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Studies of the immune network based on shape-space and distance coefficient Royer, Sophie 1993

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STUDIES OF THE IMMUNE NETWORK BASED ON SHAPE-SPACEAND DISTANCE COEFFICIENTBySophie RoyerB. Sc. (Physique) Université de Montrea’, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTERS OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1993© Sophie Royer, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of________________UThe University of British ColumbiaVancouver, CanadaDateDE-6 (2/88)AbstractA selective history of immunology is first presented that tells about the development ofthe theories of “acquired immunity”, from the early observations of the phenomenon ofimmunity in the Roman times until the long debate ending in the late 1960’s between theproponents of the theories of “cellular immunity” and those of the theories of “humoralimmunity”. Some theories of antibody formation are also reviewed up to the “clonalselection theory”. Then the “immune network hypothesis” and some of the models thatit engendered are explained, with a focus on a particular model due to Hoffmann and hisco-workers: the N-dimensional network model. A short history of the attempts to modelthe “affinity” distribution and various choices of “connectivity” matrices is presented,focussed on a particular one: a connectivity matrix based on a one-dimensional “shapespace”. The + shape-space , a shape-space formulated by Segel and Perelson, is reviewedand a “new shape-space without shape zero”, the shape-space, is introduced in which“complementarity” relationships between clones differ from the ones in the + shape-space. Some analysis and numerical simulations of the two different versions of shape-space embedded in Hoffmann’s N-dimensional network are shown, which are the firstsimulations of the model to have ever been be done with non-Boolean affinities. Theconcept of a “distance coefficient” is reviewed, analyzed and developed further for its usewith non-Boolean affinities. The first numerical simulations of the distance coefficients tohave ever been done are presented and analyzed, embedded in Hoffmann’s N-dimensionalnetwork model with Boolean and non-Boolean affinities.‘The author’s renaming of the origina’ shape-space of Segel and Perelson.11Table of ContentsAbstractTable of ContentsList of FiguresAcknowledgements1 IntroductionU111vixv12 From immunity to clonal selection2.1 The phenomenon of immunity2.2 Theories of antibody formation2.2.1 Antigen incorporation theories2.2.2 Ehrlich’s side-chain theory2.2.3 Instruction theories . . . ..2.2.4 Selection theories3 Immune network theories and models3.1 Jerne’s network hypothesis3.2 mathematical models of the network hypothesisIntroductionRichter’s modelHoffmann’s model449991015First3.2.13.2.23.2.3191922222324II’3.2.4 Adam-Weiler’s model . . . .,,.., 263.3 Later models 263.4 More on Hoffmann’s model 273.5 Hoffmann’s N-dimensional model 323.5.1 The model 323.5.2 The steady-states of the model 344 The affinity distribution 374.1 The problem of the reconstruction of the affinity distribution 374.2 Choices of connectivity matrices 395 Simulations in one-dimensional shape-space 435.1 Introduction 435.2 The + shape-space 445.2.1 Introduction 445.2.2 The non-periodic + shape-space 445.2.3 The periodic + shape-space 555.3 A new shape-space without shape zero: the Li shape-space 595.3.1 Introduction 595.3.2 The non-periodic shape-space . . . . . . 595.3.3 The periodic A shape-space . . . 616 The distance coefficient 756.1 The concept of a distance coefficient 756.2 The first model of the similarity coefficient 796.3 The limitations of the first model of the similarity coefficient 826.4 The first model of the distance coefficient is a metric 85iv6.5 The generalized distance coefficient. 896.6 The distance coefficients plane 936.7 Simulations of the distance coefficient 956.7.1 Affinities from the periodic Li shape-space 956.7.2 Random boolean affinities 1027 Conclusion 106A The N-dimensional model brought back to a 1-dimensional model 108B Complementary clones can have the same concentration 112B.1 The periodic Li shape-space 112B.2 The non-periodic ± shape-space 115C The distance coefficients of Hoffmann and Tufaro form a metric 117Bibliography 120VList of Figures2.1 Ehrlich’s side-chain theory: “the antitoxines represent nothing more thanside-chains reproduced in excess during regeneration and therefore pushedoff from the protoplasma, and so coming to exist in a free state”. (Scannedfrom [38].) 112.2 Diagram of Pauling’s theory of antibody formation, representing the fourstages in the process of formation of a molecule of normal serum globulin,the six stages in the process of formation of an antibody molecule as theresult of interaction of the globulin polypeptide chain with an antigenmolecule and the antigen molecule surrounded by attached molecules orparts of molecules and thus inhibited from further antibody formation.(Scanned from [106].) 132.3 The clonal selection theory of antibodies: antigen binds to a receptor onthe surface of a lymphocyte bearing the right specificity and triggers itsproliferation into a clone of lymphocytes of the same specificity; some ofthe daughter cells enlarge to become plasma cells that secrete antibodiesof the same specificity, others are the memory cells ready to respond tofurther stimulation by antigens of the same specificity 17vi3.4 The domain structure of an Ig molecule. L and H represent the light andheavy chains respectively. CL and VL are the “constant” and “variable”regions of the light chain. CH1, CH2, CH3 are constant regions of theheavy chains. VH1 is the variable region of the heavy chain. CDR’s arethe complementarity determining regions, that is those regions that formthe antigen-binding sites. (Scanned from [50].) 203.5 The stable states of Hoffmann’s model. (Scanned from [56].) 253.6 The interactions between cells in the plus-minus model. (Scanned from[57].) . 283.7 The eq function for flq=7. . . . . . . 303.8 Phase-plane representation of Hoffmann’s plus-minus model. The virgin,immune, anti-immune and suppressed states are labeled V.S., I.S., A.I.S.and S.S. respectively. All four stable states are attractors. Parametervalues are k1 = 0.1, k2 = 1, k = 1, k 0.01, S = 1, e = 10, c2 = 3,= 0.3, n1 = 1, n2 = 2, n3 2. (Scanned from [57].) 314.9 Two realizations of a one-dimensional shape-space. In figure (a), the shapeis defined by the height of wedge-shaped epitopes, positive for protuberances and negative for indentations. In figure (b), the shape is defined asthe charge of the homologously located patches. (Scanned from [124].) . . 425.10 The non-periodic ± shape-space. Clones are numbered from —N to N andpositioned accordingly to their similarity with each other. Neighbouringclones are the most similar and are separated by an arbitrary distanceS. Dotted lines join the clones of maximum complementarity. Note thatclone 0 has maximum complementarity with itself 45vi’5.11 The matrix of affinities of a 21-dimensional network, for the non-periodic+ shape-space. The left and right pictures are two-dimensional and three-dimensional representations. Clones are numbered from —10 to 10 forboth representations. In the two-dimensional representation, the shadeof grey represents the value of the affinity with a scale going from whiterepresenting a maximum affinity of one and black representing a minimumaffinity of zero. In the three-dimensional representation, the vertical axisrepresent the value of the affinity. Here, ci = 6 and cut-off= 10—6 455.12 Clone concentration x versus time t for the non-periodic + shape-space,with random initial conditions. 0 < x < 1, S = 10, k2 1, k3 = 0.01,= 1, C2 = 0.6, C3 = 0.7, ci = 4, cut -off= iO 485.13 Clone concentration x versus T-cell factor W[ concentration for the non-periodic + shape-space. Same parameter values as in figure 5.12 485.14 Field Y versus clone concentration x for the non-periodic + shape-space.Same parameter values as in figure 5.12 495.15 Field Y versus T-cell factor W’ concentration for the non-periodic +shape-space. Same parameter values as in figure 5.12 495.16 Effective field U2 versus clone x concentration for the non-periodic +shape-space. Same parameter values as in figure 5.12 505.17 Effective field U3 versus clone x concentration for the non-periodic +shape-space. Same parameter values as in figure 5.12 50viii5.18 Formation of clusters of clones in the virgin(V), immune(I), suppressed(S)steady-states in the non-periodic ± shape-space for a 21-dimensional network with random initial conditions. Complementary clones share thesame steady-state concentration. There is a symmetry in the configuration of clusters. Same parameter values as in figure 5.12, so a = 4 andcut -off= i0. This configuration is also the same for values of a between2and4 525.19 Same as in figure 5.18 but now a = 4.5. Clone —5 has become suppressed(S). The concentrations of complementary clones are no longeridentical. 545.20 Same as in figure 5.18 but now a = 5. Clone —6 has become immune(I).Concentrations of complementary clones are still not identical 545.21 Same as in figure 5.18 but now a = 5.5. Clone —5 and 6 have becomeimmune(I) and clone 6 has become suppressed(S). Concentrations of complementary clones are still not identical, 545.22 Same as in figure 5.18 but now a = 6. Clone 5 has become immune(I), sothat now clones —6, —5, 5 and 6 are immune(I). The situation is symmetricagain: complementary clones have the same concentration and are in thesame steady-state. The same situation holds for a = 7 555.23 The ± shape-space with periodicity 2N + 1 (left) and with periodicity 2N(right). Crones are positioned on the “circular” shape-space. Each cloneis most similar to its nearest neighbours. Dotted lines join the clones ofmaximum complementarity 575.24 The matrix of affinities for the ± shape-space, with periodicity 2N + 1,for a = 6 and cut-off= 10—6. Refer to figure 5.11 for explanations on thelegend 57ix5.25 Formation of clusters in the periodic + shape-space. This experiment wasdone for the exact same values of parameters as in figure 5.18, except fora different set of initial conditions. The oniy observed difference is theswitch between clones in the virgin(V) and suppressed(S) states 585.26 The non-periodic A shape-space. Clones are numbered from 1 to N (whereN is an even number) and positioned on the shape-space according to thesame rules of similarity with each other than for the ± shape-space. Notethat the dotted lines that join clones of maximum complementarity exhibita different pattern than for the ± shape-space. 605.27 The + shape-space with the shapes renumbered appropriately and omitting the shape zero shows the same pattern of dotted lines that join clonesof maximum complementarity as in the A shape-space 615.28 The A shape-space with periodicity N+1. Dotted lines that join the clonesof maximum complementarity are diameters of the “circular” shape-spaceand denote that those clones are also maximally dissimilar 625.29 The matrix of affinities of a 20-dimensiollal network, for the A shape-spacewith periodic boundary conditions. The left and right pictures are two-dimensional and three-dimensional representations. Clones are numberedfrom 1 to 20 for both representations. In the two-dimensional representation, the shade of grey represeilts the value of the affinity with a scale goingfrom white representing a maximum affinity of one and black representing a minimum affinity of zero. In the three-dimensional representation,the vertical axis represent the value of the affinity. Here, a = 6 and cutoff= 106 62x5.30 Clone concentration z versus time t for the periodic A shape-space, withrandom initial conditions. 0 < xj < 1, S 10, k2 = 1, k3 = 0.01, k4 = 1,C2 = 0.62, C3 = 0.65, a = 2, cut-offz= iO 645.31 Clone x versus T-cell factor W’ concentrations for the periodic A shapespace. Same parameter values as in figure 5.30 645.32 Field Y versus clone concentration x for the periodic A shape-space. Sameparameter values as in figure 5.30 655.33 Field Y versus T-cell factor 147 concentration for the periodic A shape-space. Same parameter values as in figure 5.30 655.34 Effective field U2 versus clone x concentration for the periodic A shapespace. Same parameter values as in figure 5.30 665.35 Effective field U3 versus clone x concentration for the periodic A shapespace. Same parameter values as in figure 5.30 665.36 Formation of clusters of clones in the virgin(V), immiine(I) and suppressed(S)steady-states in the periodic A shape-space for a 20-dimensional networkwith random initial conditions. 0 < x < 1, S = 10, k2 = 1, k3 = 0.01,= 1, a = 2, cut-off= iO, Left: All clones are immune(I). Complementary clones do lot share the same concentration. C2 = 0.35 and C3 = 0.6.Right: Clone 8 has become virgiri(V). C2 = 0.358 and C3 = 0.6 695.37 Same as in figure 5.36 excepted for the values of C2 and of C3. Left: Clone9 has become virgin. C2 = 0.36 and C3 = 0.6. Right: Clone 7 has becomevirgin. C2 = 0.39 and C3 = 0.6 69xia5.38 Same as in figure 5.36 excepted for the values of 02 and of 03. Left: Clone6 has become virgin. 02 = 0.44 and 03 = 0.6. Right: Drastic change in theconfiguration. There are now clusters of four virgin and suppressed clonesseparated from each other by immune clones. The situation is symmetricand complementary clones have the same concentration. 02 = 0.358 and03 = 0.6 705.39 Same as in figure 5.36 excepted that 02 = 0.59 and of 03 = 0.6. Theimmune clones have disappeared and have been replaced by suppressedclones. The situation is still symmetric and complementary shapes havethe same concentration 705.40 Clusters in the periodic shape-space for a 22-dimensional network withrandom initial conditions. The configuration that exhibits all three steady-states has complementary clones sharing the same concentration and being in the same steady-state. The pattern is different than with the 20-dimensional network. Same parameter values as in figure 5.36 exceptedforC2=0.65and030.7 715.41 The matrix of affinities of a 21-dimensional network, for the periodic L\shape-space. It is very similar to the one for a network comprised of aneven number of clones. u = 6 and cut -off= 106 735.42 Clusters in the periodic shape-space for a 21-dimensional network withrandom initial conditions. Dotted lines of maximum complementarityshow that each clone is set complementary to a hypothetical clone situated between its most complementary clones. Same parameter values asin figure 5.36 excepted that 02 = 0.6 and 03 = 0.7 73xii6.43 The fractions of C that are taken into account in the Hoffmann-Tufaro’sdefinition (eq. 6.25) of the similarity coefficient. (Adapted from [62].) . 766.44 Representation in the distance coefficient plane: a useful tool for the diagnosis of diseases. and Lay are the average normal and lupus sera.N and L are the individual normal and lupus sera that constitute theaverage sera. Plotting the distances of all these sera in the distance coefficient plane is expected to result in two different regions (illustrated withthe semi-arcs). The region into which an unknown serum would be plotted could help determine whether the unknown serum is normal or froma person with lupus. (Adapted from [62].) 786.45 The new fractions of C that are taken into account in the second definition(eq. 6.30) of the similarity coefficient. (Adapted from [62].) 806.46 The distance coefficients plane 946.47 The distance coefficient representation for the periodic A shape-space,with uniform initial conditions. The positions of the clones are fixed intime and are symmetrically positioned with respect to complementary reference clones 1 and 11. x = 0.1, S = 10, k2 = 1, k = 0.01, k1, = 1,C2 = 0.6, C3 = 0.7, a = 2, cut -off= i0 976.48 The distance coefficient representation for the periodic A shape-space of a20-dimensional network with random initial conditions. d is the distancebetween reference clones 1 and 11. S = 10, k2 = 1, k3 = 0.01, k4 = 1.a = 3, cut -off= 0.9, C2 = 0.45, C3 = 0.65 986.49 Same as in figure 6.48 but a = 3, cut -off= 0.5, C2 = 0.65, C3 = 0.7. . . . 996.50 Same as in figure 6.48 but a = 3, cut -off= 0.1, C2 = 0.65, C3 = 0.7. . . . 1006.51 Same as in figure 6.48 but a = 3, cut -off= 0.01, C2 = 0.65, C3 = 0.68. . . 101XII’6.52 The two-dimensional representation of a matrix of affinities for a 22-dimensional network, with random Boolean affinities. Clones are numbered from 1 to 22. In the two-dimensional representation, the squaresshaded white represent an afffinity value of one and the ones shaded blackrepresent an affinity value of zero 1036.53 The distance coefficient representation for a 22-dimensional network ofclones randomly connected with Boolean affinities. One can see that theregular pattern that was exhibited for the affinities derived from a onedimensional shape-space is not present anymore. 0 < x.j < 1, S = 10,k2=1,k30,4,C1 ,3 104C.54 The Venn diagrams for the fractions of C that interact with three sera X,Y and Z. (Adapted from [62].) 118xivAcknowledgementsThere are so many people that I could thank for their support and friendship throughout my stay in Vancouver and the completion of this thesis, but more particularly I amvery grateful to those that follow here.I would like to thank my supervisor Geoffrey Hoffmann for having had so much patience during the first months while I was beginning to familiarize myself with computersand trying to learn what I could about theoretical biology, and for his continuing supportthroughout the entire degree. I also have to thank Leah Keshet who always had faith inme.I would like to thank my father Antoine Royer and mother Bach-Tuyet Vo for alwayshaving had so much faith in me and for having helped me in many ways. I also hopethat my grand-father Jacques Royer who deceased toward the end of the completion ofthis thesis is proud of me, wherever he is now. He was also a great source of support andof inspiration.My very dear friends that have helped me in one way or another and that I shouldthank more particularly are Robin Barley, Gul Civelekoglu and Cord Seele,I must also thank those that have shared their knowledge on computer science withme: Alan Boulton, Roger Kemp and Henry Lee.xvChapter 1IntroductionImmunology is among the most dynamical branches of biology. An amazing number offacts are being discovered every year. But still it is realized that a lot is left to understandabout the immune system, and especially about its regulation.The history of immunology starts a long time ago, with observations of the phenomenon of “acquired immunity”, that is observations that those that have recoveredfrom a disease are “immune” to it upon a second attack. Chapter 1 of this thesis beginswith a history of those early observations and presents the first theories of acquired immunity up to the long battle between supporters of “cellular immunity” who thought thatcells were the important constituents responsible for the defense of the immune system,and the supporters of “humoral immunity” who considered “antibody” molecules to bethe defenders of the body. It also presents the development of the theories of antibodyformation up to the “theory of clonal selection”, which is now considered to be a cornerstone in immunology. The information contained in Chapter 1 is mostly derived fromthe book titled “A history of immunology” [125] by Arthur Silverstein.The first mathematical models in immunology that would try to give insight to thequestion of the regulation of the immune system were essentially triggered by the “network theory” of self-regulation. Chapter 2 presents the network hypothesis and reviewsthe first models of immune networks, focussing on a particular one: Hoffmann’s model. Italso meiltions some later work by other theorists in the field. It then presents Hoffmann’s“N-dimensional network” model which is the basis of the work presented in Chapters 41Chapter 1. Introduction 2and 5.Chapter 3 talks about a practical aspect that network theorists face when they wantto input the biological parameters “affinities” in their models. It tells about attempts tomodel the “affinity distribution” of antibodies and the eventual choices of “connectivitymatrices” (matrices of affinities) that network theorists have made.Chapter 4 presents some of the results obtained when implementing Hoffmann’s N-dimensional network into a particular setting of computed affinities presented in chapter3, the “shape-space”. It first shows a few simulations done with the original shape-spaceof Segel and Perelson which was the earliest version of shape-space. Then it presents anew shape-space without shape-zero and shows some simulations done with it.Chapter 5 contains the most analytical and principal part of the thesis. It presentsa generalization of the “distance coefficient” which is a measure of the “dissimilarity”between two substances for its use with non-Boolean affinities. Some simulations are alsoshown which are based on Hoffmann’s N-dimensional network model with affinities of thenew shape-space without shape zero. It can actually be said that all previous chapters tothis one are only introductory to it. In order to justify the generalization of the distancecoefficient for its use with non-Boolean affinities and to show some simulations of it,there was a need to use an immune network model, so the N-dimensional network modelneeded to be presented. It then seemed a good idea to present the network hypothesiswhich originated the model, as well as the background to the hypothesis. Also, since thegoal of the generalization was to enable the use of non-Boolean affinities in the conceptof a distance coefficient. it seemed sensible to mention the problem of the choice of theaffinity distribution. The affinity matrix chosen here being of a shape-space, it also madesense to show simulations in shape-space with Hoffmann’s model, since the model hadalways been used with Boolean affinities,Finally, chapter 6 contains the conclusion.Chapter 1. Introduction 3The simulations that are presented in this thesis have been done with Fortran programs created by the author, excepted for some Fortran routines obtained from the book“Numerical Recipes in Fortran” [113]. For the integration of the differential equation ofHoffmann’s N-dimensional network model, the following routines from chapter 16 wereused: the stepper routine “rkqs”which is a fifth-order Runge-Kutta step with monitoringof the local truncation error to ensure the accuracy and adjust the stepsize, the algorithmroutine “rkck” which is a fifth-order Cash-Karp Runge-Kutta method, the driver routine“odeint” with adaptive stepsize control (modified by the author). For the generation ofrandom initial conditions (variables having a value between 0 and 1), the routine “Ran2”from chapter 7 was used. It is a long period (> 2 x 1018) random number generator ofl’Ecuyer with Bays-Durham shuffle and added safeguards.The original illustrations that are part of this work were made with the graphicspackages: Gnuplot, MG, and Mathematica.Chapter 2From immunity to clonal selection2.1 The phenomenon of immunityThe latin word “immunitas” is related to the legal concept of exemption [125]. It initiallydescribed the exemption of a person from service or duty in Rome. Later, in the MiddleAges, it referred to the exemption of the Church and its properties and personnel fromcivil control. The first use of the term in a disease context can be traced to the Romanpoet Marcus Annaeus Lucanus (A.D. 39-65) when he observed the great resistance ofthe Psylli tribe of North Africa to snakebites. It was later used occasionally in the samemanner, but gained popularity only in the nineteenth century, with the discovery ofsmallpox vaccination by Edward Jenner.The phenomenon of acquired immunity had been observed long ago. Pestilence andpoison were once the most feared causes of death. Even then, some keen observers realizedthat a person who had been infected by a disease and recovered would not be affected by itagain, at least not fatally. The first related report comes from the historian Thucydides[139] of Athens, in 430 B.C. Also, it is known that in Roman times, Mithridates VI,king of Pontus, wrote in his medical commentaries about his increasing daily intakes ofpoison, to prevent himself from attempts against his life. His conqueror Pompey hadthose writings translated, and the practice became well-used, so that throughout theMiddle Ages there were mixtures called Mithridaticum used to that effect. It is worthyto note that for centuries, it was thought that many diseases were due to poison, called4Chapter 2. From immunity to clonal selection 5“virus” in latin. Also, the Greek word “pharmakeia” still means poisoning, witchcraft ormedicine!The first theory of acquired immunity was made by the best known of the Islamicphysicians, Abu Bekr Mohammed ibn Zakariya al-Razi (880-932). In his Treatise on theSmallpox and Measles [115], he gave the first modern clinical description of smallpox,stated the fact that survival from smallpox conferred lasting “immunity” and providedan explanation for it [126].The notions of a disease being caused by small seeds (seminaria) and of its contagionwas first raised by Fracastoro [46] in 1546. It is interesting to note that he thought thatthose seminaria have affinities for different objects. And with this he explained “naturalimmunity” to certain diseases. Furthermore, he stated that the seminaria of smallpoxhave an affinity for that trace of menstrual blood that each individual was supposed tohave inherited from its mother. Shortly after, Girolamo Mercuriali [92] denied the theoryof menstrual blood affinity stating that if smallpox, measles, and leprosy were all resultsof menstrual blood contamination, then getting one disease should give protection fromthe others, which obviously was not the case. One can see there a resemblance to themodern concept of “cross-immunity”.By the end of the seventeenth century, smallpox (previously a rare disease) had become a serious infection, which was going to plague the world for long after. With therise of physical sciences in the sixteenth and seventeenth centuries, new theories of smallpox were developed, the most dominant ones being those proposed by the iatrophysicistsand the iatrochemists who would explain everything in terms of physical and chemicalprocesses respectively [127).At the beginning of the eighteenth century, the practice of variolation (consisting ofinoculation of smallpox to produce mild dermal infections which prevent further moresevere attacks), that had become popular as part of the folk medicine of several EasternChapter 2. From immunity to clonal selection 6cultures, gained particular attention from the official Western medicine, through theindependent influences of Cotton Mather [88], a Boston resident, and of Lady MaryMontagu [49], the wife of a British Ambassador in Constantinople. Several interestingtheories were raised as explanations for the phenomenon of “acquired immunity” againstsmallpox [128].The humoral theory of diseases was then still prevalent, as a remnant of the last twentycenturies belief that diseases originated from a maladjustment of the four humors: theblood, the phlegm, the yellow bile and the black bile. It was only iii 1858 that RudolfVirchow [143] suggested that all pathology is based on the malfunction of cells ratherthan the maladjustment of humors. His successful theory gave birth to cellular pathology.In the late 1870’s, modern bacteriology was founded, mainly as a result of the workof Robert Koch [78] and Louis Pasteur [104]. They showed that each infectious disease isproduced by its own specific pathogenic microorganism. Studies of these organisms bothin vitro and in vivo, led to the germ theory of disease. Pasteur [103] was able to induceacquired immunity to chicken cholera, but without being able to suggest an adequatemechanism that would account for it.Whether inflammatory responses were beneficial or detrimental to one’s health wasstill controversial. Because inflammatory responses often seemed to accompany infectiousdiseases, the latter theory was most favored. As microscopy and anatomical pathologydeveloped, macrophages and microphages were identified as the important cells presentin inflammatory responses and were thought by most pathologists to carry the infectiousorganisms throughout the body. Then, in 1884, the Russian zoologist Metchnikoff [93]proposed that phagocytic cells constitute the primary line of defense of an organism.He was immediately opposed by the supporters of humoral immunity, which were notable then to rigorously disprove him, But the discovery in 1888 by Nuttall [98] that theserum of normal antibodies has a natural toxicity for some microorganisms was the startChapter 2. From immunity to clonal selection 7of further investigations in that direction. During the same year, Metchnikoff becamechef de service at the Pasteur Institute where hostility toward scientists based at Koch’sInstitute in Berlin reigned. This feeling was inherited from a violent debate that tookplace between Louis Pasteur and Robert Koch since the late 1870’s, caused by the international politics that had not long ago opposed France to Germany. Each accused theother of scientific incompetence and refuted the validity of the other’s experiments, Thus,following this, a fierce and nationalist battle lasting for two decades ensued. Some scientists grouped around Metchnikoff and his phagocytic theory at the Pasteur Institute werein favour of cellular immunity. They claimed that the phagocytic and digestive powers ofthe macrophage and of the microphage constituted the main defense of the body againstinfection. Others based at the Koch’s Institute in Berlin under the leadership of Kochwere for huinoral immunity. They argued that invading pathogens could be immobilizedand destroyed only by the soluble substances of the blood and other body fluids. Eachgroup performed experiments designed to support their own theory. Metchnikoff and hisco-workers did numerous experiments to prove that the natural bactericidal powers ofthe sera of different species are often not related to the species’ susceptibility to infectionby a given organism. His opponents would continually report that bacteria could bekilled by the cell-free fluids of normal and especially of immunized animals. The discovery in 1890 by von Behring and Kitasato [144] that immunity to diptheria and tetanusis due to antibodies without obvious interactions of any cellular elements hit hard thecellular theory of immunity. Von Behring and Wernicke [145] observed soon after thatpassive transfer of immune serum protected a healthy organism from diptheria, withoutany involvement of any cellular activity. Pfeiffer [111] then found direct bacteriolysis ofcholera microorganisms by antibodies in guinea pigs, and shortly after it was found thatthis phenomenon could also be induced in a normal animal by injection of an immuneserum. And Bordet [12] would soon observe that even the erythrocyte could be killed byChapter 2. From immunity to clonal selection 8antibodies in the absence of phagocytes.More observations were made over the 1890’s that showed the importance of antibodies for immunity. As more antibodies of different specificities were being discovered andas Ehrlich [36] was able to isolate antibodies in a test tube via the recently discoveredprecipitin reaction [80], more immunologists turned towards the humoral theory, as theyfelt more comfortable with the antibodies than with the less manageable phagocytes. In1908, the Swedish Academy tried to reconcile both camps by confering a joint Nobelprize to Metchnikolf and Ehrlich, then leader of the humoral theory “in recognition oftheir work in immunity”. It is interesting to note that in England, Wright and Douglas[153] claimed that “both humoral and cellular functions were equally important and interdependent, in that humoral antibody interacts with its target microorganism to renderit more susceptible to phagocytosis by macrophages” But apart from them, the theoryfailed to find strong supporters, and in the following decades only rarely would someone explore the role of cells in immunity or the phenomena of “bacterial allergy” and of“delayed sensitivity” which had been observed. In the 1920’s and early 1930’s, Zinsser[155, 156], investigated bacterial allergy and Dienes and co-workers [35] got interestedin delayed hypersensitivity followed by two other groups [71, 130]. The importance ofinflammatory cells in tuberculin allergy was shown by Rich and Lewis [116]. In the mid1940’s, Harris, Ehrich and co-workers [52] studied extensively the role of lymphocytes inantibody formation.Only in the 1960’s was there a switch from a chemical to a more biological approachto immunology. It was then realized that the phenomena of allograft rejection, immunological tolerance, immunity in some viral infections, pathogenesis of autoallergic diseasesand those related to immunological deficiency diseases could not be explained withinthe framework of a theory based uniquely on humoral antibodies. This was the start ofan explosion of research in cellular immunity that would try to catch up with the lastChapter 2. From immunity to clonal selection 9decades of inaction in that domain.2.2 Theories of antibody formationThe discovery of humoral antitoxic antibodies in the early 1890’s was the start of atheoretical investigation that would last for about eighty years. At that time, nothing wasknown about the nature of toxins or antitoxins and little was known about the chemistryof biological macromolecules in general. The questions that immunologists would try toanswer were about the origin and formation of antibodies within the immunized host andabout the way that they would acquire their specificity.2.2.1 Antigen incorporation theoriesThe antigen incorporation theories appeared initially, with the first one proposed in1893 by the noted German bacteriologist Hans Buchner [19]. He suggested that theantitoxin was derived from the antigen. But new experiments [121, 77] would soondisprove the hypothesis by showing that the amount of antibody formed in the immunizedanimal was far greater than the amount of antigen utilized. Despite this, there wereseveral immunologists that continued to formulate theories to support the hypothesis[55, 85, 84, 114, 86, 87], and experiments to disprove it [54, 140, 63, 64, 10].2 2 2 Ehrlich’s side-chain theoryIn 1897, Paul Ehrlich [37] described the interaction of diphteria toxin and antitoxin andhow to measure it. He suggested that a unique stereochemical relationship between theactive sites of an antigen and its antibody was responsible for immunological specificity.He also introduced the concepts of “affinity” and of “functional domains” on the antibody molecule. At the same time, he presented his side-chain theory which relied onChapter 2. From immunity to clonal selection 10intracellular digestion, like Elie Metchnikoff’s earlier phagocytic theory of immunity. Hepostulated that every cell capable of synthesizing antibodies had on its surface “sidechains” of different specificities. The antigen would react with its specific side-chain andbe taken up by the cell. The side-chain could then perform its function anew or elseit could be regenerated. However, in presence of large or repeated amounts of antigen,the cell would overproduce the specific side-chains and the excess would be released asantitoxins into the blood. This theory already had all the requirements of a natural selection theory. When it was proposed, only a limited number of antibodies were known,so it seemed possible for a cell to have side-chains of all the required specificities onits surface. But within a few years, numerous other antibodies were found, discreditingthe side-chain theory. Figure 2.1 is one of the diagrams [38] that Ehrlich presented toillustrate his side-chain theory.2.2.3 Instruction theoriesThe instruction theories were based on the assumption that the amount of informationrequired for an antibody repertoire is so large that it cannot be contained inside the bodyand hence must come from the outside. The first theories of that kind were the directtemplate theories.Direct template theoriesIn 1909, Oskar Bail and his co-workers [6] suggested that the antigen is not eliminatedafter interaction with its specific antibody. It would then free the specific antibody topursue its function which is to bind to “natural antibodies” of the normal blood andthereby imprint these with its specificity. Theories built on the same basic idea were presented by other immunologists [5, 4, 138, 99, 100] until the beginning of the 1930’s. It wasthen known that proteins were made of some kind of assembly of 20-odd types of so-calledChapter 2. From immunity to clonal selection 11Figure 2.1: Ehrlich’s side-chain theory: “the autitoxines represent nothing more thallside-chains reproduced in excess during regeneration and therefore pushed off from theprotoplasma, and so coming to exist in a free state”. (Scanned from [38].)IpChapter 2. From immunity to clonal selection 12amino acids and that antibodies were globular proteins. A new theory was then proposedalmost simultaneously by various immunologists [141, 94, 3], but most conclusively byBreiril and Haurowitz [16, 53]. They suggested that the antigen travels to the site ofprotein synthesis and acts as a template for building up the nascent antibody molecule;further, through an unknown mechanism, the stereochemical structure of the antigenicsite determines a unique amino acid sequence, which results in the complementarity andspecificity of the interaction between the antibody and the antigen.In 1940, the interaction of complementary three-dimensional configurations of atomswas formally shown by Pauling and his students to be responsible for the specificity ofthe antibody-hapten interaction, as Paul Ehrlich had suggested long ago. Their bindingenergy was apparently a combination of ionic, hydrogen-bonding and van der Waalsinteractions. Following this, Pauling [106] suggested that the specificity of an antibodymust be due to its unique tertiary structure which is determined by a unique folding ofits peptide chain. The antigen would serve as a template only at the time of coiling ofthe nascent polypeptide chain of the antibody molecule and the resulting configurationwould be stabilized by familiar interatomic bonds. Figure 2.2 shows one of the diagramsthat Pauling used to illustrate his “theory of the structure and process of formation ofantibodies”. This theory was extended by Karush [73], who proposed that the uniquefolding of the peptide chain was stabilized by multiple disulfide bridge cross-linkages,which with their different extents would produce the variety of antibodies. He alsoargued that the linearity of any template that determines a primary amino acid sequenceis mandatory, which would invalidate the Breiril-Haurowitz theory.One major problem with the direct template theories, not even mentioned by itsproponents, was that they could not account for a greater and faster antibody responsedue to a second booster encounter with an antigen.Chapter 2. From immunity to clonal selection 13lvFOUP STAGES OF POSTULATED PROCESSOF FORM4TION OF GLOBULIN MOLECULEANTIGENfBAç•‘Cffl—.---SIX STAGES OF POSTULATED PROCESSOF FOdATION OF ANTIB)Y MOLECULESATURATIO4 OF ANTIGEN MOLECULEWITH IWNIBrnON OF ANTIBODY FORMATIONFigure 2.2: Diagram of Pauling’s theory of antibody formation, representing the fourstages in the process of formation of a molecule of normal serum globulin, the six stagesin the process of formation of an antibody molecule as the result of interaction of theglobulin polypeptide chain with an antigen molecule and the antigen molecule surroundedby attached molecules or parts of molecules and thus inhibited from further antibodyformation. (Scanned from [106].)ILiChapter 2. From immunity to clonal selection 14Indirect template theoriesIn 1941, Burnet [20] presented his “adaptive enzyme theory”, which later led to the indirect template theories, At the time it was thought that all proteins were broken downand synthesized by special proteinase enzymes. Some experiments had also suggestedthat “adaptive” enzymes could appear in response to special changes or needs of thebacterial organism. Burnet suggested that the antigen would reach the cells of the reticuloendothelial system and there get in contact with the local proteinases. Dissolutionof the antigen molecule would be accompanied by an adaptive transformation of thoseenzymes, which could then synthesize a globulin molecule specific to the antigen. Theinformation carried by those enzymes would be passed on to their daughter cells, thusenlarging the number of antigen-specific enzymes, and this would explain the strongerresponse obtained with large or repeated antigen injections. Furthermore, the theorywould account for the recent observation that the affinity of the antibodies would improve after these booster injections, since the enzyme would adapt at each injection andthus become more specific. This was a net advantage over the direct template theories.Towards the end of the 1940’s, adaptive enzymes were no longer popular. It wasknown that a “genome” of undetermined composition ruled proteins synthesis. So in1949, Burnet and Fenner [23] presented a new indirect template theory that would endowthe antigen the ability to imprint its specificity on the genome, thus transfering it to theantibody. The information would persist in the genome of a cell and be passed on tothe daughter cells, which would explain an enhanced secondary response. Reexposure toantigen would also improve the quality of the genocopy. Some experiments on grafts hadjust suggested the rationale of an immune system able to distinguish between self andnon-self. So Burnet and Fenner postulated that during development, the body elementsget “self-markers” which protect them from future attacks from the immune system.Chapter 2. From immunity to clonal selection 15By 1957, it was known that DNA is the carrier of genetic information. Schweetand Owen [123] presented then their “template-inducer” theory in which they proposedthat the antigen first changes the DNA of the globulin gene, giving somatically heritableinformation for the generation of a new RNA template to produce cells “primed” forspecific antibody formation. The antigen would furthermore act as an inducer on thosecells, stimulating the formation of many templates and enhanced antibody production.2.2.4 Selection theoriesThe first of the selection theories was presented by the physicist Jordan [72] in 1940. In hisquantum-mechanical resonance theory, he suggested that the antigen, during its partialdigestion inside the body, is divided into fragments which select and interact with naturalmolecules, their specific antibodies. Molecules having identical groups get attractedto each other through quantum mechanical resonance aild this leads to autocatalyticreproduction of the antibody molecule. The structure of the daughter molecules can bealtered during reproduction, which would account for the graded cross-reactions observedby Landsteiner [81]. His theory was attacked by Pauling [105] who stated that resonanceattractions were more likely between complementary molecules thall between identicalmolecules.In 1955, Niels Jerne [66] presented his natural selection theory. He postulated thatthe antigen, after interaction of its specific determinants with their natural specific atibodies, would carry the antibodies to specialized cells able to produce them. Theselected antibodies would induce synthesis of a specific RNA or change the structure ofthe existing RNA to allow for synthesis of more specific antibody molecules. The theoryaccounted for larger booster response and for improved affinity. Jerne also explainedimmunological tolerance by suggesting that during embryogenesis the first natural antibodies made against self-antigens are integrated by the body tissues and thus becomeChapter 2. From immunity to clonal selection 16absent as stimuli for further autoantibody formation. But the theory had a problemthat Talmage [136] pointed out, due to the so-called “central dogma” of genetics laterformalized by Francis Crick [25]. It was that the instruction for protein structure camefrom the DNA via the RNA to the protein and stayed there. This process could notbe reversed. This meant that none of the antigen or the antibody could directly informthe DNA on the production of a specific antibody. It could at most trigger an existingprogram. Ehrlich had already suggested this idea and Burnet and others returned to it.The clonal selection theory of antibodies was first presented in a general wayby Talmage [136] and Burnet [21], and then more extensively by Burnet [22], Talmage[137] and Ledeberg [82]. In his book published in 1959, Burnet suggested that natural receptors or antibodies possessing a unique specificity are situated on the surface ofevery lymphoid cell. The antigen interacts selectively with its specific receptor and triggers the differentiation of antibody production and proliferation to form daughter cellswhich constitute a clone of the same specificity. This already accounted for enhancedsecondary responses and changes in quality of the antibody, the latter possibly improvedby somatic mutations. Furthermore, Burnet explained self and acquired tolerance bypostulating that clonal precursor cells might be especially susceptible to lethal action oftheir respective antigens early in ontogeny, which eliminates them from the repertoire.The antigen being “sequestered” and thus being absent for clonal elimination, as well assomatic mutations were suggested as the precursors events of auto-immune diseases.Talmage theorized further about the roles of antigen selection and antigen-induced celldifferentiation. He studied particularly the specificity and the size of the antibody repertoire and proposed that a heterogeneous immune system might show a greater specificityfor an antigen than any of its constituent antibodies, because a distinct specificity wouldbe displayed by each set of cross-reacting antibodies. Lederberg examined the geneticinferences of the clonal selection theory and suggested that somatic mutations continueChapter 2. From immunity to clonal selection 17memory cellsantigenB—lymphocytes* *Figure 2.3: The clonal selection theory of antibodies: antigen binds to a receptor on thesurface of a lymphocyte bearing the right specificity and triggers its proliferation into aclone of lymphocytes of the same specificity; some of the daughter cells enlarge to becomeplasma cells that secrete antibodies of the same specificity, others are the memory cellsready to respond to further stimulation by antigens of the same specificity.*0 ****** *plasma cellsChapter 2. From immunity to clonal selection 18to occur after fetal life . Soon, the clonal selection theory became widely accepted in itsgeneral lines and is now considered to be the cornerstone of immunology.‘This idea of the somatic generation of antibody diversity would be the center of another importantdebate between supporters of the germ-line and the somatic theories, which would be concluded by bothsides making concessions.Chapter 3Immune network theories and models3.1 Jerne’s network hypothesisIn 1974, Jerne [68] reviewed the pertinent immunological data that was available andpostulated that the immune system functions as a network with a complexity comparableto that of the nervous system. To put his postulate in its context, let us make a summaryof some basic facts of immunology which were generally accepted at the time.It was known that the antibody is a protein molecule called immmunoglobulin (Ig)which is made up of four polypeptide chains: two identical light chains and two identicalheavy chains, joined together by various disulfide bonds. The domain structure of theIg molecule is illustrated in figure 3.4 borrowed from [50]. Both types of chains contain“constant” and “variable” regions. The five classes of immunoglobulins: IgG, 1gM, IgA,IgD and IgE are determined by the constant regions of the heavy chains. The greatdiversity of antibodies within each class comes from the differences among the variableregions (V regions). These constitute the antibody combining sites (paratopes) which canreact with the antigenic determinant (epitope). The importance of cells in immunity hadfinally been recognized, leading to the discovery that antibodies are actually producedby a class of white blood cells, the lymphocytes. There are two classes of these, namelythe B cells originating from the bone marrow and the T cells derived from the thymus.Both possess specific receptors capable of recognizing antigens. The receptors of the Bcells are immunoglobulins. Originally, the B cells are in their resting state and secrete19Chapter 3. Immune network theories and models 20CDRVLLHCLV11lCnlCHZCft3Figure 3.4: The domain structure of an Ig molecule. L and H represent the light andheavy chains respectively. CL and VL are the “constant” and “variable” regions of thelight chain. C11, CH2, CR3 are constant regions of the heavy chains. VH1 is thevariable region of the heavy chain. CDR’s are the complementarity determining regions,that is those regions that form the antigen-binding sites. (Scanned from [50].)Chapter 3. Immune network theories and models 21only a relatively small number of antibodies (mainly 1gM) which they also display ontheir surface as “receptors”. When the epitopes of antigens interact with paratopes ofreceptors, the B cells become either stimulated or suppressed. If they become stimulated,they begin to reproduce into a clone of cells all possessing the same specificity. Someof them enlarge and become “plasma” cells which begin to secrete large amounts ofantibodies (mainly IgG) into the blood. The remainder reverts to the resting state andbecome “memory” cells, ready to respond if the antigen reappears. If the B cells becomesuppressed, they are no longer capable of being stimulated. This can happen, for example,with very high injections of antigens (high zone tolerance) or low injections below thethreshold for stimulation (low zone tolerance).The basis of Jerne’s network hypothesis was that antibody molecules can recognizenot only antigens but also other antibody molecules, Antibodies formed against theantigenic determinants of other antibodies had been discovered experimentally almostsimultaneously by Jacques Oudin and Mauricette Michel [101] in France and by PhilipGell and Andrew Kelus [47] in England. They were called “anti-clone” antibodies byGell and Kelus. The term “idiotype” was used by Oudin [101] to denote the antigenicspecificities of antibodies produced by an individual or a group of individuals in responseto a given antigen: that is, an idiotype is the set of epitopes on the variable regions of aset of antibodies. Jerne named each single idiotypic epitope an “idiotope”. He thoughtthat antibody molecules possess combining sites called idiotopes which can be recognizedby paratopes of other antibody molecules, and viewed the immune system as a networkinvolving stimulatory and suppressive interactions between idiotopes and paratopes. Thesystem was normally maintained in equilibrium by the dominating suppressive interactions. Injection of an antigen into the system would disturb that equilibrium by givingmore stimulation to clones with certain idiotopes. These would in turn stimulate otherclones with anti-idiotopes specific to the idiotopes, and so on, leading to a chain reactionChapter 3. Immune network theories and models 22that could potentially spread throughout the whole network. If the system then settledinto a new equilibrium, this would represent immunological memory. Jerne [67] says:“I am convinced that the description of the immune system as a functional network oflymphocytes and antibody molecules is essential to its understanding, and that the network as a whole functions in a way that is peculiar to and characteristic of the internalinteractions of the elements of the immune system itself: it displays what I call an eigeribehaviour.” This was then almost the first time [129] that the production of antibodiesagainst self-antigen was suggested to be normal rather than exceptional.It is often forgotten that the main features of this hypothesis had already been putforward implicitly by Ehrlich in his side-chain theory in 1897. Furthermore, as Kossel[79] in Germany and Camus and Gley [24] in France had claimed having discovered anti-antibodies in 1898-9, Bordet [13, 14] on one side and Ehrlich and Morgenroth [39, 40,41] on the other side had performed more experiments and speculated further on thisdiscovery. They had been imitated by Alexandre Besredka [11], Pfeiffer and Friedberger[112], August von Wasserman [146], Hans Sachs [122] and others. By 1905, it had beenrealized that data had been misinterpreted, and so had died the first series of anti-antibodies theories. But even though they had been built on misinterpreted experiments,conceptually they had not differed much from what would be presented some seventyyears later by Jerne.3.2 First mathematical models of the network hypothesis3.2.1 IntroductionThe first models of Jerne’s hypothesis [69] appeared during the year that followed his 1974paper. Before examining them, it is appropriate to mention their precursor which wasone of the first mathematical model proposed to describe clonal selection and antibodyChapter 3. Immune network theories and models 23production, presented by Bell [7, 8, 9] in 1970-1971. His model can be understood fromthe point of view of a network, because the absence of any interactions between cells orantibodies is equivalent to a network with zero strength of connections. Bell made nodistinction between different classes of antibodies and ignored the T cell activity. Heassumed that only a few lymphocytes can respond to an antigen to different degreesdepending on the average properties of the association constants of their receptors forthe antigen. Upon stimulation by an antigen, the lymphocyte can become paralyzed,or divide into two proliferating lymphocytes, or divide into either two plasma cells or aplasma cell and a memory cell, this latter being a new target cell for antigen. Antibodiessecreted by plasma cells help to eliminate the antigen. This limited model embodiedinto a set of differential equations, accounted reasonably well for experimental antibodyresponses, for high and low zone tolerance and for changes in the affinity between alymphocyte and its stimulating antigen.3.2.2 Richter’s modelThe first model subsequent to Jerne’s hypothesis including network interactions waspresented by Richter [117] in 1975 and includes excitatory, suppressive and inhibitoryinteractions between the variable regions (V-regions) of the lymphocytes and of the antibodies. It ignores the differences between B and T cells and between antibody classes.An antigen stimulates a specific set of idiotypic antibodies which, in turn, stimulates a setof anti-idiotypic antibodies and so on. Negative feedback between idiotypic populationsis assumed to occur to limit the growth of the populations. This negative feedback canhappen only subsequent to the presence of stimulation by antigen. The set of differential equations representing the model can simulate low zone tolerance, a normal immuneresponse and high zone tolerance, as well as immunological memory. It simulates severalimportant features of immune responses and leads to testable expectations.Chapter 3. Immune network theories and models 243.2.3 Hoffmann’s modelThe second model to appear in 1975 was due to Hoffmann [56]. It will be examinedin more detail since we will use its latest version as a basis for our investigations. Anew feature of this model is that it gets rid of the distinction between paratopes andidiotopes by assuming that the same surface pattern that can recognize antigen canitself be recognized by the variable regions of other lymphocytes and antibodies. Moreprecisely, it is assumed that by averaging the three-dimensional shapes of all the Vregions that are recognized by an antigen, one should obtain a shape which is somewhatcomplementary to the shape of the epitope of the antigen. The system is thereforeapproximated by two sets of lymphocytes, the “positive” set that interacts with theantigen, and the “negative” set that interacts with the positive set. The symmetricalinteraction between the two sets is postulated to happen via cross-linking of receptors.Thus the positive set can stimulate the negative set to proliferate and vice versa.The difference between T and B cells is taken into account. T cells secrete monovalentspecific factors. Specific T cell factors produced by one set can block the receptors ofthe T and B cells of the other set and thus inhibit interactions of the two sets, B cellsof one set secrete antibody molecules which can kill, in association with complement, Tand B cells of the other set. This is called “complement-mediated cytotoxicity”. Oneantibody molecule of the 1gM type or two of the IgG type bound to the cell surfaceare required for this process. B cells switch from producing mainly 1gM molecules toproducing mainly IgG molecules during the course of an immune response. Effector cells(e.g macrophages), to which the constant region of the antigen specific T cell factor canbind, were postulated to be able to kill T and B cells of the complementary specificity.This was called “indirect cell-mediated cytotoxicity”.Four stable states of the system are postulated to be: a virgin state where populationsChapter 3. Immune network theories and models 25Virgin state () ()T, T_ 8, B_ c;: c— — — —Suppressed stateinrrImmune stateJ T_ 8_,“Anti-immune” state_____________@1 ;IFigure 3.5: The stable states of Hoffmann’s model. (Scanned from [56].)of both sets are low enough that there is not a significant amount of stimulation but thereis killing (linear), an immune state where the populations of positive cells is high andthe immune effector function kills the negative population, a suppressed state wherepopulations of T cells of both sets is high and where all stimulation is inhibited by a highlevel of monovalent factors, and finally an anti-immune state which is the converse of theimmune state. Figure 3.5 illustrates the stable states of the model.Hoffmann presents a relatively complex set of differential equations that takes intoaccount the theory described above and generates the steady-states. For simplification,he assumes that the concentration of antibodies is proportional to the concentration ofthe cells that produce them, thus allowing the use of only one variable to model both ofthese. The difference between B and T ceiis is taken into account by different functionsChapter 3. Immune network theories and models 26and kinetic constants. The antigen, by modulating the dynamics of its specific cells, takescare of the difference between cells of each specificity. The model also includes the twotypes of killing, cell and complement mediated, and for this latter more specifically thedifference between 1gM and IgG killing. The switch from 1gM production to IgG is alsoincluded. The theory also deals with with several other features of the immune systemsuch as self-tolerance, and suggests experiments.3.2.4 Adam-Weller’s modelDuring the same year of 1975, Adam and Weiler [2] presented a model which aimed atexplaining the generation of antibody diversity and of a large set of different lymphocyteclones in early ontogeny.3.3 Later modelsWe have just examined the very first models of Jerne’s network hypothesis that appearedduring the following year that is was proposed. Those first modelers pursued their investigations to different extents. Adam [1] published one more paper in 1978. Richter[118, 119] published two other papers in 1978. Hoffmann still continues his research inthe field. His work will be examined in more details in the next chapter.Many other researchers have investigated the network hypothesis. Some of the mostimportant ones are mentioned here.Kaufman, Urbain and Thomas [76] produced a model in 1985 which is based on a logical analysis of the immune response, using boolean variables. In 1987, the continuationof that paper is presented by Kauman and Thomas [75]. They analyse the same modelwith continuous variables this time. Kaufman, Urbain and Thomas have published moreafter that and are still continuing their studies.Chapter 3. Immune network theories and models 27De Boer [27, 281 presents his first model in 1988. His model differs from the previousones in that he does not consider suppression to be the important mechanism for maintaining homeostasis in the virgin state. He is continuing his investigations, in conjunctionwith others [29, 30, 31, 32, 33].Here, it is certainly not the aim to do an extensive review of all the literature onstudies of immune networks. But the names of some of the other researchers in the fieldthat have persevered over the years are given here for reference (in alphabetical order)with some of their papers: Avidan Neumann [95, 96], Alan Perelson [110, 109], Lee Segel[124], Dietrich Stauffer [131, 134, 133, 132, 131, 134], Francisco Varela with AntonioCoutinho and John Stewart [142, 135], Gerard Weisbuch [149, 151, 150]. It should bewell understood that this is not an exhaustive list.3.4 More on Hoffmann’s modelHere, we will follow the development of Hoffmann’s model in the following years. Thereason for doing so is that it will help the reader to understand gradually the apparentlycomplex form of the latest version, the N-dimensional network model presented in §3.5.A symmetrical two-dimensional model: the “plus-minus” modelIn 1979, Hoffmann [57] presented the “plus-minus” model that was aimed at a better understanding of the behaviour of the immune system near the steady-states rather than themore complex events that occur during the switching between them. But his postulateswere essentially the same. He was also more interested in a qualitative rather than quantitative understanding of the steady-states. So he made several further simplifications tothe ones already made in his previous model. First, he made no difference between theChapter 3. Immune network theories and models 28I•> Stimulation (aosslinkingof receptors)—> inhibition (blocking)Elimination (killing)Figure 3.6: The interactions between cells in the plus-minus model. (Scanned from [57].)T and 13 cells in the mathematical model, on the basis that at the level of approximation of the model the same selective forces act on T± cells and B cells, and similarlyfor T_ and B_ cells, where T+, B±, T_ and B_ cells represent T and B cells of thepositive and of the negative sets. Figure 3.6 illustrates the interactions between cells inthe “plus-minus” model. Accessory cells, previously termed effector cells, were assumedto be involved in the switching between the steady-states but not in the steady-statesthemselves. They were thus no longer considered in the mathematical model; similarly,the antigen did not enter the model. The concentration of specific I cell factors waschosen to be proportional to the product of the concentration of the cells that producethem and of the concentration of cells that. specifically stimulate those cells. And 011ccagain the concentration of antibodies was treated as being simply proportional to theconcentration of cells that produce antibodies.The “pius-minus” mathematical model is presented here since we will be using aChapter 3. Immune network theories and models 29version which is simply its extension to N dimensions. If one denotes by x and x_ theconcentrations of the positive and negative sets respectively, the differential equationsgoverning their dynamics are of the form:dx+ 2——= S+kix+x_ei+k2x+x ek3( _)e3—k4x+ (3.1)dx -= S+kix+x_ei+k2x+x_e +k3(x+)2x_e—k4x_ (3.2)1where eq= nq q = 1,2,3. (3.3)1 + (:-)The first term S models the constant non-specific influx of cells into the system. Thesecond term models the mutual stimulation of the cells. The third term denotes killingby killer T-cells and/or by 1gM plus complement. It is linear in the concentration of cellsof the opposite specificity and has a rate constant equal to k2. The fourth term modelsantibody-dependent cellular cytoxicity and/or killing by IgG plus complement, It has aquadratic dependence on the concentration of cells of the opposite specificity and has arate constant equal to k3. The last term models the natural non-specific death and hasa rate constant equal to k4.The eq terms model the specific inhibition of stimulation and killing by specific T-cellfactors (figure 3.7). The interaction between positive and negative cells and antibodiesare assumed to be inhibitable by both positive and negative specific T cell factors. TheCq are constants that specify the threshold values of the product x x_ at which theinhibition becomes effective, and the riq are constants that determine the sharpness ofthe thresholds. The product of the concentrations is used (rather than, say, the sum)because the concentration of T cell factors depends on the amount of stimulation betweenpositive and negative cells. The fact that 1gM production is more important in the virginstate than TgG production and vice versa in the immune state is incorporated in themodel by choosing different values for the Cq and the flq.Chapter 3. Immune network theories and models 30-Cq2Figure 3.7: The eq function for flq=7.Hoffmann finds parameters that lead to the postulated steady-states and performssome stability analysis on them. He finds that the steady-states are all attractors. Figure3.8 is a phase-plane diagram of Hoffmann’s symmetrical plus-minus model. He alsostudies the model when only one of the two 1gM or IgG killing terms is present and findsthat one does not then get all the steady-states or that they are then not all stable.It should be noted that in a paper published in 1980, Hoffmann [58] reviews theexperimental findings which support the idea of symmetric1interaction between idiotypesand anti-idiotypes of both T and B cells, cross-linking as the mechanism of interactionand the existence of specific T-cell factors.HofFmann’s next aim is to find the minimal model that leads to the postulated set ofstable-states. Gunther and Hoffmann [51] find that a minimal model does not require apositive stimulatory term (the term with rate constant equal to k1), but only the negativesuppressive ones.‘In 1984, Jerne [70] reviewed experimental data on idiotypic interactions and came to the conclusionthat they are indeed symmetrical.Chapter 3. Immune network theories and modelslog x_log x,31Figure 3.8: Phase-plane representation of Hoffmann’s plus-minus model. The virgin,immune, anti-immune and suppressed states are labeled V.S., I.S., A.I.S, and S.S. respectively. All four stable states are attractors, Parameter values are k1 = 0.1, k2 = 1,= 1, k4 = 0.01, S = 1, c1 = 10, c2 = 3, c3 = 0.3, n1 = 1, n2 = 2, n3 = 2. (Scannedfrom [57].)—- Ip £%%%,%.SChapter 3. Immune network theories and models 323.5 Hoffmann’s N-dimensional model3.5.1 The modelA first attempt to generalize the two dimensional “plus-minus” model to N dimensions appears in 1988, presented by Hoffmann et al. [60]. A second improved version by Hoffmarinet al. [61] appears later, which gets rid of an inconsistency with regard to the symmetry ofthe interaction, overlooked in the first version, but brings unnecessary complexity to themodel. The most recent and simplest version is achieved by Mathewson and Hoffmann[90] and is presented here, since it will used as a basis for further investigations.One denotes the population of a clone i as x. For a network of N clones, thedifferential equation governing the dynamics of clone i has the form:= S + k1 x U1 e1 — x1 U2 e2 — k3 x(U)2e3 — i = 1, N (3.4)where eq= flq q=1,2,3. (3.5)One can see that equation (3.4) now replaces equations (3.2) and (3.1), with various valuesof the index i now representing clones of various specificities (in the two dimensional case,the plus and minus specificities). The only difference is the term Uqi (where q = 1, 2, 3)instead of the concentration of clones of the other specificity (plus or minus). We alsonow have equation (3.5) instead of equation (3.3) with the term W’ replacing the previousproduct of the concentrations of plus and minus clones; the term Cq is used instead ofcq2 for simplicity.Consider the term To be consistent with the two-dimensional model, it mustrepresent a measure of the concentration of T cell factors. Before giving its value inthe N-dimensional case, let us first introduce other quantities that are relevant in an Ndimensional network of interacting clones. Clones in the network will interact with eachChapter 3. Immune network theories and models 33other with different strengths of interaction (affinities), depending on their specificities.One can denote by Kk3 the affinity between clones k and j. Studies of Hoffmann’sN-dimensional network have been done up to now only with Boolean affinities, thatis = 0 for no interaction between clones i and j, and K, 1 for an interactionbetween clones i and j. The implementation of non-Boolean affinities in Hoffmann’sN-dimensional model (as will be seen later) is one of the novelties of the work performedfor this thesis. Another useful quantity is the field of a clone k, given by:Nk=1,N. (3.6)j=1It is the sum of the concentrations of all clones j that interact with clone k, weightedby the strength of each interaction. A measure of the T cell concentration of a clone i isproposed as:NW=KkxkYk. (37)k=1One can see that this expression is a generalization of the two-dimensional one in thesense that it is proportional to the product of the concentrations of two clones for N = 2.Hoffmann et al. [61] found that this formulation was creating some problems in that, intheir numerical simulations, all the clones tended towards the suppressed state. So theyadded a refinement to this formulation. They postulated that an enzyme controls thetotal amount of T cell factors. It keeps the total amount of T cell factors constant, moreprecisely equal to N times the average of the threshold constants Cq. This can be doneby first calculating W, then multiplying it by a factor equal to the average of the twothreshold constants divided by the average of all the T cell factor concentrations, so thatthe new expression for the T cell factor concentration is given by:=(C2 +C3)/2 (3.8)±wiI i=1Chapter 3. Immune network theories and models 34The other term we need to look at is the Uqi. It is called the effective field and ismodelled by:Uqi =Kjjeqjxj q= 1,2,3. (3.9)The summation takes into account the fact that clone i potentially receives stimulationfrom all other clones j in the network. Each clone j can stimulate clone i more or lessefficiently depending on its specificity, which is taken into account by the affinity Kbetween clones i and j. This stimulation can be inhibited if the level of T cell factors ofclone j is too high, reflected by eqj in equation (3.9), and/or if the level of T cell factorsof clone i is too high, reflected by eqi in equation (3.4).A last comment is that in their recent simulations of the N-dimensional network,Hoffmann and his co-workers have used effectivities eqi with a very sharp threshold.They have found that the results were not significantly affected by this choice [91]. Wehave decided to keep the same choice in order to have results that can complement theirs.For sharp threshold, that is riq = x, the effectivities become:Ii ifWiCqeqi = (3.10)0 otherwise.3.5.2 The steady-states of the modelWe will consider the minimal model for which k1 = 0. The theory predicts three steadystates for the N-dimensional model: a virgin state where the production of 1gM antibodiesis dominant, an immune state where the producion of IgG antibodies becomes moreimportant and a suppressed state where the production of both types of antibodies isinhibited. The state of a clone is determined by W’, the amount of its T cell factors.Depending on the values of the threshold constants 02 and 03, there are two manners toobtain the predicted steady-states.Chapter 3. Immune network theories and models 35Case C2 > C3. A clone is in the virgin state if C3 < W < C2. The steady-state isreached when:Sk2Xvuv4vO. (3.11)The steady-state has the value:S= 1 IT I (3.12)2 (2v + 4A clone is in the immune state if W < C3 < C2. The steady-state is reached when:= 5—k2xU —k3x1U— k4x = 0. (3.13)The value of the steady-state is given by:Xi=kUkU2k• (3.14)A clone is in the suppressed state if C3 < C2 < The steady-state is reachedwhen:= S — k4 x3. (3.15)This yields:= (3.16)Case C2 < C3. A clone is in the virgin state if W < C2 < C3. The steady-state isreached when:5—k2xU—k3xU — k4x = 0. (3.17)The value of the steady-state is given by:XV=k2Uv+k3jv+4• (3.18)A clone in the immune state is characterized by 02 < W’ <03. Then the steady-stateis reached when:(3.19)Chapter 3. Immune network theories and models 36The steady-state has the value of:1 2 (3.20)I3 U3 + k4The suppressed state happens if 02 < C3 < T4/’, Then the steady-state is reachedwhen:= S—kx8. (3.21)This yields:=-. (3.22)At this moment, further analysis on the N-dimensional network model is still unpublished [90]. But it is available in the Master’s thesis of D. Mathewson [89]. In furtherwork that is presented in this thesis, only the case 02 < 03 will be considered.Chapter 4The affinity distribution4.1 The problem of the reconstruction of the affinity distributionIntroductionA great problem that theorists face when they try to model the immune network is thelack of experimental data available for the parameter values of their models. This concernseems to be most prominent in the determination of the distribution of the equilibriumbinding constants (affinities). The affinity K between two substances A and B is definedsuch that for any reaction:A + B AB, (4.23)one has that:K= [A][B] (4.24)It is known from experimental evidence that the distribution of the affinities between an“invading” antigen and the antibodies produced by the lymphocytes is heterogeneous andvaries with time under various circumstances [107, 42, 43, 48]. Many attempts have beenmade to reconstruct the affinity distribution from the relatively few measured values,which are available only for a finite number of free hapten concentrations. One majordrawback when one attempts to do so is that several solutions can fit the data.37Chapter 4. The affinity distribution 38Gaussian and Sips distributionsPauling et al. [107] tried a Gaussian distribution in 1944, followed by Eisen and Karush[42] in 1949. The Sips distribution was proposed by NisonofF and Pressman [97] in1958 and by Karush [74] in 1962. The affinity mean value and the variance of thosedistributions were estimated from the experimental data by non-linear regression. Theinterpretation of affinites for a small range can be done using the Gaussian and the Sipsdistribution but has been found to be inappropriate in general. One good reason for thisis that the affinity distribution is now known to be asymmetric and/or bimodal [152, 120].Fourier and Stieltjes transformsOther attempts have been made to reconstruct the affinity distribution without preassigning a specific distribution. An exact analytic inverse based on the Fourier transformwas proposed by Bowman and Aladjem [15]. It was never used in practice because itrequires measurement of the experimental data with an extreme precision over the wholerange of free hapten concentrations. Its advantage though was to provide a unique solution in the analytical sense. A similar approach was used by Bruni et al. [17, 18] usingthe Stieltjes transform. They offered a way to interpolate and extrapolate the data togenerate a complete binding curve for the inversion. The problem with this is that thereconstruction of the affinity distribution can be strongly influenced by the details andmanner of this completion. Hsu [65] suggested state-variable reduced order procedures.His method was also never used in practice because it requires both the complete bindingcurve and its derivative.Chapter 4. The affinity distribution 39Histograms and delta functionsOther approaches to the problem of the reconstruction of the affinity distribution includea method based on histograms by Werblin and Siskind [152] in 1972 and another oneon delta functions by Erwin and Aladjem [44] in 1976. The problem with these twomethods is that several histograms or delta distributions can be found to fit the limiteddata available.Minimum cross-entropyIn 1991, Yee [154] proposed a method based on minimum cross-entropy. It is a highlynon-linear inversion procedure that does not require any prior assumptions about theaffinity distribution or of parts of it. It apparently provides a good recovery of thedistribution from a very limited amount of data. At this point, it is not known whetherthis method has been implemented by experimentalists.4.2 Choices of connectivity matricesFrom the previous discussion of attempts to reconstruct the affinity distribution, it can beseen that no solid ground is available for immune network theorists to work on. Underthis uncertainty, they have chosen diverse connectivity matrices (matrices of affinitiesbetween the idiotypes of the immune network) for the simulations of their models.Experimental connectivityVarela et al. [142] have chosen affinities derived from experimental findings. They havegiven the affinities values 0 or 1 taken from connectivity matrices based on cross-reactivityas measured by ELISA assays.Chapter 4. The affinity distribution 40Random connectivityOthers such as Hoffmann [60], Parisi [102], De Boer [30] have used random and symmetrical connectivity matrices. Weisbuch [150] has used a random but asymmetricalconnectivity matrix. The reasons for choosing a random connectivity matrix are wellexpressed by Hoffmann [59], and by Weisbuch [150]. As HofFmann mentions, somaticmutation processes produce a considerable part of the immune system repertoire of Vregions (variable regions), which renders their exact knowledge impossible for a particularanimal. This means that the immune system must be constructed in such a way that itis not important which V-regions are present, as long as a very diverse repertoire exists,such that for each idiotype there are some matching anti-idiotypes. He continues bystating that the actual V-regions are not completely random since they are produced bycombinations and mutations of a finite set of germ line genes. It seems that in somestrains, particular “germ line” antibodies (or idiotypes) occur reproducibly. But considering that diversity has developed under various somatic mechanisms and especially thefact that repertoires can be mixed (by crossing strains, for example) without ill effect, onemay deduce that the stability of the network should not depend on a specific structureof the connectivity matrix. The reasons that Weisbuch mentions for choosing a randomconnectivity matrix complement the ones given by Hoffmann. He suggests that the use ofa random connectivity matrix is related to not knowing the real connectivity matrix Healso mentions it as a “good choice” since one searches for general properties that shouldnot rely in any critical way on a specific structure of the matrix which might vary fromone animal to the other.Chapter 4. The affinity distribution 41Bit-string complementaritySome have decided to model the affinities using the bit-string approach [45, 32]. Theyassociated the shape of the receptors with a binary string and used the degree of cornplementarity between the bitstrings to determine the affinity between the clones.Shape-space conceptA recently popular approach is to use the concept of shape-space first introduced in animmunological context by Perelson and Oster [110] in 1979. The affinity between tworeceptors depends on factors such as the average geometric shape of the binding regionand the intermolecular attractive forces (hydrogen bonding, electrostatic forces, van derWaals bonds and hydrophobic bonds). Perelson and Oster [110] have shown that a smallnumber (five to ten) of measurements of such factors is enough to describe adequatelythe “generalized shape” of the binding region.This approach was first used by Perelson and Segel [124] in 1988. Each idiotype is associated with a vector of real numbers that describes its generalized shape. They studieda one-dimensional shape-space, in which the variable x would describe, for example, thegeometric shape of the binding region or its electric charge. Figure 4.9 is a diagram thatthey used to illustrate their “idealized” one-dimensional shape-space. Perelson and Segelnumbered all the shapes from —N to N and assigned maximum complementarity to theshapes y = —x and x. They assumed that the smaller is x—(—y) x + y, the betteris the fit of shape x with shape y. To simulate this, they used a Gaussian distributioncentered around shape —x to represent the affinities of shape x with the other shapes.More will be said about this in the next chapter.In 1988, a shape-space for which no dimension needed to be specified was studiedby Percus [108]. Then, a high-dimensional shape-space restricted to a Bethe lattice wasChapter 4. The affinity distribution 42(a) (b)Figure 4,9: Two realizations of a one-dimensional shape-space. In figure (a), the shape isdefined by the height of wedge-shaped epitopes, positive for protuberances and negativefor indentations. In figure (b), the shape is defined as the charge of the homologouslylocated patches. (Scanned from [124].)used by Weisbuch et al. [151] ill 1990. Studies of models of the immune network intwo- and/or three-dimensional shape-space frameworks were done in 1990-1 by Weisbuch[150], Weinand [147, 148], and Stewart and Varela [135], and in 1992 by De Boer etal. [33]. Higher-dimensional studies have been dolle by Stauffer and Weisbuch [134].De Boer et al. [34] have studied a model of the immune network for which there wereonly two possible states for the clones. This was an incentive for Stauffer and Sahimi[133, 132, 131] to use techniques developed from statistical physics (for example, forIsing models) to simulate a high-dimensional shape-space (up to ten dimensions) with ahigh-dimensional network (10 clones).Chapter 5Simulations in one-dimensional shape-space5.1 IntroductionIt was said previously that Hoffmanri’s models had oniy been studied with Boolean affinities. It was thus of interest to insert non-Boolean affinities in his model. The followingstudies in shape-space are not meant to be extensive, but rather to be short incursionswhich give an idea of the interesting behaviour that can arise from the implementationof an immune network model in shape-space. Since this had never been done with Hoff-mann’s model, it seemed quite relevant. But in fact, the original reason for choosingaffinities of the shape-space was that an arbitrary choice of non-Boolean affinities wasrequired to test numerically certain new analytical work about the “distance coefficient”which will be presented in the last chapter of this work. So, the reader is asked for his(her) indulgence with regard to its demands toward a more extensive study of such aninteresting topic, that is the shape-space concept. He is reminded that this work is onlymade of brief incursions in several directions which call for further work.In the following simulations, networks of dimensions approximately equal to 20 willbe done since all previous work by others has been done for that dimension. Hoffmann’sN-dimensional network model presented in 3.5 is implemented with the following fixedparameter values: S = 10, k2 = 1, k3 = 0.01, k4 = 1 and others which will be specifiedwhen relevant.43Chapter 5. Simulations in one-dimensional shape-space 445.2 The + shape-space5.2.1 IntroductionHoffmarin’s N-dimensional model is first studied using the original idea of a shape-spaceproposed by Perelson and Segel (see §4.2). This shape-space will be referred to as the+ shape-space, since positive shapes are complementary to negative shapes in it, Forthe simulations, a network of dimension 21 is studied. First, the non-periodic case isconsidered, then the periodic case. The initial concentrations of the clones are chosento be uniform or random. Some parameters are modified to observe the effects on thedynamics of the network.5.2.2 The non-periodic ± shape-spaceThe clones are numbered from —N to N and positioned on a finite one-dimensional shape-space axis accordingly to their similarity with each other (figure 5.10). The distancebetween two clones determines their similarity. Immediate neighbours on shape-space(like clones x and x + 1) have the greatest similarity and are at an arbitrary distanceof S from one another. More distant clones are less similar. For example, clones —Nand N are the least similar. Clones of opposite sign (x and —x) are assigned maximumcomplementarity (maximum affinity). This is illustrated in figure 5.10 by dotted linesjoining the clones of maximum complementarity. The complementarity of clone x withother clones situated symmetrically on each side of clone —x decreases as these are beingfurther apart from clone —x. For example, the complementarity of clone x with clones—x— 1 or —x + 1 is the second greatest. To model this, De Boer et al. [33] have useda “discrete” Gaussian centered at x. The same will be done here. Note that this shapespace representation allows for different combinations of similarity and complementaritybetween clones. For example, clone 0 has maximum complementarity with itself, as wellChapter 5. Simulations in one-dimensional shape-space 45-N -34- —x OFigure 5.10: The non-periodic + shape-space. Clones are numbered from —N to N audpositioned accordingly to their similarity with each other. Neighbouring clones are themost similar and are separated by an arbitrary distance 6, Dotted lines join the clonesof maximum complementarity. Note that clone 0 has maximum complementarity withitself.Figure 5.11: The matrix of affinities of a 21- dimensional network, for the non-periodic± shape-space. The left and right pictures are two-dimensional and three-dimensionalrepresentations. Clones are nnmbered from —10 to 10 for both representations. In thetwo-dimensional representation, the shade of grey represents the value of the affinity witha scale going from white representing a maximum affinity of one and black representinga minimum affinity of zero. In the three-dimensional representation, the vertical axisrepresent the value of the affinity. Here, a = 6 and cut-off= 10—6.1•.‘.2; Ir-10-10-50510 In-10-5 0 5 10Chapter 5. Simulations in one-dimensional shape-space 46as maximum similarity. But clone N has maximum complementarity with clone— N andminimum similarity with itself.The affinity between clones i and j is denoted by K. It is of course equal to bysymmetry of the interaction between two clones. The 2N +1 x 2N + 1 matrix of affinitiesshown for N = 10 in figure 5.11 is based on the following relationship:K3 = k (2iru)112exp[—(i +j)2 6/2a2] i,j = —N,N. (5.25)where k is a normalization constant, S is the space between shapes on the one-dimensionalshape-space axis, and u2 is the variance of the Gaussian. In all simulations, the normalization constant k was selected such that the maximum value taken by the affinity wasone, that is k = 2Kuhhl’2. The space 6 was arbitrarily given the value 1. The only parameter that was varied was ci. To include non-periodic boundary conditions, the tails ofthe discrete Gaussians are truncated when they reach the boundaries of the shape-spaceaxis. An additional parameter not shown here which appears in the Fortran programthat generates the matrix of affinities is cut -ofl the smallest possible value of the affinity;whenever K3 < cut -off it is set equal to zero. Such a simplification has been done by DeBoer et aL [33]. The matrix of affinities has a band of Gaussian distributions that arecentered on the upper-right to lower-left diagonal.The non-periodic ± shape-space with random initial conditionsThe system is studied with random initial clone populations, more specifically for 0 <x < 1 at time t=0. The characteristic parameters of the Gaussian distribution ofaffinities are chosen to be: a = 4 and cut -off= iO. The other parameters of the modelare S = 10, k2 = 1, k3 = 0.01, k4 = 1, C2 = 0.6 and C3 = 0.7. The differentialequation (3.4) and its dependents is integrated until time t=100, which allows for allclone populations to reach complete equilibrium.Chapter 5. Simulations in one-dimensional shape-space 47Dynamics of the model Figures 5.12 to 5.17 show the dynamics for different variablesof the model. Figure 5.12 represents the dynamics of the clone concentration x, versusthe time t. At time t=0, the clone concentrations are randomly distributed betweeno and 1. They increase to a value greater than the equilibrium value, decrease belowit, and then increase again to reach it and settle there. At equilibrium, the clones havethree distinct ranges of populations and complementary clones have the same population.Each pair of complementary clones has its own value of the population. Clones are in thethree steady-states of equations (3.18), (3.20) and (3.22), that is the virgin, immune andsuppressed state. The steady-state that one can easily predict is the suppressed state,since it has a unique value of S/k4 = 10. In the suppressed state, clones have the highestpopulation. In the virgin state clones have various populations determined by the cubicequation in x (3.18. The populations are the lowest and are quite similar, although onlyeach pair of complementary clones has an identical population. In the immune state,clones can have various populations determined by the cubic equation in x (3.20). Forthis set of parameters, only one pair of complementary clones (with identical population)ended up in that state. Clones in the immune state have an intermediate population.The state of a clone is defined by the amount of T-cell factors W’ that it producesat equilibrium. This is shown in figure 5.13 which depicts the dynamics of the cloneconcentration x versus the T-cell factor concentration W’. At equilibrium, one can alsosee the three regions delimited by the values of C2 = 0.6 and of C3 = 0.7. Clones that havea low amount of T-cell factors (below 0.6) are virgin, those that have an intermediateamount of T-cell factors (between 0.6 and 0.7) are immune, and clones that have anelevated amount of T-cell factors above 0.7 are suppressed. This classification was usedto label the state of the clones in figure 5.12. At equilibrium complementary clones havethe same amount of T cell factors and each pair of complementary clones has its ownvalue of the T cell factors.Chapter 5. Simulations in one-dimensional shape-space 48Suppressed clones: -4,-3,-2,-i,0,l,2,3,4Immune clones: -5,54,2 Virgin clones: -lO,-9,-8,-7,-6,6,7,8,9,l00• I I IC 20 40 60 80 100TimeFigure 5.12: Clone concentration x versus time t for the non-periodic + shape-space,with random initial conditions. 0 < x < 1, S 10, k2 = 1, k3 = 0.01, k4 = 1, C2 = 0.6,C3 = 0.7, a = 4, cut -off= i0.Suppressed clones-4,4 -3,3 -2,2 -1,1 01086 Immune clones-5,5Virgin clones:01:2 0304 05 06O7 08 09 I ii 1.2T-cell factor WIFigure 5.13: Clone concentration x, versus T-cell factor W’ concentration for thenon-periodic + shape-space. Same parameter values as in figure 5.12.Chapter 5. Simulations in one-dimensional shape-space 4980Suppressed clones -2,270-3,3-4,460 Immune clones-5,550 Virgin clones-6,640-7,730-8,820-10,1010•V. I I I I0 2 6 8 10Clone concentration xFigure 5.14: Field Y versus clone concentration x for the non-periodic + shape-space.Same parameter values as in figure 5.12.080-2211Suppressed clones-3 370-4,460 Immune clones-5,550-6,6Virgin clones20 10008 09 I ll 1.2T-cell factor WIFigure 5.15: Field versus T-cell factor W’ concentration for the non-periodic +shape-space. Same parameter values as in figure 5.12.Chapter 5. Simulations in one-dimensional shape-space 50Virgin clones7linmune clones6 -8,8-9,9,-7,7l0,l0,-6,65-5,54-4,4-3,33-2,2—1,12 0Suppressed clones0 I I0 6 8 10Clone concentration xFigure 5.16: Effective field U2 versus clone x concentration for the non-periodic +shape-space. Same parameter values as in figure 5.12.12 Virgin clones-77-66 IsumuneclonesClone concentration x1Figure 5.17: Effective field U3 versus clone x concentration for the non-periodic +shape-space. Same parameter values as in figure 5.12.Chapter 5. Simulations in one-dimensional shape-space 51Figure 5.14 represents the dynamics of the field Y versus the clone concentrationx. At equilibrium, one can see the three regions of values of the field, This illustratesthe fact that clones with a low field are virgin , those with an intermediate field areimmune, and those with a high field are suppressed. Complementary clones have thesame field. Each pair of complementary clones has its own value of the field. The fieldsare also plotted against the T-cell factor concentrations in figure 5.15. Remember thatone is somewhat dependent on the other. The graph shows a “linear” configuration ofthe equilibrium points.The last variables of the model which can be of interest are the effective fields U2 andU3. They are depicted in figures 5.16 and 5.17 with respect to the clone concentration.The pattern exhibited is much more dramatic, with sharp corners in the trajectories,that is with major changes in the continuity of the, at times, smooth behaviour. Complementary clones have the same effective fields. Each pair of complementary clones hasits own value of effective fields.The interest here was more to examine the configuration of the clones in shape-space,so the graphs of the dynamics will be left for now without further analysis.Clusters in shape-space Clones are found to form clusters of the steady-states inshape-space (figure 5.18). The clones position themselves in the clusters such that clonezero is suppressed and is the center of the unique cluster of 9 suppressed clones. Then, oneach side of this cluster, there is an immune clone, separating the cluster of suppressedclones from the two clusters of 5 complementary virgin clones. Complementary shapesend up in the same steady-state and have the same steady-state population. Each pairof virgin and immune complementary clones has its own steady-state population. Allsuppressed clones have the same steady-state population, as was shown previously in§3.5.2. It is verified analytically in appendix B.2 that the differential equation (3.4) withChapter 5. Simulations in one-dimensional shape-space 52i/ V v’ i ‘ s”s& ssi”v’v ‘v ‘vI I I I I I I I I I I I-10 -9 -8 -7 -6 -5 -4 -3 -2 -l 0 1 2 3 4 5 6 7 8 9 10Figure 5.18: Formation of clusters of clones in the virgin(V), immune(I), suppressed(S)steady-states in the non-periodic + shape-space for a 21-dimensional network with random initial conditions. Complementary clones share the same steady-state concentration.There is a symmetry in the configuration of clusters. Same parameter values as in figure5.12, so a = 4 and cut-off= iO. This configuration is also the same for values of abetween 2 and 4.which the affinities obey the relationship (5.25) has a solution such that complementaryshapes have the same concentration.Note that clone 0 is at the center of the cluster of suppressed clones, i.e. those thathave the highest concentration. De Boer et al. [33] had observed that clone 0 autostimulates itself to grow because it has maximum complementarity with itself, and influencesthe other clones around it to grow. But their model involves positive stimulation termsrather than the negative suppressive terms of Hoffmann’s model. So their explanation isvalid in their case, but not in this one. In Hoffmann’s model, the scenario is different andless obvious. The amount of T cell factors that clone 0 has initially is high enough (seefigure 5.13) that both negative terms due to suppression by other clones are zero in thedifferential equation (3.4). Therefore, clone 0 grows until it reaches its maximum value inthe suppressed state. If one looks at the matrix of affinities displayed in figure 5.11, onerealizes that clone 0 is the one clone for which the truncation of the tails of the Gaussianimposed by the non-periodic boundary conditions is the least effective. As clones arepositioned further away in shape-space from clone 0, the influence of the truncation ofChapter 5. Simulations in one-dimensional shape-space 53the tails of the Gaussian increase. Clones —10 and 10 are the ones that have the smallestamount of interactions with other clones. As one can see from equations (3.6), (3.7)and (3.8), the amount of T-cell factors produced by a clone is strongly dependent on itsaffinities with other clones. So, even when considering the influence of the clone concentration on the computation of the T-cell factors, there should be a general tendency forthe clones near clone 0 to have the greatest amount of T-cell factors. This was actuallyseen to be the case in the different simulations which were done for various sets of initialconditions.Modify the random seed generator of the initial populations of clones Therandom seed that generates the initial clone populations was changed to different values.There was no observed change in the positions of the clusters. The values of the steady-states essentially remain the same.Modify the variance The variance was given different values from 2 to 7. Majorchanges were observed. The different configurations in shape-space obtained are depictedin figures 5.18 to 5.22. For o = 2 to 4, the configuration in shape-space was as shownin figure 5.18. It is made of clones —10 to —6 being virgin, clone —5 being immune,clone —4 to 4 being suppressed, clone 5 being immune, clones 6 to 10 being virgin.Complementary pairs of clones share the same steasy-state population in all cases. Aso is increased towards the value 4.5, there is a break in the symmetry. The concentration• of the complementary clones begins to differ slightly. As it reaches 4.5, the steady-statesituation becomes more asymmetric. The disparity between the concentrations of thecomplementary clones is greater, and clone —5 switches from being immune to beingsuppressed (figure 5.19). When u = 5, the clone —6 switches from being suppressed tobeing immune (see figure 5.20). As a reaches 5.5, the clone —5 returns to its originalChapter 5. Simulations in one-dimensional shape-space 54I I I I I I I I I I I I I-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10Figure 5.19: Same as in figure 5.18 but now a = 4.5. Clone —5 has become suppressed(S).The concentrations of complementary clones are rio longer identical./ •“ 5’ S, v ‘v ‘v ‘VI p I I I I I I I-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10Figure 5.20: Same as in figure 5.18 but now a = 5. Clone —6 has become immune(I).Concentrations of complementary clones are still not identical.I I I I I I I I-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10Figure 5.21: Same as in figure 5.18 but now a = 5.5. Clone —5 and 6 have becomeimmune(I) and clone 6 has become suppressed(S). Concentrations of complementaryclones are still not identical.Chapter 5. Simulations in one-dimensional shape-space 55I I I I I I I I I I I I I I I-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10Figure 5.22: Same as in figure 5.18 but now a = 6. Clone 5 has become immune(I),so that now clones —6, —5, 5 and 6 are immune(I). The situation is symmetric again:complementary clones have the same concentration and are in the same steady-state.The same situation holds for a = 7.immune state, clone 5 switches from an immune to a suppressed state and clone 6 switchesfrom a virgin to an immune state (see figure 5.21). Then, the concentrations of thecomplementary clones get more similar, until at a = 6 clone 5 returns to its originalimmune state (see figure 5.22). Now, the configuration of the clusters is symmetric andthe concentration of complementary shapes is identical again. And we have that theimmune clusters are comprised of two clones instead of one as we had at our point ourdeparture for a = 2. For a = 7, the same situation holds.The system has not been studied for greater or smaller values of the variance, so thereis left an open question as to whether one could have other regions of the parameter spacefor which the symmetry would break and then be retrieved with a different number ofimmune clones.5.2.3 The periodic + shape-spaceThe system is now studied with periodic boundary conditions on the shape-space, whichsimulate an infinite domain. There are different ways that one could do so. One couldset a period of 2N. This was the choice of De Boer et al. [33]. This means that thereChapter 5. Simulations in one-dimensional shape-space 56is no separation in shape-space between shape —N and shape N. A period of 2N + 1 ischosen here. Figure 5.23 shows the two different periodic shape-spaces. To indicate theperiodicity, “circular” diagrams of shape-spaces are now presented, that is the ends ofthe finite one-dimensional shape-space are joined. The matrix of affinities was computedwith a Fortran program based on equation (5.25) but including several modificationsto simulate the periodicity of the shape-space. It is shown in figure 5.24 for N = 10.Note that an implication of having periodic boundary conditions is that the sum of theaffinities of one clone with the whole network is the same for every clone in the network,that is:N= constant for each clone i. (5.26)j=iThe periodic ± shape-space with uniform initial conditionsThe system was first studied with uniform initial conditions for the clones concentrations. It is seen that the N-dimensional problem becomes one-dimensional because ofthe symmetry brought about by the fact that the clones in the system encounter identical stimulation. The analytical proof for this is presented in appendix A. The mainadvantage of using uniform initial clone populations is that it permits one to find a setof parameters that constitutes a point of departure for our simulations. In the case ofrandom initial clone populations, the system is too complex to allow for this finding Sothe parameters for all the simulations in this work were based on the ones found withthe procedure indicated in appendix A. The one-dimensional is not that interesting initself, since all clones reach the same steady-state. We cannot see any difference in theconfiguration of the clusters in shape-space when we change the value of the variance,for example. We therefore restrict ourselves to non-uniform initial conditions for the restof the numerical experiments.Chapter 5. Simulations in one-dimensional shape-space 570Figure 5.23: The + shape-space with periodicity 2N + 1 (left) and with periodicity 2N(right). Clones are positioned on the “circular” shape-space. Each clone is most similarto its nearest neighbours. Dotted lines join the clones of maximum complementarity.-10-50510-10 -5 0 5 10Figure 5.24: The matrix of affinities for the + shape-space, with periodicity 2N + 1, fora = 6 and cut-off= l0_6. Refer to figure 5.11 for explanations on the legend.0•rChapter 5. Simulations in one-dimensional shape-space 58—1 1-2 2_3,/VVY 7N3‘VV-5 I. 5—6 S S 6- S.— ‘•. ‘S 7S. ci ‘S—8 8—io 10Figure 5.25: Formation of clusters in the periodic + shape-space. This experiment wasdone for the exact same values of parameters as in figure 5.18, except for a different setof initial conditions. The only observed difference is the switch between clones in thevirgin(V) and suppressed(S) states.The periodic ± shape-space with random initial conditionsThe system is examined for exactly the same values of parameters as in figure 5.18 ofthe simulations of the non-periodic case. The new steady-state configuration attainedwas composed uniquely of immune clones, all with the same concentration. There was aspontaneous restoration of symmetry, such that the system somehow became effectivelyone-dimensional. The random seed generator of the initial conditions was changed to geta situation where all three steady-states are represented and complementary clones havethe same concentration. The new configuration in shape-space is shown in figure 5.25.The difference with figure 5.18 of the non-periodic + shape-space is simply a switch inthe positions of the virgin and suppressed states.In the non-periodic case, remember that there was an asymmetry with regard to thenumber of interactions that the different clones of the network would have with others.Chapter 5. Simulations in one-dimensional shape-space 59For example, clone 0 had the greatest number of interactions with others, In the periodiccase, every clone has the same number of interactions with other clones. So, one wouldnot expect the type of special configuration in the clusters that was brought by theasymmetry in the number of interactions. And indeed, for the exact same parametervalues as in figure 5.18, it was seen that by changing the boundary conditions fromnon-periodic to periodic, one would lose that special configuration. Surprisingly, simplychanging the set of initial conditions brings a special configuration again, although thistime, the configuration has “reversed” itself; the clones that were previously suppressedare now virgin and vice versa.5.3 A new shape-space without shape zero: the L shape-space5.3.1 IntroductionIn the + shape-space, one shape had maximum complementarity with itself, that is theshape zero. It was thus of interest to construct a new shape-space in which there wouldbe no shape zero and see how this would modify the configuration of the clusters ofsteady-states in shape-space. A new shape-space without shape zero is now presented.It will referred to as the L shape-space (where stands for the Greek symbol “delta”),to illustrate the fact that the complementarity in it is assigned by means of a constanttranslation.5.3.2 The non-periodic L shape-spaceIn the L shape-space, the clones are numberered from 1 to N, where N is an evennumber. Once again, the distance separating two clones on the shape-space determinestheir similarity. But this time complementarity is assigned differently. Each clone x is setmaximally complementary with the clone positioned at a distance equal to (1 + N/2)SChapter 5. Simulations in one-dimensional shape-space 60I I ri I r ii I I I I 1.Figure 5.26: The non-periodic shape-space. Clones are numbered from 1 to N (whereN is an even number) and positioned on the shape-space according to the same rulesof similarity with each other than for the ± shape-space. Note that the dotted linesthat join clones of maximum complementarity exhibit a different pattern than for the +shape-space.from itself. The position of this latter clone is thus the center of the Gaussian thatdetermines the affinities of clone x with other clones, as was the case in the + shapespace. Figure 5.26 depicts the L shape-space.Note that this particular version of shape-space is quite different from the ± shape-space depicted in figure 5.10. One can see that the dotted lines joining the clones ofmaximum complementarity exhibit a different pattern. As was observed in §5.2.2, inthe + shape-space there is a great flexibility about the degrees of complementarity andsimilarity that can be associated with two shapes. In particular, two shapes with a givendistance in shape-space and that have a maximum amount of complementarity can haveall degrees of similarity. But in the shape-space, the rules of complementarity assigna fixed level of similarity to all pairs of maximally complementary clones.One can also see the L shape-space as a reconstruction of the + shape-space in thefollowing way’. First, take the + shape-space and remove the shape zero. Then fold the1This information was given to me at the Discussion meeting for Statistical Physicists held at theSt. Francis-Xavier University on October 3rd, 1993 by D. Stauffer, who himself had obtained it from S.Dasgupta [26].Chapter 5. Simulations in one-dimensional shape-space 61I I I I I r ‘i-—N—‘16F-- —1 N 1Figure 5.27: The + shape-space with the shapes renumbered appropriately and omittingthe shape zero shows the same pattern of dotted lines that join clones of maximumcomplementarity as in the z\ shape-space.positive part on the negative part. That is, renumber the positive part by exchangingclone 1 with clone N, then clone 2 with clone N— 1, etc... The reconfiguration is shownin figure 5.27. One can see that the dotted lines indicating the complementary clones areidentical to the ones in the L shape-space.5.3,3 The periodic L shape-spaceWe now impose periodic boundary conditions on the Z shape-space, by joining the twoends of the shape-space together. We assign a periodicity equal to N + 1 so that shapes1 and N are now immediate neighbours on the shape-space. One of the consequences ofdoing this is that now clones of maximum complementarity are also clones of minimumsimilarity (or maximum dissimilarity). The periodic shape-space is shown in figure 5.28.The dotted lines that join the clones of maximum complementarity are now “diameters”of the periodic shape-space. The interactions between clones are symmetric, so theaffinity between two clones i and j denoted by is equal to To determine theN x N matrix of affinities one thus needs only specify the affinities K3 between clones iand j for which i = 1, N and j = i, N and then symmetrize to get the rest of the matrix.Chapter 5. Simulations in one-dimensional shape-space 62Figure 5.28: The z shape-space with periodicity N+ 1. Dotted lines that join the clonesof maximum complementarity are diameters of the “circular” shape-space and denotethat those clones are also maximally dissimilar.Figure 5.29: The matrix of affinities of a 20-dimensional network, for the shape-spacewith periodic boundary conditions. The left and right pictures are two-dimensional andthree-dimensional representations. Clones are numbered from 1 to 20 for both representations. In the two-dimensional representation, the shade of grey represents the value ofthe affinity with a scale going from white representing a maximum affinity of one andblack representing a minimum affinity of zero. In the three-dimensional representation,the vertical axis represent the value of the affinity. Here, a 6 and cut-off= 10—6.b0)rChapter 5. Simulations in one-dimensional shape-space 63So the affinities are defined by:= k (2a2)_h/2exp[_(i +— j)2 6/2a] (5.27)for i = 1, N and j i, N, where k is a normalization constant, 6 is the space betweenshapes on the one-dimensional shape-space axis, and a2 is the variance of the Gaussian.This expression is a modification on the one derived by De Boer et al. [331 for theirshape-space.The affinity matrix now has two bands of Gaussian distributions (in contrast to oneband for the ± shape-space), positioned symmetrically with respect to the upper-left tolower-right diagonal, and parallel to it. It is pictured in figure 5.29 for N = 10.The periodic Li shape-space with random initial conditionsThe Li periodic shape-space has been studied only for random initial clone populations.They were randomly chosen to be between 0 and 1. The characteristic parameters of theGaussian were a = 2 and cut -off= i0. A system of 20 clones is first examined.Dynamics of the model Figures 5.30 to 5.35 show an equivalent of figures 5.12 to5.17 of the dynamics of the different variables of the model. Figure 5.30 shows thedynamics of the clone concentrations x, versus the time t. It can be seen that thereis no great difference between figure 5.12 and figure 5.30 in that the values reachedat equilibrium are essentially the same. Some difference can be observed in the waythat the virgin and immune states are reached by the clones. In figure 5.12, the cloneconcentrations increased to a greater value than the equilibrium value, then decreasedbelow it, then increased until they reached it and finally settled there. In figure 5.30,the clone concentration increases to a greater value than the equilibrium value, thendecreases to this latter and settles there. At equilibrium, complementary clones N andChapter 5. Simulations in one-dimensional shape-space 64‘:.(1 Suppressed clones: 4,14,5,15,6,16,7,176 Immuneclones:3,13,8,185)05)CCU2 Virgin clones: 2,12,9,191,11,10,20I I I I0 20 40 60 80 100Time tFigure 5.30: Clone concentration x versus time t for the periodic z shape-space, withrandom initial conditions. 0 < x < 1, S = 10, k2 = 1, k3 = 0.01, k4 = 1, C2 0.62,C3 = 0.65, a = 2, cut -off= i0.Suppressed clones10 4,7,14,17 5,6,15,168C0Immune clones6 3,8,13,18C.)5)42 Virgin clones 2,9,12,1900.3 07 08 09T-cell factor W,’Figure 5.31: Clone x versus T-cell factor W’ concentrations for the periodicshape-space. Same parameter values as in figure 5.30.Chapter 5. Simulations in one-dimensional shape-space 65405,6,15,16.../Suppressed clones 4,7,14,1735Immune clones30 3,8,13,1825 Virgin clones2,9,12,19201,10,11,2015105V I I I I0 4 6 8 10Clone concentration xFigure 5.32: Field Y versus clone concentration x for the periodic L shape-space. Sameparameter values as in figure 5.30.405,6,l5,16FSuppressed clones4,7,14,17Immune clones30 3,8,13,18252,9,12,190T-cell factor W’’Figure 5.33: Field Y, versus T-cell factor W’ concentration for the periodic shape-space.Same parameter values as in figure 5.30.Chapter 5. Simulations in one-dimensional shape-space 6665 1,10,11,20// Virgin clones/ I 2,9,12,194:33Immune clones2 4,7,14,17Suppressed clones5,6,15,160 I I I0 2 6 8 10clone concentration xFigure 5.34: Effective field U2 versus clone x concentration for the periodic z\shape-space. Same parameter values as in figure 5.30.111,10,11,2010 2,9,12,19Immune clonesVzrgm clones9 3,8,13,188 4,7,14,17Suppressed clones6 5,6,15,16U320 I I I I0 4 6 8 10clone concentration xFigure 5.35: Effective field U3 versus clone x1 concentration for the periodic Ashape-space. Same parameter values as in figure 5.30.Chapter 5. Simulations in one-dimensional shape-space 67N + 10 (for N < 10) have the same steady-state concentration, e.g. clones 2 and 12 arevirgin with identical concentration.Figure 5.31 depicts the dynamics of the clone concentration x versus the T cell factor Wi’. At equilibrium, complementary clones have the same value of T cell factors.Comparing with figure 5.13, it can be seen though that rather than each pair of complementary clones having its own equilibrium value, there are two pairs of complementaryclones having the same value. Also, at time t=0, the range of values of the T cell factorsis narrower in figure 5.31.Figure 5.32 shows the dynamics of the field Y versus the clone concentration x.The same clusters of complementary clones that were sharing the same level of T cellfactors (refer to figure 5.31) are seen here to share the same field. Comparing with figure5.13, it can be seen that the behaviour is smoother here, that is not with sharp corners.Figure 5.33 shows the dynamics of the field Y versus the T cell factors W’. The greaterclustering ocurring here than in figure 5.15 is again obvious. And once more it can beobserved that there is a “linear” configuration of the equilibrium points.Figures 5.34 and 5.35 depict the effective fields U2 and U3 versus the clone concentration x. At equilibrium, the same clusters of complementary clones of the previousfigures are seen to have the same value of effective fields. The behaviour can also be seento be much smoother than in figure 5.16 and 5.17. Once again, we will go directly to thestudy of the configuration in shape-space without examining further the graphs of thedynamical variables of the model.The conclusions that can be drawn from comparing the non-periodic ± shape-spacewith the periodic L shape-space are: more clustering and a smoother behaviour in thelatter case.Chapter 5. Simulations in one-dimensional shape-space 68Clusters in shape-space Figure 5.38(right) shows the configuration in shape-spacefor a particular set of parameters. The clones end up in the three steady-states. Theyposition themselves in a very symmetrical manner. There are two clusters of four virginand of four suppressed clones, separated from each other by clusters of one immune clone.Complementary shapes are in the same steady-state and share the same concentration.It is shown in appendix B.l that this is a solution.Modifying the initial clone populations The random seed that generates theinitial clone population was changed many times. The only observed change was thatthe configuration of clusters rotates around the shape-space. The clone populationswere also given random values between 0 and 20 for the same random seed. There wasno significant change. The same steady-states were occuring with essentially the samepopulations. The configuration of clusters stayed the same.Modifying the values of the threshold constants Figures 5.36 to 5.39 show theconfigurations of the clones in shape-space as only the values of the threshold constantsC2 and C3 are modified. Figure 5.36 left shows the configuration when C2 = 0.35 andC3 = 0.6. All clones are in the immune state. The concentrations of the complementaryclones are not the same. For the rest of the figures, the value of C3 is kept fixed andthe value of C2 is increased up to 0.59. In figures 5.36 (right), 5.37 (left), 5.37 (right)and 5.38 (left), the value of C2 has been changed to 0.358, 0.36, 0.39 and 0.44. Onecan see that virgin clones appeared one by one consecutively at positions 8, 9, 7 and 6.They positioned themselves side by side until they formed a cluster of four clones. As C2reaches the value 0.45, the configuration of the shape-space changes drastically. Not onlydo we have one cluster of four virgin clones, but two of them, as well as two other clustersof four suppressed clones. These clusters are separated from each other by one immuneChapter 5. Simulations in one-dimensional shape-space 6920 2 20 219 3 19 318 7 j 4 18 7 ,r 4-L N. \ / - 1-5 N \ / N17 5 17 516 1 0- F 6 16 1-- - 0 1 6-7 15 7-1-— ,“ I \ ‘-. ‘I- -r ,‘ i ‘-.. v14 J ‘ \ L 8 14 \ 1. 8IT_I. II L13 9 13 912 10 12 10Figure 5.36: Formation of clusters of clones iii the virgiri(V), immiine(I) and suppressed(S) steady-states in the periodic L shape-space for a 20-dimensional networkwith random initial conditions. 0 < x < 1, S = 10, k2 = 1, /c3 0.01, k = 1, o = 2,cut-off= iO. Left: All clones are immune(I). Complementary clones do not share thesame concentration. 02 = 0.35 and 03 = 0.6. Right: Clone 8 has become virgin(V).02 = 0.358 and 03 = 0.6.20 1 2 20 1 219 3 19 318 i 4 18 i 7 ,1./ / _-I -L N. \ / / _-117 5 17 5161 - 0- IE 6 161 0- --1 615 I 7 15 V-1-’ ,/ / \ ‘\ V -1 / / \ N V14 .T J \ 8 14 1’ / ‘ \K 8III IJI13 9 13 912 10 12 10Figure 5.37: Same as in figure 5.36 excepted for the values of 02 and of 03. Left: Clone9 has become virgin. 02 = 0.36 and 03 = 0.6. Right: Clone 7 has become virgin.02 = 0.39 and 03 = 0.6.Chapter 5. Simulations in one-dimensional shape-space 7020 1 2 20 1 219 3 19 318 7 ,t 4 18 y. ‘1 Y/ i. \. \ / /17I5 17s161 -0-- V 6 168- --0- S 615 - V 15 - S.j 7 1 \ S / \ \ SZJ14 8 14 813 9 13 912 10 12 10Figure 5.38: Same as in figure 5.36 excepted for the values of C2 and of C3. Left: Clone 6has become virgin. C2 = 0.44 and C3 = 0.6. Right: Drastic change in the configuration.There are now clusters of four virgin and suppressed clones separated from each otherby immune clones. The situation is symmetric and complementary clones have the sameconcentration. C2 = 0.358 and C3 = 0.6.20 1 219 318 4\. \ / 717 516 5-- - 0 -S 615 -- S8’ ,/ / ‘N V14 S / 8Vyv.13 912 10Figure 5.39: Same as in figure 5.36 excepted that C2 = 0.59 and of C3 = 0.6. The immuneclones have disappeared and have been replaced by suppressed clones. The situation isstill symmetric and complementary shapes have the same concentration.Chapter 5. Simulations in one-dimensional shape-space 7122 1 221 320 419 .. ‘\ \ / ,1’ ,8\ / /7 8’18 6:O:: -17 -<-Z’’’- V“I ‘S16 g”‘ / \ \ 5” 815 914 1013 12 11Figure 5.40: Clusters in the periodic shape-space for a 22-dimensional network withrandom initial conditions. The configuration that exhibits all three steady-states hascomplementary clones sharing the same concentration and being in the same steady-state.The pattern is different than with the 20-dimensional network. Same parameter valuesas in figure 5.36 excepted for 02 = 0.65 and C3 = 0.7.clone. The situation is symmetric and the concentrations of complementary clones areidentical. It was discussed previously. Note furthermore that the original position ofour first cluster of virgin clones has not been kept! As 02 reaches the value of 0.59, theimmune clones have vanished and are replaced by suppressed clones (figure 5.39). Thesituation is again symmetric and complementary clones share the same concentrationstill. But note that the fact that we have a symmetric situation does not involve havingcomplementary shapes sharing the same concentration. In the first case all shapes werein the immune state, which can be seen as a symmetric situation, but they did not sharethe same concentration.Changing the dimension of the network The nice symmetry encountered inthe previous cases was thought to come from having chosen a number of 20 clones iii theChapter 5. Simulations in one-dimensional shape-space 72network which would bring that symmetrical distribution of the clusters. To test thathypothesis, the model was studied with other numbers of clones in the network.First, an experiment was done with 22 clones to keep the even number number ofclones. The result is shown in figure 5.40. And indeed, the system exhibits a differentpattern. Complementary shapes still exhibit the same concentration as it is to be expected (adding an even number of clones surely does not invalidate the demonstrationpresented in Appendix B.l), and there can still be three general types of steady-states.But the number of virgin and suppressed clones grouped in clusters has changed, andthose clusters are not always separated from each other by an immune clone. There stillare some immune clones separating some of the clusters, but one is not present and hasbeen replaced by a suppressed clone.The model was then studied with an odd number of clones, that is 21 clones. Eachshape was set complementary to another hypothetical one situated between the two mostcomplementary shapes, so that the expression giving the the affinities is:== k (2ira)’Iexp[(i— j + N/2 + 8/2)2 8/2g2] (5.28)where i = 1, N and j = i, N (N is an even number), with symmetrization of the matrix.The matrix of affinities shown in figure 5.41 is very similar to the one for an even numberof clones. One now has that each clone has maximum complementarity with two otherclones. For example, clone 1 has maximum affinity equal to 1 with clones 11 and 12, andthen a smaller affinity with clones 10 and 13, etc... The result of the simulation is shownin figure 5.42. It is found that the three actual shapes mentioned above ended up in thesame virgin or suppressed state as long as the first one is part of a cluster of virgin orsuppressed shapes. If the first shape ends up being iii the immune state, then the twomost complementary shapes are found to be one in the virgin state and the other in thesuppressed state. It is believed that the hypothetical complementary shape to the actualChapter 5. Simulations in one-dimensional shape-space 73159131721_____________________________1 5 9 13 17 21Figure 5.41: The matrix of affinities of a 21-dimensional network, for the periodic Ashape-space. It is very similar to the one for a network comprised of an even number ofclones. a = 6 and cut -off= 10_6,21 1 220 . 3in1:J18 S VV V17v V 6V15/V/,:iHt\’J’5.. 814 913 12 11 10Figure 5.42: Clusters in the periodic A shape-space for a 21-dimensional network withrandom initial conditions. Dotted lines of maximum complementarity show that eachclone is set complementary to a hypothetical clone situated between its most complementary clones. Same parameter values as in figure 5.36 excepted that C2 = 0.6 andC3 = 0.7.rflflChapter 5. Simulations in one-dimensional shape-space 74shape in the immune state would also be in the immune state. The new geometry ofthe shape-space is such that one has actually doubled the number of actual shapes (byadding to it an equal number of hypothetical shapes), so that one ends up with a totalnumber of an even number of clones. Then the demonstration presented in Appendix B.1also holds for the case when one is taking into account all the actual and hypotheticalshapes in it.This ends the studies of Hoffmann’s N-dimensional network in shape-space.Chapter 6The distance coefficient6.1 The concept of a distance coefficientIn 1989, Hoffmann and Tufaro [62] presented the concept of a “similarity coefficient”,which represents the extent to which two substances A and B are similar, as seen by athird substance C. The justification for defining such a parameter follows here. Biologists can measure experimentally the degree of complementarity of two proteins, whichis defined by the affinity K. Sometimes, one wants to know how similar proteins A andB are. To answer that question, one usually compares their sequences of amino acids andDNA. But this doesn’t tell much about other characteristics like the three-dimensionalshapes of those proteins, which can vary significantly even for a small change in the sequence. Furthermore, different observers can have diverging opinions about the similarityof two proteins. One example given by Hoffmann and Tufaro is about the similarity ofthe proteins of bovine serum albumin (BSA) and of mouse serum albumin (MSA). Theseproteins, which have similar three-dimensional structure, would be considered as dissimilar from the point of view of a mouse or of a cow’s immune system; but they would beseen as quite similar by a chicken’s immune system, since chicken anti-BSA is likely tocross-react with chicken anti-MSA, the reason being the close relatedness of mice andcows from the chicken’s phylogenetic point of view.Thus it is sensible to define the similarity of two substances in the context of athird one. This can be done in various ways. One way which has been suggested by75Chapter 6. The distance coefficient 76CC .‘ AnotB BnotAADBC -+ AandBFigure 6.43: The fractions of C that are taken into account in the Hoffmann-Tufaro’sdefinition (eq. 6.25) of the similarity coefficient. (Adapted from [62].)Hoffmann and Tufaro is to define the similarity coefficient between two substances A andB in the context of C as the ratio between the fraction of C that reacts with A and Bsimultaneously and the fraction of C that reacts with either A or B or both:S1ABC—*AandBL’where C —* A arid B denotes the fraction of C that reacts with A and B, C — A not Bdenotes the fraction of C that reacts with A but not with B, C — B not A denotesthe fraction of C that reacts with B but not with A, and C —* A and B denotes thefraction of C that reacts with both A and B. Note that this definition does not takeinto account the parts of C that interact with neither A or B. Figure 6.43 illustrates thedifferent fractions of C that are taken into account in this definition It can easily beseen that the similarity coefficient defined above is a number ranging from 0 to 1, since• the denominator contains the term in the numerator plus some others. It will be zero ifthere is no part of C interacting with some parts of A and B simultaneously, in whichcase substances A and B appear completely dissimilar to substance C. It will be one ifChapter 6. The distance coefficient 77some part of C interacts to the same extent with parts of A and B only but not witheach of them separately, meaning that they are considered as being totally similar fromthe point of view of C. Hoffmann and Tufaro describe experimental ways of measuringthis coefficient by absorption of sera against others, while being aware of the practicaldifficulties that can arise from it.They also introduce the “distance coefficient” which is defined as the converse of thesimilarity coefficient:D[A,B(C] = 1— S[A,BIC} (6.30)— C—*AnotB+C--*BnotA1C-*AnotB + C—*BnotA + C-*AandB’It also is a number ranging between 0 and 1, with properties opposite to those of thesimilarity coefficients’. They prove, using Venn diagrams, that the distance coefficientsfor sera are a metric (see Appendix C), one of the consequences being that they obeythe triangle inequality such that for any three sera X, Y and Z and a reagent C, one hasthat:D[X,YC] <D[YZC]+D[Z,XC]. (6.32)The three distances between three sera can be plotted by points on a plane, Theysuggest it to be very useful for different practical purposes such as the diagnosis of certaindiseases. Take for example the disease lupus. The diagnosis method using the distancecoefficients is illustrated in figure 6.44. One measures the distance coefficient betweenan average serum obtained from normal individuals and an average serum of individualsfor lupus, using a third reagent C. Each of the sera used to form the average sera alsohas a distance coefficient with each of those average sera. Ideally, when plotted in thedistance coefficients plane, the normal sera will have positions somewhere close to theaverage normal serum and the lupus sera will be close to the average lupus serum. TheyChapter 6. The distance coefficient 78Figure 6.44: Representation in the distance coefficient plane: a useful tool for the diagnosis of diseases. Nay and Lay are the average normal and lupus sera. N and L are theindividual normal and lupus sera that constitute the average sera. Plotting the distancesof all these sera in the distance coefficient plane is expected to result in two differentregions (illustrated with the semi-arcs). The region into which an unknown serum wouldbe plotted could help determine whether the unknown serum is normal or from a personwith iupus. (Adapted from [62].)Nay LayChapter 6. The distance coefficient 79then define separate regions which can be used to diagnose unknown sera, depending onwhere the latters are located in the distance coefficients plane.6.2 The first model of the similarity coefficientThe concept of similarity coefficient sees its first use in immune network modelling withHoffmann et al. [61]. Choosing the context C to be the set of N idiotypes in thenetwork, each one represented with a certain concentration, they ask how much any twoidiotypes i and j taken from inside the network look similar from the point of view ofthe network. To answer this question, they use the original definition of the similaritycoefficient presented by Hoffmann and Tufaro [62], which for this system becomes whatwill be called here the first definition of the similarity coefficient:S[i,jC] = . . C .‘ iandj ... (6.33)C—÷znotj + C—*jnoti. + C—*zandjThey also present another definition which this time takes into account the fractionof C that reacts with neither clones i nor j. The basic argument is that if some clonesof C react with neither clones i nor j, then this is an aspect of similarity of clones iand j. They define what will be called here the second definition of the similaritycoefficient as the fraction of C that reacts with both i and j plus the fraction of C thatreacts with neither of them, divided by the whole of C:not.C—*zandj + C—*orjS [z,1IC]=. (6.34)where C i or j represents the fraction of C that reacts with neither A nor B, andC is the total population of C. Figure 6.45 illustrates the new fractions that are takeninto account in this definition of the distance coefficients. Their work until then has beenusing only affinities which are dimensionless Boolean variables, i.e. if two antibodies i andj bind to each other the affinity K:3 = 0, and if they do not interact K, = 1. They haveChapter 6. The distance coefficient 80notC —* i orCC—iandjFigure 6.45: The new fractions of C that are taken into account in the second definition(eq. 6.30) of the similarity coefficient. (Adapted from [62])written down some equations for the similarity coefficients defined by expressions (6.33)and (6.34) using those affinities. It appears that one runs into problems when wanting touse those equations for non-Boolean affinities, i.e for 0 < 1 (where the K’s havebeen normalized and non-dimensionalized). To see this, let us review the formulationsof the similarity coefficient developed by Hoffmann et al. [61], using Boolean affinities.One considers a system of N clones. One can denote the population of a clone k byxk. One then has that:NXjk=1(6.35)where xk 0. One wants to know how much the idiotypes i and j are similar from thepoint of view of the network. It makes sense that they will look similar to C if cloneswithin C react to a similar extent with them. Taking a representative clone k of thesystem, one can write its contribution to the fraction of C that reacts simultaneouslywith clones i and j as the product of its affinity with each of them, weighted by itspopulation xk. This product is zero unless clone k reacts with both clones i and j.Chapter 6. The distance coefficient 81Including the contributions from all the clones in the system yields the expression:NC —* i andj = K K3k XIV. (6.36)One represents the other terms in the similarity coefficient expression in a similar manner.One can define the fraction of C that reacts with clone i but not with clone j as:NC —* i notj K (1— K3k) xk. (6.37)k=1Each term in the summatioll is zero unless clone k reacts with clone i and not with clonej. Identically, one gets the fraction of C that reacts with clone j but not with clone i as:NC —* j not i = K (1 — Kk) Xk. (6.38)k=1The fraction of C that reacts with neither clones i nor j is expressed by:C iorj=(1— Kk) (1— Kk) Xk. (6.39)The term multiplying xk in the summation is 1 only if both Kk and K3k are zero.Finally, one can write the first expression of the similarity coefficient using(6.33) and (6.36) to (6.39), after simplifications, as:NK Kk XkS[i,jC]= Nk=1 (6.40)(K + Kk — Kk Kk) Xkk=1The second expression of the similarity coefficient is given by (6.34), (6.35),(6.36) and (6.39), as:N(1— Kk—Ajk + 2 Kk Kk) XkS’[i,jC] = k=1 N . (6.41)Xkk=1Chapter 6. The distance coefficient 826.3 The limitations of the first model of the similarity coefficientLet us here analyze these formulations. First, note that expressions (6.36) to (6.39) arecorrectly defined for Boolean affinities since each clone can contribute uniquely to oneof the fractions of C. These fractions are all non-negative and symmetric with respectto clones i and j. Their maximum and minimum values are respectively Z- Xk and 0.It is also easy to verify that the summation of the different fractions of C does give thewhole population of C, using expressions (6.35), (6.36), (6.37), (6.38) and (6.39):C ‘ iandj + C ‘ inotj + C,‘j noti + CiorjN= [K K3k + ‘ik (1 — K1k) + ‘jk (1 — Kk) + (1 — Kk)(1— I(3k)] Xk= Xkk=1= C. (6.42)Taking a close look at the first expression of the similarity coefficient given by (6.40),one realizes that it cannot be used in two cases. It will be seen that they actually do notcause any problem, since one of them appears in a situation where one is not interestedin computmg the similarity coefficient of two clones in the context of a iletwork, and theother situation is non-existing for a network. If one looks at the denominator of S[i, j IC],one realizes that it can be zero if:(i) all clones have a population equal to zero, that is xk = 0 for all k’s,(ii) clones i and j are completely disconnected from the rest of the network,that is if Kk = ‘jk = 0 for all k’s.When situation (i) occurs, the network has vanished and there is no interest anymorefor computation of the similarity coefficients in the context of a non-existing network.Chapter 6. The distance coefficient 83Situation (ii) never appears since a clone considered to be part of a network must beconnected to at least another clone in the network. Clones which are completely disconnected from the network do not affect the dynamics of the network and are therefore notconsidered to be part of it. The denominator of S[i,jC] is thus always positive for alltimes at which one needs to compute the similarity coefficient of a network.One can arbitrarily use the concept of a distance coefficient to demonstrate that theprevious formulations of the similarity coefficient are inadequate for non-Boolean affinities. For this, only the first definition of the similarity coefficient (6.33) will be consideredas a basis for deriving an expression for the distance coefficient. The conclusion derivedfrom the following demonstration applies equally to the distance coefficient derived fromthe second definition of the similarity coefficient (6.34). The first definition of thedistance coefficient derived from (6.40) and (6.30) is given by:C—*inotj+C--*jnotiD[z,jC] = . . . ... (6.43)C—znotj + C—*jnotz + C—*zandjUsing (6.37), (6.38) and (6.39), one can obtain the first expression of the distancecoefficient:N(K + K3k — 2 Kk K3k) XkD[i,jjC] = ktzl (6.44)(K + Kk — Kk Kk) Xkk=1Sillce the denominator of D[i,jC] is the same as the one of S[i,jCj, it is also positive.As pointed out by Hoffmann and Tufaro [62], the distance coefficient of a clone withitself should be zero. This makes sense biologically and is part of the requirements forthe distance coefficient to form a metric. The distance coefficient of a clone i with itselfChapter 6. The distance coefficient 84is obtained by letting j = i in equation (6.44):N2 (Kk— Kk K2k) XkD[i,iC] = k=1 (6.45)(2 Kk— ‘ik Kk) xkk=1It can be zero for any network only if the affinity Kk is confined to values 0 or 1. One cansee a problem here. As long as one is dealing with affinities that are Boolean variables,one is fine using the formulations proposed by Hoffmann et al. [61], but as soon as onewants to use more realistic affinities whose values can range anywhere from 0 to 1, onehas to find another way to express the similarity and the distance coefficients. Since onlyBoolean affinities have been used before in Hoffmanri models, the previous formulationswere suitable. The second definition of the distance coefficient derived from (6.41)and (6.30) is given here for completeness:C—*inotj+C--jnotiD[z,jlCj= . (6.46)The second expression of the distance coefficient is thus given by:N(K + Kk — 2 Kk K3k) XkD’[i,jIC] = k=1 N (6.47)>Xkk=1Note that in all of the last four formulations of the similarity and distance coefficients(6.40), (6.41), (6.44) and (6.47), the denominators are positive, for reasons mentionedabove. The numerators are non-negative, since they consist of summations of the nonnegative fractions of C. The coefficients are numbers ranging between 0 and 1 since theirdenominators consist of their numerators plus some other terms.Chapter 6. The distance coefficient 856.4 The first model of the distance coefficient is a metricNote that the above demonstration has established the fact that, as long as one is usingBoolean affinities, the first expression of the distance coefficient (6.43), using expressions (6.36) to (6.39) satisfy at least one of the requirements for them to be a metric,that is the second criterion stated in Appendix C. Since the only difference between thefirst aild the second definition of the distance coefficient appears to be the denominator,it is easily seen that the second definition of the distance coefficient also satisfies thesecond criterion. It seems appropriate to determine whether the two definitions of thedistance coefficient also satisfy the other criteria. The first of the criteria requires thatthe distance coefficient is never negative. This surely is true since it is a number rangingbetween 0 and 1. The third criterion asks for its symmetry with respect to clones i andj. This certainly is respected since the distance coefficient is composed of symmetric expressions of the fractions of C. The fourth criterion specifies that the distance coefficientbe positive for distinct clones i and j. This is true since if clones i and j are distinctthen there exists at least one k for which Kk Kk, in which case the numerator ofD[i,jC] is positive. The fifth criterion requires the distance coefficients to obey thetriangle inequality. The proof of this is analogous to the proof provided by Hoffmannand Tufaro [62] for this criterion with respect to the original formulation of the distancecoefficient. The original notation of Hoffmann and Tufaro will also be modified in thisprocedure. The total population of C can be written as:NcC=> Xk. (6.48)k=1One can write down some expressions for the different fractions of C that react to differentextents with pairs of clones i, j and j, 1 and i, 1. The expressions that give the fractionsChapter 6. The distance coefficient 86of C that react to different extents with clones i and j are given by:Nlij C —* i andj = K K3k Xkk=1Nk=iN13 C—*jnoti=>Kk(1—Kjk)xkk=1Ciorj=(1—Kk)(l—I(k)xk (6.49)so that: lj + l + 13 + lj = c. The expressions that give the fractions of C that react todifferent extents with clones j and 1 are given by:Niii C—j and 1 = K3k ‘4k Xkk=1NC—jnotl=K3k(1—Kjk)xkk=1Ni1C—*lnoti=Klk(l—K3k)xk=1i1 Cjorl= (1—K3k) (—K1k) xk (6.50)so that: iji + i3 + i1 + i1 = c. The expressions that give the fractions of C that react todifferent extents with clones i and 1 are given by:NC —* 1 arid i = K Kk Xkk=1Nj C — i not 1 = Kk (1 — Kik) xkk=1NC — lrioti = Kik (1— Kk) Xkk=1j Clori= (1—Kik) (l—Kk) Xk (6.51)so that: ji +ji +i +i = c.Chapter 6. The distance coefficient 87One can rewrite each of the fractions previously defined in (6.49), (6.50) and (6.51)with respect to pairs of clones i,j and j, 1 and i, 1 as sums of other fractions with respectto clones i,j, 1. For this, one must first define those latter fractions:Nx C—+inotjnotl= Kk(l—Kk)(l—Klk)xkk=1Ny C—*jnotlnoti= Kk(1—Kk)(1—Klk)xkk=1Nz C_1lnotinotj=ZKlk(1—Kjk)(1_Kjk)xku C —* j and 1 not i = K,, KIk (1 — Kk) Xkk=1Nv C —* iandlnotj = Kk Kik (1 —Kk) Xkk=1w C .‘ i andj not 1 = Kk Kk (1 — Kik) xkt C —* i andj and 1 = Kk K3k Kik xkCiorjorl= (1—Kk) (1—Kk) (1—Klk) xk. (6.52)It can then easily be seen that fractions (6.52) are related to fractions (6.49) to (6.51) inthe following way:(w+t)+(x+v)+(y+u)+(z+s) = lj+li+lj+lj =c(u+t)+(y+w)+(z+v)+(x+s) =3i+i+ii+i1c(v + t) + (x + w) + (z + u) + (y + s) = ji + ii + j + j = c. (6.53)Now, consider the first expression of the distance coefficient (6.44). One can rewrite thefirst expressions of the distance coefficient for pairs of clones i,j and j, 1 and i, 1 that arein terms of the fractions (6.49) to (6.51 as first expressions for clones i,j, 1 that are inChapter 6. The distance coefficientterms of fractions (6.52):88D[i,jC] = li+lJii + ii + lii— (x+v)+(y+u)— (x+v)+(y+u)+(w+t)(6.54)D[j,IIC] = ij+ilii + ii + iii— (y+w)+(z+v)— (y+w)+(z+v)+(u+t)(6.55)D[i,lC] =Ji +31 +Jii= (x + w) + (z + u) (6.56)(x+w)+(z+u)+(v+t)It has already been shown by Hoffmann and Tufaro (see Appendix C) that expressions(6.54), (6.55) and (6.56) satisfy the triangle inequality.Now, consider the second expression of the distance coefficient (6.47). The onlydifference with the first expression (6.44) is in the denominator. Actually, the situationis even simpler now since the three expressions for the distance coefficients D[i,jC],D[j, lC] and D[i, lC] now have the same denominator, that is the whole of C. Then, itsuffices to demonstrate that the sum of the numerators for two clones is greater than thenumerator of one clone. So one one wants to prove that:(l+l)+(i+i1) > (ii+ii)[(x+v)+(y+u)}+[(y+w)+(z+v)] [(x+w)+(z+u)1.But this last equation is equivalent to say that 2 (v + y) 0, which is certainly true,since v, y > 0. Thus, it is seen here that the second expression of the distance coefficientis also satisfying the requirements for it to satisfy the triangle inequality.Chapter 6. The distance coefficient 896.5 The generalized distance coefficientThe following is a proposition for the generalization of the old expressions (6.40), (6.41),(6.44) and (6.47) towards their utilization with non-Boolean variables. It will be seenthat new expressions can be found for each of the fractions of C expressed in (6.36) to(6.39). Those expressions will not be truly describing these fractions, Remember that inthe Boolean case, a clone in the system contributes uniquely to one of the fractions of C.With non-Boolean affinities, the problem has to be approached differently. There are ofcourse several ways that this can be done. One simple way that one could think of to beable to use the old formulations with non-Boolean affinities is to set all affinities smallerthan a certain threshold equal to an affinity of zero and all affinities greater or equal tothat threshold equal to an affinity of one. But if one wants some finer “tuning”, then onehas to come up with another expression which achieves it. The one presented here waschosen because it was the simplest that came to mind that could be an extension of theBoolean case and for which the general behaviour was concordant with expectations. Inthis formulation, each clone will contribute partly to each of the different new “fractions”of C, depending on its degree of binding with the clones of C. Although it is understoodthat one does not deal with real fractions anymore, they will still be denoted as such,since they turn out to be real fractions in the Boolean case.A model is considered in which the affinity of the clone i V region (variable region)for the clone j V region is a variable that can take any value in the range 0 to 1, that is:0 1. Now one has more than two different degrees to which antibodies can bindto each other. As before, two antibodies can bind to a third one with the same degreeand thus appear similar to it. For example, if KIk = Kk = 0.3, one can say that clonesi and j bind to clone k with the same degree and that they should appear similar tothe latter. But if one has the case where Kk = 0.3 and Kk = 0.8, then from the pointChapter 6. The distance coefficient 90of view of clone k, clones i and j will appear to a certain point similar or dissimilar.The extreme case is the Boolean case, when Kk 0 and Kk = 1 (or the converse).Clone k would then see clones i and j as maximally dissimilar. Thus, the contributionto C — i and j is expected to depend not only on the degree of binding of each clone ior j to clone k (see later), but also on the difference between those. If the affinities Kkand Kk are “close” to each other, in other words if the difference between them is small,then one would expect to have a bigger contribution to the similarity of clones i and jthen if they were “far” apart.The following expression is proposed to represent the extent to which clones i and jsimilarly bind to clone k:similar extent = 1— Kk — K3k I . (6.57)Notice that if the difference between the affinities is small, i.e. if clones i and j bindsimilarly to clone Ic, then the extent is closer to 1. But if the difference is big then theextent is closer to 0. The similar extent is a function whose isoclines in the K3k versusKk plane are given by: K3k = Kk + s, where — 1 s 1. Its maximum and minimumvalues are respectively 1 and 0. One here would be tempted to define the contributionof clone k to C —* i and j as simply equivalent to the similar extent. And indeed itcould make sense that all the clones that react similarly to clones i and j should havethe same contribution. But consider a specific example, with clone m which has affinitiesKim = 0.1 and Kjm = 0.4, and clone n which has affinities = 0.6 and = 0.9. Thesimilar extent is the same for both clones m and n. Nevertheless, it seems reasonablethat clone n should have a greater contribution, since it binds more strongly with clonesi and j than does clone m. To account for that fact, one can multiply the similar extentby the product of the affinities KkKk. The final expression to describe the fraction ofChapter 6. The distance coefficient 91C that reacts “similarly” to clones i and j is then given by combining (6.36) with (6.57):C i andj=K Kk (1— I K — Kk I) Xk. (6.58)This surely is not a unique way that one could rewrite C —* i and j. No biologicalbasis indicates that one should have a cubic expression to describe it. But as was mentioned previously, this expression is essentially the simplest that is consistent with theold Boolean expression (6.36).In a similar manner, we can find new ways to express C —f i not j and C—k j not i.Consider the latter. One expects the converse to the C —* i andj case to be applicable interms of similarity of clones i and j with respect to clone k. For example, if Kk is closeto Kk, one wants the contribution to C —* j not i to be small. This gives the followingexpression for the extent to which clones i and j bind dissimilarly with clone k:dissimilar extent = — Kk . (6.59)The dissimilar extent has isoclines in the Ki. versus Kk plane that look the same asthe ones of the similar extent. Its maximum and minimum values respectively are 1 and0. Now, consider clone m with affinities Km = 0.2 and Kjm = 0.9 and clone n withaffinities = 0.9 and = 0.2. The dissimilar extent is the same for both clones mand n. However, clone m should have a greater contribution to C —* j not i than clonen, firstly because it binds more strongly with clone j and secondly because it binds lessstrongly with clone i. To represent this, one can multiply the dissimilar extent of a clonek with the product K3k(1 — Kk). Combining (6.38) and (6.59), one can then write:NC—*jnoti= K (1—Kk) Kk—Kk I xk. (6.60)k=1Identically, using (6.37) with (6.59) gives:NC —* i notj = K (1— K3k) — K,k I xk. (6.61)k=1Chapter 6. The distance coefficient 92To find the fraction of C that reacts with neither i nor j is not as intuitively simple.But one can use the argument that all the fractions of C have to add up to the wholepopulation of C to calculate it. This yields:C no,t Orj [1— KIk Kk — (KIk + Kk —3 K Kk) KIk— Kjkl] Xk, (&62)The term multiplying xk in the summation takes the value 1 if both K13 and K3k are 0,and the value 0 if they are both 1 or if one is 0 and the other is 1.An a priori conceivable alternative is that that the different fractions could be writtenmore simply in terms of the similar and dissimilar extents. In this view, one would havedefined C —* i and j as identical with the similar extent, and each of C — i not j andC —* j not i as identical with the dissimilar extent. C i or j would however then haveto take the negative value of—jK1k — Kk, which is not acceptable. As defined above,expressions (6.58) and (6.60) to (6.62) are all foil-negative and symmetric with respectto clones i and j. Their maximum and minimum values respectively are Xk and 0.The generalized first expression of the similarity coefficient can then be written, after a few simplifications, as:N> KlkKjk(l—Klk—KjkI)xkS[i,jC]= Nk=1. (6.63)[K1 Kk + (K1k + K3k — 3 K1 Kk) KIk — K3k I] Xkk=1Its counterpart defining the distance coefficient gives the generalized first expressionof the distance coefficientN(K + K3k — 2 KIk Kk) KIk — Kk XkD[i,jIC]= Nk=i. (6.64)[Kk K3k + (ik + K3k — 3 ‘ K) I K11 — Kk I] Xkk=1Similarly, the generalized second expression of the similarity coefficient is givenChapter 6. The distance coefficient 93by:N[1 + (2 KIk K3k Kk— Kk) Kk — KjkIJ XkS’[i,jC] = k=1N . (6.65)Exkk=And the generalized second expression of the distance coefficient is:N[(K1, + Kk — 2 Kk Kk) Kk — Kk] XkD’[i,jC] = k=1N . (6.66)Xkk=1Note that all the above new expressions are generalizations of the old expressions in thesense that they reduce to the old expressions when one uses Boolean affinities.Since expressions (6.63) to (6.66) are derived from expressions (6.58) and (6.60) to(6.62), they are numbers ranging between 0 and 1. Therefore, the distance coefficient isnever negative. The dissimilar extent of a clone with itself being zero, so is the distancecoefficient of a clone with itself. Since the expressions of the fractions of C are symmetricwith respect to clones i and j, the distance coefficient derived from them is symmetrictoo with respect to those clones. For two clones i and j to be distinct, there must be atleast one k for which Kk K3k. Within these conditions, D[i,jIC] is always positive.The first four criteria of the metric are thus fulfilled. As for the fifth criterion which isthe triangle inequality, it still needs be verified analytically. But it was found numericallythat for all the studied cases, the generalized distance coefficients did satisfy the triangleinequality, that is, one could always plot the distance coefficients of all clones in thenetwork (which was the primary goal).6.6 The distance coefficients planeIn this section, it will be demonstrated how one can plot the positions of the clones in adistance coefficient plane, once one has calculated the distance coefficients of all clonesChapter 6. The distance coefficient 94xFigure 6.46: The distance coefficients planewith respect to two arbitrary reference clones.In the XY plane, choose the X-axis as the reference axis, Pick the two reference clonesi and j. Position clone i at the origin and clone j at a distance D[i,jC] from clone i onthe X-axis. One has thereby set the reference distance. Now one wants to determine theposition of clone k in the plane, with respect to the reference clones. Clone k could bepositioned anywhere on a circle of radius D[i, kC] centered on clone i at the origin. Theequation describing the circle is given by:x2 + y2 = D[i, kb]2. (6.67)Clone k also has to be positioned anywhere on a circle of radius D[k,jbC] centeredon clone j, described by:(x—D[i,jC])2+ y2 = D[j, kbC]2. (6.68)Chapter 6. The distance coefficient 95Solving for both equations simultaneously gives the position of clone k in the plane:— D[i, kid2— D[k,jiC]2+ D[i,jiC]2 6 69— 2 D[i,jiC]y = (D[i, kid]2 — x2)” (6.70)where the positive square root is chosen. Since the two circles intersect symmetrically attwo different places above and below the X-axis, one can choose either.6.7 Simulations of the distance coefficient6.7,1 Affinities from the periodic L shape-spaceTo test the generalized formulations of the distance coefficient, one can begin by choosinga particular set of affinities relating the clones in the network. Affinities derived from theshape-space with periodic boundary conditions were chosen. A network of 20 cloneswas studied.Uniform initial conditions The system with uniform initial conditions is one-dimensionalas described in appendix A. The distance coefficients can be plotted independently ofthe populations of the clones in this system. Consider the expression (6.64) giving thedistance coefficients. For the uniform initial conditions prevalent in the one-dimensionalsystem, the concentrations of all the clones are identical, and can be factorized in front ofthe summations. They then cancel each other in the numerator and in the denominator,and one gets the following expression for the distance coefficient:N(K. + K3k — 2 K<, K3k) Kk — ID[i,jIC]= Nk=i. (6.71)[Kg,, K3k + (Kk + K3k — 3 K1 K3k) I Kk — ‘jk I]k=1Chapter 6. The distance coefficient 96Thus in this particular system, the distance coefficient is constant in time and does notdepend on the change of the populations of the clones. For a shape-space with a highlevel of symmetry, the individual positions of the clones in a distance coefficients plotwould be expected to have a corresponding degree of symmetry. If one chooses twocomplementary shapes as reference shapes, one would expect that all the other clonesare positioned in a symmetrical way with respect to them. For example, here clones 1and 11 are chosen as reference clones. Then clones 6 and 16 are “perpendicular” to themin the circular shape-space (see figure 5.39); they are the most dissimilar from clones 1and 11 for this shape-space. They are thus most “distant” from clones 1 and 11 in thedistance coefficient representation. This is shown in figure 6.47, which shows the distancecoefficient representation of all 20 clones with complementary clones 1 and 11 as referenceshapes. The two dashed curves are part of two circles of radius 1 and of centers equalto the positions of the reference shapes. Thus, the distance coefficients have a maximumvalue of 1, as was shown in §6.5.Random initial conditions The system was studied with random initial conditions,0 < x < 1. A fixed set of initial conditions and a fixed value of the variance wasgiven, but the value of the cut-off was varied. The values of the threshold constantsC2 and C3 were chosen so that in all cases, the three steady-states are present and thecomplementary shapes have the same concentrations. The other values of the parametersare the same as for all previous simulations in shape-space: S = 10, k2 = 1, k3 = 0.01,k4 = 1.Figures 6.48 to 6.51 show the distance coefficient dynamics for the same variance= 3 and different values of the cut-off and of the threshold constants C2 and C3.The initial positions of the clones are represented with the “stars”. Their trajectories intime are along the full lines. Their final positions at equilibrium is depicted by one ofChapter 6. The distance coefficient 971- -.;-.-.-. -.sigma=2-- . - -- clones 0cut-off=0.0i .- sqrt(l-x’2) - - -.sqrt(l-(x-i)”2)— - —.6,16 0,j’ 5,17 7,15 ‘.,,8,14ø\,3,i9 9,13•2,20 lO,120_i l10 1Figure 6.47: The distance coefficient representation for the periodic z shape-space, withuniform initial conditions. The positions of the clones are fixed in time and are symmetrically positioned with respect to complementary refereilce clones 1 and 11. x = 0.1,S = 10, k2 = 1, k3 = 0.01, k4 = 1, C2 = 0.6, C3 = 0.7, a = 2, cut -off= iO.Chapter 6. The distance coefficient 98sigma=3 _- a’ *trajectoriescut-off=0.9 4,5 6.7,8,14,15,1 6,1718 virgin 04,&6P,8,14,506,17,18 immune Ela suppressed A19 — sqrt{1xA2) — - —sqrt{1-(x-d)”2) — —/. 1020 N2711 d 1111Figure 6.48: The distance coefficient representation for the periodic A shape-space ofa 20-dimensional network with random initial conditions. d is the distance betweenreference clones 1 and 11. S = 10, k = 1, Ic3 = 0.01, k = 1. a = 3, cnt-off= 0.9,C2 = 0.45, C3 0.65.Chapter 6. The distance coefficient 991— — —. -— —sigma—3— —— initial *—— trajectoriescut-off=0.5 — — virgin 0— immune Elsuppressed t18 — 8 sqrt(1-x”2) — - —sqrt{1 (x d)A2) — —191493/ 1331o2O13\20’1o2 1212ii d 1111Figure 6.49: Same as in figure 6.48 but o = 3, cut -off= 0.5, C2 = 0.65, C3 = 0.7.Chapter 6. The distance coefficient 1001—sigma3— ——‘ initial *trajectories —cut-off=O.1 —. virgin G-. immune 0%—suppressedsqrt(1-x’2) — —sqit(l-(x-d)’2) — —/// 17/•1/// 18I. 5‘4/‘ 18I/ 19 \14\/ 41• 31 1913\10202 12!1211 d liiiFigure 6.50: Same as in figure 6.48 but o = 3, cut -off= 0,1, 02 = 0.65, C3 = 0.7.Chapter 6. The distance coefficient 1011— — -—sigma=3 iilltidJ *trajectories —cut-off=O.O1-- virgin 0- immune El—-.. suppressed Asqrt(1xA2)—- —sqit(1-(x-dY’2) — . —/ S.S.// S.1617 N.-S.6 -‘/65S///ø13\120 0 10 -2 2 l2’12ii d 1111o. K1Figure 6.51: Same as ill figure 6.48 but a = 3, cut -off= 0.01, C2 = 0.65, C3 = 0.68.Chapter 6. The distance coefficient 102the three different symbols that identifies their final state: virgin, immune or suppressed.The dashed curves are part of two circles of centers equal to the final positions of thereference clones 1 and 11, and of radius 1 (the formula that generates the curve for clone11 contains the term d, which is equal to the distance between the final positions of clones1 and 11).It is observed that the similarity computed by the distance coefficient is here alsoquite in accordance with the one which is implicit in the periodic L shape-space. Forexample, clones 2 and 20 are neighbours of clone 1 in shape-space. They also are in thedistance coefficient representation. Also, there seems to be a tendency for suppressedclones to move less in the distance coefficient plane than the virgin or immune clones.It can be seen that for a wide Gaussian (which is the case in figure 6.48 where cutoff=O.O1, the dynamics are happening along a wide arc. But for a narrower Gaussian(for example, in figure 6.49, the cut-off is 0.1), the dynamics happen along a narrowerband.The system was also studied for a smaller value of the variance, that is o = 2. It wasfound that the distances between clones are thell bigger (not shown).6.7.2 Random boolean affinitiesOne could easily observe that in the previous cases where one used affinities computedin relation to a one-dimensional shape-space, the positions of the clones in the distancecoefficients were observing a somewhat smooth pattern along a more or less wide semiarc. One can imagine that this regular pattern arises from the particu’ar choice of theaffinities. To test this hypothesis, one can make a simulation with random Booleanaffinities. The matrix of affinities is shown in figure 6.52. The simulation was done for a22-dimensional network and random initial conditions, with random Boolean affinities.It is shown in figure 6.53 and does indeed verify that the previous regular pattern aroseChapter 6. The distance coefficient 1031471013161922_________________________________1 4 7 10 13 16 19 22Figure 6.52: The two-dimensional representation of a matrix of affinities for a 22-dimensional network, with random Boolean affinities. Clones are numbered from 1 to 22. Inthe two-dimensional representation, the squares shaded white represent an aiffinity valueof one and the ones shaded black represent an affinity value of zero.from the particular choice of the affinities computed in the shape-space context. It canalso be noticed that in the last simulation of the distance coefficient using affinities fromthe shape-space, that is with u = 0.9, there were many clones that would be consideredas maximally dissimilar from reference clones 1 and 11. This arose from the fact that theclones far away from the reference clones in shape-space were not very much connectedto other clones. In the random Boolean case, depending on the connectance (percentageof ones in the matrix), one can have more or less clones maximally distant from thereference clones. For a smaller connectance, most clones are not very much connectedwith others, and thus there is a great probability that they would be considered asmaximally dissimilar from other clones, including the reference clones. As one increasesthe connectance, there are more connections established. This improves the chances ofhaving clones being more similar from the point of view of the network.• ••• •• •• I••• •• • •• •••• U • •• • •• U •U• U• • •• • •••• • ••••U. • U U••• .•.•• U • UUU... U•. •• • •••U.. ••.... U • •Chapter 6. The distance coefficient 104initia1K—‘ trajectories —virgin 0immune El2Q&12sup:E’NSI sqrt{1(x1)A2) — - —,1 11,17,f1AFigure 6.53: The distance coefficient representation for a 22-dimensional network of clonesrandomly connected with Boolean affinities. One can see that the regular pattern thatwas exhibited for the affinities derived from a one-dimensional shape-space is not presentanymore. 0 <x < 1, S = 10, k2 = 1, k3 = 10, k 1, C2 = 10, C3 = 3.Chapter 6. The distance coefficient 105So the similarity between this simulation and the one done in shape-space is thatas one increases the connectance of the matrix or the variance of the Gaussian, onegets closer to the mean-field (all clones are connected to each other in the same way).And then they should all appear to be equally similar (this would be the case if theconcentrations of the clones were all identical). In the other extreme case, one couldhave a situation where there is an almost null connectance or a very narrow variance ofthe Gaussian. And then almost all clones would be considered as greatly dissimilar toeach other.This ends the work presented in this thesis.Chapter 7ConclusionThis thesis has covered a wide range of subjects. It started by recalling the first observations of immunity which gave rise to the theories of acquired immunity, then presentedthe theories of antibody formation which culminated in the theory of clonal selection.This was important so that the reader had some background for the understanding ofthe immune network hypothesis which pushed so many theorists to show interest inmodelling the immune network. Hoffmann’s N-dimensional model was then presented.It was said that this model had never been studied with affinities other than Boolean.So some non-Boolean possible matrices were presented. A particular one based on theshape-space formulated by Segel and Perelson was chosen. This led to the creation of anew shape-space without shape zero. Some simulations and analysis of Hoffmann’s N-dimensional network model with non-Boolean affinities derived from those two versionsof shape-space were shown. And finally the main part could be presented, that is thereview, analysis and generalization of the distance coefficient in terms of non-Booleanaffinities, as well as numerical simulations of the distance coefficients of clones of animmune network modelled by Hoffmann’s N-dimensional network model.A summary of the new material that was presented in this thesis can be made. Itconsists of implementing non-Boolean affinities in HofFmann’s model, more specificallyaffinities from a shape-space. Also, a new shape-space setting is presented and simulationswith various mappings onto a plane of the N-dimensional dynamics are done with boththe original and the new shape-space. A mapping using distance coefficients is presented.106Chapter 7. Conclusion 107The distance coefficient is analyzed and generalized for its use with non-Boolean affinities.Simulations are done with both Boolean and non-Boolean affinities.However, as it was mentioned several times already, because the work presented wasbroad it was not very deep to keep the length of the thesis within reasonable bounds. Itthus calls for further work, that is analysis and simulations. It is hoped, though, that itaroused the curiosity of the reader and showed what interesting work can be performedin theoretical immunology.Appendix AThe N-dimensional model brought back to a 1-dimensional modelIt is here shown that the N-dimensional system behaves like a 1-dimensional one whenperiodic boundary conditions on a shape-space and uniform initial clone populations areconsidered.Consider the field of a clone k, given by expression (3.6). Now, due to the specificinitial conditions, the concentration xk is identical for each clone. It can be characterizedby a constant x and factorized in front of the summation. Furthermore, because of theperiodicity the shape-space, the sum of the affinities relating a clone to the others in thesystem will be the same. As a consequence, the field experienced by each clone is thesame and can be simply written as:Y=xK (A.72)where it is understood that K is the summation over the affinities of one clone with theothers.The same reasoning allows to show that the parameter expressed iii (3.7) that givesa measure of the T-cell factors are identical for all clones and given by:W=xYK. (A.73)Note that in this case it would not make sense to normalize W. See what happens ifone wants to do so. Remember that in the original formulation of the model presentedby Hoffmann et al. [61], one first calculates the T-cell factor concentration for each clone108Appendix A. The N-dimensional model brought back to a 1-dimensional model 109and then multiplies it by a factor equal to the average of the two threshold factors C2 andC3 divided by the average of all the T-cell factors concentrations. Now examine whatone gets when applying the normalization to this case.The average of the T-cell factors concentrations is here simply given by W. Thenormalized value of the T-cell factors concentration is then given by:—(C2 + C3)/2= C2+C3which is a constant! It is thus independant of the change in concentration of the clones,which does not make sense. It is concluded that the the normalization of the T-cellfactors concentrations is not required when dealing with uniform initial conditions.The next step is to consider the effectivities, given by (3.5). As they only depend onthe T-cell factors concentrations, one can see that they are also identical for all clones:eq = flq q = 1,2,3. (A.75)This last finding leads to the fact that the effective fields expressed by (3.9) are alsoidentical for all clones and simply given by:UqxeqK q=1,2,3. (A.76)Finally, the basic differential equation (3.4), in respect of what has just been shown,can be written as:=S+kixeiU1—k2xU3(U)4. (A.77)It can be seen that the same equation is controlling the behaviour of the uniformlydistributed initial populations of the cells, and therefore these populations will all varyAppendix A. The N-dimensional model brought back to a 1-dimensional model 110identically. For that reason, the model can be seen as one-dimensional in this specialsituation and will later be refered as so. Well, this is not a very interesting case initself, but one of the advantages of its extreme simplicity is to enable the calculation ofparameters that yield biologically reasonable steady-states. These parameters can thenbe used for further investigations of the model.To find them, one can borrow the same idea that was used previously in the firstof paper on Hoffmann’s symmetrical two-dimensional models [57]. It is an approximatemethod of analysis that allows to find parameters that generate predetermined steady-states. One considers each of the steady-states to be essentially a balance between twodominant terms in the differential equation. Here, the case where 02 < 03 is studied and= 0.In this approach, some biological assumptions are made. First, the virgin state x ischaracterized by a balance between source and 1gM killing terms:S k2 x U2 2• (A.78)This can be solved to yield, in view of (A.76):5 1/2(2 K) . (A.79)This solution can occur only if W, < 02, i.e. if:(A.80)The immune state x, represented by a balance between source and IgG killing termsis given by:Sk3x(U)2e. (A.81)This yields:s 1/3() . (A.82)Appendix A. The I\Tdimensional model brought back to a 1-dimensional model 111This result requires that 02 < W < 03. So one must have:SK 2/302< (-i—-) <03. (A.83)Finally, the suppressed state x is a balance between source and natural death terms:S x. (A.84)This yields:(A.85)This gives the condition that W3 > C3 i.e./ S 2> 0. (A.86)Combining the above inequalities gives the conditions on the parameters which yieldthose steady-states:<02< () <03 < (v), (A.87)A similar procedure was used by Hoffmann et al. [61] when they considered their onedimensional model using Boolean affinities, It appears that the only difference betweeninequality (A.87) and the one they find is the presence of the sum of the affinities in thefirst case!Appendix BComplementary clones can have the same concentrationB.1 The periodic Li shape-spaceIt is shown here that for the periodic shape-space, a solution that satisfies the N-dimensional model exists for which complementary clones have the same concentration.An important feature that makes this solution possible appears to be the periodicityassociated with the affinities which allows to equate the expressions for the steady-statesof the two complementary clones. Indeed, the following relationships exist, at least,between the affinities:KijKjiKnmKmn (B.88)where i,j = 1,NI i+N/2 ,i<N/2m=i—N/2 ,i>N/2=j+N/2 ,j<N/2(B.89)j-N/2 ,j>N/2.The fields of two complementary clones k and k + N/2, where k = 1, N/2 are givenaccordingly to (3.6) by:Yk = Kk,l x1 + ... + Kk,N/2 XN/2+ Kk,1+N/2 X1+N/2 + ... + Kk,N XN (B.90)Yk+N/2 = Kk+N/2,1 x1 + ... + Kk+N/2,N,2 XN/2112Appendix B. Complementary clones can have the same concentration 113+ Kk+N/2,1±N/2 X1+N/2 +... + Kk+N/2,N ZN (B.91)where k = 1, N/2. But the relationships (B.89) between the affinities allow to write that:Kk,1 = Kk+N/2,1+N,2; . 1k,N/2 = Kk+N/2,N (B.92)Kk,1+N/2 = . .. ; Kk,N = Kk+N/2,N/2. (B.93)We can then rewrite (B.91), using these relationships, as:k+N/2 = Kk,1 X1+N/2 + . + Kk,N/2 ZN+ Kk,1+N/2 X1 + ... + Kk,N XN/2 (B.94)with the order of the terms in the summation of (B.91) reversed. Now assume thatcomplementary shapes that have the same concentration is a solution. One can thenreplace each cell concentration by the concentration of the complementary clone, that is,one replaces x, by Xi+N/2 for i N/2 and xj by Xi_N/2 for i > n/2. It can easily be seenthat one then has that Yk = Yk+N/2.The T-cell factor concentrations of the same two complementary clones k and k + N/2are given, accordingly to (3.7), by:Wk = Kk,1 x1 Y + ... + Kk,N/2 XN/2 YN/2+ Kk,1+N/2 X1+N/2 Y1÷N/2 + . + Kk,N XN YN (B.95)Wk+N/2 = Kk+N/2,1 X1 Y1 + .. . + Kk+N/2,N/2 XN/2 YN/2+ Kk+N/2,1+N/2 Z1+N/2 Y1+N/2 + ... + Kk+N/2,N ZN YN. (B.96)There is no need to consider the normalized T-cell factor concentrations since the presenceof the normalization constant does not invalid the present verification. The expressionfor Wk+N/2 can be rewritten in a similar way that was done for Yk+N/2, as:Wk+N/2 = Kk,l X1+N/2 Y1+N/2 + ... + Kk,N/2 ZN YN+ Kk,1+N/2 x1 Y1 +... + Kk,N XN/2 YN/2. (B.97)Appendix B. Complementary clones can have the same concentration 114Again, assuming that complementary clones have the same conceiltratioll shows that,their fields also having the same values as shown above, their T-cell concentrations arethe same, that is Wk = Wk+N/2. This also means that ej = ek+N/2, since ek is directlyproportional to Wk, accordillg to (3.5).The effective fields of the two complementary clones k and k + N/2 are given, accordingly to (3.9), by:Uk = Kk,l e1 x1 + .. + Kk,N/2 eN/2 XN/2+ Kk,1+N/2 el+N,2 X1+N/2 + .,. + eN XN (B.98)Uk+N/2 = Kk+N/2,1 e1 x1 + .. . + Kk+N,2,N/2 eN/2 XN/2+ Kk+N,2,1+N/2 X1÷N/2 + ... + Kk+N/2,N XN. (B.99)Again, the same work can be performed on the expression for Uk+N/2 to yield the newexpression for (B.99):Uk+N/2 = Kk,1 el+N,2 X1+N/2 + .. + Kk,N/2 el+N/2 XN+ Kk,1+N/2 e1 X1 + ... + Kk,N eN/2 XN/2. (B.100)It was already said above that assuming complementary clones to have the same valuemakes the effectivities have the same value, and as a direct consequence the effectivefields also are seen to take the same value.So, looking back at the expressions (3.18) and (3.20) that describe respectively thevirgin and immune states, one can see that complementary clones that have the sameconcentration and also have the same effective fields is a solution that satisfies thoseexpressions. It is obvious that in the suppressed state complementary clones that sharethe same concentration is a solution, since its value is unique.Appendix B. Complementary clones can have the same concentration 115B.2 The non-periodic ± shape-spaceIt is shown here that for the non-periodic shape-space, there exists a solution that satisfiesthe N-dimensional network for which complementary clones have the same concentration.One here numbers the clones from —N to N, where N is any number. The fields of twocomplementary clones k and —k are given accordingly to (3.6), by:Yk = Kk,_N XN + ... + Kk,o x0 + ... + Kk,N XN (B.lOl)K_k,_N X_N + . . . + x0 + ... + K_k,N XN. (B.102)But the following relationships exist between the affinities:Kk,_N = K_k,N,. . . , K_k,o, . . , Kk,N = K.k,_N, (B.103)By doing the similar type of work that was done in Appendix B.1, on can rewrite (B.102)as:= Kk,_N XN + ... + Kk,o Xo + ... + Kk,N xp. (B.104)Assuming that complementary clones having the same concentration (i.e. xk = x_k fork = —N, N) is a solution shows that in that case their fields take the same value, thatis: Yk=Ykfork=—N,N.The T-cell factors concentrations of two complementary clones k and —k are givenby:W = XN Y.N + ... + K,0xYN + ... + K XN YN (B.105)W_ = K_,_N X_N YN + ... + K_,0 x0 Yo + ... + K_,N XN YN. (B.106)This last expression can be written using relationships (B.103) as:W. = K,N ZN YN + ... + K,0 x0 YN + . .. + K XN YN. (B.107)Appendix B. Complementary clones can have the same concentration 116If complementary clones have the same concentration, then their fields have been shownto have the same value, and therefore the T-cell factor concentrations also have the samevalue, that is: Wk = W_k for k = —N, N. Again, since the effectivities depend only onthe T-cell factors concentrations, they also take the same value in that case.The effective fields of two complementary clones k and —k are given by:U = K,_jv e_N Xjj + ... + K,o e0 xo + ... + K,N GN ZN (BJO8)U_ =-N X_N + ... + K_i,o e0 xo + ... + K_,N eN ZN. (B.l09)This last expression can be rewritten using relationships (B.103) to yield:U K,_pq eN ZN + ... + K,0 e0 xo + . .. + K,N e_N X_N. (B.11O)The effective fields of complementary clones are easily shown to take the same value ifcomplementary clones that have the same concentration is assumed to be a solution,since then their effectivities have been shown to take the same value. One then has thatUk=Ukfork=—N,N.So looking back at expressions (3.18) and (3.20) that describe respectively the virginand the immune states, one can see that complementary clones that have the sameconcentration and also have the same effective fields is a solution that satisfies thoseexpressions. The same thing can be said for the suppressed state for the same reasonsthat were stated in Appendix B.1.Appendix CThe following is essentially a slightly reworded review of the demonstration done byHoffmann and Tufaro [62] that serological distance coefficients form a metric. Considerany three sera X, Y and Z, and a reagent C. In order for the distance coefficients betweenthese sera to form a metric, they must obey the following criteria [83]:(1) D[X,YIC]>O(2) D[X,XIC] = 0(3) D[X,YC] = D[Y,XC](4) If X Y, then D[X,YC] > 0(5) D[X,YC] <D[Y,ZC] + D[Z,XIC].Using Venn diagrams, one can denote the fractions of C that react with different groupsof sera the following way:x the fraction of C that reacts with X onlyy the fraction of C that reacts with Y onlyz the fraction of C that reacts with Z onlyu the fraction of C that reacts with Y and Z onlyv the fraction of C that reacts with X and Z onlyw the fraction of C that reacts with X and Y onlyi the fraction of C that reacts with X, Y and ZThe distance coefficients of Hoffmann and Tufaro form a metric117Appendix C. The distance coefficients of Hoffmann and Tufaro form a metric 118Figure C.54: The Venn diagrams for the fractions of C that interact with three sera X,Y and Z. (Adapted from [62].)where x,y,z,u,v,w,t> 0, (C.111)With respect to these, the distance coefficient between sera X and Y is given’ by:D[X,YC]= X+V+Y+U. (C.112)x+v+y+u+w+tThe distance coefficients betweell sera Y and Z, and between sera X and Y can be writtenin a similar way.The proof that the distance coefficients obey the metric criteria follows immediately.(1) The fact that the denominator of (C.112) contains all the terms in the numeratorplus some others, combined with (C.lll) automatically leads to the conclusion thatcriterion (1) is obeyed.(2) For two sera X and Y where X = Y, one has that x = v= y = u = 0. So one getsthat D[X,X(C] = 0, which satisfies criterion (2).(3) The expression giving D[Y,XC] is identical to the one giving D[X,YC], whichsatisfies criterion (3).‘using the first definition 6.31Appendix C. The distance coefficients of Hoffmann and Tufaro form a metric 119(4) First note that D[X,YCj can be zero even if X Y, since they are identical in theeyes of C if they interact identically with it. This can seem to violate criterion (4).But one has to realize that one does only consider distances from the perspective ofC. With this regard, sera X and Y are identical from the perspective of C if theyinteract identically with C and then D[X,YIC] = 0, which satisfies criterion (2).But if they interact differently with C, then they are different from the perspectiveof C and then D[X,YC] > 0, which satisfies criterion (4).(5) It needs to be shown that:D[X,YIC] + D[Y,ZC] > D[X,ZC] (C.113)which is equivalent to asking that:X+V++U+X+W+Z+U y+W+Z+V (C114)x+v+y+u+w+t x+w+z+u+v+t — y+w+z+v+u+tThis last inequality can be rewritten as:(x+v+y+u)(x+w+z+u+v+t)(y+w+z+v+u+t)+ (x+w+z+u)(x+v+y+u+w+t)(y+w+z+v+u+t)(y+w+z+v)(x+v+y+u+w+t)(x+w+z+u+v+t)(C.115)Since the left-hand side of this inequality contains all the terms contained in theright-hand side plus others, and because of (C.111), the inequality is respected.This satisfies criterion (5).Bibliography[1] 0. Adam. Theoretical models of lymphocyte network interactions in ontogenicgeneration of antibody diversity. In G. I, Bell, A. S. Perelson, and G. H. 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