pnasresearcharticle Marsland Understanding the origins and mechanisms
underlying autoimmune diseases represents a major conceptual problem in
immunology. It is now clear that regulatory T cells (Tregs) play an important
role in suppressing autoimmune responses, but we lack an algorithmic
understanding of how they function. Here we present a simple mathematical
model that organizes known biology and suggests that Tregs implement a
sophisticated optimization algorithm to prevent autoimmune responses. We
propose novel experimental tests of these ideas and discuss implications of
our results for autoimmune diseases. RM, AM and PM conceptualized the problem,
RM, OH and PM performed analytic calculation, RM and OH performed simulations,
RM and PM analyzed the results and wrote the manuscript. The authors declare
no competing interest. 1To whom correspondence should be addressed.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Tregs self-organize into a "computing ecosystem" and implement a
sophisticated optimization algorithm for mediating immune response

Robert Marsland III  Department of Physics, Boston University, Boston, MA
02215  Owen Howell  Department of Physics, Boston University, Boston, MA 02215
Andreas Mayer  Lewis-Sigler Institute, Princeton University, Princeton, NJ
08540  Pankaj Mehta  Department of Physics, Boston University, Boston, MA
02215  College of Data Science, Boston University, Boston, MA 02215

###### Abstract

Regulatory T cells (Tregs) play a crucial role in mediating immune response.
Yet an algorithmic understanding of the role of Tregs in adaptive immunity
remains lacking. Here, we present a biophysically realistic model of Treg
mediated self-tolerance in which Tregs bind to self-antigens and locally
inhibit the proliferation of nearby activated T cells. By exploiting a duality
between ecological dynamics and constrained optimization, we show that Tregs
tile the potential antigen space while simultaneously minimizing the overlap
between Treg activation profiles. We find that for sufficiently high Treg
diversity, Treg mediated self-tolerance is robust to fluctuations in self-
antigen concentrations but lowering the Treg diversity results in a sharp
transition – related to the Gardner transition in perceptrons – to a regime
where changes in self-antigen concentrations can result in an auto-immune
response. We propose a novel experimental test of this transition in immune-
deficient mice and discuss potential implications for autoimmune diseases.

###### keywords:

Adaptive immunity $|$ Tregs $|$ optimization $|$ ecology $|$ biophysics

###### doi:

www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX

This manuscript was compiled on

The adaptive immune system of humans and other mammals solves a challenging
computational problem with amazing reliability. Using only the information
contained in the binding affinities between certain macromolecules, the system
must distinguish potentially pathogenic cells from its own cells, in order to
eliminate the former without harming the latter. While understanding how the
immune system accomplishes this feat is fascinating even from a purely
theoretical point of view, this problem also has many urgent practical
implications since an increasing number of autoimmune diseases and allergies
are thought to stem from an inability to accurately distinguish self and
foreign antigens (1).

An important step forward in the effort to understand adaptive immunity came
with the discovery of regulatory T cells (Tregs) (2, 3). Like all T cell
phenotypes, Tregs express T cell receptors (TCR’s) on their surface, which
bind to antigen peptides displayed on the surface of other cells via the major
histocompatibility complex (MHC). But unlike in conventional T cells, TCR
binding and activation in Tregs have the effect of suppressing T cell
proliferation and cytokine production. This Treg-mediated suppression of self-
activation complements negative selection in the thymus against self-reactive
T cell lineages. In fact, it has been experimentally shown that even after
undergoing negative selection in the thymus, T cells can raise a full immune
response against native tissues if Tregs are artificially removed from the
immune system (2). For all these reasons, Tregs are thought to play a critical
role preventing autoimmune responses.

The problem of distinguishing self from non-self is made even harder by the
requirement that the immune system must be able to reliably respond to even
small amounts of foreign antigen but be robust to potentially large
fluctuations in the concentrations of self-antigens. Specifically, even in the
absence of foreign antigens, the immune system needs to tolerate fluctuations
in the relative abundances of self-antigens, as development, circadian cycles
and other natural variations in biological activities lead to very different
protein expression patterns even in healthy cells.

In this paper, we propose a simple dynamical model for the interactions
between conventional T cells, Tregs and antigens, which captures the essential
aspects of known Treg biology. We then use this minimal model to understand
how the immune system can achieve the stringent requirements described above,
while simultaneously providing an algorithmic interpretation for Treg-mediated
adaptive immunity. This work builds on a longstanding tradition of ecological
modeling in theoretical immunology (4, 5, 6), and to our knowledge is the
first application of this approach to Treg dynamics. Finally, we note that for
notational brevity, in this manuscript we will always use T cells to mean
conventional T cells (non-regulatory T cells).

Figure 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” network of T cell and Treg cross-reactivity
functions. Each edge represents the interaction of T cell $i$ ($p_{ix}^{c}$)
or Treg $\alpha$ ($p_{\alpha x}^{r}$) with antigen $x$. In this example, each
kind of Treg only binds to one kind of self-antigen, and no Tregs bind to the
foreign antigen. We model the growth rate of the population $\lambda_{i}$ of T
cells of type $i$ as a function of the cross-reactivities, the abundance
$v_{x}$ of each antigen and the abundances $w_{\alpha}$ of the Tregs. _(c)_
The ideal network ensures that T cell proliferation rates are insensitive to
self-antigen concentrations $v_{x}$, 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.

## Model Development

We now present a minimal model for Treg-mediated self-tolerance. As shown in
Figure 1, our model has three basic components: conventional T cells, Tregs,
and antigens. Upon binding an antigen, T cells enter an active state in which
they begin to proliferate and to release various interleukin signaling
molecules. These interleukins stimulate other immune cells, including Tregs
which in turn suppress T cell proliferation (Figure 1(_a_)). It is known that
the activation of a sufficiently large number of T cells triggers a immune
response, a process likely mediated through a quorum sensing mechanism (7). We
do not explicitly include this last process in the minimal model presented in
the main text. Instead, we focus on the initial phase of the response and ask
whether T cells will begin to proliferate in response to a foreign ligand or a
change in the concentration of self ligands. We assume that sufficient
proliferation will result in immune response through the processes downstream
of the signaling pathways we study here. In SI Appendix (Section I), we show
that the minimal model presented in the main text can be derived from a more
biologically realistic mechanistic model that includes additional components.

A typical human immune system has been estimated to contain about $N_{c}\sim
10^{6}$ distinct lineages of conventional T cells and a similar number $N_{r}$
of Treg lineages, each carrying a different TCR, specific to a different set
of antigens (8). Labeling each T cell lineage by $i$ ($i=1,2\dots N_{c}$) and
each Treg lineage by $\alpha$ ($\alpha=1,2,\dots N_{r}$), we can encode this
diversity in the cross-reactivity functions $p_{ix}^{c}$ and $p_{\alpha
x}^{r}$, which quantify the strength of interaction of conventional T cells
and Tregs with possible antigens $x$, respectively. At the most basic level,
the index $x$ simply represents a unique amino acid sequence that could be
displayed on the cell surface. But since T cells are known to respond to
antigens in a tissue-specific manner (9), $x$ can more abstractly be thought
of indexing possible tissue-antigen pairs (see SI for more details). For
brevity, we will often refer to these antigen-tissue pairs by the shorthand
antigen. One can visualize the cross-reactivity functions as an interaction
network, with nodes corresponding to T cells, Tregs, and antigens, and edges
representing the interaction strengths (see Figure 1(_b_) for a particularly
simple example).

Our aim is to use these cross-reactivity functions to model the dynamics of
the number of cells $\lambda_{i}$ of conventional T cell lineage $i$ and the
number of cells $w_{\alpha}$ of Treg lineage $\alpha$. In general, these
abundances will depend on antigen concentrations. We will use $v_{x}$ to
denote the abundance of antigen $x$. We will assume that the time scales on
which self-antigen concentrations change is much slower than the Treg/T cell
dynamics, so these $v_{x}$ will be treated as fixed quantities when we analyze
T cell and Treg dynamics or find possible steady-states of T cell and Treg
abundances. It turns out that this assumption is not essential to our main
results, as shown in Supplementary Figure S3, because the independence of
growth rates from $v_{x}$ in the emergent tiling phase implies that the same
solution exists regardless of the the variations in antigen levels.

In our minimal dynamical model, T cells of lineage $i$ can be activated at a
rate proportional to the cross-reactivity functions $p_{ix}^{c}$ times the
antigen concentration $v_{x}$. When activated, T cells proliferate at a rate
$\rho$. As shown in Figure 1(_b_), T cell proliferation is suppressed by
Tregs. Experiments indicate that Treg-mediated suppression of T cell
proliferation is highly localized (10, 11). This is incorporated in our model
by an antigen specific suppression level $Q_{x}$ that is proportional to the
abundance of Tregs activated by antigen $x$, with a constant of
proportionality $b$. With these assumptions, conventional T cell abundances
can be described using the differential equation

$\displaystyle\frac{d\lambda_{i}}{dt}$
$\displaystyle=\lambda_{i}\sum_{x}p_{ix}^{c}v_{x}(\rho-Q_{x})$ $\displaystyle
Q_{x}$ $\displaystyle=b\sum_{\alpha}p_{\alpha x}^{r}w_{\alpha},$ (1)

where the first term $p_{ix}^{c}\lambda_{i}v_{x}$ gives the abundance of T
cells activated by each tissue-specific antigen $x$, and the second term
$(\rho-Q_{x})$ is the growth rate of activated cells.

Experiments also indicate that T cell activation stimulates proliferation of
nearby Tregs (3). One potential mechanism for this interaction is the local
production of interleukin signals by activated T cells. Tregs are known to be
particularly sensitive to interleukin levels and to rapidly take up
interleukin from their environment (12). We denote the local interleukin
concentration in the vicinity of cells displaying a particular tissue-specific
antigen concentration $x$ by ILx. The change in number of Tregs from lineage
$\alpha$, $w_{\alpha}$, can be written as the product of the abundance of
Tregs bound to a tissue-specific antigen $x$ (given by $p_{\alpha
x}^{r}v_{x}w_{\alpha}$) and an interleukin-dependent local proliferation rate
$c_{r}\mathrm{IL}_{x}$ (with proportionality constant $c_{r}$). We also assume
that the in absence of interleukin Tregs die at a rate $m$. These Treg
dynamics can be summarized in the differential equation

$\displaystyle\frac{dw_{\alpha}}{dt}$
$\displaystyle=w_{\alpha}\sum_{x}p_{\alpha x}^{r}v_{x}[c_{r}{\rm IL}_{x}-m]$
$\displaystyle{\rm IL}_{x}$
$\displaystyle=\frac{a\sum_{j}p_{jx}^{c}\lambda_{j}}{c_{r}\sum_{\beta}p_{\beta
x}^{r}w_{\beta}}.$ (2)

In the SI Appendix (Section I) we show that Eq. 1 and Eq. 2 can be derived
from a realistic mechanistic model in the limit where interleukin dynamics are
assumed to be fast compared to T cell and Treg proliferation.

Surprisingly, we can rewrite these dynamics in a slightly different way that
makes no explicit reference to antigens. The central objects in this
formulation are the “overlap kernels” $\phi_{i\alpha}$ – which measure the
similarity between the activation profiles of a T cell from lineage $i$ and a
Treg from lineage $\alpha$ – and $\phi_{\alpha\beta}$ – which measures the
overlap between Tregs from lineages $\alpha$ and $\beta$ – with

$\displaystyle\phi_{i\alpha}$ $\displaystyle=\sum_{x}v_{x}p_{\alpha
x}^{r}p_{ix}^{c}$ $\displaystyle\phi_{\alpha\beta}$
$\displaystyle=\sum_{x}v_{x}p_{\alpha x}^{r}p_{\beta x}^{r}.$ (3)

Notice that these overlaps depend on the cross-reactivity functions as well as
the antigen concentrations. In SI Appendix (, we describe an approximation
that is exact in the “emergent tiling” regime discussed below, which allows
the dynamics of Eq. 1 and Eq. 2 to be written entirely in terms of the
overlaps:

$\displaystyle\frac{d\lambda_{i}}{dt}$
$\displaystyle=\lambda_{i}r_{i}\left[\rho-
br_{i}^{-1}\sum_{\alpha}\phi_{i\alpha}w_{\alpha}\right]$
$\displaystyle\frac{dw_{\alpha}}{dt}$
$\displaystyle=\frac{m}{\sum_{\beta}w_{\beta}\bar{p}_{\beta}}w_{\alpha}\left[\kappa_{\alpha}-\sum_{\beta}\phi_{\alpha\beta}w_{\beta}\right],$
(4)

with $\kappa_{\alpha}=a\sum_{j}\lambda_{j}\phi_{j\alpha}/m$,
$r_{i}=\sum_{x}v_{x}p_{ix}$ and $\bar{p}_{\beta}=\sum_{x}p_{\beta x}^{r}$.
Surprisingly the explicit dependence on antigens has completely disappeared
from Eq. 4. Instead, all information about antigen concentrations appears only
through the overlap kernels. This is similar to the “kernel trick” in Machine
Learning where all information about overlaps in a feature space can be
encoded in a kernel function (13).

These new equations also naturally lend themselves to an ecological
interpretation in terms of Consumer Resource Models and generalized Lotka-
Volterra models. We can view T cells as exponentially growing “resources”,
with growth rate $r_{i}^{-1}\rho$, that are consumed by Tregs, with the
consumption rate depending on the “resource utilization function”
$\phi_{i\alpha}$. Notice that the Treg dynamics in Eq. 4 take the form of a
Lotka-Volterra equation. Tregs grow at a rate $\kappa_{\alpha}$ that depends
how many resources they consume but also compete with other Tregs. The
strength of competition depends on the overlap kernel $\phi_{\alpha\beta}$.
Ecologically, $\phi_{\alpha\beta}$ can be thought of as the“niche-overlap”
between Tregs in antigen space. The mechanistic origins of this competition
can be traced to the fact that higher niche-overlaps mean Tregs are more
likely to be colocalize and hence more likely to compete for interleukins
produced by T cells at a given tissue/antigen (see Fig. 1).

## Results

We now analyze the implications of these dynamics in greater detail. Our
analysis exploits the ecological interpretation described above by making use
of our recently discovered mapping between ecological dynamics and constrained
optimization (14, 15, 16). This allows us to naturally give an algorithmic
interpretation of the computations performed by Tregs and identify a novel
phase transition in the behavior of Treg mediated self-tolerance as a function
of Treg diversity.

Figure 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 the 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 the first
panel, 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 $\rho$. 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 previous panel. Rate is
shown as a function of the concentration $v_{x}$ of antigen $x$, assuming that
the concentration $v_{y}$ of antigen $y$ is held fixed. _(d)_ Generic cross-
reactivity network, with non-uniform 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 previous panel as a function of $v_{x}$ at
fixed $v_{y}$. _(g)_ Generic cross-reactivity network, as above. _(h)_ Treg
coverage under “emergent tiling,” where a set of Treg abundances $w_{\alpha}$
is found that restores uniform suppression levels, despite the heterogeneity
and overlaps in the cross-reactivity network. Note that this solution is
generically only possible at much higher levels of Treg diversity for the
given number of antigens. _(i)_ Proliferation rate of T cell from previous
panel as a function of $v_{x}$ at fixed $v_{y}$.

### Tregs minimize "niche-overlap" in antigen space

If Treg and conventional T cell dynamics are fast compared to the the rate at
which antigen concentrations change, we can focus on analyzing the steady-
state abundances of Tregs and T cells. In this case, we can set the left hand
side of equations in Eq. 4 equal to zero and the resulting steady-state
equations have a natural interpretation in terms of constrained optimization.
In SI Appendix (Section II), we show that the steady states of the dynamical
equations Eq. 4 are equivalent to the solutions of the following the
constrained optimization problem:

$\displaystyle\underset{\mathbf{w}}{\rm argmin}\,$
$\displaystyle\frac{1}{2}\sum_{\alpha,\beta}w_{\alpha}\phi_{\alpha\beta}w_{\beta}$
$\displaystyle{\rm subject}\,{\rm to:}$
$\displaystyle\,\,\,r_{i}^{-1}\sum_{\alpha}\phi_{i\alpha}w_{\alpha}\geq\frac{\rho}{b}\,\,{\rm
and}\,\,w_{\alpha}\geq 0.$ (5)

The T cell concentrations $\lambda_{i}$ play the role of generalized Lagrange
multipliers in the Karush-Kuhn-Tucker (KKT) conditions that enforce the
inequality constraints above.

This optimization problem has a very beautiful biological interpretation. To
see this, note that the $\phi_{\alpha\beta}$ is a measure of the similarity
between Tregs. Thus, this optimization tells us that Treg populations self
organize to minimize overlaps in the activation profile of Tregs. However,
when performing this optimization, one must ensure that no T cell lineage $i$
can be activated in the absence of foreign ligands to prevent undesirable
autoimmune responses. This last condition is represented in the inequality
constraint $r_{i}^{-1}\sum_{\alpha}\phi_{i\alpha}w_{\alpha}\geq\frac{\rho}{b}$
which simply states that Treg must be able to suppress the proliferation of
any T cell lineage $i$. Equivalently, from an ecological perspective, this
means that Tregs must be able to cover the potential space of activated T
cells while simultaneously minimizing niche overlap between Tregs. This is
analogous to niche partitioning and species packing in ecological systems (17,
18), with Treg lineages playing the roles of species and T cells playing the
role of resources.

### Emergent tiling is required for robust self-tolerance

The previous results relied on the overlap kernel formulation of our model. We
now reanalyze these dynamics from the perspective of antigens. As discussed in
the introduction, an important property of an effective immune system is that
it must reliably respond to foreign antigens but also be robust to
fluctuations in the concentrations of self-antigens. One simple, but
biophysically unrealistic, way of achieving this robustness is depicted in
Figure 1 _(b)_. The figure depicts a simple “whitelist” scenario for Treg-
mediated self-tolerance, with a specialized Treg for each self-antigen and all
Tregs having the same binding affinity. In such a scenario, the sums over $x$
in Eqs. (1-2) are unnecessary, since each equation contains only one term.
Setting $d\lambda_{i}/dt=dw_{\alpha}/dt=0$ then yields uniform equilibrium
Treg coverage $Q_{x}=\rho$ and a uniform interleukin profile ${\rm
IL}_{x}=m/c_{r}$ over all antigens $x$ that interact with a surviving T cell
and Treg. As can be immediately seen from the equations, these conditions
guarantee that proliferation rates $d\lambda_{i}/dt$ and $dw_{\alpha}/dt$
remain zero for all choices of the antigen concentrations $v_{x}$. A direct
consequence of this is that Treg-mediated self tolerance is robust against
fluctuations in the antigen abundances. Nonetheless, if a foreign antigen is
introduced that does not interact with any of the Tregs, T cells that bind to
the new antigen will still proliferate at the original rate $\rho$ since there
is no specialized Treg to inhibit their growth.

Real immune systems, however, cannot achieve this one-to-one correspondence
between self-antigens and Tregs because of biophysical constraints stemming
from the nature of TCR-peptide interactions (19). Figure 2 illustrates three
other possible scenarios of response to fluctuations in self-antigen
concentrations (Fig. 2(_a_ -_c_)). In the first scenario, each Treg binds with
equal strength to a different set of non-overlapping antigens. These sets
perfectly tile the set of self-antigens, covering all of them without gaps or
overlaps. Since each antigen interacts with only one Treg (though Tregs will
generically interact with multiple antigens), a set of Treg abundances can
still easily be found for which the proliferation rate vanishes everywhere.
However, just as in the original whitelist example, this scenario can only
achieved by requiring a biophysically implausible fine tuning of TCR-peptide
binding. In the absence of such fine tuning, the system becomes sensitive to
fluctuations in self-antigen abundances $v_{x}$ since generically the Treg
coverage of antigen space will be uneven (Fig. 2(_d_ -_f_)).

These examples suggest that the only biophysically plausible way for the
immune system to achieve robustness to fluctuations in self-antigens while
maintaining sensitivity to foreign antigens is to tune the relative abundances
of the Tregs and T cells to produce uniform coverage and a uniform interleukin
profile, with $Q_{x}=\rho$ and ${\rm IL}_{x}=m/c_{r}$ at every antigen $x$,
despite the biophysically unavoidable overlaps between the Treg cross-
reactivity functions (Fig. 2(_d_ -_f_)). We call such a tilling of the antigen
space an “emergent tiling.” Note that emergent tiling is generically
impossible in the simple cross-reactivity network sketched here, with many
antigens and only two Tregs, and the schematic is only meant to convey the
idea of uniform coverage in the presence of overlaps. We now proceed to
investigate the conditions for emergent tiling without fine-tuning in more
complex networks.

Figure 3: A phase transition from imperfect tiling to emergent tiling as a
function of Treg diversity. _(a)_ Mean squared deviation $\langle\delta
Q_{x}^{2}\rangle$ of Treg coverage from the ideal uniform solution
$Q_{x}=\rho$ versus the ratio $N_{r}/N_{a}$ of the number of Treg lineages to
the number of antigens. Equilibrium Treg and T cell abundances were obtained
by convex optimization as described in the main text and SI. $N_{a}$ and the
number of conventional T cell lineages $N_{c}$ were held fixed at 100 and
1,000, respectively, while $N_{r}$ was swept from 100 to 300. The elements of
the cross-reactivity functions $p_{\alpha x}^{r}$ and $p_{ix}^{c}$ were
sampled independently from Bernoulli distributions with success probability
$p$ from 0.1 to 0.3. _(b)_ Mean squared deviation $\langle\delta{\rm
IL}_{x}^{2}\rangle$ of local interleukin levels from the ideal uniform
solution ${\rm IL}_{x}=m/c_{r}$, for the same set of simulations. _(c)_ Same
as first panel, but with structured cross-reactivity functions $p_{\alpha
x}^{r}$ and $p_{ix}^{c}$ encoding a one-dimensional shape space, as described
in the main text and SI. The horizontal axis indicates the ratio of the
effective number of Tregs $N_{r}^{\rm eff}$ to $N_{a}$, where $N_{r}^{\rm
eff}$ is defined as the number of singular values of the cross-reactivity
matrix $p_{\alpha x}^{r}$ that exceed a cutoff threshold of
$\epsilon=10^{-6}$. _(d)_ Mean squared deviation of local interleukin levels
for the same set of simulations.

### Emergent tiling is possible above a threshold level of Treg diversity

Inserting the full expressions for ${\rm IL}_{x}$ and $Q_{x}$ from Eqs. 1-2
into the conditions for uniform coverage discussed above ($Q_{x}=\rho$, ${\rm
IL}_{x}=m/c_{r}$), we find that sufficient conditions for a stable emergent
tiling solution can be written as

$\displaystyle\sum_{\alpha}p_{\alpha x}^{r}w_{\alpha}=\frac{\rho}{b}\hskip
14.22636pt{\rm and}\hskip 14.22636pt\sum_{j}p_{jx}^{c}\lambda_{j}={m\rho\over
ab},$ (6)

where as before $\alpha$ runs over Treg lineages, $j$ over T cell lineages,
and $x$ over possible self antigens. Note that we have used the first
condition to simplify the expression for ${\rm IL}_{x}$ in the second. When a
solution to these equations exists, it also solves the optimization problem
stated in Eqs. 5 above, but this is not immediately clear from the formulation
in terms of overlaps. In SI Appendix (Section II), we perform a series of
duality transformations to rewrite the optimization problem in such a way that
Eqs. 6 naturally arise. In general, whether these conditions can be satisfied
will depend on the dimensionality of the space we are working in (i.e. the
number of Tregs $N_{r}$, the number of T cells $N_{c}$, the number of tissue
specific antigens $N_{a}$, and the structure of the cross-reactivity
function).

The question of whether the Treg dynamics achieve emergent tiling is thus
reduced to the question of whether the equations in Eq. 6 have solutions. We
begin by investigating the simplest kind of cross-reactivity function, where
the entries of the matrix $p_{\alpha x}^{r}$ and $p_{ix}^{c}$ are
independently drawn from a Bernoulli distribution. (  As we discuss below,
real cross-reactivity functions are much more complicated than this but it is
helpful theoretically to start with this simple model to gain intuition. This
problem has been considered in various contexts including the theory of
perceptrons (a simple model of statistical learning) and more recently in the
context of ecology (20, 21, 22). In fact, it is possible to show that in high
dimensions when $N_{r},N_{c},N_{a}\gg 1$ a solution generically exists if
there are at least twice as many T cell and Treg lineages as types of
antigens: $N_{r}/N_{a}>2$ and $N_{c}/N_{a}>2$ and no solution exists if the
opposite is true. These two regimes are separated by a phase transition known
as the Gardner transition in the statistical physics literature (20, 23, 22).
We give a brief overview of this relation in SI Appendix (Section IIID).

In Figure 3 _(a)-(b)_ , we plot the equilibrium deviations from uniform Treg
coverage $\langle\delta
Q_{x}^{2}\rangle=\frac{1}{N_{a}}\sum_{x}(Q_{x}-\rho)^{2}$ and from uniform
interleukin concentration $\langle\delta{\rm
IL}_{x}^{2}\rangle=\frac{1}{N_{a}}\sum_{x}({\rm IL}_{x}-m/c_{r})^{2}$ as a
function of the ratio $N_{r}/N_{a}$, for randomly generated cross-reactivity
functions with $N_{a}=100$ antigens, with $p_{\alpha x}^{r}$ sampled from
Bernoulli distributions with three different probabilities of interaction. To
make these plots, we solved the corresponding constrained optimization problem
(Eq. 5) using standard numerical techniques for convex optimization (see SI
Appendix Section IV). We see that in all cases, the nonuniformity vanishes
near the predicted transition point $N_{r}/N_{a}=2$.

To determine the number of Treg lineages $N_{r}$ needed to achieve emergent
tiling, we must estimate the number of distinct self-antigens $N_{a}$. The
maximum number of cleavage points for creating a peptide for display on the
MHC is the total number of codons ($\sim 10^{7}$) in the coding regions of the
human genome. Since some of these peptides are redundant, and only a fraction
can be successfully cleaved and loaded onto MHC’s, a conservative upper bound
would have 10 percent of the possible cleavage sites result in displayed
peptides, yielding $N_{a}\sim 10^{6}$. This number is the same order of
magnitude as the observed Treg diversity, which has been estimated at
$3.5\times 10^{6}$ (8). It is therefore not implausible that $N_{r}/N_{a}>2$
in real immune systems.

However, biologically realistic cross-reactivity functions differ
significantly from the i.i.d cases described above since antigens with similar
shapes bind to similar sets of receptors.  Receptor affinities in TRegs and
conventional T cells are shaped through a complex process of positive and
negative selection in the Thymus. For example, experiments suggest that Tregs
may in fact have higher affinity for self antigens than conventional T cells
(24). In general, quantitatively understanding cross-reactivity functions in
different cell types is an important open problem. In light of this incredible
and unknown complexity, we sought to ask if the intuition above also holds for
slightly more biologically realistic scenarios. To do so, we used a toy-models
for cross-reactivity based on antigen low-dimensional shape spaces that have
been extensively used in the statistical physics literature to model TCR
repertoires (25). Fig. 3 _(c)-(d)_ shows numerical simulations of a one-
dimensional shape space with $N_{a}=5,000$ antigens. Each of $N_{r}=500$ Tregs
and $N_{c}=500$ conventional T cells binds to a group of similar antigens,
with $p_{\alpha
x}^{r}=e^{-(x-x_{\alpha})^{2}/2\sigma^{2}},\,p_{ix}^{c}=e^{-(x-x_{i})^{2}/2\sigma^{2}}$,
where the center of the group $x_{\alpha}$ or $x_{i}$ is randomly chosen for
each Treg $\alpha$ and T cell $i$, and the width $\sigma$ is the same for all
lineages in a given simulation run (see SI Appendix for details). Since the
antigens are now “correlated”, instead of the absolute number of antigens it
is useful do define an effective antigen dimension $N_{a}^{\rm eff}$. We
define $N_{a}^{\rm eff}$ as the effective rank of the cross-reactivity matrix
$p_{\alpha x}^{r}$, that is, the number of singular values that exceed a
cutoff threshold $\epsilon=10^{-6}$. The effective number of antigens
decreases from $N_{a}^{\rm eff}=484$ to 97 as $\sigma$ varies from 10 to 100.
When we plot the nonuniformity $\langle\delta Q_{x}^{2}\rangle$ and
$\langle\delta{\rm IL}_{x}^{2}\rangle$ as a function of $N_{r}/N_{a}^{\rm
eff}$, we see a very similar pattern to the i.i.d. case, with reliable
emergent tiling for $N_{r}/N_{a}^{\rm eff}>2$. As a further check on our
model, we also ran simulations when the cross-reactivities are drawn from a
five-dimensional shape space (see SI Appendix Section IV, Fig. S4 and Fig.
S5). Once again, we found that an emergent tiling phase when $N_{r}/N_{a}^{\rm
eff}>2$. We also ran simulations to check what happens when we allow the
“antigen concentrations” $v_{x}$ to vary in time. SI Appendix, Fig. S3 show
simulations from our full model when $v_{x}$ is allowed to rapidly oscillate
in time. Somewhat surprisingly, we find that the dynamics still self-organizes
into an emergent tiling phase despite the fact that Treg and Tcell abundances
no-longer reach a steady-state. Collectively, these results suggest that the
existence of the emergent tiling phase is a robust feature of this class of
models. Finally, we note that for more biologically realistic choices of the
cross-reactivity matrices while the exact ratio of Treg to antigen diversity
at which the transition to the emergent tiling phase occurs may change,
general arguments from statistical mechanics suggest that such a phase will
exist even in these more complex settings.

Figure 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 $Q_{x}$. 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 the self-antigens.

### Proposed experimental test of emergent tiling transition

A key prediction of our previous analysis is that lowering the Treg diversity
results in a sharp transition to a regime where changes in self-antigen
concentrations can result in an auto-immune response (see Fig. 3). We now
propose an experimental test of this prediction by repurposing a classical
immunological experimental design previously used to discover Treg function
(see Fig. 4). In the original experiments T cells (including both Tregs and
conventional T cells) were transferred from the spleen and/or lymph nodes of a
mouse with a functioning thymus to another mouse from a strain (“athymic
nude”) homozygous for a mutation that renders it congenitally incapable of
producing any kind of T cell (2, 10). If all the T cells are transferred
together, the recipient mouse remains healthy. But if the Tregs are eliminated
from the population, with only the conventional T cells injected into the
recipient, severe autoimmune syndromes result in multiple organs.

In the decades since these original studies were carried out, new techniques
have been developed that make it possible to modulate and assess the impact of
Treg repertoire diversity. First of all, it was shown that Tregs can undergo
clonal expansion _in vitro_ after co-culture with dendritic cells (26).
Specific Treg clones can be expanded using known receptor-antigen pairs, or
all the Tregs can be expanded together, nonspecifically, by loading the
dendritic cells with antibodies against CD3, one of the components of the TCR,
which effectively acts like a universal antigen. Secondly, repertoire
diversity can be directly measured with high-throughput sequencing of the TCR
genes (8, 27, 28).

These two techniques together allow one to envision experiments with Treg
ecology analogous to existing methods in microbial ecology (29). In
particular, one could reduce the Treg repertoire diversity by imposing a
population bottleneck. Clonal expansion could then be applied to bring the
total cell count back to the original level, before injecting the cells into
the recipient organism. Receptor sequencing of the post-expansion cells makes
it possible to quantify the final Treg diversity, and calibrate the bottleneck
to achieve a range of diversity levels.

Our theory predicts that emergent tiling is required for robust self-
tolerance, and that emergent tiling is only possible if the Treg diversity
exceeds a certain threshold, which is larger than the bare minimum level
required to simply cover all the self-antigens. If the diversity is reduced to
sufficiently low levels, the Treg populations can only achieve imperfect
tiling, and conventional T cell proliferation can be induced by natural
fluctuations in self-antigen concentrations. Measures of autoimmune activity,
such as concentrations of antibodies against self-peptides from various organs
(10), should therefore show a strong negative correlation with the number of
distinct Treg clones injected into the athymic mouse, with a sharp Treg
diversity threshold below which there is an autoimmune response.

### Comparison with Existing Data

Our model also makes several qualitative predictions that can be compared with
existing experimental observations. First of all, we predict that autoimmune
disorders can be caused by genetic mutations that significantly restrict the
Treg TCR repertoire but otherwise leave the immune system intact. This
prediction has been confirmed in non-obese diabetic (NOD) mice, a standard
animal model for type I diabetes (30). These mice spontaneously develop an
autoimmune disorder whereby the immune system destroys the insulin-producing
$\beta$ cells in the pancreas. An assay of thymic TCR repertoires from these
mice revealed that the diversity of insertions/deletions in one subfamily of
$\alpha$-chain V domains was between 5 and 8 times smaller for Tregs than for
conventional T cells, while the diversity levels of the two lineages were
similar (within a factor of 2) for wild-type (C57BL/6) mice. Furthermore, the
NOD mice only expressed between five and eight of the 49 functional
$\alpha$-chain J domains in the mouse genome, as compared to more than 20 in
each of the wild-type mice, resulting in a further repertoire restriction.
This data suggests that the genetic defects of the NOD mice lead to
excessively stringent thymic selection criteria for commitment to the Treg
lineage, resulting in a Treg repertoire size below the threshold for emergent
tiling.

The relationship between Treg diversity and self-tolerance has also been
tested using transgenic mice engineered to eliminate all variation in the TCR
$\beta$ chain (31). In a similar setup to the one proposed above, a strain of
mice that is congenitally deficient in Tregs (due to deletion of the Treg
interleukin receptor) is injected with Tregs purified from either a wild-type
mouse or one of the low-TCR-diversity transgenic mice. While the Tregs from
the wild-type mouse reliably prevented autoimmune pathologies, most of the
grafts from the low-diversity transgenic mice resulted in some autoimmune
symptoms. The low-diversity Treg injection was sufficient, however, to keep
five out of 16 mice healthy for the duration of the experiment. This suggests
that the constricted Treg repertoire of the transgenic mice may have been
sufficient to provide full coverage of self-antigens but too small to achieve
emergent tiling. Our model predicts that diversity levels in this range can
provide some short-term protection against autoimmunity, especially in a
highly controlled laboratory setting, but that this protection is easily lost
due to changes in relative antigen abundances $v_{x}$.

In this same series of Treg transfer experiments, the authors also explored
some features of the Treg population dynamics. By quantifying the frequencies
of individual V$\alpha$2 CD3 sequences before injection into the host and at
several time points afterwards, they showed that the relative population sizes
of Treg clones changed dramatically over the course of three months (31). Many
of the dominant clones at the end of the experiment were extremely rare at the
beginning, while initially dominant clones were nearly driven to extinction.
This repertoire reshaping agrees with the predictions of our model, where the
high interleukin levels in Treg-deficient mice lead to rapid proliferation of
the exogenous Tregs, with the proliferation rates of different clones highly
dependent on the distribution of displayed self-antigens within the host.

Finally, our model makes predictions about the results of injecting exogenous
Tregs into an animal whose own Treg populations are already established. If
the recipient has high enough Treg diversity to achieve emergent tiling, the
uniform interleukin profile barely meets the minimum level required for
proliferation. This means that no new Tregs will be able to proliferate,
regardless of their binding specificity. If the recipient is not in the
emergent tiling phase, however, some of the local interleukin levels will be
sufficient to support net proliferation, and novel Tregs specific for the
corresponding antigens will be able to invade. This prediction is consistent
with experiments on TCR transgenic mice with Treg diversity reduced to about
half the level of the wild-type (32). When Tregs from wild-type mice were
injected into these transgenic mice, they proliferated and eventually made up
more than 20% of the overall Treg population in the recipient. But when Tregs
from the transgenic mice were injected into the wild-type, no proliferation
was observed.

All these qualitative observations admit of many possible explanations, but
together they demonstrate the power of our model for providing a single
underlying biologically realistic mechanism that is parsimonious with a wide
range of experimental data on Treg function.

## Discussion

In this work, we have constructed a minimal model of Treg-mediated self-
tolerance that is consistent with known biological and experimental facts.
Specifically, our model reflects the empirical observation that Tregs suppress
the proliferation of conventional T cells via a highly local mechanism (11,
10), which depends on the binding specificity of the Treg TCR (33, 34). This
feature of the biology makes the overlap $\phi_{i\alpha}$ between binding
profiles of Tregs and conventional T cells a crucial quantity in the analysis
of Treg function. Our model also accounts for the dependence of Treg
proliferation on local concentrations of interleukins generated by activated
conventional T cells (3, 12). It has long been recognized that this
stimulatory effect of conventional T cells on Tregs completes a homeostatic
feedback loop (see Fig. 2 of (3)). But the fact that interleukin is
internalized and degraded after binding to a Treg’s interleukin receptor (12)
allows us to formulate a more specific hypothesis: that Treg population
dynamics can be modeled using classical resource competition theory, with
local interleukin pools acting as scarce limiting resources. This analogy
highlights the role of the overlap $\phi_{\alpha\beta}$ between Treg binding
profiles as a second crucial quantity, and enables us to analyze the Treg
behavior in terms of the ecological concepts of species packing and niche
partitioning (17).

We have shown that the resulting immune dynamics have a natural interpretation
in terms of optimization. We also find that for sufficiently high Treg
diversities, this simple dynamics allows Tregs to self-organize into a state
that allows the immune system to retain sensitivity to foreign ligands while
simultaneously being robust to fluctuations in the concentrations of self
antigens.

The high Treg diversity required for emergent tiling helps explain the
otherwise surprising fact that Tregs contain a similar number of distinct
TCR’s as conventional T cells, even though the Tregs only need to interact
with peptides from the human genome, while the conventional T cells must cover
all possible pathogens (8). Even though a much smaller number of Treg lineages
would be sufficient to cover all the self-antigens, emergent tiling  in our
simple models requires $N_{r}/N_{a}>2$, with $N_{a}\sim 10^{6}$ in the human
immune system.

This theory raises a number of further questions, which we do not address
here. The first concerns the specific requirements for maintaining pathogen
sensitivity. While the repertoire of conventional T cells in an ideal immune
system should cover all possible foreign antigens, the repertoire of Tregs
must have some gaps in coverage in order to leave the system free to respond
to at least some subset of foreign antigens. It has been observed that high-
affinity antigen binding is associated with commitment to the Treg phenotype,
suggesting a difference in positive and/or negative selection thresholds
between Tregs and other phenotypes (3, 35, 36). This seems to be a natural way
of achieving this difference in coverage, and understanding the details of how
this might be achieved is an important area of future research.  Doing so will
require us to move beyond the simple shape space models for cross-reactivity
used in this work and consider more biophysically realistic models for
antigen-TCR binding.

Another interesting question is about the acquisition of tolerance to foreign
peptides. The immune system is tolerant to many things that are not presented
in the thymus when the Tregs are generated, such as peptides from various
foods, and commensal microbes (37). Tolerance can also be acquired later in
life. There is evidence that Tregs can be generated from other T cell lineages
in the periphery, and not just in the thymus (37, 3). This raises the
possibility that the Treg repertoire may be adaptively repopulated on a slow
timescale with new lineages that interact with such non-self peptides.

Our theoretical framework allows us to generate hypotheses about causes of
natural autoimmune disorders. One robust observation that has yet to be
explained is the de novo onset of autoimmune syndromes in people with
persistent viral infections (38). A possible reason for this is that the
characteristics of a persistent infection somehow do not allow for adaptation
via generation of new Tregs that specifically bind to the viral antigen, and
so existing Treg populations end up clonally expanding to inhibit the
responding conventional T cells with which they share a partial overlap. Since
the conventional T cells can proliferate on binding to viral antigen without
local Treg-mediated suppression, larger populations of partially overlapping
Tregs are required to maintain a global net proliferation rate of zero. But
this process naturally breaks the emergent tiling needed for robustness, and
thus could produce increased sensitivity to fluctuations in self-antigen
levels. In the future, it will be interesting to further explore this scenario
to better understand if it can increase our understanding of autoimmune
diseases.

Please see SI Appendix for detailed Materials and Methods. Briefly, we
constructed a detailed, biologically realistic model for the immune dynamics
of Tregs, conventional T cells, and interleukins. Our model relied of
experimentally supported biological assumptions that we outline in detail in
the SI. We then derived the minimal model in Eq. 1 and Eq. 2 presented in the
main text by assuming interleukin dynamics were fast compared 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 ecological
interpretation of this minimal model, we derived a dual representation of this
model in terms of optimization (Eq. 5). We then used Eq. 5 to derive
sufficient conditions for emergent tiling using a series of duality
transformations. To check these analytic results, we performed numerical
simulations of both the dynamical models and corresponding optimization
problems. All code in the accompanying github repository
https://github.com/Emergent-Behaviors-in-Biology/immune-svm.

We are extremely grateful to Arup Chakraborty for many useful conversations
and suggestions on this work, and to Rustom Antia for advice and encouragement
in the early phases of the project. This work was funded by a Simons
Investigator in MMLS award and NIH NIGMS R35GM119461 grant to PM.

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