referring to "blocking units" of antibody, rather than to Ab

92

Z. GROSSMAN ET AL.

molecules and to their "avidity". Fortunately, the qualitative nature of the results of simulations is quite insensitive to the particular choice of the parameters. The ·number of antibody molecules, or units, produced per plasma cell rer second is estimated here to be around 1000 (26), so that ')I = 2·5 x 10 mol/day. e, the rate of antibody removal from the control site, is larger than the natural death rate of IgM molecules (life time of a few days), and thus is approximated by () = 1 (daysf i . Finally, the dissociation rate of AbAg complex, yielding the order of magnitude of p and I/J, is estimated by p = I/J = 10 (days)-I. In some examples blocking of antigen was switched off for the first few days, to account for the possible buffer effect of the antigen in the circulation (see captions to Figs 6 and 7). It is convenient to renormalize the parameter set in such a way as to render Ag(O) and X(O) of the order of unity and to make concentrations non dimensional. We keep the same notation as before, but the meaning of symbols is changed according to the following set of transformations: X ~ X/X, Y -+ Y/ X, Z -+ Z/ X, Ag ~ Ag/Ag, Ab ~ Ab/ Ag. The renor~ malized parameters then become S~S/X=0'05 day-1; a=a'-+aAg= 10day-l; al~a"Ag=(20-160)day-1; ,B~,BAg=7500day-l; y-+ ')IX/ Ag = 0-005 day-t, where X SE 10 5 cells/l and Ag X5 x 10 15 mol/l. The "initial set" of parameters referred to in text, then have the following values: X(O) = 1, Ag(O) = 1, e' = S = 0·05 day-I, a = 10 day-I, a' = 4 day-t, a"=40day-1; d=5day-t, e=20day-t, 7'=7'1=0,3 days, 7'2=0, 7"3= 1 day, 7"4=2 days; ()= 1 day-I p=10day-1, ,B=7500day-t, y= 0·005 day-1. Because of the qualitative nature of the model and the experimental uncertainties we felt free to change some of these values within a factor of 10 or even more. The most fluid estimates are those of e' and s, and of ,B and 1'. The rate e' may be larger for cells at the site of interaction than the background death rate, and ,B and l' depend strongly on the details of interaction between antigen on the one hand and antibody and cell receptors on the other.