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 degree freely at the available copying of department publication this of in partial fulfilment University of British Columbia, for this or thesis reference thesis by this for his thesis and scholarly or for her Department The University of British Columbia Vancouver, Canada (2/88) I I further purposes gain the shall requirements agree that agree may representatives. financial permission. DE-6 study. of be It not that the be Library an advanced shall permission for granted is for by understood allowed the make extensive head that without it of copying my my or written ii 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 monovalent receptor between the bivalent receptor case and case. This determination can be made the 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 A manner. second network model described as an ecological competition model of steady state lymphocyte populations is presented. T h i s model, known numerically as the symmetrical network theory by integration of the differential provide a reasonable qualitative picture is analysed equations and shown to of the immune system's stable steady states, and offer a glimpse of state switching. iii Table of Contents Abstract ii Acknowledgement iv Introduction Chapter 1 1 Cross-linking in the A cell system 4 1.1 Introduction 4 1.2 C r o s s - L i n k i n g Calculation 11 1.3 Cross-linking Curves Chapter 2 N-Dimensional 27 Network Dynamics 2.1 Introduction to the Model _ 5 0 — 5 0 2.2 Discussion of the Model 65 2.3 Analysis of the M o d e l 7 3 Conclusion 97 Appendix 100 Glossary of Terms 10 6 Bibliography 110 iv Acknowledgement I want first to thank my supervisor Geoff Hoffmann for his superhuman patience and his creative suggestions. Thanks are also due to Julia L e v y , whose understanding and clarity helped me to understand. Tracy K i o n and R o b Forsyth endured innumerable questions, and M i k e L y o n s 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 various aspects of the immune secrets of life with a few models depicting system. T o attempt to unlock the differential equations and a computer appear absurd to some, especially to biologists. However, more familiar completely natural. H o w and predict complexity to those with-the tools of physical science, modelling is else are we to understand the behaviour the system as a whole, isolate the important interpret experimental of the immune interactions, results? Indeed, given the system, and of cellular interactions to study the behaviour of the entire immune of and general, this modelling approach is perhaps the best tool we available may in have 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 the immune response is clonal selection [1], whose receptors are specific for a particular a process whereby clones antigen or invader selected from the body's large and diverse repertoire are of clones by the presence of that antigen. These selected clones divide and differentiate and the resulting cells, whose receptors are identical those of their original parent cells, are responsible for the [2], does not however to immune response against the invader. T h i s theory, which has been borne by experiment of out 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 the test of withstood verification, though some have certainly experimental 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] variable regions (or V in 1974. Jerne's hypothesis was that the 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 regulation responsiveness, and perhaps even for of immune fundamental that is responsible for self/non-self discrimination. This may seem at the the 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 lends itself have regions. T h i s matrix of interactions marvelously to modelling, and several network models 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 is important population system. In particular, it that such models account for the existence of stable states, immune discrimination since these system memory are key and self/non-self experimentally observed phenomena. Hoffmann dubbed and his collaborators have developed a theory the symmetrical of these important into mathematical network theory, properties. The theory model form. However, which [13-17], demonstrates some has been gradually cast 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 i n 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 not familiar with the jargon, a glossary has been included. those 4 Chapter 1; U Cross-linking in the A cell system 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). T h e 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. T h e second theory (see figure l b ) 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: T h e 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. W i t h 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 This is based firstly cells. 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 figure 2). system cells (see Secondly, there is considerable experimental evidence in favour of the cross-linking model. F i g u r e 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 [18] the B cell domain this experimental and [14] experiment, evidence, as reviewed in is particularly compelling. In the most suggestive which has been repeated in several systems (see [14]), B cells are challenged with monovalent F b fragments which as a indicated in figure 3 are single arms of antibody molecules bearing only one binding site. N o activation of the B cells is observed (figure 6 4a) which is consistent with the cross-linking model since F b a fragments cannot cross-link and therefore However, when antibody the suspension activation cannot activate would necessary activation Figure 3: cell. specific for the F b fragments was added to a was observed (figure 4b). This is again consistent with the cross-linking model, since multivalent antibodies the cross-link bound F b a fragments, anti(F b) providing a the signal. U s i n g 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 b fragment' refers to a single arm of the antibody molecule- a single a 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 is based on the cross-linking hypothesis as the mast cells which [21]. Based on all of this evidence, it has been postulated in the network theory of the immune system [16] symmetric that the cross-linking of receptors is a necessary first signal for the activation of B cell, T cell and myeloid c e l l s . ' T o 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. F i g u r e 4: In B fragments do combination with in a cell stimulation experiments, not activate B cells (a), but anti(F b) antibodies do (b). favour of the cross-linking a b it is found that F b F b fragments in This provides evidence model. a a 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). A s with all myeloid cells, the activation of the A cell has not yet been studied definitively. cross-linking model that we are proposing here is thus purely The justified on the basis of analogy with other immunocompetent cells where there exists solid evidence. Within the symmetrical network theory, the initial stages of immune response are envisioned as follows: the 1) antigen encounters cells whose receptors are specific for certain epitopes on the 2) these T cell receptors are cross-linked by the antigen, T 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 monovalent F b fragments a modelled as, (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 they bind via their which 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] detailed description). for a more 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, concerned only with the above-mentioned first step, namely we A are cell activation. CD ^ T cell factors O Figure S : T h e initial stages of the immune response according to the Specifically, we factor-antigen symmetrical network theory calculate the concentration [16] of cross-linked receptor- 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. experiment, underway of which preparing is currently in our laboratory, macrophagie-factor-monoclonal and then assaying for IL-1 The presence of IL-1 antibody consists suspensions secretion using the standard assay [22]. would be strong evidence in support of the cross-linking hypothesis, especially if with the calculated One such concentration the IL-1 secretion correlates of cross-linked cell-surface aggregates. T h i s latter result would also be highly suggestive of an important role for specific factors, and would provide for this aspect of the symmetrical network theory system. strong support of the immune 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. F o r a chemical equilibrium governed by the equation: A + B * ^ (1) y where k+ is the forward rate constant and k. the reverse, the law of mass action reads: K = r a \ F t 0 [A].[B] = — = constant (2) 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. receptors and antibodies) factors, cell and the rate constants characterising various reactions which built up the aggregate. It the 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 constants supplied by experiment, equilibrium 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 to compare with IL-1 is this total cross-linking which we seek 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. W e do not attempt to model IL-1 secretion here. a-fTaV B &V e a VB R R R R c Figure 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 qualitative of the mathematical experiment model is evident. Whereas a could only determine whether or not IL-1 has been produced, a comparison of the model with quantitative experimental examine data allows the differences us to determine one to probe somewhat between these two deeper and cases, perhaps allowing 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. equilibrium calculation is an 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 approximation). 5. T h e 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. A n y steric effects are included in the constant. equilibrium 14 8. The binding of free factor to bound antibody is characterised by the same equilibrium constant as the binding free antibody of to bound factor. H a v i n g made these approximations, we begin the calculation by defining the equilibrium constants for the binding reactions occur on the cell-surface. First there is the (single receptor which site)- (single factor) binding constant K defined as follows: 2k + 0 + C5> k. k + V R 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 two of binding sites. The second constant is the (single receptor site)-(chain-bound factor) binding constant K , defined as follows: x single 15 (2) 2k x + 2k R e (2) AA R x+ \ R 2 k R x - 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. F o r the monovalent case, this superscript is clearly unneccessary since we can only form 1 and 2 receptor chains. F o r the bivalent case, we have an infinite set of binding constants k (n) (n) x+ (4) However, by the 5th approximation listed above, we have that .(1) = (2) k x+ ~ k x+ ( n ) K (5a) 16 k x0 ~ ) K (5b) k x- ( 2 ) K and so we can define a unique receptor to chain-bound factor binding constant k *x+ ( n ) Kv (6) = k ^ *xNext, there is the (single antibody binding site)-(single factor) binding constant L , which we define by: e 21 (7) F i n a l l y , there is the closure equilibrium constant for the ring, n-membered J , which corresponds to the reaction shown below for the case n 17 AAA R 3+ 6j N o w we are once again dealing with an infinite set of equilibrium constants: Jn Jn + = (8) Jn- 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]: P (r=0) a n 1 d£l (9) n where a indicates proportionality order of 1. W e therefore ring and d is a scaling exponent of the assume that the forward rate constant for closure also scales with the number of steps: 18 Jn+ = UJ J m + n > m (10) where m is an index to be chosen. Given that J f , the equilibrium constant for one receptor rings or bridges w i l l 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, whereby (2\ d Jn+ = n J2+ n £ 2 (11) F i g u r e 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 onereceptor bridge (possible, but unlikely). Higher order rings depend on diffusion of receptors through the cell membrane and so w i l l 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: (13) 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 , L , J i , J2 and d are treated as parameters in the model, and their x values are assumed to come from experiment. H a v i n g defined the equilbrium constants, we now proceed to enumerate all possible cell-surface aggregates and calculate concentrations using these constants and the law their of mass action. W e work first with the case of the bivalent receptor. Consider the set of all n-receptor complexes. A s depicted in figure 8, this set consists of the n-membered ring along with 10 distinct n-link chains. U s i n g the law of mass action, we can express the concentration of the nmembered 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 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 F o r chain #2, rate constants. the equilibrium constant is (14b) Where we have defined F=[ free factor ] W e will also use R=[ receptor ] and A=[ free antibody ]. F o r chain #3, the equilibrium constant is given by free 22 (15b) T h e concentration of all of the other n-receptor aggregates can be calculated in a similar manner. A d d i n g these concentrations, we that the total concentration of n-receptor R n = [chain #1] aggregates, R n find is given by {l+2KF+K F2+4K IJF2A+4K L F A2+8K2L F3A + 2 2 2 2 3 2 4K2L4F4A +4KLFA+4KL F A+4KL F2A^- + 4 K L F A } zn J 2 2 2 2 2 2 3 (16) 2 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: I -4K I K •2K KL X •4KKL n-1 4K KL X X (17a) 24 We thus have that R = N (4K RKL F A) ~* R 2 2 N X {1+2KF+K2F2+4K2LF2A+4K2L2F2A2+8K2L3F3A2+ 4 2L4F4A2 4KLFA+4KL2F2A+4KL2F2A^ + 4 K L F A ) L K 2 2 3 + (17b) W e 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 free I + n F n (18a) n=l oo A T = A free + 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-t4K2L2F2A +8K2L3F3A2+4K2L4F4A244KLFA+ 2 oo 4KL2F2A+4K2L2F3 A } X n(4K RKL2F A) " 2 n 1 x n=l oo +4RKL2F2A^y(4K RKL2F2A) " n 1 x n=l (19) Substituting equation (12) and evaluating the sums using 25 oo Sn(B) 1 1 - = 1 (20a) 1 2 (1-B) n=l 2 and r oo n=l ~ n = (t>(x,z) (20b) z where 0 is the Truesdell function, a standard numerical function [24], we R'p find = that R {1+2KF+K2J^+4K2LF2A+4K2L2J^A2+8K2L3F3A2+4K L4F4^ 2 +4KL.2F2A+4K2L2F3A) ^ , (1-4K RKL2F2A) + 4RKL2F2A{^- 2 1 0 J 2 J ) D _ 1 2 2 X 2 D _ +-^— 1 h <t>(4K RKL2F2A,d) X (21) G i v e n values this equation, for the together various parameters with the factor and total concentrations, and antibody equations, can be solved in a self-consistent manner, conservation the result being the concentrations of free receptors, factor and antibody (R, A , F ) . W i t h these, we aggregated can calculate the equilibrium concentration receptor complex, and in particular, concentration we of cross-linked receptors, with which of any can calculate we are the most concerned. R -Rj T Fraction of cross-linked receptors = f j c = — ^ — (22) 26 For the monovalent case, the situation is much simpler, since there are only 5 possible cell surface complexes: A R R 6 CD > e A' R R .A 6 R R Thus, we have only Rj = R + RKF + 2RKLFA + 2 R K L F A R = K R KL F A 2 2 2 2 (23a) 2 (23b) 2 X and the conservation equation (there are once again companion equations for conservation of factor and antibody which we do not use) R T reads: = R + RKF + 2RKLFA + 2 R K L F A + 2K R K L F A 2 R -R! 2R Rj Rj T cl = 2 X and for cross-linking we f 2 2 2 (24) get 2 (25) 27 1.3 We Cross-linking Curves 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 X = F + n n F ( i 8 a ) ( i 8 b ) n=l oo A T = A + X n A n n=l (A and F represent free antibody and factor respectively and F A refer to cell surface complexes with n antibodies or factors), N N and 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 o f any particular compound, i n 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 estimated i n the absence of direct experimental can be measurements. K and L are both receptor-ligand equilibrium constants and as such they may typically be of the order of 1 0 - 1 0 8 9 M " , but could also be 1 28 as low as IO M * 5 A or as high as I O 1 cell has approximately 1 1 M [23]- If we assume that the _ 1 the same concentration of receptors as other immunocompetent cells, for instance B cells which bear « receptors on their » 400 u,m of the order of 1 0 / u . m 2 2 2 surface [24], then R j and in the range 10 5 would be typically 10-10 /u.m . 3 2 Due to the adherent nature of the A celL we expect that K , J i and h x 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 the non-dimensional parameter on a geometrical model [31] x is chosen so that K R x *s o f order unity [31-34]. Based x where the Teceptors were treated as hard discs and the hapten as a smooth path segment, one of these authors estimated K =10 x - 1 u.m 2 which is consistent with this ansatz given the above range of values for R T - W e will take K R T = 10 as a X typical value, and allow a range of 1-100. 29 The ring closure constants are more difficult the difficulty estimated Now to measure and given of direct cell-surface measurements, are probably best from equilibrium constants for ring closure in solution. the factors we are working with are poorly characterised at best and such information is not available therefore we must estimate. W e take J i , J 2 e [0.01,1000] which is consistent with other linking models where solution data is available [31], previously mentioned, the J [34]. cross- As are dimensionless. n N o w for the bivalent case, we first consider the simplified model in which J i = J 2 = 0 . W e also define and substitute r _ Ri ~ R x R " R P (29a) T and a = (29b) -4K R KL2F2A X T where P = {l+2KF+K2F244K2LF2A+4K L2r^ 2 4 K L 2 F 2 A + 4 K 2 L 2 F 3 A } (30) Note that Rn+1 = Rn ( 4 K R K L 2 F 2 A ) = R X n (a r) (31) 30 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. W i t h these substitutions, the conservation equation reduces to: 1 = (1-ocr) 2 (32) which implies that (l+2oQ- V(l+2a)2-4q2 1 " 2a2 (33) T h e 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 condition reduces to: r-->l as K A ->°o (==> o>->0) (34) this 31 R R a V R R F i g u r e 9: A t 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. T h e above substitutions also give rise to a particularly simple equation for total cross-linking: for " 1 r (35) 32 and it is thus clear that r represents the non-cross-linked fraction. F o r the monovalent case, the receptor conservation equation is a quadratic which is readily solved for R: (1+KF+2KLFA+2KL F A) 2 R % 2 4K R KI2F2A = X + T -\j (1 + K F + 2 K L F A + 2 K L F A ) + 8 K R 2 2 2 X T K I ^ A 4K R KL2F2A X ' T (36) (here the positive root is clearly the only possible solution) and for cross-linking, f c l = 2 K X we R ^ 2 have K L 2 F 2 A (37) W i t h 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 mentioned above, we rather show monovalent receptor cross-linking. A s display the cross-linking information than as surfaces since experimental presented in this manner, the experiment manipulation of one variable. as curves data is commonly usually consisting of the 33 10n log (K A) b Figure 10: Graphs showing (a) bivalent and (b) monovalent receptor cross-linking for K=L=10 , Ji=J =0, F=10-7. 8 2 34 1.0n log (LA) b Figure 11: C r o s s - l i n k i n g versus antibody values of the factor concentration, (a) concentration monovalent receptor. Parameters as in figure In figure for bivalent receptor 10, with various (b) K RT=100. 12, we depict the behaviour of the model under X 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 R T is fixed at 100. Ji=J2=0. In (a) and (b), L=108. In (c) and (d), K=10 . Note the different abcissa scale in (c) and (d). X 8 36 Figure 12 (cont'd^ caption on previous page 37 b Figure 13: The dependence of cross-linking on K F - 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 dependence of f i c 10 through 13 indicate the complicated 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 but based on the which analysis is more easily accomplished), similarity between the bivalent and monovalent equations, we expect our analysis to be suggestive of the behaviour of the monovalent equations as well. W e 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). G i v e n that f i = l - r ( a ) is a monotonic increasing function, as illustrated c in figure 14, it follows that the maximum in cross-linking occurs at the maximum value of a . N o w in the expression for a , (29b), K R T X equation appears only as a scale parameter and is thereby proportional to the maximum in a cross-linking. It and related to the maximum does not however affect in the ( F , A ) co-ordinates at which the maximum occurs (as we would expect from figure 10, 39 where the position of the maximum is constant with respect to variation of the K R T X o.o I • 0 Figure parameter). i i 2 i 4 | 1 6 | i | 8 10 alpha Cross-linking as a function of a 14: i for the bivalent case with Jl=J =0 2 Secondly, with respect to the behaviour of the curves under adjustment argument. of the parameters, we can provide a loose plausibility Since f i is a monotonic increasing function of a , the c maximum value of f i will occur at the maximum value of a . c Furthermore, it is clear that the greater the width of the curve a ( K , F , A , K R j ) the greater the F W H M of the f i curve. W e are x c L, therefore interested in the behaviouT of the function a ( K , L , F , A , K R T ) as X defined by equation (29b) a = where ^4K R KL2F2A X T and (30): (29b) 40 p = {l+2KF+K2F2+4K2LF2A+4K2L2^ 4KLFA+4KL2F2A+4K L F A } 2 2 3 (30) Figures 10 through similarity between 13 reveal the bivalent one immediate feature: a strong and monovalent cases. Not only do the curves have the same shape, they also behave in much the same way under surprising (33) adjustment given the of the parameters, similarity between neither the and monovalent receptor equations (36), reveals however that there is actually fact bivalent (37). A being receptor (29b), closer look a substantial difference terms of the width of the curves (see figure in 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] reasonable first guess) then the F W H M and is in any case a 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 based on the width of the IL-1 curve and on figure 15. valency 41 6 -i CD £ O o> c _c r <o S2 o o o 5CO 'c 3 u E 2 *c cd . o oo> 3 f 4- • B • • Monovalent Bivalent 3- FW I • 1 -5 1 I • 0 5 10 log (K) -•—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 l o g ( K ) - l o g ( K . ) = l o g ( K + / K . ) and is therefore dimensionless. The other equilibrium constants are as in figure 10, with K R T = 1 0 0 . + X In order for this to work, we would of course need to have accurate values for all of the equilibrium constants, including the parameter (see K RT X 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 R T , one can easily make a distinction between the two cases. X If, however, J * 0 , then the bivalent curve acquires a different n character, as illustrated in figures 17 and 18. For this case, the selfconsistency 1 equations reduce to (f-2d-lj ) = 2 + 2 d - l J <l>(ar,d) 2 K X R T (42) and cross-linking is given by = 1-r ( 1 + ; K L 2 F 2 A Ji) 2 (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 <|>(x) = (44) -In (1-x ) 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 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 . The J2=0.01 and J2=10 curves cannot be resolved from each other at this scale. 2 1.0 n log(KA) Figure 18: Cross-linking for various values of J j . The curves are more sensitive to J i 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 45 curve then assuming simple proportionality between IL-1 and linking, we can conclude that one-membered rings are being crossformed 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 w i l l 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 measurements of the A data and direct cell receptor will be required to determine valency. In making a direct measurement, it will of course be helpful to know exactly what one expects to see on the cell surface. U s i n g the above model, we can in fact predict which aggregate will have the highest concentration. T o this end, we define oo I nR n "=j RT (45) and Rn j " Rn n f 46 (46) which represent the fraction of receptors which are in (j greater)-membered aggregates *2+ f = aggregates respectively. and the fraction in and (j)-membered Clearly, d (47) and the rest of the fj+ can be expressed in closed form by using equation (17b). F o r instance, f 3 + = f c i - 2 a r 2 - ^ - 2 r 2 a (48) an expression that we use to plot the fraction of one, two and threeplus receptor aggregates, f j , f If we are therefore or detect 2 and m figures 19 and 20. unable to draw conclusions based on curve width one-membered ring formation in our IL-1 measurements, w i l l then be necessary to make direct measurments of cell-surface aggregates in order to make the bi/mono discrimination. Such measurements would admittedly be difficult, but they have direct the added benefit of providing data which could be used to develop models of IL-1 linking secretion, since one could relate IL-1 measurements. data to cross- it 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 antibody at saturation to estimate the epitope concentration from 'standard counts per number of epitopes' curve. In would have to modify secondary a our case, we 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) Figure A s we vary A continuously, we see the emergence of 19: different cross-linked aggregates, starting with one receptor aggregates, and then two, and so on. A s A increases even further, the equilibrium we conditions favour eventually re-emerge as in figure with fewer antibodies per one receptor aggregates. 10 (K=L=10 ; F=10-; Ji=J =0), 8 7 2 aggregate, with Parameters K R =100. X and T 48 W i t h this experimental method, we could determine R T and the number of antibodies bound to the cell at saturation A verification, A m a x - For x could also be estimated from electron microscope images of gold-labelled antibodies [36]. proportionality m a between The coefficient of the two would then be the valency of the receptor. log(KA) F i g u r e 20: Same as figure 19, but with J i = 5 0 0 . T h e 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. 49 -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 W i t h the receptor valency known, IL-1 data could be compared with the above model with an eye towards developing models of secretion. This could provide valuable information expression. of the IL-1 IL-1 concerning the gene as a function of external conditions. O f course it w i l l be necessary to consider other possible A cell activation mechanisms and other experimental (including experimentally tests of our A a determination evaluate their effectiveness, and cell model must be performed of the equilibrium constants), but have made a beginning here. A n d "Were (we) (this) book would never be published" [37]. we to await perfection, 50 Chapter 2; 2.1 As Introduction mentioned N-Dimensional Network Dynamics to the Model in the introduction, several authors have models based on Jerne's Network that of P. Richter[9]. interesting of the V developed Hypothesis. The first of these was His model, while it demonstrated many physical features, relied on a distinction between region that recognized antigen (the paratope) the part and the part that was recognized by other receptors (the idiotope). T h i s distinction, which was in fact part of Jerne's original hypothesis, has never been observed in characterization therefore experiments of receptor V despite detailed regions, and Richter's model can be discounted. Contemporary network models include Hoffmann model as well as the models of Segel and Perelson Kaufman and Thomas [11], and DeBoer [12]. the [10], 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 Thomas' model is astute but relies on purely go. Kaufmann stimulatory network interactions. This is difficult to reconcile with the fact that interactions experimental are generally assumed to be suppressive, and fact that antibodies in the network will kill and network the anti- idiotypic cells in the presence of complement. Finally, in the D e B o e r models, the author demonstrates that a certain set of assumptions" do not yield a functional network. his assertion that network theory is thereby "reasonable A s evidence against discredited, we would 5 1 offer this latest version of the symmetrical network other 'reasonable assumptions' yield a highly functional In the context of the symmetrical network theory a theory, where network. 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. C l o n e 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 „ o r kills cell B then, provided A and B are of the same cell type, the converse is also true (see figure 22). T h i s , 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 crosslink 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 preserved. 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 w i l l also lead to the death of i cells since i cells w i l l cross-link anti(i) cells, causing anti(i) antibody to be produced. T h e requirement for interleukins necessitates the presence of the cell (which as mentioned in chapter 1 A 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 b fragments as in chapter 1, are a assumed to be secreted by T cells when their receptors are crosslinked. 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 receptor regions (see figure 23), and inhibiting idiotypic blocking the associated 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. Figure 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 1+ (49c) q=2,3 X+X- 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). T h e 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 . T h e 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 v i a cross-linking by cell receptors or antibodies of the opposite specificity. T h e k 2 term models killing by killer T cells and/or by I g M antibody plus complement. T h e k3 term models antibody-dependent cellular cytoxicity ( A D C C ) and/or killing by I g G antibody plus complement. T h e 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. T h e factors are assumed to inhibit when their concentration exceeds a certain 55 threshold concentration (see figure 24). Ci, C that specify this threshold value, and n i , n determine the sharpness of the thresholds. 2 2 and C3 are constants and n3 are constants that The concentration of factor is taken to be the product of the cell secreting it (x+, x.)and stimulation that that cell receives (taken as proportional concentration of the anti-idiotypic clone (x., to the the the x+). 1e+0 Oe+0 — l 10 <~ 20 30 40 50 Fl Figure 24: Graph showing e i as a function of fi q cases are depicted: n=2 (smooth) q Two q and n=«» (sharp). This two clone model was analyzed in detail in [41] for n for C = 2 0 . and was shown large to have 4 stable steady states under two exclusive conditions: ^ S E < c < c 3 < 2 < E kd r s < c 2 < c 3 < (50a) < vk , (50b) 4 These 4 states are depicted in the phase plane picture below 25) and have the following characteristics: (figure 56 1) the virgin state. L o w 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. T h i s 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. W h i l e 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 model to a greater number of clones. In to extend the a higher dimensional model such as this, a particular clone i w i l l interact with many other clones, the strength of the interaction being dependent on the between V region i and the V region of the other clones. affinity 57 Figure 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 =10, k =100, D = l , S=10, C i = 1 0 , 2 C3=V0.1 3 C=VTo , 2 , n i = n = n 3 = 5 . This choice of parameters corresponds to 2 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 Yi = N X (affinity)ji (population)j j=l = £ KJJXJ j=l ( 5 1 ) 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. W i t h this replacement, the equations read: dx - ~d7 = Qi = e s + kiXiYjen - k2XiYje2i - k3XiYj2e3i - Dxi )xu 1 + Nn q =U.3; i = 1,...,N (52a) i=l,-,N (52b) 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 i Y i (52c) In the latest published version of the model [16], Vi = fj = XiYi we took (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 i n conflict with the 59 symmetry assumption in the sense that it represents inhibition by i factors (factors secreted by clone i) but not by anti(i) factors. T o 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). these inhibition Equation (52d) represents only the first mechanisms and must therefore of be modified to take the second one into account. Figure 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). W e 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). W e _ L i j u.(I n J) " u(I where u let _ J) u.(I n u(I the denominator - J) + u(I n J) J) + u ( J - ( 5 3 ) has been selected to ensure that L y is a number between zero and 1, and that La F i g u r e 27: I) is identically 1. A V e n n 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 u(InJ) = xk: N X ikKkjx K k=l k (54) 61 To calculate u,(I - J) ( u.(J - I) is similar) we sum the product Kik(lKkj) - 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. H(I-J) = N SK (l-Kkj)x k=l ik (55) k We substitute (54) and (55) into (53) and we arrive at the final expression for the similarity co-efficients: N X KikKkjXk Lij= (56) k = 1 N X (Kik Kkj-K Kik)xk k=l + jk 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 j=l j=l j=l (57) Which we simplify by defining the relatedness matrix R: R = L+ K (58) 62 Whereby N = X Rij jYj j=l (59) x T h i s change complete, symmetry This modification however, model was integrated in section 2.3) is once again restored to the model. necessitated another one. When numerically, it was found (as w i l l be the detailed 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 by equation (59). the clones which is brought about 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. e qi = 1 - Equation (52b) A*., sn. n, v ) was modified q= 1,2,3; to read: i = l,...,N (60a) where (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 i n the next paragraph): i ^ f f i i t a i 1 k=l z ( 6 1 ) 63 W e thus arrive at an expression for v : _ = v N (C 2 + C ) 3 (62) N X^Pi 2 k=l W e 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 w i l l 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 (C concentration exceeds 2 factor + C3) ^— , 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 w i l l 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] minimal dx •jf we have found that a model given by 1 = S - k x i Y i e i - k x i Y i 2 e i - D xi 2 2 3 3 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 > C i . 2 Our approach is now to integrate these coupled non-linear numerically (see the program in appendix varying the parameters 1), equations to study the effects on the model, and to analyse the of general qualitative properties of the model. But first, we discuss the model and its strengths and weaknesses. 65 2.2 Discussion of the Model 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, proliferation and begin to eliminate also stimulates the invading antigen. T h i s 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 system to evolve into another which act as negative the network, causing steady state. The idiotypic the interactions, feedback, ensure that the perturbation damped out as it propagates, producing the specific memory is state. Some have put to question this picture of the immune response 5]), ([42- arguing that receptors cannot distinguish between, for instance, a cell-bound virus particle (indicating killed-figure 27a), an infected cell which should be 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 killedfigure 27b). This is obviously true, but what is unclear is this renders the network determined is whether non-functional. What or not the whether needs to be idiotype/non-idiotype background effects such as those depicted in figure 27b swamp out the idiotype/idiotype thereby rendering regulatory the effects network as depicted in figure 27 a, non-functional. T h i s question can 66 only be resolved by experiment, experiments but given the number of which are suggestive of network regulation, it seems unlikely that the answer w i l l be a solid affirmative. However, even if b Figure 27: A receptor cannot distinguish between bound virus (a) and a receptor bound virus a membrane (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 antiidiotypes), 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 / N S 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 S/NS or not network theory can account for or even accomodate discrimination. A c c o r d i n g to the symmetrical network theory, clones directed against the host are in the suppressed state, and are therefore not damaging for the organism. U s i n g arguments similar to the above, some have vehemently argued however ([42],[44]) that the network cannot account for S / N S , 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). T h e 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 regulatory idiotype-idiotype swamp out the interaction?' and once again, a positive answer would indicate that the network is not the final word. O f course, even i f network theory cannot completely account for the S / N S , 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, i n 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]). 68 F i g u r e 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 / N S 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 / N S (S/NS not accomplished by deletion mechanisms) via suppression, it w i l l 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 o f 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 b fragments - can induce a 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 i n [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. F o r these reasons, we feel 70 that network theory is still the best theory of immunoregulation, albeit one that is still being developed. H a v i n g defended the principles of network theory, there things that should be said about the particular network represented by equation (63). are several model 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) dealing with cellular stimulation/killing), (when applying (63) antibody to T cell populations, or when as a B cell population to B cell populations or when dealing with stimulation/killing) and as an antibody population dealing with antibody killing/stimulation (when of cells). T h e 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. A l s o 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 secreting antibody, these processes depending heavily and 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 secreting B cells them. A s a final comment, we would like to clarify the notion of steady state. It may seem absurd to speak of steady states when immune system is constantly responding to foreign antigens and when there are many fluctuations in system behaviour the course of these responses (see figure 29). the throughout T h e 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 T h e justification physical system. for this approach is thus purely empirical: it what is observed in experiments, or more simply: it works. evolving slowly through a succession of steady states, systems seem to lend themselves to mathematical we are carrying out in this thesis. reflects In natural treatment such as 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 / N S 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 antiimmune 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. W e now repeat the analysis performed in [16] confirm that this is still the case for the modified model represented by to N-dimensional (63). A s in [16], we consider dynamics in the x i / Y i 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/Yi 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 1 0 0 system (see figure 31). F 1 10 i 11 0. 1 i 0.01 i 0 . 0 0 1 ~i—• 0.001 1 1 1 " " i — ' 0.01 ' ' i ' ' '' " r n — I ' I I 0.1 1 nm—' i 10 111III 100 X; F i g u r e 3 0 : A trajectory of a single clone in the x i / Y i phase plane. T h e time increments are small enough that the points form an apparently continuous line. 75 100 | 10 i 11 0. 1 0.01 i i 0.001 1 J 0.001 Figure 31: Trajectory time increments l • .in.l 0.01 1 . • M I M , 0.1 , , , , M M | 1 , , , , . , „ • 10 100 of 20 clones in the X i / Y i phase plane. The are small enough that the points form an continuous apparently line. W e 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 (63) the system as described by w i l l move into these steady states. In equation order to compute the dynamics of the system more easily, we simplify the model by using step functions for the e i : q e i q = 1 if 0 if ¥f <C ¥i' > C q q q = 2,3; i=l,N this substitution being equivalent to taking n= , co. T h i s sharp transition constitutes a high degree of non-linearity system, but (64) in the 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. 76 100 10 4 1 0 .1 0.01 0 . 001 i I 0.001 I 0.01 I I I I I I I l| I I I I M I I | i i 1 0.1 . . i - • • i 10 100 *i 100 10 i 1 E 0 . 1 i 0.01 = 0.001-1 "Tl 0.001 0.01 1 | 1 0.1 I \ ! I 1 I I l| 10 1 II ,1 I I I I I ' 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 =1, k = 0.1, D = 2, C = 10, C3 = 30. For (b) n =n =5. 2 3 2 2 3 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 > N for d x S ij j j < C R x Y 3 < C2, C3, we have . "Jf = 0 w h e n j=l xi = 3 2 N for (65) k Yj + k Yi2 + D d x C3 < X ij j j R x < 2> Y . "df c = 0 w h e n j=l X i k Yi + D = ( 6 6 ) 2 C3 < C and for N 2 dx- X u J j' < R x "HT Y = 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: for dx- N S ij J J R x < Y C 2 C3, < "df = 0 w h e n j=l xi = — k Y i + k Yj2 + D 3 2 for C N 2 < X ij j j R x (68) K Y < C3, dx"dT = 0 w h e n J 78 Xi = ( 6 9 ) k Yj2 + D 3 N and for C 2 < C 3 < X iJ J J» R j=l x Y dxi dT = 0 when S (70) D 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 < C 2 in figure 33, and for a set of parameters with C3 C 2 < C3 in figure 34. IMMUNE/ANTIIMMUNE 100 SUPPRESSED IT 10 = 0.1 = 0.01 = 0.001 0.001 Figure 33: The three equilibrium loci in the x i / Y j by equations (65)-(67) with the parameters D = 1, C3= 3, C2- 10. plane, as specified S = 10, k2 = 1, k3 =10, V . S . - virgin state; I.S./A.I.S. - anti-immune state; S . S . - suppressed state. 79 0.01 = -I 0.001 1 — i i 1 1 1 1 1 [ 0.01 0.001 Figure i m 0.1 1—i i i 11111 1 1 — i i 111111 1 — i i 10 M i i i 100 34: T h e three equilibrium loci in the xJYi plane for the parameters corresponding to figure 32. When integrating important 1 — i i I I differential equations numerically, it is of course 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: < Xi constant i=l,..,N (65) 80 Based upon an initial survey of the dynamics, we determined 0.5% change per step was sufficiently the dynamics of the system (meaning an order of magnitude periodically throughout verified to faithfully replicate that decreasing ( A x / x ) produced the same dynamics). that this mesh was indeed fine m a x by We enough the course of our study. The two regions of parameter space different small that a physical situations. The C 2 > C 3 and C 3 > C 2 correspond to former represents an immune system where there is more factor in the virgin state than there is in the immune state, (see figure 35). and the latter represents the opposite situation W e 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 challenged with is antigen. Eqi 1 virgin suppressed immune 0 Fi C2 C3 a 81 suppressed 0 Fi C2 C3 F i g u r e 3 5 : A graphic illustrating the different cases C 2 > 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) value. were all set equal to the same 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 yielded similar dynamics. In figure 37, system with a random initial condition. 100) we display trajectories for a 0.01 i 0 . 00 1 I 0.001 i i i i MII| 0.01 i i i i 11111 i i i i i ni| 0.1 „ 1 i IM IIIII 10 i i i 11 III 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 F i g u r e 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). 84 100-r 1 101 1 i 0.1 i 0.01 0.001 — ' i — 1 1 0.001 1 1 1 1 n i — i i 1 1 1 , H I — i 0.01 i 111 m i — i — i 11 1 0.1 i H I i—i 111 10 m i 100 a 0. 01 i 0 . 00 1 1 0.001 1 1 I I Mll| 1 I T 0.01 I Mll| 1 1 I I I H | 1 0.1 1 1 I I I 10 lll| 1 1 I I I III 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 and k3 to 10 and 100 respectively gives us a final state 2 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. 1 100 J ^ N ^ ^ 10! o. 0 . 00 W 11 0.01 1 1 \ i I 0.001 ' I M 1 0.01 I I II I ' l l IMI| 1 0.1 a 1 I I I Mill 10 1 I I I Mil 100 X, 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 0.01 0.001 0.001 0.001 Figure '''' 0.01 r 39(cont'd): caption on previous page. 87 F i g u r e 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 k 3 = 1 0 0 . 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. A s 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. 100^ ^ — j ^ ^ ^ w 10i 0. 1 j j ^ 0.01 i 0.001 ! 1 0.001 — i 1111111—i 1111111 0.01 i—i 1 1 1 1 n i i 1 0.1 i 1111HI, i—i 1 1 1 1 I I 10 100 Xi a 100 j 1 lOi 11 0 . 1! 0.01 i 0 . 00 1 ~ ~ i — ' 0.001 " i — 0.01 i 1 1 1 m i — i — i — i 1 0.1 11111111—i—1111m 10 100 Xj b Figure 41: The dynamics for several values of C . 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. 2 Eqi Fi 0 C2 average C3 factor concentration 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 = v i r g i n state, I=immune state, S=suppressed state. 90 B y studying the factor concentration, we can see i f the steady stable states of the system correspond to our phenomenological picture discussed in the opening paragraph of this section. T y p i c a l results are shown i n 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 antiidiotypic 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_. O f 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 i n 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. O n the whole however, the factor concentrations correspond quite well with differences our intuitive picture of between the factor the steady concentrations i n the 2-dimensional and N-dimensional models, but these hardly there are states. There are some surprising given that many more connections between the clones in the N - 91 dimensional model, and more complicated stabilization patterns are possible. final population 24355 0.8438 6.0000 2.8623 1.0441 1.7973 0.8619 42181 22758 0.5790 1.9010 1.9149 6.0000 1.1319 1.1069 2.5164 92696 6.0000 0.6995 3.4505 Figure fir*I stele V I S I I I V I I I Y V S I I I I s I Y like factor 8.0 192 33.6 14.9 152 19.8 9.9 13.1 11.5 22.0 9.8 7.3 31.3 19.6 19.1 12.4 17.3 33.1 16.6 3.1 unlike factor 1.7 7.3 9.6 3.3 5.1 3.8 2.4 1.4 1.8 7.8 12 2.0 9.5 5.1 5.0 2.5 6.7 9.6 6.3 0.5 j 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). 92 Furthermore, for a given set of parameters and K y , the system can adopt many steady states as illustrated in figures 36 and 37, coincides with the many system can which states that an individual's physical immune adopt. 0.01 0 . 00 E 1 H 0.00J T—TTTTTTTl 0.01 1 I l l l Il'| 0.J 1 . I I I *I n | . I I I I to 111 100 b Figure 44: Different Kjj give qualitatively similar but different dynamics and steady states. quantitatively 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 ^ S - k XiYie = 2 2 perturbation: - k x i ( Y i ) 2 e - Dxi + RiAg(t) 3 (71) 3 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. course, our model does not intend to simulate switching Of 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 94 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 10i 1 = 0.1 = 0.01 = 0.001 I 0.001 I I I 111II 0.01 I—I I I I I 111— 0.1 -i—i i 1111 n 1 T 1 I II II 10 F i g u r e 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. 100 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 w e envisron the immune response as being characterized by the presence of a substantial amount of factor, it is probable that we w i l l have to rethink the normalisation when it comes time to to deal with switching dynamics. 0.001 0.01 1 0.1 10 *i F i g u r e 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 t o their original l o c i of equilibrium. Parameters again as for figure 45. 100 96 W e 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 S S S , memory, and the presence of the suppressed state which may contribute towards the S / N S 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 w i l l 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 w i l l 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 / N S 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, with improved experiments together w i l l hopefully raise the veil away the subject of immuno-regulation and reveal the deeper layer truth that presently reveals to us only its shadow. from of 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 T extracellular conditions, namely the concentration of specific 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 crosslinked 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. W e 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 magnitude between the two cases, provided one had previously obtained accurate values for the equilibrium characterize the by a full order of constants system and provided that activation which 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). T h i s one-receptor bridge formation only takes place for the case of the bivalent receptor, and 98 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 Program Immune; The 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; VAR Numeral 1 Integration RonfinP (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) 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 FORREAD(kmatrix,K(x1.yj]); yj := 1 TO dim DO 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 101 Lfx1.yj):=0; FOR zk:=1 TO dim DO Ux1yjJ:=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 Integer; outInd BEGIN rewr te(d1,*f1le=od1 rewr te(d2.'f11e=od2 rewr te(d3.'f11e=od3 rewr te(d4.'f1le=od4 rewr te(d5,'f1le=od5 rewr te(d6.'f1le=od6 rewr te(d7,*f11e=od7 rewr te(d8."f 1le=od8 rewr te(d9.'f1le=od9 rewr tefdlO,"f 11e=od10 rewr te(d11,"f1le=od11 rewr te(d12.'file=od12 rewr te(d13,fi le=od13 rewr te(d14.'f1le=od14 rewr te(d15,'f1 le=od15 rewr te(d16,*f1le=od16 rewr te(d17,*f 11e=odl7 rewr tetd18,'f1le=od18 rewr te(d19,'f1le=od19 rewr te(d20.'f11e=od20 rewr tefefHe,'fi1e=effectlvltie '); rewr te(f1na1_posi t ion,f1le=eqloci'); END; ( 1 1 PROCEDURE dataout; VAR outtime, 1 : integer; BEF G OIN R 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 1985-present T Reference cell-factors Literature: antigen-specific A. PNAS 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 C . A n n . Inst. Pasteur 137D:391. Rawat I et al 138:815. Dieli et al 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 antigen specific suppressor cells x x x x x antigen specific suppressor cells antigen specific suppressor cells x X X X 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 i n the immune response. aliasing-an effect produced when sampling rate leads to high frequency effects appearing as low frequency ghosts. anti-as i n anti-i or anti-factor. Refers to an antibody or cell directed against the entity i n question (eg. clone i or specific factor) anti-immune state-the converse of the immune dimensional network: low x+ and high x.. antibody-dependent immunity antigen-a that requires cellular cytoxicity state in the 2- (ADCC)-cellular the presence of antibody. virus or bacteria that 'invades' the host's immune system. factors-proteins secreted by T cells when antigen specific T cell they are activated by the presence of antigen. W e 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 a n e r g y - t h e 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 E h r l i c h which is the cornerstone of Theoretical Immunology. T h e 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 conformational change-a change in the orientation backbone of a large organic molecule. of the carbon c o n s t a n t end-the part of a factor or antibody molecule that is not antigen specific. cross-linking of r e c e p t o r s - r e f e r s 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 o f forward and reverse rate constants i n 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 dX 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 F a b 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 o f internal stability v i a feedback mechanisms. idiotypic interactionsinteractions 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 o f f 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 i n 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 o f the myeloid cells. Responsible for allergic reactions. membrane--refers to the membrane memory state-see immune that surrounds all cells state. M H C - t h e major histocompatibility complex, refers to the D N A encodes for the proteins that serve as a marker of 'self, and colloquially to these same proteins. monoclonal antibodies-antibodies which 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 o f 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 o f a chemical reaction. It is a constant by the law of mass action. receptor-a protein structure that is anchored in the c e l l 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 i n experiments. self-consistency equation-refers to an equation which cannot be solved exactly due to its complexity. It is used to search for solutions satisfying R H S = L H S . self-avoiding random walk-refers to a walk i n 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 i n 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 w i l l 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 perturbed. steric repulsion-repulsion between two chemical entities due to the proximity of their electron clouds. suppressed dimensional state-refers to the unresponsive state in the model. 2- symmetric matrix-a matrix K that has K i 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 [ 4 5 ] . V regions-see variable regions v a l e n c y - i n 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 m o d e l . 110 Bibliography [I] K i m b a l l , J . W . 1986. Introduction to Immunology. Macmillan. N e w York, ppl65-94. [2] Ibid, pp 168-94 [3] K i m b a l l , J . W . 1986. Introduction to Immunology. Macmillan. N e w 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 W h i c h Govern Immune Responsiveness. In: The Cellular Basis of Immune Modulation. Proceedings of the 19th International Leucocyte Culture Conference. Edited by Kaplan, J . G . A l a n L i s s . New Y o r k , 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] C e l l Cascade and Antigen Elimination: H o o d , L . E . , Weissman, I. L . , W o o d , W . B . and W i l s o n , 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. A n n . 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 C e l l to C e l l Signalling. Edited by Goldbeter, A . Academic Press. New Y o r k . [II] Kaufman, M and Thomas, R. 1987. M o d e l Analysis of the Bases of Multistationarity in the Humoral Immune Response. J . Theor. B i o l . 129:141 [12] D e 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 . A d d i s o n Wesley, pp 265-289. [13] Hoffmann, G . W . 1975. A Theory of Regulation and Self-Nonself Discrimination in an Immune Network. Eur. J . Immunol. 5:683. [14] Hoffmann, G . W . 1980. O n Network Theory and H-2 Restriction. In: Contemporary Topics in Immunobiology. Edited by Warner, N . Plenum. New York. 11:185. [15] Hoffmann, G . W . 1978. Incorporation of a Non-specific T cell Dependent Factor into a Network Theory of the Regulation of the 111 Immune Response. In: Theoretical Immunology. Edited by B e l l , G . I. et al. Marcel Dekker. New Y o r k , pp 571-602. [16] Hoffmann, G . W . , K i o n , T . A . , Forsythe, R. B., Soga, K. G . , CooperW i l l i s , A . 1988. The N-Dimensional Network. In: Theoretical Immunology, Part 2, SFI Studies in the Science of Complexity. Edited by Perelson, A . Addison Wesley. Menlo Park, C A . pp 291-320. [17] Hoffmann, G . W . and L y o n s , M . 1989. Immunological Measures of Similarity (draft in preparation). [18] H o o d , L . E . , Weissman, I. L . , W o o d , W . B. and Wilson, J . H . 1984. Immunology. Benjamin/Cummings. M e n l o Park, C A . pg 298-9 [19] K a y e , J . and Janeway, C . A . 1984. The Fab Fragment of a Directly Activating M o n o c l o n a l Antibody that Precipitates a D i s u l f i d e - L i n k e d Heterodimer froma Helper T C e l l Clone Blocks Activation by either Allogeneic la or Antigen and Self-la. J . E x p . M e d . 159:1397. £20] T e h , H . 1989. T Lymphocytes (manuscript). U B C Department of M i c r o b i o l o g y . Vancouver. [21] Goldstein, B . Desensitization, Histamine Release and the Aggregation of IgE an Human Basophils. In: Theoretical Immunology, Part 1, SFI Studies in the Science of Complexity. Edited by Perelson, A . Addison Wesley. Menlo Park, C A . pp 3-40. [22] Paul, W . E . editor. 1984. Fundamental Immunology. Raven Press. New Y o r k , pg 563. [23] M c K e n z i e , D. S. 1976. Polymers and Scaling. Phys. Reports. 27:35. [24] True'sdell, C . 1945. Annals of Math 46:144. [25] H o o d , L . E . , Weissman, I. L., W o o d , W . B . and Wilson, J . H . 1984. Immunology. Benjamin/Cummings. M e n l o Park, C A . pg 58 [26] T i z a r d , I. 1984. Immunology, an Introduction. Saunders College Publishing. Philadelphia, pg 164 [27] Reif, F . 1965. Statistical and Thermal Physics. M c G r a w H i l l . N e w Y o r k . pp. 322-362. [28] Glasstone, S. 1946. Handbook of Physical Chemistry. D. van Nostrand. Princeton, pp 816-7. [29] Jerne, N . July 1973. The Immune System. Scientific American. New Y o r k , pg 52. [30] Reif, F. 1965. Statistical and Thermal Physics. M c G r a w H i l l . New Y o r k , pp.262-269. [31] Goldstein, B . 1978. Theory of Equilibrium Binding of Symmetric Bivalent Haptens to C e l l Surface Antibody: Application to Histamine Release from Basophils. J . of Immunol. 121:345. [32] D e L i s i , C . and Thackur A . K. 1977. Antigen Binding to Receptors on Immunocompetent Cells. C e l l . Immunol. 28:416. 112 [33] Perelson, A . S. 1984. Some Mathematical Models of Receptor Clustering by Multivalent Ligands. In: C e l l Surface Dynamics. Edited by Perelson, A . S. and D e L i s i , C . Marcel Dekker. New York, pp 223276. [34] Schweitzer-Stenner, R., Licht, A . , Liischer, I., and Pecht, I. Oligomerization and Ring Closure of Immunoglobulin E Class Antibodies by Divalent Haptens. B i o c h e m . 26: 3602. [35] Jefferies,W. A . , Riither, U . , Wagner, E . F. and Kvist S. 1988. Cytolytic T Cells Recognize a Chimeric M H C Class I Antigen Expressed in Influenza A Infected Trangenic M i c e . E M B O J . 7:3423. [36] Stryer, L . 1988. Biochemistry. W , H . Freeman. New Y o r k , pg 64. [37] T U n g T . A s referenced in: Stryer, L . 1988. Biochemistry. W . H . Freeman. New Y o r k , pg xxxii. [38] Kimball, J. W . New York, pg 193 [39] Ibid pg 308 1986. Introduction to Immunology. Macmillan. [40] Barisas, B . A . , Singer, S . G . , Sturtevant. 1975. Kinetics of Zbinding of 2,4-Dinitrophenyl and 2,4,6-Trinitrophenyl Haptens to Homologous and Heterologous Rabbit Antibodies. Immunochem. 12: 411. [41] Gunther, N . and Hoffmann, G . W . 1982. Qualitative Dynamics of a Network M o d e l of Regulation of the Immune System: A Rationale for the I g M to IgG Switch. J . Theor. B i o l . 94:815. [42] C o h n , M . 1989. The A Priori Principles W h i c h Govern Immune Responsiveness. In: T h e Cellular Basis of Immune Modulation. Proceedings of the 19th International Leucocyte Culture Conference. Edited by Kaplan, J . G . A l a n Liss. New York, pp 11-44. [43] C o h n , M . 1988. Idiotype Network V i e w s of Immune Regulation: For W h o m the B e l l Tolls. In: Idiotypy and Medicine. Edited by Capra, J . D . Academic Press. New Y o r k , pp 321-399. [44] Langman, R. E . 1987. the Self-Nonself Discrimination is Not Regulated by Suppression. C e l l . Immunol. 108:214. [45] C o h n , M . 1981. Commentary: Conversations with Niels Kaj Jerae on Immune Regulation: Associative versus Network Regulation. C e l l . Immunol. 61:425. [46] K i m b a l l , J . W . 1986. Introduction to Immunology. Macmillan. New Y o r k , pg 352 [47] Nossal, G . J . V . 1983. A n n . Rev. Immunol. 1: 5. [48] [49] Miller, R. G . 1986. Immunology Today. 7:112. H o o d , L . E . , Weissman, I. L . , W o o d , W . B . and Wilson, J . H . Immunology. Benjamin/Cummings. Menlo Park, C A . pg 375. 1984.
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Mathematical models of immunity Mathewson, Donald Jeffrey 1990
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Title | Mathematical models of immunity |
Creator |
Mathewson, Donald Jeffrey |
Publisher | University of British Columbia |
Date Issued | 1990 |
Description | 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 FWHM 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. |
Subject |
Immune response -- Mathematical models Immune system Idiotypic networks |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-10-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085003 |
URI | http://hdl.handle.net/2429/29575 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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