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The and R.M., R.M. tools; and reagents/analytic data; P.M. new analyzed and P.M. contributed O.H., and P.M. R.M., and research; R.M. designed P.M. research; and performed A.M., O.H., R.M., contributions: Author main the in presented model minimal the include through explicitly in mediated not process do likely last We trig- process this (7). cells a mechanism T quorum-sensing of response, a number immune large known is sufficiently an It a gers 1A). of (Fig. activation proliferation the cell that T suppress turn molecules. 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IMMUNOLOGY AND PHYSICS INFLAMMATION Fig. 1. A minimal model of Treg-mediated self-tolerance. (A) Binding to antigen stimulates proliferation of conventional T cells (“T cell”), which in turn initiates an immune response. Antigen binding also stimulates proliferation of Tregs, which locally inhibit T cell proliferation, with inhibition strength Q. Treg proliferation also requires a sufﬁcient local concentration of interleukin, which is produced at rate a by activated T cells. (B) Idealized “whitelist” c r network of T cell and Treg cross-reactivity functions. Each edge represents the interaction of T cell i (pix ) or Treg α (pαx ) with antigen x. In this example, each kind of Treg binds only to one kind of self-antigen, and no Tregs bind to the foreign antigen. We model the growth rate of the population λi of T cells of type i as a function of the cross-reactivities, the abundance vx of each antigen, and the abundances wα of the Tregs. (C) The ideal network ensures that T cell proliferation rates are insensitive to self-antigen concentrations vx and remain zero as levels of various proteins naturally ﬂuctuate. At the same time, foreign antigens that do not bind to the Tregs can cause net proliferation of T cells and produce an immune response.

text. Instead, we focus on the initial phase of the response This is incorporated in our model by an antigen-specific sup- and ask whether T cells will begin to proliferate in response pression level Qx that is proportional to the abundance of Tregs to a foreign ligand or a change in the concentration of self- activated by antigen x, with a constant of proportionality b. ligands. We assume that sufficient proliferation will result in With these assumptions, conventional T cell abundances can be immune response through the processes downstream of the sig- described using the differential equation naling pathways we study here. In SI Appendix, section I, we show that the minimal model presented in the main text can be dλi X c derived from a more biologically realistic mechanistic model that = λi p vx (ρ − Qx ) dt ix includes additional components. x A typical human immune system has been estimated to con- X r 6 Qx = b pαx wα, [1] tain about Nc ∼ 10 distinct lineages of conventional T cells and α a similar number Nr of Treg lineages, each carrying a different TCR, specific to a different set of antigens (8). Labeling c each T cell lineage by i (i = 1, 2 ... Nc ) and each Treg lineage where the first term pix λi vx gives the abundance of T cells acti- by α (α = 1, 2, ... Nr ), we can encode this diversity in the cross- vated by each tissue-specific antigen x, and the second term c r reactivity functions pix and pαx , which quantify the strength of (ρ − Qx ) is the growth rate of activated cells. interaction of conventional T cells and Tregs with possible anti- Experiments also indicate that T cell activation stimulates pro- gens x, respectively. At the most basic level, the index x simply liferation of nearby Tregs (3). One potential mechanism for represents a unique amino acid sequence that could be displayed this interaction is the local production of interleukin signals on the cell surface. But since T cells are known to respond to by activated T cells. Tregs are known to be particularly sen- antigens in a tissue-specific manner (9), x can more abstractly sitive to interleukin levels and to rapidly take up interleukin be thought of as indexing possible tissue–antigen pairs (see SI from their environment (12). We denote the local interleukin Appendix for more details). For brevity, we often refer to these concentration in the vicinity of cells displaying a particular tissue- antigen–tissue pairs by the shorthand antigen. One can visu- specific antigen concentration x by ILx . The change in number alize the cross-reactivity functions as an interaction network, of Tregs from lineage α, wα, can be written as the product of with nodes corresponding to T cells, Tregs, and antigens and the abundance of Tregs bound to a tissue-specific antigen x r edges representing the interaction strengths (see Fig. 1B for a (given by pαx vx wα) and an interleukin-dependent local prolif- particularly simple example). eration rate cr ILx (with proportionality constant cr ). We also Our aim is to use these cross-reactivity functions to model the assume that the in absence of interleukin Tregs die at a rate dynamics of the number of cells λi of conventional T cell lin- m. These Treg dynamics can be summarized in the differential eage i and the number of cells wα of Treg lineage α. In general, equation these abundances will depend on antigen concentrations. We use v x x to denote the abundance of antigen . We assume that the dwα X r = wα p vx [cr ILx − m] timescales on which self-antigen concentrations change are much dt αx slower than the Treg/T cell dynamics, so these vx will be treated x P c as fixed quantities when we analyze T cell and Treg dynamics a j pjx λj or find possible steady states of T cell and Treg abundances. It ILx = P r . [2] cr p w turns out that this assumption is not essential to our main results, β βx β as shown in SI Appendix, Fig. S3, because the independence of growth rates from vx in the emergent tiling phase implies that In SI Appendix, section I we show that Eqs. 1 and 2 can be derived the same solution exists regardless of the variations in antigen from a realistic mechanistic model in the limit where interleukin levels. dynamics are assumed to be fast compared to T cell and Treg In our minimal dynamical model, T cells of lineage i can be proliferation. activated at a rate proportional to the cross-reactivity functions Surprisingly, we can rewrite these dynamics in a slightly dif- c pix times the antigen concentration vx . When activated, T cells ferent way that makes no explicit reference to antigens. The proliferate at a rate ρ. As shown in Fig. 1B, T cell proliferation is central objects in this formulation are the “overlap kernels” suppressed by Tregs. Experiments indicate that Treg-mediated φiα—which measure the similarity between the activation pro- suppression of T cell proliferation is highly localized (10, 11). files of a T cell from lineage i and a Treg from lineage α—and

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IMMUNOLOGY AND PHYSICS INFLAMMATION Fig. 2. Emergent tiling of self-antigen space required for robust self-tolerance. (A) Cross-reactivity network with perfect tiling of self-antigens by Tregs. Each self-antigen binds to exactly one Treg receptor, and all nonzero affinities are equal. (B) Local Treg-mediated suppression in this scenario, with the height of each colored region representing the suppression level contributed by a given Treg in the vicinity of a given self-antigen. The two Treg lineages are shown in the same colors as in A, and antigens are arranged along the horizontal axis. Perfect tiling ensures that the suppression strength is uniform across all self-antigens, exactly canceling the basal proliferation rate ρ. Also shown is a conventional T cell (black circle) that binds to two of the antigens (x and y). (C) Net proliferation rate of T cell from B. Rate is shown as a function of the concentration vx of antigen x, assuming that the concentration vy of antigen y is held fixed. (D) Generic cross-reactivity network, with nonuniform affinities and overlap between the cross-reactivity functions of different Tregs. (E) Possible result for Treg-mediated suppression, which is no longer uniform across antigens. (F) Proliferation rate of T cell from E as a function of vx at fixed vy .(G) Generic cross-reactivity network, as in D. (H) Treg coverage under emergent tiling, where a set of Treg abundances wα is found that restores uniform suppression levels, despite the heterogeneity and overlaps in the cross-reactivity network. Note that this solution is generically possible only at much higher levels of Treg diversity for the given number of antigens. (I) Proliferation rate of T cell from H as a function of vx at fixed vy .

achieved by requiring a biophysically implausible fine tuning of also solves the optimization problem stated in Eq. 5, but this TCR–peptide binding. In the absence of such fine tuning, the sys- is not immediately clear from the formulation in terms of over- tem becomes sensitive to fluctuations in self-antigen abundances laps. In SI Appendix, section II, we perform a series of duality vx since generically the Treg coverage of antigen space will be transformations to rewrite the optimization problem in such uneven (Fig. 2 D–F). a way that the equations in Eq. 6 naturally arise. In general, These examples suggest that the only biophysically plausible whether these conditions can be satisfied will depend on the way for the immune system to achieve robustness to fluctuations dimensionality of the space we are working in (i.e., the number in self-antigens while maintaining sensitivity to foreign antigens of Tregs Nr , the number of T cells Nc , the number of tissue- is to tune the relative abundances of the Tregs and T cells to pro- specific antigens Na , and the structure of the cross-reactivity duce uniform coverage and a uniform interleukin profile, with function). Qx = ρ and ILx = m/cr at every antigen x, despite the biophys- The question of whether the Treg dynamics achieve emergent ically unavoidable overlaps between the Treg cross-reactivity tiling is thus reduced to the question of whether the equations in functions (Fig. 2 D–F). We call such a tiling of the antigen Eq. 6 have solutions. We begin by investigating the simplest kind r space an emergent tiling. Note that emergent tiling is generi- of cross-reactivity function, where the entries of the matrix pαx c cally impossible in the simple cross-reactivity network sketched and pix are independently drawn from a Bernoulli distribution. here, with many antigens and only two Tregs, and the schematic (As we discuss below, real cross-reactivity functions are much is meant only to convey the idea of uniform coverage in the more complicated than this but it is helpful theoretically to start presence of overlaps. We now proceed to investigate the con- with this simple model to gain intuition.) This problem has been ditions for emergent tiling without fine tuning in more complex considered in various contexts including the theory of percep- networks. trons (a simple model of statistical learning) and more recently in the context of ecology (20–22). In fact, it is possible to show that Emergent Tiling Is Possible above a Threshold Level of Treg Diver- in high dimensions when Nr , Nc , Na 1 a solution generically sity. Inserting the full expressions for ILx and Qx from Eqs. 1 exists if there are at least twice as many T cell and Treg lineages 2 and into the conditions for uniform coverage discussed above as types of antigens: Nr /Na > 2 and Nc /Na > 2 and no solution (Qx = ρ, ILx = m/cr ), we find that sufficient conditions for a exists if the opposite is true. These two regimes are separated by stable emergent tiling solution can be written as a phase transition known as the Gardner transition in the statis-

X r ρ X c mρ tical physics literature (20, 22, 23). We give a brief overview of p wα = and p λj = , [6] αx b jx ab this relation in SI Appendix, section IIID. α j In Fig. 3 A and B, we plot the equilibrium deviations from 2 1 P 2 uniform Treg coverage hδQx i = (Qx − ρ) and from uni- where as before α runs over Treg lineages, j over T cell lin- Na x 2 1 P 2 form interleukin concentration hδIL i = (ILx − m/cr ) eages, and x over possible self-antigens. Note that we have x Na x used the ﬁrst condition to simplify the expression for ILx in as a function of the ratio Nr /Na , for randomly generated cross- r the second one. When a solution to these equations exists, it reactivity functions with Na = 100 antigens, with pαx sampled

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IMMUNOLOGY AND PHYSICS INFLAMMATION (SI Appendix, section IV and Figs. S4 and S5). Once again, we pairs, or all of the Tregs can be expanded together, nonspecif- eff found an emergent tiling phase when Nr /Na > 2. We also ran ically, by loading the dendritic cells with antibodies against simulations to check what happens when we allow the “antigen CD3, one of the components of the TCR, which effectively concentrations” vx to vary in time. SI Appendix, Fig. S3 shows acts like a universal antigen. Second, repertoire diversity can be simulations from our full model when vx is allowed to rapidly directly measured with high-throughput sequencing of the TCR oscillate in time. Somewhat surprisingly, we find that the dynam- genes (8, 27, 28). ics still self-organize into an emergent tiling phase despite the These two techniques together allow one to envision exper- fact that Treg and T cell abundances no longer reach a steady iments with Treg ecology analogous to existing methods in state. Collectively, these results suggest that the existence of the microbial ecology (29). In particular, one could reduce the Treg emergent tiling phase is a robust feature of this class of mod- repertoire diversity by imposing a population bottleneck. Clonal els. Finally, we note that for more biologically realistic choices of expansion could then be applied to bring the total cell count back the cross-reactivity matrices while the exact ratio of Treg to anti- to the original level, before injecting the cells into the recipient gen diversity at which the transition to the emergent tiling phase organism. Receptor sequencing of the postexpansion cells makes occurs may change, general arguments from statistical mechanics it possible to quantify the final Treg diversity and calibrate the suggest that such a phase will exist even in these more complex bottleneck to achieve a range of diversity levels. settings. Our theory predicts that emergent tiling is required for robust self-tolerance and that emergent tiling is possible only if the Proposed Experimental Test of Emergent Tiling Transition. A key Treg diversity exceeds a certain threshold, which is larger than prediction of our previous analysis is that lowering the Treg the bare minimum level required to simply cover all of the self- diversity results in a sharp transition to a regime where changes antigens. If the diversity is reduced to sufficiently low levels, the in self-antigen concentrations can result in an autoimmune Treg populations can only achieve imperfect tiling, and conven- response (Fig. 3). We now propose an experimental test of this tional T cell proliferation can be induced by natural fluctuations prediction by repurposing a classical immunological experimen- in self-antigen concentrations. Measures of autoimmune activity, tal design previously used to discover Treg function (Fig. 4). In such as concentrations of antibodies against self-peptides from the original experiments T cells (including both Tregs and con- various organs (10), should therefore show a strong negative cor- ventional T cells) were transferred from the spleen and/or lymph relation with the number of distinct Treg clones injected into nodes of a mouse with a functioning thymus to another mouse the athymic mouse, with a sharp Treg diversity threshold below from a strain (“athymic nude”) homozygous for a mutation that which there is an autoimmune response. renders it congenitally incapable of producing any kind of T cell (2, 10). If all of the T cells are transferred together, the recipi- Comparison with Existing Data. Our model also makes several ent mouse remains healthy. But if the Tregs are eliminated from qualitative predictions that can be compared with existing exper- the population, with only the conventional T cells injected into imental observations. First, we predict that autoimmune disor- the recipient, severe autoimmune syndromes result in multiple ders can be caused by genetic mutations that significantly restrict organs. the Treg TCR repertoire but otherwise leave the immune system In the decades since these original studies were carried out, intact. This prediction has been confirmed in nonobese dia- additional techniques have been developed that make it pos- betic (NOD) mice, a standard animal model for type I diabetes sible to modulate and assess the impact of Treg repertoire (30). These mice spontaneously develop an autoimmune disor- diversity. First, it was shown that Tregs can undergo clonal der whereby the immune system destroys the insulin-producing expansion in vitro after coculture with dendritic cells (26). Spe- β cells in the pancreas. An assay of thymic TCR repertoires cific Treg clones can be expanded using known receptor–antigen from these mice revealed that the diversity of insertions/deletions

Fig. 4. Proposed experimental test for emergent tiling transition. (A) T cells (circles) are purified from a wild-type mouse, and Tregs are selectively depleted from the sample following standard protocols (e.g., antibody plus complement). The number of Treg clonotypes (open circles) remaining after depletion will depend on the size of the resulting population bottleneck, allowing a variety of levels of Treg diversity to be generated. The total Treg cell count is then restored to its original level using standard protocols for clonal expansion of Tregs. These Tregs, along with the rest of the T cells from the original sample, are then introduced into athymic nude mice, which lack native T cells. After an appropriate waiting period, the mouse is evaluated for an autoimmune response, for instance by measuring levels of antibodies against various self-peptides. (B) Schematic of Treg coverage before and after the population bottleneck. As in Fig. 2, each colored region represents a different Treg, and the height of the region represents the contribution of that Treg to the total suppression Qx . The wild-type mouse has sufficient Tregs to achieve emergent tiling (Left), but this is no longer possible if the Treg diversity is driven below the threshold (Right). (C) Schematic of predicted autoimmune activity (e.g., self-antibody levels) as a function of Treg diversity. Our model predicts a sharp transition at a critical diversity level, which is significantly higher than what is required simply to cover all of the self-antigens.

6 of 8 | PNAS Marsland et al. https://doi.org/10.1073/pnas.2011709118 Tregs self-organize into a computing ecosystem and implement a sophisticated optimization algorithm for mediating immune response Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 rg efognz noacmuigeoytmadipeetaspitctdoptimization sophisticated response a immune implement mediating and for ecosystem algorithm computing a into self-organize Tregs al. et Marsland si ehns hti asmnoswt ierneof range wide a with of real- parsimonious function. biologically Treg power is on underlying data that the single experimental a mechanism demonstrate providing istic they for was model together our proliferation but no explanations, type, transgenic wild the the pop- from Treg into Tregs overall when observed. injected the But were of recipient. mice mice 20% the wild-type than and in from proliferated more ulation they Tregs up mice, When made the transgenic (32). eventually half these into type about injected to wild were to reduced the TCR diversity able of on Treg be level experiments with with will mice consistent antigens novel transgenic is and corresponding prediction proliferation, This the net invade. for support specific to in sufficient not Tregs interleukin be is local proliferate, the will recipient to of the levels some able however, If phase, be tiling specificity. will emergent binding the Tregs their prolifera- new of for no required regardless that level means minimum This the tion. meets interleukin enough barely uniform high the profile tiling, has emergent recipient achieve to the diversity If Treg popu- established. Treg already own whose are animal lations an into Tregs exogenous injecting self-antigens displayed of host. distribution the the clones within the of different on of proliferation dependent rates interleukin rapid proliferation highly high the to the with lead Tregs, where mice exogenous model, Treg-deficient our in of levels predictions agrees the clones reshaping repertoire dominant with This initially extinction. were to experiment while driven the nearly beginning, were of the end at the rare at extremely clones (31). mo dominant 3 the of of course sizes the Many population over dramatically relative changed the clones Treg time that of several showed at they and afterward, host the points Vα2 into individual injection of before frequencies sequences CD3 the quantifying popula- Treg By the dynamics. of features tion additional some explored also 31 ref. antigen relative in but changes setting, to due abundances laboratory lost easily controlled is autoim- protection highly this against that a this protection in in short-term especially levels some munity, diversity provide that achieve can predicts to range model small too Our but sufficient tiling. self-antigens been emergent of for have coverage healthy may full mice provide mice transgenic to 16 the constricted of of the that repertoire 5 suggests Treg This keep experiment. injec- the to of Treg duration however, the low-diversity sufficient, The was symptoms. tion autoimmune resulted mice some transgenic low-diversity in the from grafts the of pathologies, most from autoimmune is prevented Tregs reliably the receptor) mouse While wild-type the interleukin mice. one transgenic or Treg low-TCR-diversity mouse wild-type the the a of either of from purified deletion Tregs with to injected deficient (due congenitally is Tregs that mice in of strain a above, proposed one lobe etduigtasei ieegnee oeliminate TCR to the engineered in mice variation transgenic all using tested been emergent also for threshold the result- below lineage, size thymic Treg repertoire stringent tiling. the Treg excessively a to in to commitment ing lead fur- for genetic a mice criteria the in NOD that selection resulting func- the suggest mice, 49 of data wild-type These defects the the restriction. of of repertoire each 8 ther in 20 and the than 5 Furthermore, more a between mice. 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IMMUNOLOGY AND PHYSICS INFLAMMATION of autoimmune syndromes in people with persistent viral infec- of Tregs, conventional T cells, and interleukins. Our model relied on exper- tions (38). A possible reason for this is that the characteristics of imentally supported biological assumptions that we outline in detail in SI a persistent infection somehow do not allow for adaptation via Appendix. We then derived the minimal model in Eqs. 1 and 2 presented generation of new Tregs that specifically bind to the viral anti- in the main text by assuming interleukin dynamics were fast compared gen, and so existing Treg populations end up clonally expanding to T cell and Treg proliferation rates and that interleukin consumption of Tregs is much larger than those of conventional T cells. By exploiting the to inhibit the responding conventional T cells with which they ecological interpretation of this minimal model, we derived a dual repre- share a partial overlap. Since the conventional T cells can pro- sentation of this model in terms of optimization (Eq. 5). We then used Eq. liferate on binding to viral antigen without local Treg-mediated 5 to derive sufficient conditions for emergent tiling using a series of dual- suppression, larger populations of partially overlapping Tregs ity transformations. To check these analytic results, we performed numerical are required to maintain a global net proliferation rate of zero. simulations of both the dynamical models and corresponding optimization But this process naturally breaks the emergent tiling needed for problems. robustness and thus could produce increased sensitivity to fluctuations in self-antigen levels. In the future, it will be interesting Data Availability. Data, code, and scripts have been deposited in GitHub to further explore this scenario to better understand whether it (https://github.com/Emergent-Behaviors-in-Biology/immune-svm). can increase our understanding of autoimmune diseases. ACKNOWLEDGMENTS. We are extremely grateful to Arup Chakraborty for Materials and Methods many useful conversations and suggestions on this work and to Rustom Antia for advice and encouragement in the early phases of the project. This Please see SI Appendix for detailed materials and methods. Briefly, we con- work was funded by a Simons Investigator in Mathematical Modeling of structed a detailed, biologically realistic model for the immune dynamics Living Systems award and a NIH NIGMS R35GM119461 grant (to P.M.).

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