J. theor. Biol(1977)
65, 609-631
Mathematical Model of Immune Processes B. F. DIBROV, M. A. LIVSHITS and M. V. VOLKENSTEIN Institute of Molecular
Biology, Academy of Sciences of the USSR
(Received 14 November 1975, and in revised,form
10 May 1976)
In the model the time lags of the antibody production and immune memory formation are taken into account explicitly. The antibody-antigen reaction is supposed to be very fast. The cases of a reproducing antigen as well as that of a non-reproducting antigen are considered. The conditions of the infinite increase of the antigen quantity and of the antigen elimination are obtained. For the rapidly reproducing antigen the latter condition includes the requirement for the time lag of the immune response to be not too short or not too long. In the case of the poorly catabolized nonreproducing antigen the cyclic appearance of the antibody producing cells due to the immune memory is described in the frame-work of the model. The mathematical structure of the model is similar to that of the
Volterra-Lotka equations. The only difference is the presence of the time lags in the non-linear terms. The time lags lead to the instability of the stationary state. In the prolonged reaction the antigen quantity may perform several oscillations before the elimination of the antigen.
1. rntrodnction The vertebrate organism responds on the introduction
of the foreign material
(antigen) by the production of specitic gamma globulines (antibodies). This response is the humoral one. The other kind of response, the cell-mediated one, consists of the production of the specific reactive cells. Both the antibodies and the reactive cells can circulate in the organism and react with the antigen. As a result of these reactions the antigen can be killed, inactivated or phagocytosed by the cells of the reticula-endothelial system. The contemporary ideas concerning immunity are based on the clonal selection theory (Burnet, 1969), which suggests that independently of the antigen the lymphocytes are produced in the organism, and every one of them is sensitive to one antigen or several similar antigens. That is determined by the presence of the antigen-specific receptors at the cell surfaces. If an antigen is introduced into organism it acts in a selective way at those lymphocytes (target cells), which possess already the receptors for this antigen. The interaction of antigen with receptors activates the cells (cf. T.B. 609 40
610
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
Burnet, 1969; Raff, 1973). We consider here only the humoral immune reaction. The general scheme of the development of the humoral immune response is given in Fig. 1. In this case target cells are B-lymphocytes. They are formed as the result of differentiation of the stem cells. The B-cells themselves do not divide and they obtain the ability for proliferation only being transformed into blast cells (Y). The blast-transformation is caused by the antigen action and occurs after the latent period of 24-48 h. The Y-cells proliferate intensively. A part of the Y-cells transforms into the plasmablasts (Zi), which after a series of mytoses transform into the mature plasma cells (Z,). The definite clone of the Z-cells produces antibodies of the same specificity as that of the receptors of the target cells (cf. Nossal, 1964; Chertkov & ____-___------I
I Memory
i I I
/-
1
I
Ahbodles
FIG. 1. The scheme of the development of the antibody-producing line symbolizes the proliferation.
cells. The oscillating
Friedenstein, 1972). The mature plasma cells are the main source of the antibodies. They are not capable of further division. Their life-span is some tens of hours. The antigen-stimulated B-lymphocytes can produce also the so-called memory cells which are responsible for the change of immune reactivity of the organism to the recurrent antigen expositions. Since no differences between memory cells and virgin target cells are known it is reasonable to believe that the immune memory effect is due to the increased number of target cells. In the case of the cell-mediated immune response the antigen stimulates the so-called T-lymphocytes. They are the progeny of the stem cells like B-lymphocytes but during their development T-lymphocytes migrate through the thymus. After contact with the antigen, T-lymphocytes proliferate and differentiate into the specific reactive cells. T-lymphocytes and their progeny
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
611
are not capable of producing antibodies, but they can play an important role in the development of the humoral response because of the so-called T-B co-operation. It can be suggested that the T-lymphocytes stimulated by antigen produce some non-specific factors which lower the threshold of sensitivity of the B-lymphocytes to antigen (cf. Raff, 1973). T-B co-operation is not observed for every kind of antigen. There are the so-called thymusindependent antigens which are capable of stimulating the normal humoral response without the help of the T-lymphocytes (cf. Chertkov & Friedenstein, 1972). Several mathematical models of the immune reaction have been described. The model of Jilek & Sterzl (1970) is devoted to description of the development of the antibody-producing cells and does not take into account the antibody-antigen interaction. A more complete model is presented by Smirnova & Stepanova (see Romanovsky, Stepanova & Chemavsky, 1975). The model of Molchanov (1971) is formulated in very general terms. It allows many different regimes of reaction depending on the parameters. The most advanced models are those of Bell (1970, 1971, 1973). The first of Bell’s models presents a detailed description of the development of the antibody-producing cells and of the antibody-antigen interaction with a great deal of the computer calculations. Unfortunately the complicated character of these models did not allow qualitative analysis. In another model (Bell, 1973) the immune reaction is described by two differential equations for antigen and antibody quantities, the rate of antibody production being taken as a function of the current antigen and antibody quantities. This implies the constant number of the target cells for the given antigen during reaction. In fact, the variation of number of the target cells can be considerable. The differentiation of antigen stimulated B-lymphocytes into antibody-producing cells takes some time and therefore the rate of the antibody production is determined not by the current antigen quantity but by the previous values of the variables. Such a time lag in the immune response due to lags in the triggering of the cells in other states is not taken into account in the models listed above (except that of Jilek & Sterzl, 1970). The aim of this work is the construction of the mathematical model of the thymus-independent reaction of the organism on the introduction of antigen. Such a model cannot describe all the details of the process, which have not yet been sufficiently studied. The purpose of the model is the qualitative and semi-quantitative treatment of the main features of the process. The important features of immune processes are the time lags in the immune response and in the immune memory formation. In the present model these time lags are taken into account explicitly. The cases of the
612
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTRTN
self-reproducing antigen (infection) as well as that of the non-reproducing antigen are considered. We assume that the antibody-antigen reaction is very fast and this allows us to consider the relatively simple equations. Such a simplification is correct until the quantity of antigen becomes very small. In section 3 the immune reaction is considered, without the participation of the new-formed memory cells. Such consideration is valid if the total antigen is eliminated before the formation of the considerable quantity of memory cells or in the case of a weak immune memory formation. Section 4 deals with the reactions in which the memory cells take part. In the case of the non-reproducing and poorly-catabolizing antigen the cyclic and synchronized appearance of the antibody-producing cells is described. Such phenomena have been observed for example after the single injection of the aggregated human IgG in rabbits (Romball & Weigle, 1973). According to our model the analogous dynamics can take place in the case of infections as well. In sections 3 and 4 the conditions of the total elimination of the antigen and of the unlimited propagation are obtained. These conditions show to what extent the immune process depends on the different parameters. In particular the important role of time lags is demonstrated. 2. TheModel For the sake of simplicity we shall assume all target cells to be equivalent and all corresponding antibodies to be identical. The differentiation of a single target cell after an antigen stimulation at the moment t = 0 is supposed to be independent of the antigen and of other cells. So the number of the mature plasma cells [z(t)] at a time t can be represented as a sum of responses of the individual target cells stimulated previously : z(t)
=
j F(t0
t'j@(t')
dt’
(1)
where Q(t) dt is the number of the target cells stimulated by the antigen during the time from t to t+dt. F(t) is the linear response function. For a single target cell which has been stimulated at the time t = 0, z(t) coincides with F(t). The qualitative shape of F(t) is shown in Fig. 2. After some time which is necessary for the differentiation of a stimulated target cell, the number of the plasma cells increases. The subsequent decrease of the number of plasma cells is due to their natural removal. Equation (1) represents the retardation in the formation of the mature plasma cells with respect to the stimulation of the target cells. For the sake
MATHEMATICAL
FIG.
2.
MODEL
OF
IMMUNE
PROCESSES
613
The qualitative shape of the response function F(t).
of simplicity we shall take into account this retardation in the form of the discrete time lag. Such a simplification corresponds to the substitution of the realistic response function F(t) by the delta function, F(t) N @f-T,) where T, is constant. In this case equation (1) leads to z(f)
where
6(f)
N cD(f--
T,)bqf-
T,>
(2)
is the step function, that is
t?(t) =1 tt 0, if 1, if
< 0 2 0.
Since the main source of the antibodies is the mature plasma cells, we shall assume the rate of the antibody production to be proportional to the current number of the plasma cells. From equation (2) we get
Nqt- r,>qt- T,>.
(3)
Here a(f) is the quantity of the antibodies. The rate of the target cells stimulation is supposed to be proportional to the number of the target cells [x(f)] and to the quantity of the antigen [g(f)]. Then = &(t
The bimolecular antibody-antigen the antigen and the free antibody
- T,)g(t - T,)O(t - T,>.
reaction is supposed to decrease both quantities. Thus the equation for the
614
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
quantity of the free antibodies is:
-WO = A$(tdt
T,)g(t- zp(t-
T,)--Rg(t)a(t)-J%(t).
Here -Rg(t)a(t) describes the antibody binding by the antigen, -h(t) describes the decay of the antibodies. The dynamics of the antigen quantity [g(t)] follows the equation: F
= Kg(t)-Qg(t)a(t).
Here K is the rate of increase (K > 0) or decrease (K -=c0) of the antigen quantity determined by the reproduction of the antigen and by the interaction with the non-specific protective factors of the organism. We shah call fir the rate of the antigen reproduction. In reality K is the rate of the development of infection (if K > 0) in the organism before the appearance of the specific antibodies. The term - Qg(t)u(t) describes the decrease of the antigen quantity due to the interaction with specific antibodies. We consider the memory cells as identical with the virgin target cells. The retarded appearance of the memory cells can be described by the expression analogous to equation (5). The corresponding rate is proportional to x(t- T,)g(t - T,), where T, is the constant representing the time lag in the immune memory formation. We can write
MO = J - ‘+ - Px(t)g(t)+A,x(t-Tm)g(tdt
T,)e(t-T,).
(8)
Here J is the rate of appearance of the specific target cells as the result of differentiation of stem cells. The term -x(t)/7 describes the natural decay of the target cells (z is the average life-time). The term -Px(t)g(t) describes the decrease of the number of target cells due to the interaction with the antigen. Both the differentiation of the target cells and their destruction due to the interaction with the antigen are included here. The term A,x(t- T,)g(t- TJ describes the retarded appearance of the memory cells after the stimulation of the target cells by the antigen. The affinity of the specific antibodies to the antigen is high. If the antigen is in excess the process rapidly reaches the steady state du(t)/dt N 0 because of the large value of the “coefficient” Ag(t). This means that the antibodies, produced by the plasma cells, bind the antigens rapidly. The experimental data show that in reality the considerable quantities of antibodies are observed only after the elimination of the antigen (see Boyd, 1966). This allows the consideration of a mere simple system of equations for the description of the main part
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
61.5
of the process: wo = Kg(t) - A,x(t___ dt
W) __
dt
T,)g(t-- T,yqt-
40 = J - 7 - Px(t)g(t) + A,x(t-
T,) (9) T,)g(t-
T,)t?(t- 7-J
where
[We have omitted here also the last term of the equation (6) that is correct if Rg(r) B E.] These equations describe “the struggle” between the target cells and antigen. The rapidly acting “mediators”, the antibodies do not enter explicitly into the equations. The equations are similar to those of Volterra and Lotka if A, > P. In our case the role of the prey is played by the antigen. The substitution of equations (6)-(8) by the system of the two equations (9) corresponds formally to the infinite rate of the antibody-antigen interaction. As the real rate is finite this substitution is not correct when the antigen quantity is close to zero. However, the calculations show that equations (9) represent the main features of the process. The detailed analysis of the effects connected with the finite rate of the antibody-antigen interaction will be presented in subsequent papers. Equations (9) formally allow the antigen quantity g(t) to cross zero. In the more realistic equations (6)-(8) this corresponds to the rapid elimination of the antigen [g(t) rapidly tends to zero]. All cases when in the framework of equations (9) g(t) tends to zero asymptotically or crosses zero will be interpreted as the antigen elimination.
3. The Immune Reaction Without Participation Cells
of the Memory
In the case of reaction which leads to the total antigen elimination before the formation of a considerable number of memory cells or in the case of the weak immune memory formation (A, m 0), we can consider equations (9) without the term describing the formation of the new memory cells [Amx(t-TJg(t-Tm)]. In addition to the short time intervals we can omit the terms describing the natural decay [ - x(f)/z] and the increase of the number of target cells due to the differentiation of stem cells (J). Equations (9)
616
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
then become
-ddt) = Kg(t) - A,x(t- T,)g(t- T;)a(t- Tr) dt WO = -Px(t)g(t). dt
(10)
The same equations may describe a reaction on a repeated introduction of a particular antigen. In such a case the initial number of the target cells can be increased. Equations (10) do not possess steady solutions with a non-zero quantity of the antigen. If the natural decay or the non-specific removal of the antigen provide K < 0 the quantity of the antigen tends to zero. It can be shown that if K > 0, the quantity of the antigen g(t) can only increase infinitely or decrease towards zero. Let us consider first the case when the change of the number of the target cells (x) during the reaction can be neglected [P = 0, x(t) = x0 = x(O)]. Instead of equations (10) we obtain -MO = Kg(t) - A,x,g(t - T,)O(t - T,). dt It is easy to see that if K -> 4x0
1,
(12)
the rate of the antigen elimination is always smaller than the rate of its reproduction and the equation describes the unlimited increase of the antigen quantity g(t). In the opposite case (K/&x0 < 1) the unlimited increase of the antigen quantity is possible only if A,x,T,.exp(-KT,+l)
1.
(13)
A,x,T, exp (-KT,+l)
< 1, KT, < 1,
(14)
If then g(t) tends to zero asymptotically;
if
A,x,T,exp(-KT,fl)>
1,
(15) then the antigen vanishes at the finite time, i.e. g(t) crosses zero (cf. Appendix A). These conditions can be reformulated in terms of the time lags. Let T’ < T” be the roots of the equation A,xoT exp(-KTfl)
= 1,
WI
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
617
then the conditions (13) (14), (15) become T,>T" (13) T,< T' (14) T' < Tr < T". (15) Now let us consider equations (10) for P > 0. The expenditure of the target cells during reaction (P > 0) weakens the protective ability of the organism. Therefore conditions (12) and (13) of an infinite increase of the antigen quantity which are sufficient at P = 0 are also sufficient at P > 0. But conditions (14) and (15) of the antigen elimination at P = 0 can prove to be not sufficient at P > 0. For any moment of time t” it is possible to find the conditions of the total antigen elimination before t* [in the framework of equations (lo)]. In particular the condition of the total antigen elimination before t * = 2T, is : A,x,T,exp
1
&lo -KT,-K(eKrv-l)
> 1 [ where go is the initial quantity of the antigen [ig(O)]. Somewhat weaker condition of the antigen elimination before 3T, is A,x,T,exp
pgo'p(KT,) - KT, - K
I
>2--J2
where q(s) = (d-1)
2-Ji e*: -I- F(e2r-e3-1.
Conditions (12)-(19, (17) and (18) demonstrate the role of the different factors and in particular the role of the time lag in the immune reaction. It should be noted that not all the relevant parameters are independent. Some part of the time lag is spent on the proliferation of the precursors of the plasma cells. If the rate of the proliferation (x) can be considered as constant the dependence of the immune response intensity parameter (A,) on the time lag T, is exponential: A, = B exTp. (19) The conditions (12) and (13) of the infinite increase of the antigen quantity in the framework of equations (10) become -Bx0 exTr < 1 wo K Bx,T,exp[(x-K)T,+lJ 1. w According to equation (20) if K > Bx, then for the small time lag T, the antigen quantity increases infinitely. According to equation (21) if K > 1c
618
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
then for the sufficiently large T, the antigen quantity increases infinitely. Thus the successful suppression of the rapidly reproducing infectious micro-organisms (K > x, K > Bx,) needs the time lag to be not too small and not too large. The model considered in this section [equations (lo)] does not take into account the immune memory formation. Such simplification is justified in particular if the memory cells which can be stimulated by antigen appear only after the total antigen elimination. In this case the number of the memory cells formed as a result of the reaction can be calculated as dt. %I = A,,, ji(t)g(t) 0 Here x(t) and g(t) are the solutions of the equations (lo), tr is the time of the total antigen elimination [g(tf) = 0; g(t) > 0 for all t in (0, tf)] according to equations (10). Such calculations can help choose the best way of immunization. We shall not discuss this problem here.
4. Immune Reaction with the Participation
of the Memory Cells
.The newly-formed memory cells can participate in the immune reaction if the antigen persists in tissues for long periods of time. Romball & Weigle (1973) observed that the injection of the poorly catabolized non-reproducing antigen (aggregated serum protein) causes the cyclical appearance of the antibody-producing cells. The first peak in the number of the antibody-producing cells was reached on the Mth day after injection of the aggregated human IgG in rabbits. The second peak occurred on day 13. The third peak was detected on day 21. So the interval between peaks was again 8 days. The dynamics of the antibody-producing cells depends slightly on the dose of the antigen. The interpretation suggested by the authors is that the initial peak of the antibody-producing cells results from the stimulation of the virgin precursor cells, succeeding peaks are the results of stimulation of the memory cells. This suggestion agrees well with our model. The model is represented by equations (9). If the natural decay and the appearance of the specific target cells due to differentiation of the stem cells during reaction can be neglected, equations (9) become ddt) = Kg(t) dt
- A,x(t-
-d-40 = -Px(t)g(t) dt
T,)g(t-
T,)O(t-
T,) (22)
+ A,x(t-T,)g(t-7;,)6(t-T,).
MATHEMATICAL
MODEL
The equations imply
the number
OF
IMMUNE
PROCESSES
of the antibody-producing
619 cells to be
40 au x0 - ml(f - m (23) (cf. section 2). Figure 3 shows the number of antibody-producing cells as a function of time obtained by the help of equations (22) and (23). The parameters are
f (days) FIG. 3. The dynamics of the number of the antibody-producing cells z(t) N x (t-T;)g. (t-T,) after the injection of the non-reproducing poorly catabolized antigen (K = 0). The parameters T, = 5 days; T, = 7 days; Pg,,T, = 4.5; A,x,T, = 1.2; A,/P = 3 are chosen to fit the experimental curves (Romball & Weigle, 1973). The scale on the vertical axis is arbitrary.
chosen to fit the corresponding experimental curves (Romball & Weigle, 1973). The very good fit which can be achieved with reasonable values of parameters (cf. legend to Fig. 3) can be considered as evidence in favour of our model. The fact that the initial point of the calculated curve z(t) is z = 1, and not z = 0, is due to the discrete form of the time lag used here. This discrepancy would be removed if one takes the continuous time lag (1). As will be seen later the cyclic dynamics can also take place in the case of infections (K > 0). However, the over-large rate of antigen reproduction leads to the infinite increase of the antigen quantity according to equations (9). So in the framework of equations (22), which are the simplified versions of (9), the antigen quantity increases infinitely if +f exp [(4,x,-K)T,]
< 1,
A+ > 1. r 0
620
B. F. DIBROV,
M. A. LIVSHITS
On the other hand, the fulfillment $exp
AND M. V. VOLKENSTEIN
of the condition (-ITT,)
> 1
(25)
provides the total antigen elimination independently of the values of all other parameters. The conditions (24) and (25) demonstrate in particular the essential role of the time lag (TJ. The intensity of the immune memory formation (A,) obviously favours the antigen elimination. Figure 4 gives the examples of the phase trajectories of equations (22) calculated for different values of the intensity of the immune memory formation.
FIG. 4. Phase trajectories of the system (22) in dimensionless variables for #T, = 2; = 5 and for differentvaluesof A,,,g.,T,; A, O-5; B, 3; C, 5; D, 10; The duration of the reactiont, in T, unitsis shown.
PgoTr = O-1; A&T,
5. The Model of a Prolonged linmnne Reaction The treatment of the immune reaction in the preceding sections did not take into account the fact that the target cells of a particular specificity are supplied as the result of differentiation of the stem cells, and that the lifetime of these cells is finite. If the reaction is so long that these effects are important then the immune reaction must be described by the complete
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
421
equations (9).
ig$ = Kg(t) - A,x(t-K)g(t--zy(t--T,) -WO = J - 40 - - Px(t)g(t) + A,X(E- T,)g(t- T,y?(t- T,). dr
(9
z
Equations (9) imply that before the antigen introduction target cells of particular specificity is stationary: x = allow more complicated kinds of behaviour than in the the equations (10) and (22). Equations (9) in contrast and (22) possess the non-zero stationary point
the quantity of the JT. Equations (9) cases described by to equations (10)
The stationary point has a sense not at any values of the parameters, since x(t) and g(t) must be non-negative. With positive x,, and gst it is convenient to use the dimensionless variables c(t) = x(t)/+ q(t) = g(t)/gSt. Equations (9) become
Wt) -=q(t)-gt-?;)r/(t-T#l(t--T,) dt d&O -
dt
(27) = n - t(t) - b- 1 - exp W, tl(O- 1 N exp (A?) of the linearized equations are stable, namely if all characteristic numbers have the negative real parts. The
622
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
characteristic numbers are the roots of the quasipolynome (l-I/K)(l+bf~~)-(l+lz)e-“T~-c(l-~/~e-”r~=O. (30) With the zero time lags (T, = T, = 0) and a > 0 both roots have the negative real parts, and the stationary point is asymptotically stable. If a = 0 then both roots are purely imaginary (centre). If a > 0 the stability is also preserved for the sufficiently small non-zero time lags T, and T, (cf. Elsgoltz & Norkin, 1971). The increase of the time lags of T, or T, or both of them breaks the stability of the stationary point. In particular, if a is small (a 4 1) and the retardations are small
I 0
I
I IO
I
I
\I 20
I 30
I
I 40
I 50
W-r
5. The solutions of equations (32) in dimensionkss units for KT, = T,/r = 0.3; C(O) = l-4; q (0) = O-75 and for different Tm: A, l-5 T,; B, 2.5 T,. FIG.
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
623
(T, -+ l/K, T, 4 l/K) the roots of the quasipolinome a-KT,-cKT, 1 /I=Ifi: ZT;J(a-KT,-cKTm)2-4KT(l-a) (31) 22 at T,+ T,c > a/K are located on the right half-plane of the complex 1 plane. In the case of a = 0 any non-zero time lag leads to non-stability. At a = b = 0 equations (27) become
1 MO --=~(t)-~(t-7&o-~)f?(t-T,)
K dt (32) dtV) T--=-&t)+&t-T,)q(t-T,)O(t-T,). dt These equations are analogous to the Volterra-Lotka equations, the only difference is the retardations in the non-linear terms. The oscillatory character of the system which is also preserved with the non-zero time lags may manifest itself in a cyclical kinetics of an immune reaction. Figure 5 shows examples of the calculated solutions of equations (32). It is seen that the increasing instability (with increasing time lag) may accelerate the antigen elimination. It can be shown (cf. Appendix D) that if x(t) 2 eKTr/AcT, at least within the time interval T,, then according to equations (9) the antigen will be eliminated not later than during the next time interval of the same length T,. Note that the equations (32) do not allow the antigen quantity to increase infinitely [this is due to the neglect of expenditure of the specific target cells: P = 0 in equations (9)]. 6. Conclusion The model of the immune response is formulated where the antibodies, produced after some time T, after the stimulation of the target cells (B-lymphocytes), immediately react with the antigen. According to such a model, the free antibodies can be observed only after the antigen inactivation. The conditions are obtained for the antigen elimination and for the infinite increase of the antigen quantity in reactions without the participation of the newly-formed memory cells. In particular these conditions demonstrate the role of the time lag T, in the immune reaction. For the elimination of the rapidly self-reproducing antigen it is necessary that the time lag is not too long and not too short. If the immune system does not cope with the antigen before the formation of a considerable number of the memory cells, then the newly-formed memory cells can take part in the reaction. Such a retarded replenishment of the specific target cells can be responsible for the cyclic appearance of
624
B. F. DIEROV,
M. A.
LIVSHITS
AND
M. V. VOLKENSTEIN
the antibody-producing cells after a single injection of the poorly-catabolized non-reproducing antigen. The conditions (25) of the elimination of the selfreproducing antigen and of the infinite increase of itsquantity show that the time lag (TJ in the immune memory formation can play an essential role in the reaction. The complete equations (9) take into account the natural decay of the specific target cells and their supply due to the differentiation of the stem cells. The qualitative investigation of the non-linear equations with time lags presents great difficulties. Such equations in contrast to the analogous equations without time lags do not possess the “phase portraits” in the usual sense of the word. The motion of the system from any point depends on the way which has led it to this point. In particular because of this dependence on the prehistory the phase trajectories can cross themselves (cf. Fig. 4). Equations (9) are similar to the Volterra-Lotka equations which describe the predator-prey interaction. In our case the role of the prey plays the self-reproducing antigen. The time lag leads to the instability of the stationary point. The oscillatory character of the system can be manifested by several rounds of the phase trajectories about the stationary point before the antigen quantity vanish. In other words, such a regime of an infectious disease is possible when the antigen quantity in an organism performs several oscillations even with increasing amplitudes before the total antigen elimination (cf. Fig. 5). REFERENCES Biol. 29, 191. J. theor. B&I. 33, 339, 379. BELL, G. I. (1973). Math. Biosci. 16, 291. Bovn, W. C. (1966). Fundamentals of Immunology. New-York, London, Sydney: Interscience Publishers. BURNJXT, F. M. (1969). Cellular Immunology, Melbourne: Melbourne University Press. CHERTKOV, I. L. & FRIEDENSTJIIN, A. YA. (1972). Uspekji Sovremennoy Biologii 74, 292 (in Russian). ELSOOLTZ, L. E. & NORKM, S. B. (1971). Introduction to the Theory of di&rentialEquations with Deviating Argument. Moscow: Nauka (in Russian). HLEK, M. & STERZL, J. (1970). Developmental Aspects of Antibody Formation and Structure, vol. 2, p. 963, Academia, Praga, New York: Academic Press. MOLCHANOV, A. M. (1971). Biophysics. 16, 482 (in Russian). MY~HKIS, A. D. (1972). Linear Diferential Equations with a Retarded Argument. Moscow: Nauka (in Russian). MYSHKIS, A. D. (1951). Mathematicheskiy sbornik 28,70, 15. (in Russian). NOSSAL, G. J. V. (1964). Sci. Am. 211, N6, 106. RAFF, M. C (1973). Nature, Lond. 242, 19. ROMANOVSKY, Yu. M., STEPANOVA, N. V. & CHERNAVSKY, D. S. (1975). Mathematical Modelling in Biophysics, Moscow: Nauka (in Russian). ROMBALL, C. G. & WEIGLE, W. 0. (1973). J. exp. Med. 138, 1426. BELL, BELL,
G. I. (1970). G. I. (1971).
J. fhw.
MATHEMATICAL
MODEL
OF
IMMUNE
APPENDIX Investigation
625
PROCESSES
A
of Equation (11)
We are interested only in the case
4x0 -> K
1,
since in the opposite case it is immediately increases infinitely. The change of the variable
(AlI seen that the antigen quantity
g(t) = eKt u(t)
642)
in equation (11) leads to: Wt) = --Bu(t-TJ,
t 2 T,
dt
u(t) = go
for 0 I t I T,,
go > 0
where B = ArxO evKTr.
(A4) The linear differential equations with time lags were considered by Myshkis (1951, 1972). It was found that if in equation (A3) BT, > l/e
(A5) = 0 and u(t) > 0
then there exists the moment t* > T, such that I for all t in [0, t *). In our case this means that condition (15) A,x,T,exp(-KT,+l)> provides the elimination If
1
(15)
of the antigen during a finite time. BT, < l/e
@6)
then u(t) > 0 for all t 2 T, and also u(t) 2 go exp [-B(t-
T,)e],
t 2 T,.
647)
In our case this means that g(t) 2 go exp [Kt-
A,+ e -KTr*l(t-
T,)],
for all t 2 T,
w9
if ApoT, T.B.
exp (-KT’,+l)
< 1.
(A% 41
626
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
It is easily seen that if in addition to equation (A9) the condition K > ArxO exp (-KT,+l) (AlO) is fulfilled then the antigen quantity g(t) increases infinitely. Taking into account equation (Al) one can replace equations (A9) and (AlO) by (A9) together with KT,>l. (All) Thus the antigen quantity increases infinitely if conditions (AS) and (All) are fulfilled. It can be shown that if the conditions (A9) and KT, 0 according to inequality (A9).
APPENDIX B
Investigation The formal integration
of Equations (10)
of equations (10) after the change of the variable g(t) = eKr u(t) (Bl)
leads to du(t) u(t)
= -B(t)@-
dt = go for 0
0
where B(t)
= ArxO exp
- KT,
- P’iTveKr’
dt’i(t)
-
dt
= -B”J(t
- T,),
1
u(t’) dt’ .
0 [ Now we outline the main stages of the investigation In addition to equation (B2) equation
(B3j
of equation
(B2).
t 2 T,
E(t) = go
for 0 I t I T,, with constant B” can be considered. According to the comparison (Myshkis, 1951), if B(t) 2 B and u(t) 2 0 for ali t in [0, t*]
theorem 035)
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
627
then u(t) 5 ii(t)
for all t in [0, t*]
036)
and so if equation (B5) is fulfilled and u”(l,) = 0, then there is the moment t, < f1 such that u(t,) = 0. Now let us consider the condition for u(t) to cross zero before t, = 2T,. The solution of equation (B4) in the time interval [T,, 223 is u”(t) = gJ1 --B’(t- TJ] and hence ii(t) crosses zero at the moment
tf
(Tr
0 for
t
2 T
and y(t) > 0 for 0 s
I T
t
then the inequality y(t) 2 min y(t),
t
Wl)
2 T
OStST
holds. In addition it can be shown that if WO --&- I Ku(t)
(K > 0)
ww
and
then one of the statements is valid: or
(1) a(t) tends to zero or crosses zero (2) y(t) increases infinitely.
Proof: let u(t) > 0 for all t 2 T, i.e. a(t) does not cross zero. The change of variable y(t) = z(t) exp i e(t’) dt’ tT 1
Wl3)
leads to 32
-[u(t)+z(t)]z(t)+j?(t)
exp -
J e(t’) dt’ t-T
1
z(t- T).
(C14)
If a(t) does not tend to zero, c(t) can always be chosen such that 0 < u(t) +8(t) 2 p(t) exp - t jT 40 dt ‘]
(CI5)
and 1 E(t’) dt’ + + co when t++oo.
W6)
630
B.
F.
DIBIiOV,
M.
In its turn inequality
A.
LIVSHITS
AND
(C15) with condition
M.
V.
VOLKENSTEIN
(Cl 1) leads to
y(t) 2 C exp
(C17)
where
and so according to condition
(C16) y(t) increases infinitely.
Consider now equations (C4) and (C5) [equations equations (C4) and (C5) it follows that
(11) in text]. From
where 40 = J??(t), If condition
j?(t) = A, emKTn* g(t)
(25) A, pexp
(-KT,)
> 1
(25)
holds, then, according to the auxiliary statement made above, g(t) tends to zero or crosses zero (since in the framework of equations (C4) and (C5) the infinite increase of x(t) is impossible (cf. Appendix D).
APPENDIX D
The Condition of the Total Antigen Elimination If there is L 2 T, that x(t) 2 eKTr/A, T,
then according CL L-i- Tl.
to equations
for all t in [L - T,, L]
(9) g(t)
CD11
crosses zero in the time interval
Proof: in addition to W) -
dt
= Kg(t)-A,x(tg(t) = go e”’
T,)g(t-
T,),
for 0 < t I T,
t 3 T,
(W
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
631
consider the equation 47(t) - dt = m(t)--Bg(tg(t) = g(L) exp [K(t -L)]
T,),
t >L
(D3)
for L- T, I t < L
where B = A,
min x(t) IL- TV,Ll
since g”(t)lg(t)
forL-T,
65, 609-631
Mathematical Model of Immune Processes B. F. DIBROV, M. A. LIVSHITS and M. V. VOLKENSTEIN Institute of Molecular
Biology, Academy of Sciences of the USSR
(Received 14 November 1975, and in revised,form
10 May 1976)
In the model the time lags of the antibody production and immune memory formation are taken into account explicitly. The antibody-antigen reaction is supposed to be very fast. The cases of a reproducing antigen as well as that of a non-reproducting antigen are considered. The conditions of the infinite increase of the antigen quantity and of the antigen elimination are obtained. For the rapidly reproducing antigen the latter condition includes the requirement for the time lag of the immune response to be not too short or not too long. In the case of the poorly catabolized nonreproducing antigen the cyclic appearance of the antibody producing cells due to the immune memory is described in the frame-work of the model. The mathematical structure of the model is similar to that of the
Volterra-Lotka equations. The only difference is the presence of the time lags in the non-linear terms. The time lags lead to the instability of the stationary state. In the prolonged reaction the antigen quantity may perform several oscillations before the elimination of the antigen.
1. rntrodnction The vertebrate organism responds on the introduction
of the foreign material
(antigen) by the production of specitic gamma globulines (antibodies). This response is the humoral one. The other kind of response, the cell-mediated one, consists of the production of the specific reactive cells. Both the antibodies and the reactive cells can circulate in the organism and react with the antigen. As a result of these reactions the antigen can be killed, inactivated or phagocytosed by the cells of the reticula-endothelial system. The contemporary ideas concerning immunity are based on the clonal selection theory (Burnet, 1969), which suggests that independently of the antigen the lymphocytes are produced in the organism, and every one of them is sensitive to one antigen or several similar antigens. That is determined by the presence of the antigen-specific receptors at the cell surfaces. If an antigen is introduced into organism it acts in a selective way at those lymphocytes (target cells), which possess already the receptors for this antigen. The interaction of antigen with receptors activates the cells (cf. T.B. 609 40
610
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
Burnet, 1969; Raff, 1973). We consider here only the humoral immune reaction. The general scheme of the development of the humoral immune response is given in Fig. 1. In this case target cells are B-lymphocytes. They are formed as the result of differentiation of the stem cells. The B-cells themselves do not divide and they obtain the ability for proliferation only being transformed into blast cells (Y). The blast-transformation is caused by the antigen action and occurs after the latent period of 24-48 h. The Y-cells proliferate intensively. A part of the Y-cells transforms into the plasmablasts (Zi), which after a series of mytoses transform into the mature plasma cells (Z,). The definite clone of the Z-cells produces antibodies of the same specificity as that of the receptors of the target cells (cf. Nossal, 1964; Chertkov & ____-___------I
I Memory
i I I
/-
1
I
Ahbodles
FIG. 1. The scheme of the development of the antibody-producing line symbolizes the proliferation.
cells. The oscillating
Friedenstein, 1972). The mature plasma cells are the main source of the antibodies. They are not capable of further division. Their life-span is some tens of hours. The antigen-stimulated B-lymphocytes can produce also the so-called memory cells which are responsible for the change of immune reactivity of the organism to the recurrent antigen expositions. Since no differences between memory cells and virgin target cells are known it is reasonable to believe that the immune memory effect is due to the increased number of target cells. In the case of the cell-mediated immune response the antigen stimulates the so-called T-lymphocytes. They are the progeny of the stem cells like B-lymphocytes but during their development T-lymphocytes migrate through the thymus. After contact with the antigen, T-lymphocytes proliferate and differentiate into the specific reactive cells. T-lymphocytes and their progeny
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
611
are not capable of producing antibodies, but they can play an important role in the development of the humoral response because of the so-called T-B co-operation. It can be suggested that the T-lymphocytes stimulated by antigen produce some non-specific factors which lower the threshold of sensitivity of the B-lymphocytes to antigen (cf. Raff, 1973). T-B co-operation is not observed for every kind of antigen. There are the so-called thymusindependent antigens which are capable of stimulating the normal humoral response without the help of the T-lymphocytes (cf. Chertkov & Friedenstein, 1972). Several mathematical models of the immune reaction have been described. The model of Jilek & Sterzl (1970) is devoted to description of the development of the antibody-producing cells and does not take into account the antibody-antigen interaction. A more complete model is presented by Smirnova & Stepanova (see Romanovsky, Stepanova & Chemavsky, 1975). The model of Molchanov (1971) is formulated in very general terms. It allows many different regimes of reaction depending on the parameters. The most advanced models are those of Bell (1970, 1971, 1973). The first of Bell’s models presents a detailed description of the development of the antibody-producing cells and of the antibody-antigen interaction with a great deal of the computer calculations. Unfortunately the complicated character of these models did not allow qualitative analysis. In another model (Bell, 1973) the immune reaction is described by two differential equations for antigen and antibody quantities, the rate of antibody production being taken as a function of the current antigen and antibody quantities. This implies the constant number of the target cells for the given antigen during reaction. In fact, the variation of number of the target cells can be considerable. The differentiation of antigen stimulated B-lymphocytes into antibody-producing cells takes some time and therefore the rate of the antibody production is determined not by the current antigen quantity but by the previous values of the variables. Such a time lag in the immune response due to lags in the triggering of the cells in other states is not taken into account in the models listed above (except that of Jilek & Sterzl, 1970). The aim of this work is the construction of the mathematical model of the thymus-independent reaction of the organism on the introduction of antigen. Such a model cannot describe all the details of the process, which have not yet been sufficiently studied. The purpose of the model is the qualitative and semi-quantitative treatment of the main features of the process. The important features of immune processes are the time lags in the immune response and in the immune memory formation. In the present model these time lags are taken into account explicitly. The cases of the
612
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTRTN
self-reproducing antigen (infection) as well as that of the non-reproducing antigen are considered. We assume that the antibody-antigen reaction is very fast and this allows us to consider the relatively simple equations. Such a simplification is correct until the quantity of antigen becomes very small. In section 3 the immune reaction is considered, without the participation of the new-formed memory cells. Such consideration is valid if the total antigen is eliminated before the formation of the considerable quantity of memory cells or in the case of a weak immune memory formation. Section 4 deals with the reactions in which the memory cells take part. In the case of the non-reproducing and poorly-catabolizing antigen the cyclic and synchronized appearance of the antibody-producing cells is described. Such phenomena have been observed for example after the single injection of the aggregated human IgG in rabbits (Romball & Weigle, 1973). According to our model the analogous dynamics can take place in the case of infections as well. In sections 3 and 4 the conditions of the total elimination of the antigen and of the unlimited propagation are obtained. These conditions show to what extent the immune process depends on the different parameters. In particular the important role of time lags is demonstrated. 2. TheModel For the sake of simplicity we shall assume all target cells to be equivalent and all corresponding antibodies to be identical. The differentiation of a single target cell after an antigen stimulation at the moment t = 0 is supposed to be independent of the antigen and of other cells. So the number of the mature plasma cells [z(t)] at a time t can be represented as a sum of responses of the individual target cells stimulated previously : z(t)
=
j F(t0
t'j@(t')
dt’
(1)
where Q(t) dt is the number of the target cells stimulated by the antigen during the time from t to t+dt. F(t) is the linear response function. For a single target cell which has been stimulated at the time t = 0, z(t) coincides with F(t). The qualitative shape of F(t) is shown in Fig. 2. After some time which is necessary for the differentiation of a stimulated target cell, the number of the plasma cells increases. The subsequent decrease of the number of plasma cells is due to their natural removal. Equation (1) represents the retardation in the formation of the mature plasma cells with respect to the stimulation of the target cells. For the sake
MATHEMATICAL
FIG.
2.
MODEL
OF
IMMUNE
PROCESSES
613
The qualitative shape of the response function F(t).
of simplicity we shall take into account this retardation in the form of the discrete time lag. Such a simplification corresponds to the substitution of the realistic response function F(t) by the delta function, F(t) N @f-T,) where T, is constant. In this case equation (1) leads to z(f)
where
6(f)
N cD(f--
T,)bqf-
T,>
(2)
is the step function, that is
t?(t) =1 tt 0, if 1, if
< 0 2 0.
Since the main source of the antibodies is the mature plasma cells, we shall assume the rate of the antibody production to be proportional to the current number of the plasma cells. From equation (2) we get
Nqt- r,>qt- T,>.
(3)
Here a(f) is the quantity of the antibodies. The rate of the target cells stimulation is supposed to be proportional to the number of the target cells [x(f)] and to the quantity of the antigen [g(f)]. Then = &(t
The bimolecular antibody-antigen the antigen and the free antibody
- T,)g(t - T,)O(t - T,>.
reaction is supposed to decrease both quantities. Thus the equation for the
614
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
quantity of the free antibodies is:
-WO = A$(tdt
T,)g(t- zp(t-
T,)--Rg(t)a(t)-J%(t).
Here -Rg(t)a(t) describes the antibody binding by the antigen, -h(t) describes the decay of the antibodies. The dynamics of the antigen quantity [g(t)] follows the equation: F
= Kg(t)-Qg(t)a(t).
Here K is the rate of increase (K > 0) or decrease (K -=c0) of the antigen quantity determined by the reproduction of the antigen and by the interaction with the non-specific protective factors of the organism. We shah call fir the rate of the antigen reproduction. In reality K is the rate of the development of infection (if K > 0) in the organism before the appearance of the specific antibodies. The term - Qg(t)u(t) describes the decrease of the antigen quantity due to the interaction with specific antibodies. We consider the memory cells as identical with the virgin target cells. The retarded appearance of the memory cells can be described by the expression analogous to equation (5). The corresponding rate is proportional to x(t- T,)g(t - T,), where T, is the constant representing the time lag in the immune memory formation. We can write
MO = J - ‘+ - Px(t)g(t)+A,x(t-Tm)g(tdt
T,)e(t-T,).
(8)
Here J is the rate of appearance of the specific target cells as the result of differentiation of stem cells. The term -x(t)/7 describes the natural decay of the target cells (z is the average life-time). The term -Px(t)g(t) describes the decrease of the number of target cells due to the interaction with the antigen. Both the differentiation of the target cells and their destruction due to the interaction with the antigen are included here. The term A,x(t- T,)g(t- TJ describes the retarded appearance of the memory cells after the stimulation of the target cells by the antigen. The affinity of the specific antibodies to the antigen is high. If the antigen is in excess the process rapidly reaches the steady state du(t)/dt N 0 because of the large value of the “coefficient” Ag(t). This means that the antibodies, produced by the plasma cells, bind the antigens rapidly. The experimental data show that in reality the considerable quantities of antibodies are observed only after the elimination of the antigen (see Boyd, 1966). This allows the consideration of a mere simple system of equations for the description of the main part
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
61.5
of the process: wo = Kg(t) - A,x(t___ dt
W) __
dt
T,)g(t-- T,yqt-
40 = J - 7 - Px(t)g(t) + A,x(t-
T,) (9) T,)g(t-
T,)t?(t- 7-J
where
[We have omitted here also the last term of the equation (6) that is correct if Rg(r) B E.] These equations describe “the struggle” between the target cells and antigen. The rapidly acting “mediators”, the antibodies do not enter explicitly into the equations. The equations are similar to those of Volterra and Lotka if A, > P. In our case the role of the prey is played by the antigen. The substitution of equations (6)-(8) by the system of the two equations (9) corresponds formally to the infinite rate of the antibody-antigen interaction. As the real rate is finite this substitution is not correct when the antigen quantity is close to zero. However, the calculations show that equations (9) represent the main features of the process. The detailed analysis of the effects connected with the finite rate of the antibody-antigen interaction will be presented in subsequent papers. Equations (9) formally allow the antigen quantity g(t) to cross zero. In the more realistic equations (6)-(8) this corresponds to the rapid elimination of the antigen [g(t) rapidly tends to zero]. All cases when in the framework of equations (9) g(t) tends to zero asymptotically or crosses zero will be interpreted as the antigen elimination.
3. The Immune Reaction Without Participation Cells
of the Memory
In the case of reaction which leads to the total antigen elimination before the formation of a considerable number of memory cells or in the case of the weak immune memory formation (A, m 0), we can consider equations (9) without the term describing the formation of the new memory cells [Amx(t-TJg(t-Tm)]. In addition to the short time intervals we can omit the terms describing the natural decay [ - x(f)/z] and the increase of the number of target cells due to the differentiation of stem cells (J). Equations (9)
616
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
then become
-ddt) = Kg(t) - A,x(t- T,)g(t- T;)a(t- Tr) dt WO = -Px(t)g(t). dt
(10)
The same equations may describe a reaction on a repeated introduction of a particular antigen. In such a case the initial number of the target cells can be increased. Equations (10) do not possess steady solutions with a non-zero quantity of the antigen. If the natural decay or the non-specific removal of the antigen provide K < 0 the quantity of the antigen tends to zero. It can be shown that if K > 0, the quantity of the antigen g(t) can only increase infinitely or decrease towards zero. Let us consider first the case when the change of the number of the target cells (x) during the reaction can be neglected [P = 0, x(t) = x0 = x(O)]. Instead of equations (10) we obtain -MO = Kg(t) - A,x,g(t - T,)O(t - T,). dt It is easy to see that if K -> 4x0
1,
(12)
the rate of the antigen elimination is always smaller than the rate of its reproduction and the equation describes the unlimited increase of the antigen quantity g(t). In the opposite case (K/&x0 < 1) the unlimited increase of the antigen quantity is possible only if A,x,T,.exp(-KT,+l)
1.
(13)
A,x,T, exp (-KT,+l)
< 1, KT, < 1,
(14)
If then g(t) tends to zero asymptotically;
if
A,x,T,exp(-KT,fl)>
1,
(15) then the antigen vanishes at the finite time, i.e. g(t) crosses zero (cf. Appendix A). These conditions can be reformulated in terms of the time lags. Let T’ < T” be the roots of the equation A,xoT exp(-KTfl)
= 1,
WI
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
617
then the conditions (13) (14), (15) become T,>T" (13) T,< T' (14) T' < Tr < T". (15) Now let us consider equations (10) for P > 0. The expenditure of the target cells during reaction (P > 0) weakens the protective ability of the organism. Therefore conditions (12) and (13) of an infinite increase of the antigen quantity which are sufficient at P = 0 are also sufficient at P > 0. But conditions (14) and (15) of the antigen elimination at P = 0 can prove to be not sufficient at P > 0. For any moment of time t” it is possible to find the conditions of the total antigen elimination before t* [in the framework of equations (lo)]. In particular the condition of the total antigen elimination before t * = 2T, is : A,x,T,exp
1
&lo -KT,-K(eKrv-l)
> 1 [ where go is the initial quantity of the antigen [ig(O)]. Somewhat weaker condition of the antigen elimination before 3T, is A,x,T,exp
pgo'p(KT,) - KT, - K
I
>2--J2
where q(s) = (d-1)
2-Ji e*: -I- F(e2r-e3-1.
Conditions (12)-(19, (17) and (18) demonstrate the role of the different factors and in particular the role of the time lag in the immune reaction. It should be noted that not all the relevant parameters are independent. Some part of the time lag is spent on the proliferation of the precursors of the plasma cells. If the rate of the proliferation (x) can be considered as constant the dependence of the immune response intensity parameter (A,) on the time lag T, is exponential: A, = B exTp. (19) The conditions (12) and (13) of the infinite increase of the antigen quantity in the framework of equations (10) become -Bx0 exTr < 1 wo K Bx,T,exp[(x-K)T,+lJ 1. w According to equation (20) if K > Bx, then for the small time lag T, the antigen quantity increases infinitely. According to equation (21) if K > 1c
618
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
then for the sufficiently large T, the antigen quantity increases infinitely. Thus the successful suppression of the rapidly reproducing infectious micro-organisms (K > x, K > Bx,) needs the time lag to be not too small and not too large. The model considered in this section [equations (lo)] does not take into account the immune memory formation. Such simplification is justified in particular if the memory cells which can be stimulated by antigen appear only after the total antigen elimination. In this case the number of the memory cells formed as a result of the reaction can be calculated as dt. %I = A,,, ji(t)g(t) 0 Here x(t) and g(t) are the solutions of the equations (lo), tr is the time of the total antigen elimination [g(tf) = 0; g(t) > 0 for all t in (0, tf)] according to equations (10). Such calculations can help choose the best way of immunization. We shall not discuss this problem here.
4. Immune Reaction with the Participation
of the Memory Cells
.The newly-formed memory cells can participate in the immune reaction if the antigen persists in tissues for long periods of time. Romball & Weigle (1973) observed that the injection of the poorly catabolized non-reproducing antigen (aggregated serum protein) causes the cyclical appearance of the antibody-producing cells. The first peak in the number of the antibody-producing cells was reached on the Mth day after injection of the aggregated human IgG in rabbits. The second peak occurred on day 13. The third peak was detected on day 21. So the interval between peaks was again 8 days. The dynamics of the antibody-producing cells depends slightly on the dose of the antigen. The interpretation suggested by the authors is that the initial peak of the antibody-producing cells results from the stimulation of the virgin precursor cells, succeeding peaks are the results of stimulation of the memory cells. This suggestion agrees well with our model. The model is represented by equations (9). If the natural decay and the appearance of the specific target cells due to differentiation of the stem cells during reaction can be neglected, equations (9) become ddt) = Kg(t) dt
- A,x(t-
-d-40 = -Px(t)g(t) dt
T,)g(t-
T,)O(t-
T,) (22)
+ A,x(t-T,)g(t-7;,)6(t-T,).
MATHEMATICAL
MODEL
The equations imply
the number
OF
IMMUNE
PROCESSES
of the antibody-producing
619 cells to be
40 au x0 - ml(f - m (23) (cf. section 2). Figure 3 shows the number of antibody-producing cells as a function of time obtained by the help of equations (22) and (23). The parameters are
f (days) FIG. 3. The dynamics of the number of the antibody-producing cells z(t) N x (t-T;)g. (t-T,) after the injection of the non-reproducing poorly catabolized antigen (K = 0). The parameters T, = 5 days; T, = 7 days; Pg,,T, = 4.5; A,x,T, = 1.2; A,/P = 3 are chosen to fit the experimental curves (Romball & Weigle, 1973). The scale on the vertical axis is arbitrary.
chosen to fit the corresponding experimental curves (Romball & Weigle, 1973). The very good fit which can be achieved with reasonable values of parameters (cf. legend to Fig. 3) can be considered as evidence in favour of our model. The fact that the initial point of the calculated curve z(t) is z = 1, and not z = 0, is due to the discrete form of the time lag used here. This discrepancy would be removed if one takes the continuous time lag (1). As will be seen later the cyclic dynamics can also take place in the case of infections (K > 0). However, the over-large rate of antigen reproduction leads to the infinite increase of the antigen quantity according to equations (9). So in the framework of equations (22), which are the simplified versions of (9), the antigen quantity increases infinitely if +f exp [(4,x,-K)T,]
< 1,
A+ > 1. r 0
620
B. F. DIBROV,
M. A. LIVSHITS
On the other hand, the fulfillment $exp
AND M. V. VOLKENSTEIN
of the condition (-ITT,)
> 1
(25)
provides the total antigen elimination independently of the values of all other parameters. The conditions (24) and (25) demonstrate in particular the essential role of the time lag (TJ. The intensity of the immune memory formation (A,) obviously favours the antigen elimination. Figure 4 gives the examples of the phase trajectories of equations (22) calculated for different values of the intensity of the immune memory formation.
FIG. 4. Phase trajectories of the system (22) in dimensionless variables for #T, = 2; = 5 and for differentvaluesof A,,,g.,T,; A, O-5; B, 3; C, 5; D, 10; The duration of the reactiont, in T, unitsis shown.
PgoTr = O-1; A&T,
5. The Model of a Prolonged linmnne Reaction The treatment of the immune reaction in the preceding sections did not take into account the fact that the target cells of a particular specificity are supplied as the result of differentiation of the stem cells, and that the lifetime of these cells is finite. If the reaction is so long that these effects are important then the immune reaction must be described by the complete
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
421
equations (9).
ig$ = Kg(t) - A,x(t-K)g(t--zy(t--T,) -WO = J - 40 - - Px(t)g(t) + A,X(E- T,)g(t- T,y?(t- T,). dr
(9
z
Equations (9) imply that before the antigen introduction target cells of particular specificity is stationary: x = allow more complicated kinds of behaviour than in the the equations (10) and (22). Equations (9) in contrast and (22) possess the non-zero stationary point
the quantity of the JT. Equations (9) cases described by to equations (10)
The stationary point has a sense not at any values of the parameters, since x(t) and g(t) must be non-negative. With positive x,, and gst it is convenient to use the dimensionless variables c(t) = x(t)/+ q(t) = g(t)/gSt. Equations (9) become
Wt) -=q(t)-gt-?;)r/(t-T#l(t--T,) dt d&O -
dt
(27) = n - t(t) - b- 1 - exp W, tl(O- 1 N exp (A?) of the linearized equations are stable, namely if all characteristic numbers have the negative real parts. The
622
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
characteristic numbers are the roots of the quasipolynome (l-I/K)(l+bf~~)-(l+lz)e-“T~-c(l-~/~e-”r~=O. (30) With the zero time lags (T, = T, = 0) and a > 0 both roots have the negative real parts, and the stationary point is asymptotically stable. If a = 0 then both roots are purely imaginary (centre). If a > 0 the stability is also preserved for the sufficiently small non-zero time lags T, and T, (cf. Elsgoltz & Norkin, 1971). The increase of the time lags of T, or T, or both of them breaks the stability of the stationary point. In particular, if a is small (a 4 1) and the retardations are small
I 0
I
I IO
I
I
\I 20
I 30
I
I 40
I 50
W-r
5. The solutions of equations (32) in dimensionkss units for KT, = T,/r = 0.3; C(O) = l-4; q (0) = O-75 and for different Tm: A, l-5 T,; B, 2.5 T,. FIG.
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
623
(T, -+ l/K, T, 4 l/K) the roots of the quasipolinome a-KT,-cKT, 1 /I=Ifi: ZT;J(a-KT,-cKTm)2-4KT(l-a) (31) 22 at T,+ T,c > a/K are located on the right half-plane of the complex 1 plane. In the case of a = 0 any non-zero time lag leads to non-stability. At a = b = 0 equations (27) become
1 MO --=~(t)-~(t-7&o-~)f?(t-T,)
K dt (32) dtV) T--=-&t)+&t-T,)q(t-T,)O(t-T,). dt These equations are analogous to the Volterra-Lotka equations, the only difference is the retardations in the non-linear terms. The oscillatory character of the system which is also preserved with the non-zero time lags may manifest itself in a cyclical kinetics of an immune reaction. Figure 5 shows examples of the calculated solutions of equations (32). It is seen that the increasing instability (with increasing time lag) may accelerate the antigen elimination. It can be shown (cf. Appendix D) that if x(t) 2 eKTr/AcT, at least within the time interval T,, then according to equations (9) the antigen will be eliminated not later than during the next time interval of the same length T,. Note that the equations (32) do not allow the antigen quantity to increase infinitely [this is due to the neglect of expenditure of the specific target cells: P = 0 in equations (9)]. 6. Conclusion The model of the immune response is formulated where the antibodies, produced after some time T, after the stimulation of the target cells (B-lymphocytes), immediately react with the antigen. According to such a model, the free antibodies can be observed only after the antigen inactivation. The conditions are obtained for the antigen elimination and for the infinite increase of the antigen quantity in reactions without the participation of the newly-formed memory cells. In particular these conditions demonstrate the role of the time lag T, in the immune reaction. For the elimination of the rapidly self-reproducing antigen it is necessary that the time lag is not too long and not too short. If the immune system does not cope with the antigen before the formation of a considerable number of the memory cells, then the newly-formed memory cells can take part in the reaction. Such a retarded replenishment of the specific target cells can be responsible for the cyclic appearance of
624
B. F. DIEROV,
M. A.
LIVSHITS
AND
M. V. VOLKENSTEIN
the antibody-producing cells after a single injection of the poorly-catabolized non-reproducing antigen. The conditions (25) of the elimination of the selfreproducing antigen and of the infinite increase of itsquantity show that the time lag (TJ in the immune memory formation can play an essential role in the reaction. The complete equations (9) take into account the natural decay of the specific target cells and their supply due to the differentiation of the stem cells. The qualitative investigation of the non-linear equations with time lags presents great difficulties. Such equations in contrast to the analogous equations without time lags do not possess the “phase portraits” in the usual sense of the word. The motion of the system from any point depends on the way which has led it to this point. In particular because of this dependence on the prehistory the phase trajectories can cross themselves (cf. Fig. 4). Equations (9) are similar to the Volterra-Lotka equations which describe the predator-prey interaction. In our case the role of the prey plays the self-reproducing antigen. The time lag leads to the instability of the stationary point. The oscillatory character of the system can be manifested by several rounds of the phase trajectories about the stationary point before the antigen quantity vanish. In other words, such a regime of an infectious disease is possible when the antigen quantity in an organism performs several oscillations even with increasing amplitudes before the total antigen elimination (cf. Fig. 5). REFERENCES Biol. 29, 191. J. theor. B&I. 33, 339, 379. BELL, G. I. (1973). Math. Biosci. 16, 291. Bovn, W. C. (1966). Fundamentals of Immunology. New-York, London, Sydney: Interscience Publishers. BURNJXT, F. M. (1969). Cellular Immunology, Melbourne: Melbourne University Press. CHERTKOV, I. L. & FRIEDENSTJIIN, A. YA. (1972). Uspekji Sovremennoy Biologii 74, 292 (in Russian). ELSOOLTZ, L. E. & NORKM, S. B. (1971). Introduction to the Theory of di&rentialEquations with Deviating Argument. Moscow: Nauka (in Russian). HLEK, M. & STERZL, J. (1970). Developmental Aspects of Antibody Formation and Structure, vol. 2, p. 963, Academia, Praga, New York: Academic Press. MOLCHANOV, A. M. (1971). Biophysics. 16, 482 (in Russian). MY~HKIS, A. D. (1972). Linear Diferential Equations with a Retarded Argument. Moscow: Nauka (in Russian). MYSHKIS, A. D. (1951). Mathematicheskiy sbornik 28,70, 15. (in Russian). NOSSAL, G. J. V. (1964). Sci. Am. 211, N6, 106. RAFF, M. C (1973). Nature, Lond. 242, 19. ROMANOVSKY, Yu. M., STEPANOVA, N. V. & CHERNAVSKY, D. S. (1975). Mathematical Modelling in Biophysics, Moscow: Nauka (in Russian). ROMBALL, C. G. & WEIGLE, W. 0. (1973). J. exp. Med. 138, 1426. BELL, BELL,
G. I. (1970). G. I. (1971).
J. fhw.
MATHEMATICAL
MODEL
OF
IMMUNE
APPENDIX Investigation
625
PROCESSES
A
of Equation (11)
We are interested only in the case
4x0 -> K
1,
since in the opposite case it is immediately increases infinitely. The change of the variable
(AlI seen that the antigen quantity
g(t) = eKt u(t)
642)
in equation (11) leads to: Wt) = --Bu(t-TJ,
t 2 T,
dt
u(t) = go
for 0 I t I T,,
go > 0
where B = ArxO evKTr.
(A4) The linear differential equations with time lags were considered by Myshkis (1951, 1972). It was found that if in equation (A3) BT, > l/e
(A5) = 0 and u(t) > 0
then there exists the moment t* > T, such that I for all t in [0, t *). In our case this means that condition (15) A,x,T,exp(-KT,+l)> provides the elimination If
1
(15)
of the antigen during a finite time. BT, < l/e
@6)
then u(t) > 0 for all t 2 T, and also u(t) 2 go exp [-B(t-
T,)e],
t 2 T,.
647)
In our case this means that g(t) 2 go exp [Kt-
A,+ e -KTr*l(t-
T,)],
for all t 2 T,
w9
if ApoT, T.B.
exp (-KT’,+l)
< 1.
(A% 41
626
B.
F.
DIBROV,
M.
A.
LIVSHITS
AND
M.
V.
VOLKENSTEIN
It is easily seen that if in addition to equation (A9) the condition K > ArxO exp (-KT,+l) (AlO) is fulfilled then the antigen quantity g(t) increases infinitely. Taking into account equation (Al) one can replace equations (A9) and (AlO) by (A9) together with KT,>l. (All) Thus the antigen quantity increases infinitely if conditions (AS) and (All) are fulfilled. It can be shown that if the conditions (A9) and KT, 0 according to inequality (A9).
APPENDIX B
Investigation The formal integration
of Equations (10)
of equations (10) after the change of the variable g(t) = eKr u(t) (Bl)
leads to du(t) u(t)
= -B(t)@-
dt = go for 0
0
where B(t)
= ArxO exp
- KT,
- P’iTveKr’
dt’i(t)
-
dt
= -B”J(t
- T,),
1
u(t’) dt’ .
0 [ Now we outline the main stages of the investigation In addition to equation (B2) equation
(B3j
of equation
(B2).
t 2 T,
E(t) = go
for 0 I t I T,, with constant B” can be considered. According to the comparison (Myshkis, 1951), if B(t) 2 B and u(t) 2 0 for ali t in [0, t*]
theorem 035)
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
627
then u(t) 5 ii(t)
for all t in [0, t*]
036)
and so if equation (B5) is fulfilled and u”(l,) = 0, then there is the moment t, < f1 such that u(t,) = 0. Now let us consider the condition for u(t) to cross zero before t, = 2T,. The solution of equation (B4) in the time interval [T,, 223 is u”(t) = gJ1 --B’(t- TJ] and hence ii(t) crosses zero at the moment
tf
(Tr
0 for
t
2 T
and y(t) > 0 for 0 s
I T
t
then the inequality y(t) 2 min y(t),
t
Wl)
2 T
OStST
holds. In addition it can be shown that if WO --&- I Ku(t)
(K > 0)
ww
and
then one of the statements is valid: or
(1) a(t) tends to zero or crosses zero (2) y(t) increases infinitely.
Proof: let u(t) > 0 for all t 2 T, i.e. a(t) does not cross zero. The change of variable y(t) = z(t) exp i e(t’) dt’ tT 1
Wl3)
leads to 32
-[u(t)+z(t)]z(t)+j?(t)
exp -
J e(t’) dt’ t-T
1
z(t- T).
(C14)
If a(t) does not tend to zero, c(t) can always be chosen such that 0 < u(t) +8(t) 2 p(t) exp - t jT 40 dt ‘]
(CI5)
and 1 E(t’) dt’ + + co when t++oo.
W6)
630
B.
F.
DIBIiOV,
M.
In its turn inequality
A.
LIVSHITS
AND
(C15) with condition
M.
V.
VOLKENSTEIN
(Cl 1) leads to
y(t) 2 C exp
(C17)
where
and so according to condition
(C16) y(t) increases infinitely.
Consider now equations (C4) and (C5) [equations equations (C4) and (C5) it follows that
(11) in text]. From
where 40 = J??(t), If condition
j?(t) = A, emKTn* g(t)
(25) A, pexp
(-KT,)
> 1
(25)
holds, then, according to the auxiliary statement made above, g(t) tends to zero or crosses zero (since in the framework of equations (C4) and (C5) the infinite increase of x(t) is impossible (cf. Appendix D).
APPENDIX D
The Condition of the Total Antigen Elimination If there is L 2 T, that x(t) 2 eKTr/A, T,
then according CL L-i- Tl.
to equations
for all t in [L - T,, L]
(9) g(t)
CD11
crosses zero in the time interval
Proof: in addition to W) -
dt
= Kg(t)-A,x(tg(t) = go e”’
T,)g(t-
T,),
for 0 < t I T,
t 3 T,
(W
MATHEMATICAL
MODEL
OF
IMMUNE
PROCESSES
631
consider the equation 47(t) - dt = m(t)--Bg(tg(t) = g(L) exp [K(t -L)]
T,),
t >L
(D3)
for L- T, I t < L
where B = A,
min x(t) IL- TV,Ll
since g”(t)lg(t)
forL-T,
