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Mathematical models of immunity Mathewson, Donald Jeffrey 1990

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Mathematical Models of Immunity by Donald Jeffrey Mathewson BSc. McGill University, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1990 © Donald Jeffrey Mathewson, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) i i Abstract A cross-linking model for the activation of the A cell or immune accessory cell as a function of certain extracellular conditions is developed to determine the valency of the specific factor receptor on the A cell surface. It is found that such a determination can be made based on the F W H M of cross-linking curves which differ by a full order of magnitude between the bivalent receptor case and the monovalent receptor case. This determination can be made provided one can obtain accurate values for the equilibrium constants which characterize the system and provided that activation and IL-1 secretion is a linear function of cross-linking. It is also found that a determination of valence can be made if the equilibrium constants are such that substantial one receptor bridge formation takes place (one antibody molecule bound on both ends by the same receptor). This one-receptor bridge formation only takes place if the receptor is bivalent, and it presents itself in the cross-linking curve in a very distinctive manner. A second network model described as an ecological competition model of steady state lymphocyte populations is presented. This model, known as the symmetrical network theory is analysed numerically by integration of the differential equations and shown to provide a reasonable qualitative picture of the immune system's stable steady states, and offer a glimpse of state switching. i i i Table of Contents Abstract ii Acknowledgement iv Introduction 1 Chapter 1 Cross-linking in the A cell system 4 1.1 Introduction 4 1.2 Cross-Linking Calculation 1 1 1.3 Cross-linking Curves 27 Chapter 2 N-Dimensional Network Dynamics _ 5 0 2.1 Introduction to the Model — 5 0 2.2 Discussion of the Model 6 5 2.3 Analysis of the Model 7 3 Conclusion 97 Appendix 100 Glossary of Terms 10 6 Bib l iography 110 i v Acknowledgement I want first to thank my supervisor Geoff Hoffmann for his superhuman patience and his creative suggestions. Thanks are also due to Julia Levy, whose understanding and clarity helped me to understand. Tracy K ion and Rob Forsyth endured innumerable questions, and Mike Lyons offered sage advice at several crux points. I warmly thank all of these people. M y deepest thanks however, go to my mother and my father, since it is their vision that has enabled me to see farther. 1 Introduction This thesis is a study of several mathematical models depicting various aspects of the immune system. To attempt to unlock the secrets of life with a few differential equations and a computer may appear absurd to some, especially to biologists. However, to those more familiar with-the tools of physical science, modelling is completely natural. How else are we to understand the behaviour of the system as a whole, isolate the important interactions, and interpret and predict experimental results? Indeed, given the complexity of the immune system, and of cellular interactions in general, this modelling approach is perhaps the best tool we have available to study the behaviour of the entire immune system. This being the case, we borrow from physics, or import people familiar with the tools of physics to attempt to solve biological problems. One such problem is the question of how the immune response is regulated. It has long been established that the central mechanism of the immune response is clonal selection [1], a process whereby clones whose receptors are specific for a particular antigen or invader are selected from the body's large and diverse repertoire of clones by the presence of that antigen. These selected clones divide and differentiate and the resulting cells, whose receptors are identical to those of their original parent cells, are responsible for the immune response against the invader. This theory, which has been borne out by experiment [2], does not however address the question of what happens to the cells after they have been clonally selected; the 2 question of regulation is still very much an open one. The existence of homeostatic control mechanisms is generally accepted [3], but thus far the details of these mechanisms remain unclear to us. There have been several theories of regulation [4-8], however not one of these theories has completely withstood the test of experimental verification, though some have certainly done better than the others. Amongst these theories, one of the most elegant, successful and certainly one of the most controversial is the Network Theory, of which one example is this present work. Network Theory has developed from the Network Hypothesis put forward by Niels Jerne [8] in 1974. Jerne's hypothesis was that the variable regions (or V regions) of an antibody receptor are viewed as antigens by the host's immune system, and that it is this recognition and pursuant immune system activity that is responsible for the regulation of immune responsiveness, and perhaps even for the fundamental self/non-self discrimination. This may seem at first glance to confuse matters even more, for now there are even more antigens in question. However, this is actually a large step forward, for it allows us to characterise specific pairwise interactions between cells in terms of one parameter, which is the strength of the interaction between their receptor V regions. Instead of focussing on an enormous number of molecular interactions at the cellular level, we can instead look at the dynamics of populations of various cell clones which interact via their V regions. This matrix of interactions lends itself marvelously to modelling, and several network models have been developed [9-13]. 3 In such models, we look for correlation between the behaviour of a model network of interacting cells and the observed stimulus response behaviour of the physical immune system. In particular, it is important that such models account for the existence of stable population states, immune system memory and self/non-self discrimination since these are key experimentally observed phenomena . Hoffmann and his collaborators have developed a theory [13-17], dubbed the symmetrical network theory, which demonstrates some of these important properties. The theory has been gradually cast into mathematical model form. However, the mathematical modelling still lags behind the phenomenology and the theory is far from a complete picture. This thesis represents an attempt to bring the modelling closer to par with the phenomenology by constructing and examining one model for cellular activation and by improving the previously published [16] model of the overall immune system. The former is detailed in Chapter 1, whereas the latter work is found in Chapter 2. In writing this thesis, we have assumed some familiarity with mathematics, immunology and physical chemistry, but for those not familiar with the jargon, a glossary has been included. 4 Chapter 1; Cross-linking in the A cell system U Introduction There are two theories that seek to explain the manner in which the nucleus, as the 'nerve center' of the cell , is signalled or activated by conditions which are external to the cell (for instance the presence of hormones). The first (see figure la) is known as the receptor cross-linking or aggregation model, and in this model the binding of several receptors to an external agent and the consequent aggregation of the receptors signals the cell. The second theory (see figure lb) is known as the conformational change model and as the name suggests, this theory postulates that once engaged, a cell receptor undergoes a conformational change which propagates down to the membrane-bound end of the receptor and signals the cell. a b Figure 1: The two models of trans-membrane signalling: (a) cross linking and (b) conformational change. In (b), the dark arrow indicates the propagation of the conformational change. 5 These theories have been applied on an individual basis in many separate systems, and in some cases proposed as competing models within the same system. With respect to the immune system, there are some who argue in favour of the conformational change (for instance [5]), however the cross-linking model is more generally accepted as the first step in the activation of immunocompetent cells. This is based firstly on the argument that a conformational change is not likely to be propagated past the flexible hinge region at the middle of the antibody-like receptors on immune system cells (see figure 2). Secondly, there is considerable experimental evidence in favour of the cross-linking model. F igure 2: A schematic of the B cell receptor. It is argued that a conformational change could not be propagated across the flexible hinge region of the molecule (indicated by the dark arrows). Within the B cell domain this experimental evidence, as reviewed in [18] and [14] is particularly compelling. In the most suggestive experiment, which has been repeated in several systems (see [14]), B cells are challenged with monovalent F a b fragments which as indicated in figure 3 are single arms of antibody molecules bearing only one binding site. No activation of the B cells is observed (figure 6 4a) which is consistent with the cross-linking model since F a b fragments cannot cross-link and therefore cannot activate the cell . However, when antibody specific for the F a b fragments was added to the suspension activation was observed (figure 4b). This is again consistent with the cross-linking model, since multivalent anti(F ab) antibodies would cross-link bound F a b fragments, providing the necessary activation signal. F igure 3: Using appropriate enzymes, the antibody molecule can be broken down into the above subunits. (a) provides a structural representation of these subunits, whereas (b) represents a schematic representation used in subsequent figures. 'Monovalent F a b fragment' refers to a single arm of the antibody molecule- a single oval in figure 3 b. Though the evidence in the T cell domain is somewhat less clear due to the relatively recent characterization of the T cell receptor, there is some initial evidence in favour of cross-linking [19, 20]. Furthermore, in the myeloid cell domain, there is a very successful 7 activation model for a sub-population known as the mast cells which is based on the cross-linking hypothesis [21]. Based on all of this evidence, it has been postulated in the symmetric network theory of the immune system [16] that the cross-linking of receptors is a necessary first signal for the activation of B cell, T cell and myeloid ce l l s . ' To be noted is the fact that both T and B cells require a second signal for activation: soluble proteins known as interleukins which are secreted by myeloid as well as by activated T and B cells. Whether or not a second signal is required for myeloid cells is unclear at this point, however in the model we are presenting here, we assume that it is not. a b F igure 4: In B cell stimulation experiments, it is found that F a b fragments do not activate B cells (a), but F a b fragments in combination with anti(F ab) antibodies do (b). This provides evidence in favour of the cross-linking model. 8 This model concerns the activation of non-specific accessory cells, which play a vital role in the immune system. In our model, these non-specific cells, which are of the myeloid lineage, are grouped under the title of A cells ("A" for accessory). As with all myeloid cells, the activation of the A cell has not yet been studied definitively. The cross-linking model that we are proposing here is thus justified purely on the basis of analogy with other immunocompetent cells where there exists solid evidence. Within the symmetrical network theory, the initial stages of the immune response are envisioned as follows: 1) antigen encounters T cells whose receptors are specific for certain epitopes on the antigen; 2) these T cell receptors are cross-linked by the antigen, activating the cell and causing it to secrete soluble proteins known as T cell antigen specific factors; 3) these specific factors, which have molecular weight similar to, and are therefore modelled as, monovalent F a b fragments (see Chapter 2 for a more detailed discussion of the specific factors), are cytophilic for A cells and bind via their constant end to receptors on the A cells[16]; 4) the receptor bound specific factors are then cross-linked by the antigen, to which they bind via their variable end (thus the name antigen specific factor); 5) the A cell, whose receptors are now cross-linked, is activated and secretes interleukin-1 (or IL-1), a T cell stimulating factor vital to the immune response.(see figure 5 and [15] for a more detailed description). 9 The sequence described above and in figure 5 is of course the mere beginning of a complex cascade of cell-cell interactions that initiate and mediate the immune response. In this chapter however, we are concerned only with the above-mentioned first step, namely A cell activation. CD ^ T cell factors O Figure S : The initial stages of the immune response according to the symmetrical network theory [16] Specifically, we calculate the concentration of cross-linked receptor-factor-antigen complexes for a certain model system using the law of mass action in the hope that this concentration will correlate with A cell activation. The model system consists of macrophages (a subset 10 of the set of A cells), T cell factors which are cytophilic for these cells, and bivalent anti-factor monoclonal antibodies. Thus in the model system, it is the monoclonals which cross-link, and not an invading antigen. Despite this simplification, this system is still of great interest since it can be readily studied experimentally. One such experiment, which is currently underway in our laboratory, consists of preparing macrophagie-factor-monoclonal antibody suspensions and then assaying for IL-1 secretion using the standard assay [22]. The presence of IL-1 would be strong evidence in support of the cross-linking hypothesis, especially if the IL-1 secretion correlates with the calculated concentration of cross-linked cell-surface aggregates. This latter result would also be highly suggestive of an important role for specific factors, and would provide strong support for this aspect of the symmetrical network theory of the immune system. 11 1.2 Cross-Linking Calculation A s mentioned in the introduction, the law of mass action allows one to calculate the equilibrium concentration of cell surface aggregates. For a chemical equilibrium governed by the equation: A + B * ^ y (1) where k+ is the forward rate constant and k. the reverse, the law of mass action reads: K = r a \ F t 0 = — = constant (2) [A].[B] In an approach similar to that taken in [21], we use this relation to express the concentration of a cross-linked aggregate in terms of the equilibrium concentration of its constituents (ie. factors, cell receptors and antibodies) and the rate constants characterising the various reactions which built up the aggregate. It is important to note that since the cross-linking antibody has a valency of two, the only receptor complexes that can be formed in the model system are receptor rings and chains (see figure 6). With the equilibrium constants supplied by experiment, one can thus sum over all complexes and so obtain cross-linking (the percentage of receptors in cross-linked complexes) as a function of the concentration of factors, antibodies and receptors. It is this total cross-linking which we seek to compare with IL-1 secretion, and correlation would be consistent 12 with our hypothesis that IL-1 secretion is proportional to activation and activation is proportional to cross-linking. We do not attempt to model IL-1 secretion here. a - f T a B & e a B R V V V R R R c F igure 6: Cross-linked aggregates for a bivalent cross-linking agent can give only rings and chains (a,b), but more complicated configurations are possible when the cross-linking agent is multivalent (c) The calculation is done for two cases: a monovalent A cell factor receptor and a bivalent A cell factor receptor, and it is here that the 13 predictive power of the mathematical model is evident. Whereas a qualitative experiment could only determine whether or not IL-1 has been produced, a comparison of the model with quantitative experimental data allows one to probe somewhat deeper and examine the differences between these two cases, perhaps allowing us to determine the valency of the receptor. The assumptions made in the model are the following: 1. The factors, receptors and antibodies are treated as stable molecules. 2. Equilibrium is established very quickly (ie. an equilibrium calculation is valid). 3. The antibody has no affinity for the A cell receptor, so that receptors can only be cross-linked via factor. 4. There is no co-operative binding, meaning that the equilibrium constant for a single binding site of a receptor (in the bivalent case) or antibody is not dependent on the state of the other binding site which may be engaged or free (the indepedent site approximat ion) . 5. The equilibrium constant for the reaction which adds one link to a chain of receptor molecules is independent of the length of the original chain. 6. Receptor ring closure can be treated as a self-avoiding (due to steric repulsion) random walk. 7. Any steric effects are included in the equilibrium constant. 14 8. The binding of free factor to bound antibody is characterised by the same equilibrium constant as the binding of free antibody to bound factor. Having made these approximations, we begin the calculation by defining the equilibrium constants for the binding reactions which occur on the cell-surface. First there is the (single receptor site)-(single factor) binding constant K defined as follows: + C 5 > R 2k+ 0 k. k + V The factor of two for the bivalent receptor is a statistical factor indicating that the forward reaction, characterised by rate constant k+ proceeds at twice the rate in that case due to the availability of two binding sites. The second constant is the (single receptor site)-(chain-bound single factor) binding constant K x , defined as follows: 15 R 2k (2) x + 2k (2) AA R e x + \ R 2 k x - R R with the statistical factors arising as above. The superscript (2) indicates that a link is being added to a 2 receptor chain or removed from a 3 receptor chain. For the monovalent case, this superscript is clearly unneccessary since we can only form 1 and 2 receptor chains. For the bivalent case, we have an infinite set of binding constants (n) (n) kx+ (4) However, by the 5th approximation listed above, we have that .(1) = k (2) x+ - k( n ) ~ Kx+ (5a) 16 k 0 ) - k ( 2 )  K x - ~ K x - (5b) and so we can define a unique receptor to chain-bound factor binding constant K v = k ( n )  *x+ k ^ * x -(6) Next, there is the (single antibody binding site)-(single factor) binding constant L , which we define by: 21 e (7) Finally, there is the closure equilibrium constant for the n-membered ring, J n , which corresponds to the reaction shown below for the case 17 AAA 3 + R 6j Now we are once again dealing with an infinite set of equilibrium constants: Jn = Jn + J n -( 8 ) However, as before, we reduce this to a finite set by means of the approximations of the model. Specifically, we have assumed that such rings can be treated as self-avoiding random walks which start and end at the origin. The walks are self avoiding in the sense that the ring intersects itself only where it closes ie. at the origin. The probability of the ring closing or of returning to the origin (r=0) after a sufficiently long n-step random walk scales with the number of steps n [23]: Pn(r=0) a 1 d £ l (9) n where a indicates proportionality and d is a scaling exponent of the order of 1. We therefore assume that the forward rate constant for ring closure also scales with the number of steps: 18 Jn+ = n > m (10) U J J m + where m is an index to be chosen. Given that Jf , the equilibrium constant for one receptor rings or bridges will depend more on geometry than on diffusion of the receptors through the cell membrane (see figure 7), we leave it as a special case and take m=2, w h e r e b y Jn+ = (2\d n J2+ n £ 2 (11) Figure 7; One receptor bridge formation depends on the geometry of the cross-linking agent and the receptor binding sites. In the figure, for instance, both molecules would have to flex considerably in order to bridge the gap indicated by the arrow and form a one-receptor bridge (possible, but unlikely). Higher order rings depend on diffusion of receptors through the cell membrane and so will be characterized by different equilibrium constants. Since the reverse reaction, which is merely a dissociation, is essentially independent of n, we take 19 J2- = = Jn- =-• (12) and hence, we are left with 2 equilibrium constants: J i and J2, since the above imply that: Note that J n so defined is dimensionless and also that there are no ring closure constants for the monovalent A cell receptor case since there are no rings in this case. This completes the list of equilibrium constants. These constants: K , K x , L , J i , J2 and d are treated as parameters in the model, and their values are assumed to come from experiment. Having defined the equilbrium constants, we now proceed to enumerate all possible cell-surface aggregates and calculate their concentrations using these constants and the law of mass action. W e work first with the case of the bivalent receptor. Consider the set of all n-receptor complexes. As depicted in figure 8, this set consists of the n-membered ring along with 10 distinct n-link chains. Using the law of mass action, we can express the concentration of the n-membered ring and the last nine n-link chains in terms of the least complicated chain, chain #1. A s an example, we calculate the concentration of two chains at the top of figure 8 in terms of rate constants and the concentration of chain #1. The calculation is actually quite simple: the concentration of the reaction products is (13) 21 determined by multiplying the concentration of reactants by the equilibrium constant for the process, which is merely the ratio of the forward and reverse rate constants. For chain #2, the equilibrium constant is (14b) Where we have defined F=[ free factor ] We will also use R=[ free receptor ] and A=[ free antibody ]. For chain #3, the equilibrium constant is given by 2 2 (15b) The concentration of all of the other n-receptor aggregates can be calculated in a similar manner. Adding these concentrations, we find that the total concentration of n-receptor aggregates, R n is given by R n = [chain #1] { l+2KF+K 2 F2+4K 2 IJF2A+4K 2 L 2 F 2 A2+8K2L 3 F3A 2 + 4 K 2L 4F 4 A 2 + 4 K L F A + 4 K L 2 F 2 A + 4 K L 2 F 2 A ^ - + 4 K 2 L 2 F 3 A } z n J (16) 23 The final step in the calculation is to use the law of mass action to express the concentration of the first chain in terms of the concentration of free factor, antibody and receptors. The procedure is the same as that employed in the above calculation: -4K K I I •2K KL •4KKL 4K KL X X n-1 X (17a) 24 We thus have that R N = ( 4 K X R K L 2 F 2 A ) N ~ * R {1+2KF+K2F2+4K2LF2A+4K2L2F2A2+8K2L3F3A2+ 4 K 2L4F4A2 + 4KLFA+4KL2F2A+4KL2F2A^ L + 4 K 2L 2F 3A ) (17b) We now sum over n and write out conservation equations for the total number of receptors (it is also possible to write out equations for the conservation of factors and antibodies: oo F T = F f ree + I n F n (18a) n= l oo A T = A f r e e + S n A n (18b) n= l but these will not be of any use to us as will be explained later). oo R T = X n R n n=l = R {l+2KF+K2F2-f4K2LF2A-t4K2L2F2A2+8K2L3F3A2+4K2L4F4A244KLFA+ oo 4KL2F2A+4K2L2F3 A } X n(4K x RKL2F 2 A) n " 1 n= l oo +4RKL2F2A^y (4K x RKL2F2A) n " 1 n=l (19) Substituting equation (12) and evaluating the sums using 25 oo S n ( B ) 1 1 - 1 = 1 2 (20a) n=l ( 1 - B ) 2 and oo r ~ = (t>(x,z) (20b) n z n=l where 0 is the Truesdell function, a standard numerical function [24], we find that R 'p = R {1+2KF+K2J^+4K2LF2A+4K2L2J^A2+8K2L3F3A2+4K 2L4F4 ^ +4KL.2F2A+4K2L2F3A) 1 ^ , 0 + 4 R K L 2 F 2 A { ^ - 2 D _ 1 J 2 ) J ( 1 - 4 K X R K L 2 F 2 A ) 2 2 2 D _ 1 +-^— h <t>(4K XRKL2F2A,d) ( 2 1 ) Given values for the various parameters and total concentrations, this equation, together with the factor and antibody conservation equations, can be solved in a self-consistent manner, the result being the concentrations of free receptors, factor and antibody (R, A, F). With these, we can calculate the equilibrium concentration of any aggregated receptor complex, and in particular, we can calculate the concentration of cross-linked receptors, with which we are most concerned. R T - R j Fraction of cross-linked receptors = f c j = — ^ — (22) 26 For the monovalent case, the situation is much simpler, since there are only 5 possible cell surface complexes: A A ' .A e 6 > CD 6 R R R R R R Thus, we have only R j = R + R K F + 2 R K L F A + 2 R K L 2 F 2 A (23a) R 2 = K X R 2 K L 2 F 2 A (23b) and the conservation equation (there are once again companion equations for conservation of factor and antibody which we do not use) reads: R T = R + R K F + 2 R K L F A + 2 R K L 2 F 2 A + 2 K X R 2 K L 2 F 2 A (24) and for cross-linking we get R T - R ! 2 R 2 f c l = R j R j (25) 27 1.3 C r o s s - l i n k i n g C u r v e s We now use the self-consistency equations derived in the last section to plot cross-linking curves. These equations, oo R T = X n R n (19) n=l oo F T = F + X n F n ( i 8 a ) n = l oo A T = A + X n A n ( i 8 b ) n=l ( A and F represent free antibody and factor respectively and F N and A N refer to cell surface complexes with n antibodies or factors), together with the values of FT, A T , and the equilibrium constants will , when solved provide values for R, F and A. With these values we can determine the concentration of any particular compound, in particular the concentration of cross-linked receptors. In practise, however the concentrations of free antibody (A) and factor (F) are experimental variables which can be easily manipulated by dilution or addition of reagent. It is thus useful (and much easier) to plot cross-linking as a function of these variables. This can be done using the conservation equation for receptors (this is why the other conservation equations are unneccessary). Several of the equilibrium constants and parameters can be estimated in the absence of direct experimental measurements. K and L are both receptor-ligand equilibrium constants and as such they may typically be of the order of 1 0 8 - 1 0 9 M " 1 , but could also be 28 as low as IO 5 M * 1 or as high as I O 1 1 M _ 1 [23]- If we assume that the A cell has approximately the same concentration of receptors as other immunocompetent cells, for instance B cells which bear « 10 5 receptors on their » 400 u,m 2 surface [24], then R j would be typically of the order of 10 2 /u .m 2 and in the range 10-10 3 /u .m 2 . Due to the adherent nature of the A celL we expect that K x , J i and h will differ somewhat from the values for. other immunocompetent cells since these constants depend on the fluidity of the cell membrane. In several other cross-linking models, K x is chosen so that the non-dimensional parameter K x R x *s of order unity [31-34]. Based on a geometrical model [31] where the Teceptors were treated as hard discs and the hapten as a smooth path segment, one of these authors estimated K x = 1 0 - 1 u.m 2 which is consistent with this ansatz given the above range of values for R T - We will take K X R T = 10 as a typical value, and allow a range of 1-100. 2 9 The ring closure constants are more difficult to measure and given the difficulty of direct cell-surface measurements, are probably best estimated from equilibrium constants for ring closure in solution. Now the factors we are working with are poorly characterised at best and such information is not available therefore we must estimate. We take J i , J 2 e [0.01,1000] which is consistent with other cross-linking models where solution data is available [31], [34]. As previously mentioned, the J n are dimensionless. Now for the bivalent case, we first consider the simplified model in which J i = J 2 = 0 . We also define and substitute r _ R i R ~ R x " P R T (29a) a n d a = - 4 K X R T K L 2 F 2 A (29b) w h e r e P = {l+2KF+K2F244K2LF2A+4K 2L2r^ 4 K L 2 F 2 A + 4 K 2 L 2 F 3 A } (30) Note that Rn+1 = Rn ( 4 K X R K L 2 F 2 A ) = R n (a r) (31) 3 0 and so a is a recurrence factor. B on the other hand is a degeneracy factor which expresses the concentration of all n-chain complexes (other than the ring) in units of the concentration of the simplest chain, #1. With these substitutions, the conservation equation reduces to: 1 = (1 -ocr ) 2 (32) which implies that ( l+2oQ- V( l+2a)2-4q2 (33) 1 " 2 a 2 The negative root has been selected to satisfy the physical requirement that as the antibody concentration becomes large, cross-linking goes to zero since eventually each receptor binding site is occupied by a distinct antibody (see figure 9). Mathematically this condition reduces to: r-->l as K A ->°o (==> o>->0) (34) 3 1 R R a V R R Figure 9: At high factor concentrations (a), all factor receptors are occupied, and there is very little cross-linking. Similarly, at high antibody concentrations (b), there is one antibody molecule per factor, and hence no cross-linking. The above substitutions also give rise to a particularly simple equation for total cross-linking: for1 "r (35) 32 and it is thus clear that r represents the non-cross-linked fraction. For the monovalent case, the receptor conservation equation is a quadratic which is readily solved for R: R ( 1 + K F + 2 K L F A + 2 K L 2 F 2 A ) % = 4 K X R T K I 2 F 2 A + -\j (1 + K F + 2 K L F A + 2 K L 2 F 2 A ) 2 + 8 K X R T K I ^ A 4 K X R T K L 2 F 2 A ' (36) (here the positive root is clearly the only possible solution) and for cross-linking, we have f c l = 2 K X R ^ 2 K L 2 F 2 A (37) With these equations, we can plot cross-linking as a function of F and A . Figures 10a and 11a show bivalent receptor cross-linking and figures 10b and l i b show monovalent receptor cross-linking. A s mentioned above, we display the cross-linking information as curves rather than as surfaces since experimental data is commonly presented in this manner, the experiment usually consisting of the manipulation of one variable. 33 1 0 n log (K A) b Figure 10: Graphs showing (a) bivalent and (b) monovalent receptor cross-linking for K=L=108, Ji=J2=0, F=10-7. 34 1.0n log (LA) b F igure 11: Cross-linking versus antibody concentration for various values of the factor concentration, (a) bivalent receptor (b) monovalent receptor. Parameters as in figure 10, with K X R T = 1 0 0 . In figure 12, we depict the behaviour of the model under variation with respect to K and L and in figure 13, we show a typical slice in the A=constant direction. 35 Figure 12: Cross-linking curves for various values of K (12a,b) and L (12c,d). (a) and (c) depict the bivalent receptor, (b) and (d) the monovalent. K X R T is fixed at 100. Ji=J2=0. In (a) and (b), L=108. In (c) and (d), K=108. Note the different abcissa scale in (c) and (d). 36 Figure 12 (cont'd^ caption on previous page 37 b Figure 13: The dependence of cross-linking on KF - a slice of the cross-linking surface in the (A= constant) direction for various values of A. As before (a) represents bivalent, (b) represents monovalent. Other parameters as in figure 10. 38 The curves in figures 10 through 13 indicate the complicated dependence of f c i on the parameters of the model. The general nature of the curves is exactly what we would expect from a common sense perspective: as the equilibrium constants or the concentrations go to zero or to infinity, the cross-linking goes to zero; the former limit being obvious and the latter following from figure 9. The precise behaviour of the model however, is buried in the intricacy of the model, and is best revealed in the cross-linking plots such as we have displayed here. There are some features that we can draw out of the equations despite their complexity. Our analysis will concentrate on the bivalent equations (for which analysis is more easily accomplished), but based on the similarity between the bivalent and monovalent equations, we expect our analysis to be suggestive of the behaviour of the monovalent equations as well. We first note that for a bivalent A cell receptor, the cross-linking is purely a function of a , as can be seen from equations (33) and (35). Given that f c i= l - r (a ) is a monotonic increasing function, as illustrated in figure 14, it follows that the maximum in cross-linking occurs at the maximum value of a . Now in the expression for a , equation (29b), K X R T appears only as a scale parameter and is thereby proportional to the maximum in a and related to the maximum in cross-linking. It does not however affect the (F,A) co-ordinates at which the maximum occurs (as we would expect from figure 10, 3 9 where the position of the maximum is constant with respect to variation of the K X R T parameter). o.o I • i i i 1 | i | i | 0 2 4 6 8 10 alpha Figure 14: Cross-linking as a function of a for the bivalent case with Jl=J2=0 Secondly, with respect to the behaviour of the curves under adjustment of the parameters, we can provide a loose plausibility argument. Since f c i is a monotonic increasing function of a , the maximum value of f c i will occur at the maximum value of a . Furthermore, it is clear that the greater the width of the curve a ( K , L , F, A , K x R j ) the greater the F W H M of the f c i curve. We are therefore interested in the behaviouT of the function a (K, L , F , A , K X R T ) as defined by equation (29b) and (30): a = ^ 4 K X R T K L 2 F 2 A (29b) w h e r e 4 0 p = {l+2KF+K2F2+4K2LF2A+4K2L2^ 4KLFA+ 4KL 2F 2A+ 4K 2L 2F 3A } (30) Figures 10 through 13 reveal one immediate feature: a strong similarity between the bivalent and monovalent cases. Not only do the curves have the same shape, they also behave in much the same way under adjustment of the parameters, neither fact being surprising given the similarity between the bivalent receptor (29b), (33) and monovalent receptor equations (36), (37). A closer look reveals however that there is actually a substantial difference in terms of the width of the curves (see figure 15) in the two cases. Depending on the exact value of the equilibrium constant K, the F W H M s of the cross-linking curves differ by at least an order of magnitude in concentration units. If IL-1 secretion is a linear function of cross-linking (an assumption that is successful in one exhaustively studied cross-linking system [21] and is in any case a reasonable first guess) then the F W H M of the experimental IL-1 curve would be identical to the F W H M of the cross-linking curve as shown in figure 15 (since log(ax)= log(a) + log(x)). It is possible then that we could draw a conclusion with respect to receptor valency based on the width of the IL-1 curve and on figure 15. 41 6 -i 5 -4 -o 2 3-CD £ O o> CO c 'c _c 3 r <o S2 u E  o *c .f cd o o> o 3 I FW -5 • B • Monovalent • Bivalent 1 1 I • 0 5 log (K) • 10 -•—I 15 Figure IS: F W H M of the cross-linking curves for various values of the equilibrium constant K . The F W H M is read off of the appropriate graph as log(K + ) - log(K.)=log(K+/K.) and is therefore dimensionless. The other equilibrium constants are as in figure 10, with K X R T = 1 0 0 . In order for this to work, we would of course need to have accurate values for all of the equilibrium constants, including the K X R T parameter (see figure 16) which as previously indicated would be difficult to measure directly. If such proves to be impossible, we would have to make direct measurements of cell-surface cross-linking, perhaps by methods discussed in [21] in order to make the bi/mono distinction. 42 0.8 b c Figure 16: In order to make the bi/mono distinction, we need accurate values for the equilibrium constants. In (a) we see that the curves for the bivalent and monovalent cases have practically the same width if we allow for an order of magnitude error in the K x R j parameter; hence we could not distinguish between the two cases if we did not know this equilibrium constant to within an order of magnitude. In (b) and (c) we see that with m accurate value for K X R T , one can easily make a distinction between the two cases. If, however, J n *0, then the bivalent curve acquires a different character, as illustrated in figures 17 and 18. For this case, the self-consistency equations reduce to 1 = (f-2d-lj2) + 2 d - l J 2 <l>(ar,d) K X R T (42) and cross-linking is given by = 1-r (1+ 2 ;KL2F2A Ji) (43) the one-unit rings being included in the non-cross-linked fraction as specified by equations (17b) and (22). We again solve for r, and then for cross-linking, by using a root finder program. In order to simplify the computations, we use d=l, for which With this approximation we are treating ring formation as a random walk with possible self-intersections (see [21]) and while steric repulsion dictates that we should not allow self-intersecting random <|>(x) = -In (1-x ) (44) 44 walks (rings), d=l is certainly a reasonable approximation since even for self avoiding walks, d is of the order of 1 [23]. 1.0n log (K A) Figure 17: Cross-linking for various values of J 2 . The J2=0.01 and J2=10 curves cannot be resolved from each other at this scale. 1.0 n log(KA) Figure 18: Cross-linking for various values of J j . The curves are more sensitive to Ji than they are to J 2 . As already mentioned, the monovalent case displays no rings, and as such the curves are not affected by this choice of parameter. It is thus clear that if we see wings such as those in figure 18 in our IL-1 4 5 curve then assuming simple proportionality between IL-1 and cross-linking, we can conclude that one-membered rings are being formed and that the factor receptor is bivalent. There are of course steric and simple geometric restrictions on one-receptor ring formation (as illustrated in figure 7), and these effects which are embodied in the equilibrium constant J i will affect the extent to which one-receptor rings form. Clearly, if J i £ 500, the effects of rings would be visible in the cross-linking (and in IL-1 measurements, if linearity holds). If this is not the case and if we are unable to use the F W H M plot above, then we can make no conclusion on the basis of IL-1 data and direct measurements of the A cell receptor will be required to determine va lency . In making a direct measurement, it will of course be helpful to know exactly what one expects to see on the cell surface. Using the above model, we can in fact predict which aggregate will have the highest concentration. T o this end, we define oo I n R n "=j  R T (45) a n d n R f j " Rn n 46 (46) which represent the fraction of receptors which are in (j and greater)-membered aggregates and the fraction in (j)-membered aggregates respectively. Clearly, *2+ = f d (47) and the rest of the fj+ can be expressed in closed form by using equation (17b). For instance, f 3 + = f c i - 2 a r 2 - ^ - a 2 r 2 (48) an expression that we use to plot the fraction of one, two and three-plus receptor aggregates, f j , f 2 and m figures 19 and 20. If we are therefore unable to draw conclusions based on curve width or detect one-membered ring formation in our IL-1 measurements, it wil l then be necessary to make direct measurments of cell-surface aggregates in order to make the bi/mono discrimination. Such direct measurements would admittedly be difficult, but they have the added benefit of providing data which could be used to develop models of IL-1 secretion, since one could relate IL-1 data to cross-l inking measurements. 47 One method for making such a direct measurement would be to use immunofluorescence techniques such as those described in [35] to count the number of molecules on the cell surface. This method uses epitope specific monoclonal antibodies and radioactive secondary antibody at saturation to estimate the epitope concentration from a 'standard counts per number of epitopes' curve. In our case, we would have to modify the procedure slightly and ensure that the factor concentration was also at saturation since the antibodies are specific for the factors and not the receptor molecules we want to count. -5 -4 -3 -2 -1 0 1 2 3 4 5 log (KA) F igure 19: A s we vary A continuously, we see the emergence of different cross-linked aggregates, starting with one receptor aggregates, and then two, and so on. A s A increases even further, the equilibrium conditions favour fewer antibodies per aggregate, and we eventually re-emerge with one receptor aggregates. Parameters as in figure 10 (K=L=108; F=10-7; Ji=J2=0), with KXRT=100. 48 With this experimental method, we could determine R T and the number of antibodies bound to the cell at saturation A m a x - For verification, A m a x could also be estimated from electron microscope images of gold-labelled antibodies [36]. The coefficient of proportionality between the two would then be the valency of the receptor. log(KA) Figure 20: Same as figure 19, but with Ji=500. The one-receptor rings are at a high concentration when other one receptor complexes are at low concentration (compare figures 19 and 20). This substantially reduces total cross-linking. 4 9 -5 -4 -3 -2 -1 0 1 2 3 4 5 log(KA) Figure 21: Same as figure 19, but for the monovalent case where we have only fj and f 2-With the receptor valency known, IL-1 data could be compared with the above model with an eye towards developing models of IL-1 secretion. This could provide valuable information concerning the expression. of the IL-1 gene as a function of external conditions. Of course it will be necessary to consider other possible A cell activation mechanisms and experimentally evaluate their effectiveness, and other experimental tests of our A cell model must be performed (including a determination of the equilibrium constants), but we have made a beginning here. And "Were (we) to await perfection, (this) book would never be published" [37]. 5 0 Chapter 2; N-Dimensional Network Dynamics 2.1 Introduction to the Model A s mentioned in the introduction, several authors have developed models based on Jerne's Network Hypothesis. The first of these was that of P. Richter[9]. His model, while it demonstrated many interesting physical features, relied on a distinction between the part of the V region that recognized antigen (the paratope) and the part that was recognized by other receptors (the idiotope). This distinction, which was in fact part of Jerne's original hypothesis, has never been observed in experiments despite detailed characterization of receptor V regions, and Richter's model can therefore be discounted. Contemporary network models include the Hoffmann model as well as the models of Segel and Perelson [10], Kaufman and Thomas [11], and DeBoer [12]. Segel and Perelson's model, in which receptors are represented as real numbers in 'shape space', is appealing but highly esoteric and lacking in any real predictive power insofar as current experiments go. Kaufmann and Thomas' model is astute but relies on purely stimulatory network interactions. This is difficult to reconcile with the fact that network interactions are generally assumed to be suppressive, and the experimental fact that antibodies in the network wil l ki l l anti-idiotypic cells in the presence of complement. Finally, in the DeBoer models, the author demonstrates that a certain set of "reasonable assumptions" do not yield a functional network. A s evidence against his assertion that network theory is thereby discredited, we would 5 1 offer this latest version of the symmetrical network theory, where other 'reasonable assumptions' yield a highly functional network. In the context of the symmetrical network theory a mathematical model of the immune system has been formulated which traces the time evolution of various clones or cell lines in the absence of external antigen. A clone interacts with other clones via its receptor V regions and via non-specific stimulatory or inhibitory proteins secreted under certain conditions by these other clones. Clone populations are also influenced by natural cell influx and death. The main modelling assumption that is made is that interactions are symmetrical: if cell A stimulates cell B„or kills cell B then, provided A and B are of the same cell type, the converse is also true (see figure 22). Th is , assumption is based on the cross-linking model of trans-membrane signalling (discussed in Chapter 1) and the observation that if cell A cross-links receptors on cell B, then cell B can also cross-link receptors on cell A . O f course, this merely tells us that the first signal for activation (cross-linking) is symmetrical. However, given that the second signal, which as mentioned in chapter 1 comes in the form of interleukins, and is non-specific, symmetry is likely to be p r e s e r v e d . 52 stimulation v killing ' Figure 22: Complementarity leads to the symmetry postulate whereby inter-cellular interactions are all two-way. We assume therefore that the aggregation of receptors in the presence of interleukins leads to the proliferation of a clone i and antibody secretion. Consequently, by symmetry, it will also lead to the death of i cells since i cells will cross-link anti(i) cells, causing anti(i) antibody to be produced. The requirement for interleukins necessitates the presence of the A cell (which as mentioned in chapter 1 is the principal source of IL-1). However, given that there is far less cell stimulation and proliferation involved in the maintenance of steady states than there is in the response to antigen (when populations are changing comparatively quickly), we treat the low level of interleukins necessary to maintain the steady states as a second order effect, and ignore it in our model at this point. In the model, the A cell is thus not involved in the maintenance of the steady states, serving instead 53 to perturb the antigen confronted system from one steady-state to another. In the model, we also assume that the idiotypic (V region) interactions are inhibitible by specific T cell factors. These factors, which are modelled as monovalent F a b fragments as in chapter 1, are assumed to be secreted by T cells when their receptors are cross-linked. T cell clone i will secrete factors of specificity i which are specific for V-regions of anti(i) clones. These factors will then bind to these anti(i) V regions (see figure 23), blocking the associated receptor and inhibiting idiotypic interactions (both stimulation and killing). A s in chapter 1, these factors would also bind to the A cell, where the factor receptors could be cross-linked by the V regions on anti(i) cells. However, this effect is assumed to be small in the absence of antigenic perturbations and is consequently ignored as another second order A cell effect. F igure 23: Factor secreted by an anti-A clone binds to A receptors and inhibits cross-linking and hence idiotypic interactions. Based on these assumptions, the following initial version of the symmetrical network theory was proposed: 54 (49a) (49b) where 1 q=2,3 (49c) X+X-1+ In these equations, x+ represents the population of a particular clone (the idiotypic clone) and x. that of a complementary clone (the anti-idiotypic clone). The influx of new cells, assumed identical for both clones, is represented by S. Natural cell death is taken to be linear in the population size with strength D. The ki terms model the V region or idiotypic interactions and are of the usual form: rate constant (k) times concentration of reactants (cell or antibody "+" reacting with cell or antibody "-"). The k i term models stimulation via cross-linking by cell receptors or antibodies of the opposite specificity. The k 2 term models killing by killer T cells and/or by IgM antibody plus complement. The k3 term models antibody-dependent cellular cytoxicity (ADCC) and/or killing by IgG antibody plus complement. The k i and k 2 terms are taken to be first order in the reactants (which is justified according to [40]), but since A D C C is a complex several-body phenomenon (as discussed in [41]), we take that process to be second order in the immunoglobulin. A l l of the idiotypic terms contain e q factors which represent the inhibition of these interactions by specific T cell factors. The factors are assumed to inhibit when their concentration exceeds a certain 55 threshold concentration (see figure 24). C i , C 2 and C3 are constants that specify this threshold value, and n i , n 2 and n3 are constants that determine the sharpness of the thresholds. The concentration of the factor is taken to be the product of the cell secreting it (x+, x.)and the stimulation that that cell receives (taken as proportional to the concentration of the anti-idiotypic clone (x., x+). 1e+0 Oe+0 — l <~ 10 20 30 40 50 Fl Figure 24: Graph showing e q i as a function of fi for C q =20. Two cases are depicted: n=2 (smooth) and n=«» (sharp). This two clone model was analyzed in detail in [41] and was shown for n q large to have 4 stable steady states under two exclusive condit ions: ^ < c 3 < E < c 2 < S kd E < c 2 < r s < c 3 < v k 4 , (50a) (50b) These 4 states are depicted in the phase plane picture below (figure 25) and have the following characteristics: 56 1) the virgin state. Low levels of both x+ and x. corresponding to a system that has not yet been confronted by antigen. In this state, the influx of new cells is balanced by the linear (IgM) killing term. 2) the suppressed state: Elevated levels of both x+ and x. cells, and suppression of both populations by high levels of specific factor. The system is unresponsive due to the high factor levels, and clones specific for self-antigens could be in this state. In this state, the influx of new cells is balanced by the cell death term. 3) the immune state: x+ cells are at a high level and x. are at a low level. This state corresponds to the immune memory state: an antigen has entered the system, leading to proliferation of x + cells specific for the antigen and elimination of some x. cells. The high level of x+ ensures a vigorous response at the next encounter with the antigen. In this state, the influx of new cells is balanced by the cell death term and both idiotypic terms. 4) the anti-immune state: the mirror-image of the immune state, corresponding to an antigen for which the x. clone was specific. While this model is clearly quite successful in its representation of a wide range of phenomena, it is limited in the sense that it describes only two clones; consequently, we have attempted to extend the model to a greater number of clones. In a higher dimensional model such as this, a particular clone i wil l interact with many other clones, the strength of the interaction being dependent on the affinity between V region i and the V region of the other clones. 57 F igure 25: Phase plane picture for the two dimensional model (equations 49 a, b, c) showing the four steady states. Reprinted from [41]. Parameters: ki=3, k 2=10, k 3=100, D= l , S=10, Ci=10, C2=VTo , C3=V0.1 , ni=n 2=n3=5. This choice of parameters corresponds to condition (50b). In changing the x+ equation to an equation for a general clone xj, it would therefore seem reasonable to replace the anti-idiotypic x. term with an affinity weighted sum of all the other clones. If we let the matrix K be such that Kij is the normalized affinity of receptor i for receptor j - hence a number between 0 and 1 - then we can write 58 down an expression for the affinity weighted sum - which we call the connectivity - of clone i: N N Y i = X (affinity)ji (population)j = £ KJJXJ ( 5 1 ) j=l j=l Note that K is a symmetric matrix via the symmetry postulate and that Kii=0 since a V region is assumed to not bind to itself. With this replacement, the equations read: dx -~d7 = s + kiXiYjen - k2XiYje2i - k3XiYj2e3i - Dxi i = 1,...,N (52a) e Qi = )xu Nn q =U.3; i = l , - , N (52b) 1 + fej where *F i i s the concentration of factor which inhibits the killing and stimulation of clone i. Once again, we take the concentration of factor secreted by clone i, fi, to be equal to the product of the concentration of the cell producing it and the stimulation this cell receives (which we take to be the connectivity): fi = x iYi (52c) In the latest published version of the model [16], we took Vi = fj = XiYi (52d) It was noted in that article however, that this should be changed in order to be consistent with the assumptions of the model, which is what we have sought to undertake with this thesis. The changes were required since equation (52d) is actually in conflict with the 5 9 symmetry assumption in the sense that it represents inhibition by i factors (factors secreted by clone i) but not by anti(i) factors. To elaborate, idiotypic interactions are inhibited in two ways: factors from clone i (and i-like clones) which block anti(i) receptors, and also factors from anti(i) clones (and anti(i)-like clones) which block i receptors (see figure 26). Equation (52d) represents only the first of these inhibition mechanisms and must therefore be modified to take the second one into account. F igure 26: The two pathways for inhibition: anti(i) factor blocking and i factor blocking. 60 T o accomplish this goal, we defined a quantity known as the similarity co-efficient matrix, L , whose elements L y represent the likeness of clone i and clone j within a particular immune system (the context is important since likeness is defined with respect to all of the other clones). We calculate L y by making an affinity weighted sum u. over the set I of clones which bind i and the set J that bind j (see figure 27). We let _ u.(I n J) _ u.(I n J)  L i j " u(I u J) u( I - J ) + u( I n J) + u ( J - I) ( 5 3 ) where the denominator has been selected to ensure that L y is a number between zero and 1, and that La is identically 1. F igure 27: A Venn diagram of the set of "i-binding" clones and the set of "j-binding" clones used to define the similarity co-efficient L y . The two sets can be decomposed into 3 disjoint sets: I-J, J-I and I n J . T o calculate u.(I n J), we sum the products K i k K k j . each term representing the extent to which k binds to both i and j and therefore the extent to which i and j are similar - weighted by the population xk: N u ( I n J ) = X K i k K k j x k (54) k=l 61 To calculate u,(I - J) ( u.(J - I) is similar) we sum the product Kik(l-Kkj) - which since the Ky are affinities between zero and 1 represents the extent to which clone k binds to i and not to j - again weighted by the population Xk. N H(I-J) = SK i k(l-Kkj)x k (55) k=l We substitute (54) and (55) into (53) and we arrive at the final expression for the similarity co-efficients: N X KikKkjXk Lij= N k = 1 (56) X (Kik+Kkj-K j kKik)xk k=l We are now in a position to represent both inhibition pathways in our model. To determine the amount of factor inhibiting idiotypic interactions with clone i, we sum over all factor concentrations XJYJ weighted by the extent to which clone j is 'i-like' OUj) and the extent to which clone j is 'anti(i)' (Kjj). The result is: N N N ¥ i = X KijXjYj + X LijXjYj = X (Kij+Lij)xjYj (57) j=l j=l j=l Which we simplify by defining the relatedness matrix R: R = L + K (58) 62 W h e r e b y N = X Rij x jYj j=l (59) This change complete, symmetry is once again restored to the model. This modification however, necessitated another one. When the model was integrated numerically, it was found (as will be detailed in section 2.3) that the level of specific factor in the system increased to a point where the entire system was suppressed. This is probably due to the extra coupling between the clones which is brought about by equation (59). Since such a paralysed and unresponsive system is clearly not reasonable from a biological point of view, we sought to limit the total concentration of factor in the system by introducing a normalization. Equation (52b) was modified to read: eqi = A * . , sn . q= 1,2,3; i = l,...,N 1 -where v ) n, (60a) (60b) where v is defined in such a way that the average factor concentration of the normalized system does not exceed the midpoint of the two thresholds C 2 and C 3 (C i is not considered since we will drop it in the next paragraph): i ^ f f i i t a i ( 6 1 ) 1 k=l z 63 We thus arrive at an expression for v: _ N ( C 2 + C 3 ) v = N (62) 2 X^Pi k=l We set this limit on the average factor concentration since we expect on a biological basis that the steady state of the system will be such that there are clones in all of the steady states - and therefore that the average factor concentration will be approximately at the midpoint of the thresholds for the two principal interactions (again, see the next paragraph). Furthermore, since we model systems which are 'nearly' steady state, we apply this normalization throughout the phase plane evolution of the various clones. One might argue that this normalisation should only be applied when the average factor ( C 2 + C3) concentration exceeds ^ — , as would be the case for an enzyme that is activated at this threshold. However, in this case, there is no way to ensure that an arbitrary system will finish with clones in all of the steady states, since such an arbitrary system might not exceed the threshold. This is the advantage of using the above formulation, as represented by equation (62). A s the final step in the development of the model we note that in previous work with similar equations [16] we have found that a minimal model given by dx 1 •jf = S - k 2 x i Y i e 2 i - k 3 x iY i2e 3 i - D xi i=l,...,N (63) 64 is sufficient to represent a number of physically significant details, and so we adopt it once again. This is tantamount to making the approximations k 2 » k i and C 2 > C i . Our approach is now to integrate these coupled non-linear equations numerically (see the program in appendix 1), to study the effects of varying the parameters on the model, and to analyse the general qualitative properties of the model. But first, we discuss the model and its strengths and weaknesses. 65 2.2 Discussion of the M o d e l In this model, as in most network models, we envision the immune response proceeding as follows: the antigen perturbs a certain number of clones from their initial values via clonal selection. These clones proliferate, and begin to eliminate the invading antigen. This proliferation also stimulates the clones idiotypically connected to the antigen-selected clones, which act to reduce and stabilise the population of the antigen-selected clones. O f course, anti-anti idiotypic clones (and so on) also become involved and the perturbation thus propagates throughout the network, causing the system to evolve into another steady state. The idiotypic interactions, which act as negative feedback, ensure that the perturbation is damped out as it propagates, producing the specific memory state. Some have put to question this picture of the immune response ([42-5]), arguing that receptors cannot distinguish between, for instance, a cell-bound virus particle (indicating an infected cell which should be killed-figure 27a), and a receptor bound virus particle (indicating that a cell has bound the virus and is going to participate in the response against the invader, and as such it should not be killed-figure 27b). This is obviously true, but what is unclear is whether this renders the network non-functional. What needs to be determined is whether or not the idiotype/non-idiotype background effects such as those depicted in figure 27b swamp out the idiotype/idiotype regulatory effects as depicted in figure 27 a, thereby rendering the network non-functional. This question can 66 only be resolved by experiment, but given the number of experiments which are suggestive of network regulation, it seems unlikely that the answer will be a solid affirmative. However, even if b F igure 27: A receptor cannot distinguish between a membrane bound virus (a) and a receptor bound virus (b). this does prove to be the case, the network is not ruled out (since there could still be a vital immuno-regulatory role for anti-idiotypes), it merely indicates a role for other regulatory mechanisms. A s mentioned in the introduction, an important point that needs to be addressed is just how this model accounts for self/non-self discrimination (S/NS discrimination). This process is obviously very 67 fundamental: the healthy immune system does not attack itself, but responds actively to foreign invaders, and as such all models of the immune system have to account for it in detail. There has actually been a certain amount of controversy as to whether or not network theory can account for or even accomodate S/NS discrimination. According to the symmetrical network theory, clones directed against the host are in the suppressed state, and are therefore not damaging for the organism. Using arguments similar to the above, some have vehemently argued however ([42],[44]) that the network cannot account for S /NS, again reasoning that a receptor cannot tell the difference between a membrane bound self-antigen and a self-antigen bound to the receptor of an anti-self cell (see figure 28). The argument goes on to conclude that the network could not be in the suppressed state for a self-antigen lest it be in a suppressed state for all antigens. Once again, this argument reduces to the question: 'does the epitope-idiotype interaction swamp out the regulatory idiotype-idiotype interaction?' and once again, a positive answer would indicate that the network is not the final word. O f course, even if network theory cannot completely account for the S /NS, it can certainly accomodate it. There are two important mechanisms which are often considered to eliminate anti-self clones during development: thymic education [46] and clonal anergy [47], and it is quite possible that these act as precursors to network regulation. However, in this case, the network would still have to regulate the anti-self clones which would arise due to somatic mutation, these being a non-negligible consideration (see [44]). 6 8 Figure 28: A n idiotype cannot distinguish between a self component bound to an anti(self component) receptor (a), a self component bound to the cell membrane (b), and a receptor whose idiotype resembles the self component (d). If the network is to provide the S/NS discrimination, then clone c (or clone a, which is the same thing) must be kept at a low population by clone d and other anti-idiotypic clones. This will be difficult since clone c will receive much stimulation by virtue of its anti(self component) specificity. Furthermore, autoimmunity attests to the imperfection of the above-mentioned deletion mechanisms, and so it is clear that unless the network can account for a residual S/NS (S/NS not accomplished by deletion mechanisms) via suppression, it will have to share the spotlight with other regulatory mechanisms, for example the veto cell ([48]). A final weakness in this model in particular has been pointed out by several authors ([11] and [43]). This is the fact that T cell antigen specific factors have remained somewhat of an enigma even in the days of detailed characterization of many other cytokines. These factors are defined both on the basis of a adoptive transfer and in vitro experiments. In these experiments, proteins with a molecular weight of approximately 50-75 k D -approximately one-third of the mass of an antibody molecule, which is the basis for our modelling the factors as the T cell equivalent of F a b fragments - can induce a state of specific non-responsiveness to a particular antigen. It would seem logical to conclude that these proteins suppress the response to this antigen, and it was on the basis of these experiments that specific factors are deemed to exist. These factors have also been observed by many researchers (see the list of references in [15] and a short update of this list in appendix 2) in the course of other types of experiments. In support of the network model, we would point out that network theories provide the most detailed models of the immune response: competing theories (the most prominent of which is associative recognition: [4] and [42]) are characterized by qualitative arguments and not by mathematical models. There is also the phenomenon of adoptive transfer which indicates that suppression is a real and important mode of the immune system. For these reasons, we feel 70 that network theory is still the best theory of immunoregulation, albeit one that is still being developed. Having defended the principles of network theory, there are several things that should be said about the particular network model represented by equation (63). First of all, we are using the label xj in three different ways at different times: as a T cell population (in the T cell factor term, when applying (63) to T cell populations, or when dealing with cellular stimulation/killing), as a B cell population (when applying (63) to B cell populations or when dealing with antibody stimulation/killing) and as an antibody population (when dealing with antibody killing/stimulation of cells). The rationale for this is that at this level of modelling, the T and B cells are subject to the same selective forces, and therefore we can think of their labels as being interchangeable. It is therefore perfectly consistent to use xi as the population of a particular B cell clone and to have xi appear in the expression for stimulation for T cell factors. Also to be noted is the fact that in using (63) to refer to cells being stimulated/killed by antibody, we are taking the antibody population to be simply proportional to the clone xi which is secreting it; by contrast, the factor concentration fi is taken to be a function of the stimulation and the population of the clone which is secreting it, which is a more reasonable assumption. The antibody assumption is hoped to be adequate for modelling the steady states of the system, where comparatively little killing is taking place (only enough to stabilise certain steady states), and hence antibody concentrations 71 are low. It will clearly be inadequate for modelling an immune response, since at that point, many B cells are differentiating and secreting antibody, these processes depending heavily on the specific stimulation the cell receives. It. is thus clear that to move to a more precise model, the labelling degeneracy will have to be removed by introducing separate labels for T cells, B cells, and antibodies, and the concentration of antibodies will need to be written as a function of the stimulation of the B cells secreting them. As a final comment, we would like to clarify the notion of steady state. It may seem absurd to speak of steady states when the immune system is constantly responding to foreign antigens and when there are many fluctuations in system behaviour throughout the course of these responses (see figure 29). The same criticism can be levied at many calculations log [plasma cells] to antigen injected at time - 0 time (weeks) Figure 29: Antibody concentration as a function of time in a typical immune response (taken from [49]). 72 throughout the sciences and indeed there is no a priori reason why such steady states considerations should apply to a physical system. The justification for this approach is thus purely empirical: it reflects what is observed in experiments, or more simply: it works. In evolving slowly through a succession of steady states, natural systems seem to lend themselves to mathematical treatment such as we are carrying out in this thesis. 2.3 Analysis of the Model Based on analogy with the 2-dimensional model (49a, b, c), we anticipate the existence of four qualitatively distinct classes of stable steady states in the N-Dimensional model: the virgin state, the immune state, the suppressed state and the anti-immune state. In the virgin state we expect low but not insignificant levels of both i and anti(i) clones for a given specificity, and a state characterised by a balance between influx of cells and the linear idiotypic term (killing by IgM). The immune state is the memory state in which, due to killing by cells of one specificity (i cells) we have an elevated level of i cells and a low level of cells of the complementary specificity (anti(i) cells). In the suppressed state we have elevated levels of both i and anti(i) clones, and mutual stimulation between i and anti(i) T cells (leading to inhibition by specific T cell factors) as the main network interaction. It is the N-dimensional suppressed state which might account for the residual S/NS discrimination by the network which was discussed in the previous section; the anti-self clones which slip through the deletion mechanisms stimulate anti-idiotypic clones leading to mutual inhibition by factor. The anti-immune state is the converse of the immune state: elevated anti(i) and low i population levels. In our initial analysis of the N-dimensional model [16], it was found that such states did indeed exist and that they corresponded to our intuitive picture of an N-dimensional generalization of the 2-74 dimensional model. We now repeat the analysis performed in [16] to confirm that this is still the case for the modified N-dimensional model represented by (63). A s in [16], we consider dynamics in the x i /Yi phase plane. This method of analysis provides us with strong visual images of the behaviour of the model and seems to be a suitable extension of the x+/x. phase plane studied in the 2-dimensional model. In the Xi /Y i phase plane, we represent the population of a clone on the abcissa and the connectivity on the ordinate. A s time evolves, the set of points corresponding to one clone traces out a trajectory on the phase plane (see figure 30). N such trajectories can be simultaneously displayed and in such a way we can observe the evolution of the entire N-dimensional system (see figure 31). 1 0 0 F 1 10 i 11 0 . 1 i 0 . 0 1 i 0 . 0 0 1 ~i—• 1 1 1 " " i — ' ' ' i ' ' '' " r n — I ' I I n m — ' i 111 I I I 0.001 0.01 0.1 1 10 100 X; F i g u r e 30: A trajectory of a single clone in the x i / Y i phase plane. The time increments are small enough that the points form an apparently continuous line. 75 100 | 10 i 11 0. 1 i 0.01 i 0.001 J 1 l • .in.l 1 . • M I M , , , , , M M | , , , , . , „ • 0.001 0.01 0.1 1 10 100 Figure 31: Trajectory of 20 clones in the X i / Y i phase plane. The time increments are small enough that the points form an apparently continuous line. We now analyze the system of equations defined by (63) to see if steady states such as those mentioned above exist in the modified N -dimensional model and whether the system as described by equation (63) will move into these steady states. In order to compute the dynamics of the system more easily, we simplify the model by using step functions for the e q i : e q i = 1 if ¥ f < C q 0 if ¥ i ' > C q q = 2,3; i = l , N , (64) this substitution being equivalent to taking n= co. This sharp transition constitutes a high degree of non-linearity in the system, but experience with such systems suggests that the dynamics of a system with a sharp transition will be similar to that of a system with a smooth transition. Dynamical studies of the system, such as the one indicated in figure 32 confirm this. 76 1 0 0 1 0 1 4 0 . 1 0 . 0 1 i 0 . 0 0 1 I I I I I I I I I l| I I I I M I I | i i . . i - • • i 0.001 0.01 0.1 1 10 100 * i 1 0 0 1 0 i 1 E 0 . 1 i 0 . 0 1 = 0 . 0 0 1 - 1 0.001 0.01 " T l 1 | I \ ! I 1 I I l| 1 II ,1 I I I I I 0.1 1 10 ' 100 * i b Figure 32: Sharp (a) versus smooth (b) thresholds. The dynamics are similar in both cases. The parameters used for both cases were: S = 12, k 2 =1, k 3 = 0.1, D = 2, C 2 = 10, C3 = 30. For (b) n2=n3=5. 77 The advantage of using n= «» is that it permits us to solve explicitly for the loci of equilibrium, which we now do for the two cases C 2 > C3 and C2 < C3. In the first region of parameter space, where C2 > C3, we have N d x . for S Rij xj Yj < C 3 < C2, " J f = 0 w h e n j=l xi = (65) k 2 Y j + k 3 Y i 2 + D N d x . for C3 < XRijxjYj < c2> "df = 0 w h e n j=l X i = k 2 Y i + D ( 6 6 ) N dx-and for C3 < C 2 < X R u x J Y j ' "HT = 0 w h e n j=l xi = 5 (67) In the other region of parameter space of interest, with C3 > C 2 , we have the following equilibrium lines: N dx-for S R i j x J Y J < C 2 < C3, " d f = 0 w h e n j=l xi = — (68) k 2 Y i + k 3 Y j 2 + D K J N dx-for C 2 < X Rij xj Yj < C3, "dT = 0 w h e n 78 Xi = k 3 Y j 2 + D ( 6 9 ) N and for C 2 < C 3 < XRiJxJYJ» j=l S D dxi dT = 0 when (70) In the x j /Y i phase plane there are thus three equilibrium loci at which clones can stabilise, as shown for a set of parameters with C3 < C2 in figure 33, and for a set of parameters with C2 < C3 in figure 34. 100 I T 10 = IMMUNE/ANTIIMMUNE SUPPRESSED 0.1 = 0.01 = 0.001 0.001 Figure 33: The three equilibrium loci in the x i /Y j plane, as specified by equations (65)-(67) with the parameters S = 10, k2 = 1, k3 =10, D = 1, C3= 3, C2- 10. V . S . - virgin state; I.S./A.I.S. - anti-immune state; S.S. - suppressed state. 79 0.01 = 0.001 -I 1 — i i 1 1 1 1 1 [ 1 — i i I I i m 1—i i i 11111 1 — i i 111111 1 — i i M i i i 0.001 0.01 0.1 1 10 100 Figure 34: The three equilibrium loci in the xJYi plane for the parameters corresponding to figure 32. When integrating differential equations numerically, it is of course important to set the step size sufficiently small so as to avoid artifacts. In our algorithm, we accomplish this by setting the step size so as to allow the fastest evolving population to change by at most a fixed percentage: < constant i=l,..,N Xi (65) 80 Based upon an initial survey of the dynamics, we determined that a 0.5% change per step was sufficiently small to faithfully replicate the dynamics of the system (meaning that decreasing ( A x / x ) m a x by an order of magnitude produced the same dynamics). We periodically verified that this mesh was indeed fine enough throughout the course of our study. The two regions of parameter space C 2 > C 3 and C 3 > C 2 correspond to different physical situations. The former represents an immune system where there is more factor in the virgin state than there is in the immune state, and the latter represents the opposite situation (see figure 35). We treat these two cases equally in what follows, but it should eventually be possible to determine which of the cases is physically correct. This could be done by measuring a specific factor concentration before and after the clone secreting the factor is challenged with antigen. E q i 1 virgin immune suppressed 0 F i C 2 C 3 a 81 0 suppressed F i C3 C2 Figure 35: A graphic illustrating the different cases C2 > C3 (a) and C 2 < C 3 (b). In (a), the concentration of factor is higher in the virgin state, and in (b) it is higher in the immune state. Figure 36 shows the trajectories in the x i / Y i plane for a system in which the initial clone sizes xj(0) were all set equal to the same value. Here various clones converge to the virgin state, the suppressed state, the immune (memory) state or the anti-immune state. Simulations run with a larger number of clones (up to 100) yielded similar dynamics. In figure 37, we display trajectories for a system with a random initial condition. 0.01 i 0 . 00 1 I i i i i MII| i i i i 11111 i i i i i n i | i I M IIIII i i i 11 III 0.001 0.01 0.1 „ 1 10 100 Figure 36: Trajectories in the xJY\ phase plane of 20 clones that are randomly connected with a connectance of approximately 0.3, and which are all given initial clone sizes of 0.1. The non-zero Ky are random numbers in the range 0.0 to 1.0. (a) represents C 2 > C 3 with parameters as in figure 33 and (b) represents C 3 > C 2 with parameter values as specified in figure 34. 83 Figure 37: A random start where some clones start out with an initial population of 1, and the others with random initial values. A s was mentioned in the previous section, the level of specific factor is being normalized so as to keep it at a biologically reasonable level. Without the normalisation, the system often ends up with all clones in the suppressed state (as in figure 38a). When the same system is simulated with the normalization procedure, we find clones in all 3 of the steady states (as in figure 38b). 8 4 100-r1 101 1 i 0.1 i 0 . 0 1 i 0.001 ' — 1 — 1 1 1 1 1 n i — i i 1 1 1 , H I — i i 1 1 1 m i — i — i 11 i H I i—i 1 1 1 m i 0.001 0.01 0.1 1 10 100 a 0. 01 i 0 . 00 1 1 1 1 I I M l l | 1 I T I M l l | 1 1 I I I H | 1 1 I I I l l l | 1 1 I I I III 0.001 0.01 0.1 1 10 100 Xi b Figure 38: The N-dimensional system without factor normalization (a) is often completely suppressed. Since this is not reasonable from a biological point of view, we normalize the factor, and the result is a system with clones in all of the steady states (b) We conducted a random survey of parameter space, studying the effects of varying the parameters of the model. Based upon this survey, we can conclude that: 85 1) Changing the S, D and ki terms alters not only the form of the equilibrium loci, as one can see from (65)-(70) and figure 39 but it also affects the number of steady states for the system. For instance, in. figure 40a, we see that all clones are in the suppressed state. Changing k 2 and k3 to 10 and 100 respectively gives us a final state with clones in all three loci (as in figure 40b). Therefore, while it has not been possible to derive in detail conditions like those expressed in equations (60a, b), it appears as though similar relationships are present in the N-dimensional case as well. 1 0 0 J 1 1 1 0 ! ^ N ^ ^ o. 11 W \ 0 . 0 1 i 0 . 0 0 1 I ' I M 1 I I II I ' l l IMI| 1 I I I M i l l 1 I I I M i l 0.001 0.01 0.1 1 10 100 X, a Figure 39: Varying ks from 0.01 (a) to 0.1 (b) b to 1.0 (c) does not affect the qualitative dynamics, but only the quantitative nature of the steady states. The rest of the parameters are as in figure 32. 86 o.oH 0.001 0.001 0.01 0 . 0 0 1 ' ' ' ' r 0.001 0.01 Figure 39(cont'd): caption on previous page. 87 Figure 40: L o c i and equilibrium for D=20 and (a) parameters as in figure 32 (b) parameters as in figure 32, except k2=10 and k3=100. 2) Adjusting the thresholds has the effect of altering the number of clones on a particular locus. See for instance figure 41, where the effects of varying C 2 from 0.1 to 28 while C 3 is fixed at 30. As C 2 88 increases, more clones go into the suppressed state and less into the immune state, which is what we might expect based on figure 42. 1 0 0 ^ ^ — j 1 0 i ^ ^ ^ w 0 . 1 j ^ j 0 . 0 1 i 0 . 0 0 1 ! 1 — i 1 1 1 1 1 1 1 — i 1111111 i—i 1111ni i i 1111HI , i—i 1111II 0.001 0.01 0.1 1 10 100 X i a 100 j 1 l O i 11 0 . 1 ! 0 . 0 1 i 0 . 00 1 ~ ~ i — ' " i — i 1 1 1 m i — i — i — i 11111111—i—1111m 0.001 0.01 0.1 1 10 100 Xj b Figure 41: The dynamics for several values of C 2 . In (a) C2=0.1, in (b) C2=13.0, and in (c) C2=28.0. The rest of the parameters are as in figure 32. Eqi 0 F i C 2 average factor concentration C 3 Figure 42: A s we increase C 2 the average factor concentration as set by the normalization increases, and eventually we get clones in the suppressed state. V=virgin state, I=immune state, S=suppressed state. 90 B y studying the factor concentration, we can see if the steady stable states of the system correspond to our phenomenological picture discussed in the opening paragraph of this section. Typical results are shown in figure 43. In the virgin state, we see that there are low levels of factor with idiotypic and anti-idiotypic factor concentrations approximately equal. In the suppressed state, the factor concentrations are much higher as we would expect (based on the thresholds - figure 42), however there is less anti-idiotypic factor than idiotypic factor, and this does not correspond to the 2-dimensional model where in the suppressed state x+ = x., which implies that f+ = f. since in the 2-dimensional model f=x+x_. Of course this inequality is exactly what we want for the immune state since more "like factor" than "unlike factor" implies less suppression of the "like" clones, ie. a primed immune memory state. T o be noted however is the fact that in this tabulation, there is in general more "i-like" factor than there is "anti-i" factor. This is probably reflective of the fact that there are more non-zero entries in L than there are in K and it suggests that we might want to change the normalization for the L j j since intuitively we expect that the levels of "i-like" and "anti-i" factor should be the same. On the whole however, the factor concentrations correspond quite well with our intuitive picture of the steady states. There are some differences between the factor concentrations in the 2-dimensional and N-dimensional models, but these hardly surprising given that there are many more connections between the clones in the N-91 dimensional model, and more complicated stabilization patterns are possible. final population fir*I stele like factor unlike factor 24355 V 8.0 1.7 0.8438 I 192 7.3 6.0000 S 33.6 9.6 2.8623 I 14.9 3.3 1.0441 I 152 5.1 1.7973 I 19.8 3.8 0.8619 V 9.9 2.4 42181 I 13.1 1.4 22758 I 11.5 1.8 0.5790 I 22.0 7.8 1.9010 Y 9.8 12 1.9149 V 7.3 2.0 6.0000 S 31.3 9.5 1.1319 I 19.6 5.1 1.1069 I 19.1 5.0 2.5164 I 12.4 2.5 92696 I 17.3 6.7 6.0000 s 33.1 9.6 0.6995 I 16.6 6.3 3.4505 Y 3.1 0.5 j Figure 43: A table of representative steady state factor concentrations As shown in figure 44, different Ky yield different dynamics, which is what we would hope for from a network model. There is much variation between individual immune systems and this is represented in the Ky (and also in the magnitude of the parameters). 9 2 Furthermore, for a given set of parameters and K y , the system can adopt many steady states as illustrated in figures 36 and 37, which coincides with the many states that an individual's physical immune system can adopt. 0 . 0 1 E 0 . 0 0 1 H T — T T T T T T T l 1 I l l l I l ' | . I I I * I n | . I I I I 111 0 . 0 0 J 0.01 0.J 1 to 100 b F igure 44: Different Kjj give qualitatively similar but quantitatively different dynamics and steady states. 93 Our model thus displays many of the qualitative characteristics of the physical immune system's steady states as detailed at the beginning of this section. T o illustrate that these steady states are in fact stable steady states, we add to the equation a term that can be thought of as an antigenic perturbation: ^ = S - k 2 X i Y i e 2 - k 3 x i (Y i)2e 3 - Dxi + RiAg(t) (71) Here Ag(t) is the concentration of antigen as a function of time t, and R i is the strength of interaction between the antigen and clone i. O f course, our model does not intend to simulate switching between stable states, since as mentioned in Chapter 1, we believe that switching involves the A cell , which has not been included in the model of steady states. Nonetheless, it is instructive to examine these perturbative stability tests with state switching in mind, since it is interesting to see what we can get out of the model as it is; this might even give us an idea of what terms or effects we need to include in the equations so as to model the dynamics of the immune response. If we start with a set of clones that are all at steady state levels and perturb the system with a transient pulse of antigen, we obtain trajectories in the x i / Y i phase plane as shown in figure 45. Each of the trajectories returns to its starting point, and the steady state is thus a stable one. This effect can also be seen as a model for the perturbation caused by a T independent antigen, which typically stimulates each of many clones a little, but does not induce memory. Many different clones responding simultaneously to the antigen 9 4 cause the elimination of the antigen without any of the clones proliferating enough to leave their original steady state. In figure 4 6 we see that a larger pulse of antigen results in some of the clones being switched from one locus to another. This will happen when the perturbation is sufficient to push a clone into the region of attraction of the next locus in the phase plane. While we did observe a state switch on the part of several clones in many trials, it was impossible to get more clones to switch (more than 2-5%). 100n 1 0 i 1 = 0.1 = 0.01 = 0.001 I I I I 111II I—I I I I I 111— 0.001 0.01 0.1 -i—i i 1111 n T 1 I II I I 1 10 100 Figure 45: A small perturbation of each of many clones that are all in the virgin state results in a transient perturbation of the network. A l l of the clones return to their original loci, and there is no memory associated with the response. This is a model of what happens with T-independent antigens which do not induce a memory state. 95 This may or may not be physical, but in any event, the source of the effect is most likely the factor normalisation. "With the normalisation in place, the overall factor level carmot increase when the antigen is introduced, and so to see state switching, the factor must be redistributed amongst the clones. Since we envisron the immune response as being characterized by the presence of a substantial amount of factor, it is probable that we wil l have to rethink the normalisation when it comes time to to deal with switching dynamics . 0.001 0.01 0.1 1 10 100 * i F igure 4fo A larger pulse of antigen lakes some clones into the zone of attraction of a different locus of equilibrium. A new equilibrium state results for clones that return to their original loci of equilibrium. Parameters again as for figure 45. 96 We have thus shown that our model displays many of the characteristics of the physical immune system's stable steady states (SSS) such as the presence of multiple SSS, memory, and the presence of the suppressed state which may contribute towards the S/NS discrimination. Our model even offers a glimpse of state switching, even though it does not include some of the physiologically important details. Future efforts on this model will concentrate on quantifying the model, introducing the interleukins, and modelling T and B cells as separate populations. This final modification will also necessitate the introduction of very important molecules known as the M H C antigens. This will hopefully lead to more detailed predictions and more interaction with experiment and thus provide us with an opportunity to subject our hypotheses to verification. The questions of the S/NS discrimination and "epitope/idiotype" competition will also have to be addressed both in this model, and within the larger family of network models. Such efforts, together with improved experiments wil l hopefully raise the veil away from the subject of immuno-regulation and reveal the deeper layer of truth that presently reveals to us only its shadow. Conclusion We have presented several network models depicting various aspects of the immune system. Firstly we developed a model for the activation of the A cell or immune accessory cell as a function of certain extracellular conditions, namely the concentration of specific T cell factors and monoclonal antibodies directed against these factors. The factors were assumed to bind to receptor molecules on the surface of the A cell with the receptors being subsequently cross-linked by the binding of antibody to the factor-receptor complex. We sought to determine the valency of the specific factor receptor and to this end, we used the law of mass action to calculate the equilibrium distribution of complexes on the A cell-surface. We had hoped that qualitative differences might exist between the case of a bivalent receptor and the case of a monovalent receptor and that these might permit the distinction to be made on the basis of indirect measurements of activation, such as IL-1 secretion. It was found that such a distinction could be made based on on the F W H M of the Gaussian-like cross-linking curves which differ by a full order of magnitude between the two cases, provided one had previously obtained accurate values for the equilibrium constants which characterize the system and provided that activation and IL-1 secretion is a linear function of cross-linking. A distinction could also be made if the equilibrium constants are such that substantial one receptor bridge formation takes place (one antibody molecule bound at both ends by the same receptor). This one-receptor bridge formation only takes place for the case of the bivalent receptor, and 9 8 it presents itself in the cross-linking curve in a very distinctive manner. In the event that neither of these possibilities manifest themselves, we also presented ways in which one might go about making a direct measurement to resolve this question of receptor valency, and how in this case the calculation we did could be used to further analyze cellular activation. The second network model which we presented was an ecological competition model of steady state lymphocyte populations. As background to this model we discussed the underlying assumptions, the limitations and the implications of network theory in general. We then developed a specific network model, the symmetrical network theory. This model was analysed numerically by integration of the differential equations, and shown to provide a qualitative picture of the immune system's stable steady states, and even offer a glimpse at state switching. Ultimately, it remains to be seen to what extent these models and network models in general have to be modified in order to provide a full picture of the immune response and the self/non-self discrimination (and even if they can). What is clear is that network theory is an elegant, quantifiable attempt at solving the riddle of immune regulation. Since we do not have access to 'God's thoughts', we can only write down what seems to us to be the best answer, our 99 best attempt at quantifying and representing nature. It is in this spirit that this work is presented. 100 Appendix 1 Program I m m u n e ; T h e N u m e r a l I n t e g r a t i o n R o n f i n P VAR mint, tota1_factor : real; x, y, z : array [1..dim,1..max] of real; K, L, R : array [ 1. .dim,1..dim] of real; stim, f, timeinc, e: array [L.dim] of real; t, m, n : integer; ab.lmatrix, rmatrix, kmatrix, stimvector, initial : text; d11, d12. d13, d14, d15, d16, d17, d18, d19, d20,efile : text; d1, d2, d3, d4, d5, d6, d7, d8, d9, dIO, final_position : text; e2. e3 : real; FUNCTION pow(x,y:real)-.real; BEGIN IF (xoO.O) THEN pow:=exp(y* In(x )) ELSE IF (y<> 0.0) THEN pow:=0.0 ELSE pow:=1.0; END; PROCEDURE external in; VAR xi. yj, zk: integer; BEGIN reset(kmatrix,'fi le=K '); resettinitial.'file=startvector'); reset(st imvector,'f ile=nost1mvector'); FOR x1 := 1 TO dim DO BEGIN PROGRAM immune; CONST delta = 0.002; dim = 20; max = 5000; k3 = 0.100; k4 = 2; s = 12; c2 = 13.0; c3 - 30.0; stimstep =10; c =0.0000; (dimension of network) {number of integration steps) (IgG killing) (non-spec death) (source) (effectivity threshold(50)) {effectivity threshold(2)) (number of stimulus steps) (strength of perturbation) FOR yj := 1 TO dim DO READ(kmatrix,K(x1.yj]); REA0LN(1nit1al,x[xi.1]); READLN(stimvector,st im[x1]); END; (compute static similarity coefficients) { FOR x1: = 1 TO dim DO FOR yj:=1 TO dim DO BEGIN 1 0 1 Lfx1.yj):=0; FOR zk:=1 TO dim DO Ux1(yjJ:=L[x1.yj]*K[x1,zkl'K[zk,yj]; R[x1,yj]:=Ux1,yj]+K[xi.yj]; END;) closeUmatr Ix); closet Initial); close(stlmvector); END; PROCEDURE externa lout; VAR outInd Integer; BEGIN rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr rewr END; te(d1,*f1le=od1 te(d2.'f11e=od2 te(d3.'f11e=od3 te(d4.'f1le=od4 te(d5,'f1le=od5 te(d6.'f1le=od6 te(d7,*f11e=od7 te(d8."f 1le=od8 te(d9.'f1le=od9 tefdlO,"f 11e=od10 te(d11,"f1le=od11 te(d12.'file=od12 te(d13,1fi le=od13 te(d14.'f1le=od14 te(d15,'f1 le=od15 te(d16,*f1le=od16 te(d17,*f 11e=odl7 tetd18,'f1le=od18 te(d19,'f1le=od19 te(d20.'f11e=od20 tefefHe,'fi1e=effectlvltie '); te(f1na1_posi t ion,1f1le=eqloci'); PROCEDURE dataout; VAR outtime, 1 : integer; BEGIN FOR outtime := 1 TO max DO (output data-only 1 point In 10 so as to minlmze size of data files} IF (outtime MOD 10) = 0 THEN BEGIN WRITELNJdl.xfl.outtlnel.yf1.outtime]); *WITELN(d2.xU.outt 1ae] ,yf 2,outtime]); WRITELN(d3,x[3.outtime],y(3,outtIme]); WRITELN(d4.x{4,outt1meJ,y(4.outt1mel); *RITELN(d5.x[5.outt1meJ.yt5,outt1mel); NRITELN(d6.x[6.outt1me],y(6.outt1me]); 102 WRITELN(d7,x[7,outtime),yI 7,outtime]); WRITELN(d8,x[8,outtime], y[8,outtime]); WRITELN(d9,x[9,outtime],yt9,outtime]); WRITELN(d10,x[10,outt ime],y[10,outt ime]); WRITELN(d11,x[11.outtime],y[11,outtime]); WRITELN(d12.x[12.outtime].y112.outtime]); WRITELN(d13,x[13,outtime],y[13,outtime]); WRITELN(d14,x[14,out time],y[14,outtime]); WRITELN(d15,x[15,outtime],y[15.outtime]); WRITELN(d16,x[16.outt ime],y[16,outtime]); WRITELN(d17,x[17,outtime],y[17,out time]); WRITELN(d18,x[18,outt ime],y[18.outtime]); WRITELN(d19,x[19,outtime],y[19,outtime]); WRITELN(d20,xI20.outtime],y[20,outtime]); END; FOR 1:=1 TO 20 DO WRITELN(f ina1_posit ion,x[i.max],y[ i ,max]); END; PROCEDURE calculate.!.; VAR 1,j,zk : integer; num,denom:rea1; BEGIN FOR 1:=1 TO dim DO FOR j:=1 TO dim DO BEGIN num:=0.0; denom:=0.0; FOR zk:=1 TO dim DO BEGIN num:=num+K[1,zk]*K[zk,j]*x[zk,t]; denom:=denom+(K[i,zk]+K[zk,j]-K[i,zk]*K[zk,j]) •x(zk.t); END; U[i,j1:=num/denom; R[i,j]:=K[i.j]*L[1.j]; END END; PROCEDURE principal loop; VAR 1. j, n : INTEGER; IC, JC, enorm ,1_1ike.j_like: REAL; BEGIN calculate_L; FOR 1 := 1 TO dim DO BEGIN yti.t] := 0; FOR j := 1 TO dim DO yM.tl := yfi.t] • x[ j, t ] 'KI i. j]; END; 103 total_factor:=0; FOR 1:= 1 TO dim DO (calculate the Tab cone'n=factor concentration) BEGIN •111:= 0; FOR j ;s 1 TO dim DO e[i] := e(i)*R[1,j]*x[j.tJ*y(j . t ]; total_factor:=tota1_factor*e[i J; END; IF (t=max-1) THEN FOR 1:=1 TO dim DO BEGIN 1_11ke:=0.0;j like:=0.0; FOR j:=1 TO dim 00 BEGIN - . 1_11ke: = 1_1ike*L( 1. j)*x( j.t)*y[ j.t); , j_Hke: = j_like*K(i. j)'x( j,t)'y[ j.t]; END; (write out i factor, j factor to compare and study steady S X31 CS} WRITELNfefile.i.'i, anti-i factor = \ i 1 ike,','.j_like); END; (renormalise average factor concentration Rxy by multiplying by an envelope function.Set average factor concentration at the midpoint of the two thresholds) enorm:=dim*(c2*c3)/(2*total_factor); FOR j:=1 TO dim DO e[j):= e(jJ • enorm; IF (t=max-1) THEN WRITELNtefi 1e,'final enorm=\ enorm); IF ((t-1) MOD 10)=0 THEN (print out the Tab conc'ns every 10 steps so as to follow the time evolution of the system) BEGIN FOR j:=1 TO dim DO WRITE(ef11e.j:3,e(j]:7:2); WRITEtef 1 le,'enorm=',enorm:7:2); WRITELN(efile); ENO; FOR 1:=1 TO dim DO BEGIN (determine the level of suppression with respect to the thresholds, set effectivities) IF e[1] < c2 THEN e2 := 1 ELSE e2 := 0; IF e|i] < c3 THEN e3 := 1 ELSE e3 := 0; (compute time evolution of system by integration. First compute Dx/Dt, then calculate delta t so that max(delta x/ x) < delta, which is a constant. Finally, compute delta x = Dx/Dt * delta t) (compute Ox/Ot) IF t < stimstep THEN f(1]:=8-xl1.t)*(y[i.t]*e2*k3'sqr(y[1,t])'e3*k4-c*stiml1]) 104 ELSE f [ 1] :=s-x[i,t]*(y[1.t]*e2*k3*sqr(yM.t])*e3+k4); (compute delta t) IF f[i] = 0 THEN timeincM) := 200 ELSE timeincM! := abs( x( 1, t)'del ta/f 11J); END; mint := t imeinc[1]; FOR i := 1 TO dim DO IF timeinc[i] < mint THEN mint := timeincM]; (compute delta x) FOR i := 1 TO dim DO xM . t +1 ] :=x[ i. t ]+mint * f I i ]; END; ( MAIN PROGRAM B L O C K ) BEGIN externa 1 in; externalout; FOR t := 1 to (max-1) DO begin pr inc ipa1 loop; end; FOR m := 1 TO dim DO BEGIN y(m,max] := 0; FOR n := 1 TO dim DO y[m,max] := y[m,max] + k[m,nJ*x[n.max]; END; dataout; END. ( program } 105 Append^ 2 A Survey of References to T cell Factors in the Literature: 1985-present Reference T cell-factors antigen-specific A . P N A S 86:3758. Zheng et al B.EJI 15:282. 15:351. 15:773. 15:873. 16:198. 16:252. 17:575. McCaughan et al Castalogni et al Simon et al Sujimura et al Morimoto et al Callard et al DeSantis et al antigen specific suppressor cells x x x x x antigen specific suppressor cells antigen specific suppressor cells C . Ann. Inst. Pasteur 137D:391. Rawat I et al 138:815. Dieli et al x X X X D. J . of Immunol. 139:346*. Ferguson et al 139:2130. Gulwani et al 139:2629. Steele et al 141:64. Pierce et al 141:2206. Miller et al 143:66. Vandebriel et al 143:818. Taub et al 143:3909 Iwata et al x x X X X X 106 Glossary of Terms A or accessory cell-a necessary accessory to the immune response-colloquial term for all non-specific myeloid cells that participate in the immune response. aliasing-an effect produced when sampling rate leads to high frequency effects appearing as low frequency ghosts. anti-as in anti-i or anti-factor. Refers to an antibody or cell directed against the entity in question (eg. clone i or specific factor) anti-immune state-the converse of the immune state in the 2-dimensional network: low x+ and high x.. antibody-dependent cellular cytoxic i ty ( A D C C ) - c e l l u l a r immunity that requires the presence of antibody. antigen-a virus or bacteria that 'invades' the host's immune system. antigen specific T cell factors-proteins secreted by T cells when they are activated by the presence of antigen. We model these proteins as having an antigen binding end and a constant end which binds to receptors on the A cell surface. B cell-the cells responsible for antibody secretion. clonal anergy-the process -which occurs during development and which may or may not be reversible - whereby certain B cell clones are rendered unresponsive (anergic). clonal selection-the theory of Ehrl ich which is the cornerstone of Theoretical Immunology. The presence of the antigen exerts a positive selective force on cells which are specific for the antigen, clone-a group of genotypically and phenotypically identical cells, c o m p l e m e n t - a serum protein that lyses antibody-bound cells, c o m p l e m e n t a r i t y - r e f e r s to a 3-dimensional chemical lock and key. conformational change c o n f o r m a t i o n a l change-a change in the orientation of the carbon backbone of a large organic molecule. constant end-the part of a factor or antibody molecule that is not antigen specific. cross-linking of receptors-refers to the aggregation of receptors on the surface of a cell by an external agent. c y t o k i n e s - a broad class of soluble proteins (including the interleukins, interferons,...) secreted by white blod cells which greatly enhance the immune response c y t o p h i l i c - h a s an affinity for cells. D a l t o n - the mass of 1 hydrogen atom. 107 De Broglie wavelength-an indicator of the extent of the wave-like behaviour of a particle or object. differentiation-refers to the process whereby blood cells, which all arise from a common parent cell develop into different mature cells, effector function-refers to the function carried out by a particular cell . epitopes-surface determinants on a molecule or cell , equilibrium constants-the ratio of forward and reverse rate constants in a chemical reaction. It is a constant by the law of mass action. equilibrium loci-refers to the set of points which are stationary for d X a particular system - ie. if X describes the system, then - j - ^ = 0. Fab fragment-refers to an arm of the Y-shaped antibody molecule which contains one of the variable regions. A n antibody has two Fab segments. factor receptor-refers to the hypothetical molecule on the A cell surface which acts to bind the specific factor to the cell. Full Width Half Maximum (FWHM)-the width of a curve where the ordinate is at half of its maximum value. homeostasis-the maintenance of internal stability via feedback mechanisms. idiotypic interactions- interactions between the variable regions of receptors or immunoglobulins. IgG antibody, IgM antibody-particular families of antibodies. A l l antibodies within the same family have the same constant region, immune state-refers to the state in the 2-dimensional model populated by clones which have killed off the anti-idiotypic clones. These survivor cells are memory cells, which will respond vigorously to antigen at next encounter. immune system memory-refers to the ability of the immune system to retain cells which are primed for a certain antigen so that in future encounters the system can deal with the invader much more efficiently. immunocompetent-participating in the immune response immunoglobulin-antibody. interleukin-proteins secreted by B cells and A cells that stimulate lymphocytes to divide. ligand-that section of a molecule to which a receptor binds, lymphocyte-a white blood cell . macrophage-a subset of the myeloid cells that is crucial to the immune response. 108 mast cell-a basophil. Subset of the myeloid cells. Responsible for allergic reactions. membrane--refers to the membrane that surrounds all cells memory state-see immune state. M H C - t h e major histocompatibility complex, refers to the D N A which encodes for the proteins that serve as a marker of 'self, and colloquially to these same proteins. monoclonal antibodies-antibodies which are specific for a certain antigen. monotonic increasing function-a function that does not decrease as the dependent variable increases. myeloid cells-those blood cells that are not lymphocytes, network theory-Jerne's theory that a hosts own V regions are antigenic, and give rise to immune responses which regulate the behaviour of the immune system, non-specific accessory cell-see A cell phenomenology-the attempt to describe observed phenomena. Approach whereby theories are developed to explain experimental results. rate constant-number denoting the rapidity of a chemical reaction. It is a constant by the law of mass action. receptor-a protein structure that is anchored in the cell membrane and extends out from the cell into the medium surrounding it. It is one of the methods whereby the cell communicates with the world outside its membrane. saturation-refers to a state where binding is at a maximum, secondary antibody response-the rapid and potent antibody response of an immune animal. secondary antibody-antibody used to amplify signals in exper iments . self-consistency equation-refers to an equation which cannot be solved exactly due to its complexity. It is used to search for solutions satisfying RHS=LHS. self-avoiding random walk-refers to a walk in which the direction of a step does not depend on the previous step, except that the path cannot cross itself. self/non-self discrimination-the process whereby the immune system discerns invader from native, responding to the former and not responding to the latter. similarity coefficient matrix-a matrix whose entries provide a quantitative measure of the extent to which two variable regions are similar in the context of a particular immune repertoire. 109 somatic mutation-a mutation in the D N A of non-reproductive cells - ie. a mutation that will not be passed on to offspring. specific factors-see antigen specific T cell factors. specific T cell factors-see antigen specific T cell factors. stable steady states-refers to a state of the system which does not change after a certain period of time, and one to which the system will return if just slightly perturbed. state switching-refers to a process whereby the system which was in one steady state evolves into another steady state after being per tu rbed . steric repulsion-repulsion between two chemical entities due to the proximity of their electron clouds. suppressed state-refers to the unresponsive state in the 2-dimensional model. symmetric matrix-a matrix K that has Ki j = K j i . T cell-the cells responsible for cellular immunity. T independent antigen-refers to antigens that are eliminated without the participation of T cells, ie eliminated by B cells only, thymic education-the process whereby T cells are rendered unresponsive to self. A famous set of experiments demonstrated that this occurs in the thymus organ in mammals [ 45 ] . V regions-see variable regions valency-in this work, the word has the sense of: the number of chemically active sites on the molecule - simply put, the number of receptors. variable end-the part of a factor or antibody molecule that is antigen specific. variable regions-regions of receptor or immunoglobulin encoded by segments of D N A that are subject to great variability due to genetic rearrangement, hence regions that can be made up of any of a vast number of gene products, great binding potential veto cell- a cell which can exert a veto on the immune response. A possible mechanism for self/non-self discrimination. virgin state-refers to the unchallenged or naive steady state in the 2-dimensional model. 110 Bibliography [I] Kimbal l , J . W. 1986. Introduction to Immunology. Macmillan. New York , ppl65-94. [2] Ibid, pp 168-94 [3] Kimbal l , J . W . 1986. Introduction to Immunology. Macmillan. New York , ppl64-5 . [4] Associative Recognition: Bretscher, P. and Cohn, M . 1970. A Theory of Self-Nonself Discrimination. Science 169:1042. [5] Associative Recognition: Cohn, M . 1989. The A Priori Principles Which Govern Immune Responsiveness. In: The Cellular Basis of Immune Modulation. Proceedings of the 19th International Leucocyte Culture Conference. Edited by Kaplan, J . G . Alan Liss. New York, pp 11-44. [6] Network Suppression: Miller, J . F. 1980. Immunoregulation by T lymphocytes. In: Strategies of Immune Regulation. Edited by Sercarz, E . E . and Cunningham, A . J . pp 63-76. [7] Cel l Cascade and Antigen Elimination: Hood, L. E . , Weissman, I. L. , Wood, W . B. and Wilson, J . H. 1984. Immunology. Benjamin/Cummings. Menlo Park, C A . pg 368-9. [8] Network Suppression: Jerne, N. K. 1974. Towards a Network Theory of the Immune System. Ann . Immunol. Inst. Pasteur. 125C:373. [9] Richter, P. H. 1975. A Network Theory of the Immune Response. Eur. J . Immunol. 5:350. [10] Segel, S. and Perelson A . 1989. Shape Space Analysis of Immune Networks. Submitted to: Theorectical Models for Cel l to Cel l Signalling. Edited by Goldbeter, A . Academic Press. New York. [II] Kaufman, M and Thomas, R. 1987. Model Analysis of the Bases of Multistationarity in the Humoral Immune Response. J . Theor. B io l . 129:141 [12] De Boer, R. J . 1988. Symetric Idiotypic Networks: Connectance and Switching, Stability, and Suppression. In: Theoretical Immunology, Part 2, SFI Studies in the Science of Complexity. Edited by Perelson, A . Addison Wesley, pp 265-289. [13] Hoffmann, G . W. 1975. 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Qualitative Dynamics of a Network Model of Regulation of the Immune System: A Rationale for the IgM to IgG Switch. J . Theor. Biol . 94:815. [42] Cohn, M . 1989. The A Priori Principles Which Govern Immune Responsiveness. In: The Cellular Basis of Immune Modulation. Proceedings of the 19th International Leucocyte Culture Conference. Edited by Kaplan, J . G . Alan Liss. New York, pp 11-44. [43] Cohn, M . 1988. Idiotype Network Views of Immune Regulation: For Whom the Bell Tolls. In: Idiotypy and Medicine. Edited by Capra, J . D. Academic Press. New York, pp 321-399. [44] Langman, R. E . 1987. the Self-Nonself Discrimination is Not Regulated by Suppression. Ce l l . Immunol. 108:214. [45] Cohn, M . 1981. Commentary: Conversations with Niels Kaj Jerae on Immune Regulation: Associative versus Network Regulation. Ce l l . Immunol. 61:425. [46] Kimbal l , J . W . 1986. Introduction to Immunology. Macmillan. New York, pg 352 [47] Nossal, G . J . V . 1983. Ann. Rev. Immunol. 1: 5. [48] Miller, R. G . 1986. Immunology Today. 7:112. [49] Hood, L . E . , Weissman, I. L. , Wood, W . B. and Wilson, J . H. 1984. Immunology. Benjamin/Cummings. Menlo Park, C A . pg 375. 


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