5B Stochastic models of T cell activation and signaling
T cells, which mature in the thymus, are one of two key cell types of the adaptive
immune system, and whose basic function is the detection and destruction of intracellular pathogens such as certain bacteria and all viruses [6, 2, 10]. (The other cell
type consists of B cells, which mature in the bone marrow, and are mainly concerned
with the detection and destruction of extracellular pathogens.) In order to execute
their function, T cells scan the surfaces of cells for molecular markers of infection.
Detection of the appropriate marker activates the T cell which then responds to the
pathogen, either by killing the infected cell (effector T cells) or by signaling other
parts of the immune system such as B cells (helper T cells). Since T cells only scan
the surface of other cells, it is necessary that some cells are able to present information regarding their internal contents to the surface. This is achieved by cutting
intracellular proteins into peptide fragments, and transporting these fragments to
the surface for surveillance by T cells. If a pathogen is present within the cell, then
signature peptide groups known as antigens will be made accessible. A major challenge for the pathogen recognition machinery is that the vast majority of peptides on
a given antigen presenting cell do not signify the presence of a pathogen. Thus, a T
cell has to recognize an antigen against a noisy background of these so-called selfpeptides, just as a ribosome has to recognize the correct tRNA during each stage of
protein synthesis. Both processes implement a so-called kinetic proofreading mechanism for error correction in biochemical processes, see Sect. 6.7. The proofreading
mechanism increases specificity of biochemical interactions by including a number
of intermediate steps that can undo errors at the cost of increased reaction time and
free energy expenditure. Here we consider a kinetic proofreading model for T-cell
activation introduced by McKeithan [11].
The model of McKeithan considers the interaction of a T cell receptor (TCR)
with a ligand consisting of a peptide fragment that is bound to a specialized molecule
in the surface of an antigen presenting cell, known as a major histocompatibility
complex molecule (MHC), see Fig. 5B.1(a). The peptide-MHC complex that forms
the ligand is denoted by pMHC. There are two basic assumptions of the model:
(i) In order to respond to an antigen, a TCR in an inactive state T has to undergo
a sequence of N modifications to form the activated state BN . (ii) Dissociation of
pMHC from the TCR can occur at any stage, after which the receptor quickly returns
to its inactive state, see Fig. 5B.1(b). Suppose that the off rate back to the inactive
state T is the same for all intermediate states. We then have the following hierarchy
of kinetic equations for the concentrations [T ], [B j ], j = 0, . . . , N:
1
a
antigen
T-cell
antigen
presenting
cell
TCR
MHC
b
koff
koff
kon
T+pMHC
koff
koff
B0
kp
B1
kp
B2
kp
BN
Fig. 5B.1: Kinetic proofreading model of T-cell activation. (a) Schematic diagram of a T-cell receptor (TCR) binding to an antigen that is attached to a major histocompatibility complex molecule
(MHC) in the surface of an antigen presenting cell. (b) Reaction diagram, see text for details.
N
d[T ]
= kon [T ][P] + koff  [Bi ],
dt
i=0
d[B0 ]
= kon [T ][P] koff [B0 ] k p [B0 ],
dt
d[Bi ]
= k p ([Bi 1 ] [Bi ]) koff [Bi ],
dt
d[BN ]
= k p [BN 1 ] koff [BN ],
dt
(5B.1a)
(5B.1b)
(5B.1c)
(5B.1d)
where [P] is the concentration of a specific pMHC complex. Solving these equations
in steady state shows that the fraction of activated complexes is (see Ex. 6.10)
[BN ]
=
N
Âi=0 [Bi ]
✓
kp
k p + koff
◆N
.
(5B.2)
Note that k p /(k p + koff ) is the probability that any intermediate step i, the T cell
is modified before dissociation of the pMHC. Assuming that k p is independent of
the particular substrate, it follows that the off rate koff is the only parameter whose
variation can distinguish between peptides. Even for small values of N, the fraction
of activated T cells is sensitive to small changes in koff , reflecting the objective of the
2
kinetic proofreading mechanism. However, it comes at a cost, namely that the actual
value of the activity level in response to the correct antigen reduces as N increases,
so that an increase in selectivity coincides in a decrease in sensitivity.
5B.1 Stochastic model of serial triggering mechanism for T cell
activation
So far we have considered how the recognition of an antigen involves the binding
of a T cell receptor (TCR) to a peptide-MHC (pMHC) complex on the surface of
an antigen presenting cell (APC). All TCRs on a mature T cell are of the same
type. (However, there can be up to 107 different types of TCR, and hence types of
T cell, present in any human individual.) On the other hand, every APC displays a
large variety of different types of self-peptide (roughly 1013 ) and, possibly, a few
types of antigen. The contact between a T cell and an APC occurs via a temporary
bond between the cells known as an immunological synapse, see Fig. 5B.2. If a T
cell recognizes a foreign peptide then it is activated to reproduce, with the resulting
T cell clones initiating an immune response. A major challenge for the adaptive
immune system is that there are many more types of self-peptides compared to types
of T cell, so how does a T cell distinguish between self-peptides and antigens.
Part of the solution is kinetic proofreading, which operates at the level of a single
T cell receptor (TCR); for a T cell to be activated, the TCR has to remain bound
to a peptide-MHC (pMHC) complex for sufficient time to undergo a sequence of
modifications [11]. However, T cell activation typically requires a sustained signal
lasting several hours, whereas TCR affinity to its antigen is low and its activation
only elicits a brief spike of intracellular signals. This indicates that sustained signaling requires the concerted action of multiple TCRs that are sequentially engaged
and triggered by a pMHC - the so-called serial triggering mechanism [15, 16]. When
TCR
Fig. 5B.2: Formation of an immunological synapse.
3
there are only a few pMHC complexes present, dissociation from a TCR must be
rapid enough to engage several TCRs so that the threshold for T cell activation can
be reached. Combining this with the kinetic proofreading mechanism, suggests that
there should be an optimal range of half-lives for TCR-pMHC binding, which are
long enough to allow a bound TCR to pass through all the stages of kinetic proofreading, and yet are short enough to allow a pMHC to interact with many TCRs,
as required by the serial triggering mechanism. Another consequence of the selftriggering mechanism is that the TCRs of each T cell sample from a large distribution of self-peptides, which introduces a level of randomness. This has motivated
the development of a stochastic model of kinetic proofreading and serial triggering
that takes into account the random pattern of pMHCs present in an APC [17, 18].
We will follow the particular formulation presented by Zint et al. [19]
Kinetic model
Consider an immunological synapse between a T cell of fixed type and an APC.
Suppose that the T cell receives an intracellular signal each time contact between
a TCR and a pMHC lasts longer than a critical period t ⇤ . The T cell sums up the
stimulation rates of all its receptors and if this sum exceeds a threshold g⇤ , then the
T cell is activated. Let R denote an unbound TCR of a given T cell, M j an unbound
pMHC of type j, and C j the TCR-pMHC complex. (For notational simplicity, we
drop any indices indicating the type of T cell.) The formation of the complexes
consists of the reversible reaction
aj
R + Mj ⌦ Cj,
bj
(5B.3)
where a j and b j are the association and dissociation rates, respectively. Ignoring
the spatial structure of the cell and assuming that the copy numbers of R and M j are
large, we can describe the kinetics of the synapse using the law of mass action (see
also Chap. 2):
dc j
= a j r(t)m j (t) b j c j (t), c j (0) = 0,
dt
dm j
= a j r(t)m j (t) + b j c j (t), m j (0) = z j ,
dt
dr
= Â a j r(t)m j (t) + Â b j c j (t), r j (0) = r.
dt
j
j
(5B.4a)
(5B.4b)
(5B.4c)
Here r(t), m j (t) and c j (t) are the surface densities of R, M j and C j , respectively.
The kinetic equations are supplemented by the conservations laws
r(t) + Â c j (t) = r,
m j (t) + c j (t) = z j ,
j
4
(5B.5)
where r is the total density of receptors and z j is the total density of type- j pMHCs.
Note that the sums over j are restricted to the finite set P of pMHCs that have a
relatively high affinity for the given type of TCRs, that is, b j /a j ⌧ r.
Since the time-scale for T cell activation is much longer than the receptor kinetics, it is assumed that the reactions are in quasi-equilibrium. It follows from (5B.4a)
that in steady-state,
aj
c j = rm j , 8 j 2 P.
bj
Combining this equation with the conservation equations leads to the implicit set of
equations
r Âk ck
cj = zj
.
r Âk ck + b j /a j
Under the further assumption that the concentration of TCRs is not rate-limiting,
that is, r
 j z j , we see that c j ⇡ z j . Given a constant dissociation rate b j , the
⇤
triggering probability that a complex C j exceeds the signaling threshold t ⇤ is e t /t j
with t j = b j 1 . At equilibrium, the triggering rate of a given complex C j can be taken
to be the turnover rate of the complexes, which is c j /t j , multiplied by the triggering
probability. Adding together the contributions from all of the pMHC species and
setting c j = z j , we obtain the following expression for the rate at which the TCRs
of a given T cell are triggered:
g=Â
j
zj
e
tj
t ⇤ /t j
.
(5B.6)
For an individual complex type with dissociation rate t j 1 , the mean rate of trigger⇤
ing is given by the function w(t j ) = t j 1 e t /t j , which is a unimodal function of t j
⇤
with the peak at t j = t , see Fig 5B.3. The existence of an optimal waiting time reflects the competing requirements of kinetic proofreading (large t j ) and many serial
engagements (small t j ).
Probabilistic formulation
So far it has been assumed that the set P of pMHC types processed by a given T
cell is fixed, as are the concentration z j and dissociation time t j of each type j 2 P.
However, as highlighted by van den Berg et al. [17], the diversity of the possible
complexes C j on any APC makes it impossible to specify the particular pMHCs
sampled during a single interaction between a T cell and an APC. Therefore, in
order to determine the overall probability of T cell activation, it is necessary to
adopt a probabilistic approach. In the following it will be convenient to reinterpret
densities as copy numbers.The simplest version of the probabilistic model assumes
that the number of each pMHC type on an APC is the same, that is, z j = z for
all j 2 P. Moreover, a randomly selected APC is taken to have nM self-peptides
5
0.3
0.2
w(τ)
t*
0.1
0
0
4
2
time τ
6
8
10
Fig. 5B.3: Average triggering rate of an individual type of TRC-pMHC complex as a function of
the waiting time t.
in contact with the T cell (via the immunological synapse), whose corresponding
waiting times t j are taken to be independent, identically distributed (i.i.d.) random
variables generated from a given probability density r(t), which is taken to be an
exponential,
1
r(t) = e t/t̄ .
(5B.7)
t̄
The total stimulation rate of a T cell in contact with a randomly chosen APC is now
a sum of random variables with
nc
G(0) = z  w(t j ),
j=1
nc =
nM
z
(5B.8)
Now suppose that a single type of foreign peptide is presented by an APC withe corresponding number of foreign MHC complexes given by z f . Impose the constraint
that the total number of MHCs is the same as the case z f = 0 by reducing the number of self pMHCs according to n0M + z f = nM . Then, the total stimulation rate in
the presence of the antigen is
nc
G(z f ) = qz  w(t j ) + z f w(t f ),
j=1
q=
nM z f
.
nM
(5B.9)
Proper functioning of the immune system requires two conditions: (I) if a foreign
peptide is present, that at least one T cell is activated; (II) no T cell is activated when
only self-peptides are present. This can be recast into the framework of hypothesis
testing. Let H0 denote the null hypothesis that z f = 0 and HA denote the alternative
6
hypothesis z f > 0. Suppose that the test is performed via N independent encounters
between a T cell and an APC. HA is assumed if at least one of the encounters results
in the total stimulation rate exceeding the threshold g⇤ , G g⇤ . It follows that the
probability of a false positive is
r+ ⌘ P(HA assumed | H0 true ) = 1
P(G(0)
(1
g⇤ ))N ,
and the probability of a false negative is
r ⌘ P(H0 assumed | HA true ) = 1
P(G(z f )
g⇤ )
N
.
Note that P(G(z f ) g⇤ ) ⌧ 1 and P(G(0) g⇤ ) ⌧ 1, since they represent rare
events. We conclude that in order for r± to be small,
P(G(z f )
g⇤ )
P(G(0)
g⇤ )
(5B.10)
and N cannot be too small nor too large. The issue is then whether or not a threshold
value g⇤ can be found for which the above conditions hold. This requires handling
rare events using large deviation theory (see also Chap. 10).
Large deviations
Let
n
Sn = Â Xi ,
i=1
where X1 , X2 , . . . is an i.i.d. sequence of random variables with zero mean and a finite
moment generating function
h i
MX (q ) = E eq X .
Furthermore, let a > 0 and assume P[X1 > 0] > 0. The Bahadur-Rao (BR) theorem
states the following [14, 5]:
lim P[Sn
n!•
an] = p
1
e
2pns qa
nI(a)
,
(5B.11)
where I(a) is the so-called large deviation rate function
I(a) = qa a
ln[MX (qa )],
(5B.12)
with qa satisfying the implicit equation
a=
MX0 (qa )
MX (qa )
7
(5B.13)
The variance s 2 is defined by
s2 =
MX00 (qa )
MX (qa )
a2 .
(5B.14)
We can directly apply the above theorem to the stochastic T cell activation model
by identifying X j =R z[w(t j ) w], where t j is generated from the exponential density r(t) and w = w(t)r(t)dt. Using equation (5B.9), the threshold probability
P(G(z f ) g⇤ ) becomes
!
P(G(z f )
nc
g⇤ ) = P q  [X j + zw] + z f w(t f )
j=1
=P
=P
nc
 Xj
j=1
nc
 Xj
j=1
g⇤
!
z f w(t f )
nc zw
q
!
 ⇤
g z f w(t f )
nc a , a = z
nM z f
g⇤
The moment generating function for the random variable w(t) is
!
⇤
Z
1 •
q e t /t t
M(q ) =
exp
dt
t̄ 0
t
t̄
so that
h i
h
MX (q ) = E eq X = E eq z(w
w)
The BR theorem then implies that for large nc ,
P(G(z f )
with rate function
g⇤ ) ⇡ p
a + zw = z
= M(zq )e
1
e
2pnc s qa
I(a) = qa (a + zw)
and qa determined from
i
nc I(a)
,
ln[M(zqa )],
M 0 (zqa )
M(zqa )
The corresponding variance s 2 is given by
"
✓ 0
◆2 #
00 (zq )
M
M
(zq
)
a
a
s 2 = z2
.
M(zqa )
M(zqa )
q zw
w .
(5B.15)
.
(5B.16)
(5B.17)
(5B.18)
(5B.19)
(5B.20)
It is straightforward to extend the above analysis to two classes of self-peptides,
which occur at different levels of abundance [17]. Suppose that when z f = 0 there
are nc types in the first class with abundance zc , and nv types of the second class
8
with abundance zv . (One could also treat the abundances as random variables, see
[19].) The total number of MHCs that encounter TRCs is still taken to be nM so that
q(nc zc + nv zv ) + z f = nM .
(5B.21)
Proceeding along similar lines to the case of a single class of self peptides, we find
that for nT = nc + nv
P(G(z f )
where,
a=

g⇤
g⇤ ) ⇡ p
z f w(t f )
nM z f
1
e
2pnT s qa
w
and modified rate function
nc
ln[M(zc qa )]
nT
I(a) = qa (a + zw)

nT I(a)
,
nc zc nv zv
+
nT
nT
nv
ln[M(zv qa )],
nT
(5B.22)
(5B.23)
(5B.24)
Now qa is determined from the equation
a + zw =
nc zc M 0 (zc qa ) nv zv M 0 (zv qa )
+
nT M(zc qa )
nT M(zv qa )
(5B.25)
1
P(G(zf) ≥ g*)
10-3
2500
10-6
10-9
0
200
400
1000
600
g*
1500
2000
800 1000
Fig. 5B.4: Activation curves (activation probability versus threshold g⇤ ) for different values of the
abundance z f of foreign peptides and an exponential distribution r(t). Parameter values are as
follows: t ⇤ = 1, t = 0.04t ⇤ , nc = 50, nv = 1500, nM = 105 , zc = 5 ⇥ 10 3 nM and zv = 5 ⇥ 10 4 nM .
Adapted from [17].
9
and the variance s 2 is given by
"
"
✓ 0
◆2 #
2 M 00 (z q )
2 M 00 (z q )
n
z
M
(z
q
)
n
z
c
c
a
c
a
v
v a
c
v
s2 =
+
nT
M(zc qa )
M(zc qa )
nT
M(zv qa )
◆ #
M 0 (zv qa ) 2
.
M(zv qa )
(5B.26)
In Fig. 5B.4 we show example activation curves for different levels z f of foreign
peptide, that is, plots of the activation probability P(G(z f ) g⇤ ) as function of
the activation threshold g⇤ . Results are shown for r(t) given by an exponential
distribution. It can be seen that the activation curves for z f = 0 and z f > 0 are well
separated provided that g⇤ > 500 and z f > 1000.
✓
Negative selection
One of the potential dangers of the immune system is that there could be circulating
T cells that are activated by self-peptides, resulting in an autoimmune response. The
risk of autoimmunity is an inevitable consequence of the fact the immune system
has to cope with diverse and quickly evolving pathogens. The latter is dealt with by
having a vast and diverse range of antigen receptors, which are generated randomly.
However, in order to achieve such diversity, it is inevitable that receptors selective
for body antigens are also made. This raises the issue of show these so-called autoreactive T cells (and B-cells), which could cause autoimmune diseases, are prevented
from being activated in healthy individuals. One suggested mechanism for eliminating dangerous T cells is negative selection during maturation of young T cells
in the thymus. That is, if an immature T cell is exposed to a large number of APCs
containing only self-peptides, then a T cell exhibiting a stimulation rate that exceeds
some threshold gthy could be induced to die. One can model the process of negative
selection along similar lines to that of foreign antigen detection in mature T cells
[17]. However, negative selection cannot fully account for the down regulation of
autoreactive T cells, since one finds significant numbers of autoreactive cells circulating in healthy individuals [12, 13]. Other mechanisms for regulating T cells are
discussed in the next section.
5B.2 Mechanisms for regulating the immune response
There are a number of different hypotheses for how circulating autoreactive T cells
are dealt with in healthy individuals, as reviewed by Carneiro et al. [1]. One possibility is that specific regulatory T cells control autoreactive T cells in such a fashion that they prevent the latter from proliferating and causing an immune response
[9, 3, 8]. There is growing experimental evidence that so-called regulatory CD4+ T
cells, which express forehead box protein 3 (FoxP3) play a critical role in the development of natural tolerance to self-antigens and the prevention of autoimmune
10
responses [13]. A typical experimental protocol is to extract circulatory CD4+ T
cells from a healthy animal and to inject these cells into an animal without T cells.
One finds that the animal does not develop an autoimmune response when also exposed to autoreactive T cells, in contrast to the case where the regulatory cells are
absent. An alternative hypothesis is that autoreactive T cells become unresponsive
to self-peptides by modification of their cell signaling machinery - the resulting unresponsiveness is known as anergy. A simple mechanism for the induction of anergy
is that T-cels increase their activation threshold in response to frequently recurring
stimuli [7]. In this section we consider population models of these processes, following along the lines of Ref. [1].
Tunable activation thresholds
The basic idea underlying a tuning activation threshold is that every interaction between a T cell and an APC results in an intracellular competition between activation
and suppression signaling pathways that modify the threshold for activation. A simple reaction diagram is shown in Fig. 5B.5 [4]. Let TF denote the density of free T
cells, AF the density of free APCs and C the density of T cell-APC complexes. The
corresponding population dynamics is given by the pair of equations
dTF
= (1 + a)dC cTF AF
dt
dC
= cTF AF dC,
dt
d TF
(5B.27a)
(5B.27b)
antigen presenting cell
resting T cell
δ
activated T cell
c
d(1-α)
dα
Fig. 5B.5: Reaction diagram of a minimal model of the population dynamics of T cells with tunable
activation thresholds. T cells degrade at a rate d , bind to APCs at a rate c, and dissociate at a rate
d. Finally, a denotes the probability that following dissociation, the T cell is in an activated state
and subsequently undergoes cell division.
11
where d is the degradation rate, c is the association rate, d is the dissociation rate,
and a is the probability that a T cell is in an activated state following dissociation.
Note that there is effective growth of the T cell population at a rate da due to cell
division. This pair of equations is supplemented by the conservation equation
A = AF +C,
(5B.28)
where the total APC density A is fixed. It is experimentally difficult to separately
count free and associated T cells, so one usually tracks variations in the total T cell
concentration T = TF +C, which evolves according to
dT
= daC
dt
d (T
C).
(5B.29)
Finally, it is assumed that the density of complexes is in quasi-steady-state so that
c(T
C)(A
C)
dC = 0,
which has one root that ensures all densities are non-negative:

q
1
C=
c(T + A) + d
(c(T + A) + d)2 4AT c2 .
2c
(5B.30)
It is clear from the above model that the threshold for activation is determined
by the probability a, that is, the threshold increases as a decreases. One now has to
specify how a depends on the signaling machinery of an individual T cell. One conceptual model of T cell activation is that downstream of TCR signaling are kinase
and phosphotase molecules that act on adapter proteins [7]. A simplified model of
the signaling pathway is described by the kinetic equations [1]
dK
= rK [K0 (1 + s ) K]
dt
dP
= rP [P0 (1 + s ) P],
dt
(5B.31a)
(5B.31b)
where K is the concentration of kinase, P is the concentration of phosphotase, rK
is the turnover rate of kinase, rP is the turnover rate of phosphotase, K0 is the basal
concentration of kinase, and P0 is the basal concentration of phosphotase. The parameter s represents the magnitude of the upstream signal from the TCRs, with
s = 0 if the T cell is free and s = s0 > 0 if the T cell is bound to an APC. During
T cell conjugation with an APC, it is assumed that all adapter proteins become fully
phosphorylated if the kinase activity is higher than the phosphotase activity, otherwise the adapter proteins are fully dephosphorylated. The adapter proteins thus act
like a molecular switch which determines whether or not the T cell enters the cell
cycle following dissociation from the APC. TCR activity can temporarily switch on
the adapter and promote cell division if P0 > K0 (higher basal phosphotase activity)
whereas rK > rP (higher turnover of kinase).
12
As shown by Carneiro et al. [1], it is possible to formulate the above model of T
cell activation in terms of a population-based stochastic hybrid system by assuming
that the kinetics of kinase activity is much faster than other process (rK
d, rP ) so
that the kinase activity is in quasi-steady-state. That is, K = K0 (1 + s0 ) if the T cell
is bound to an APC and K = K0 if the T cell is free. We can then just keep track of
the phosphotase activity, whose dynamics switches between two cases depending
on whether a T cell is bound to an APC (n = 1) or free (n = 0):
dP
= Fn (P),
dt
(5B.32)
with F0 (P) = rP (P0 P), F1 (P) = rP (P0 [1 + s0 ] P). At the population level, let
rn (P,t) denote the probability density that the phosphotase concentration of a T cell
at time t is P, given that the T cell is in state n. Since the transition rates are w = d
for n = 1 ! n = 0 and w+ = c(A C) for n = 0 ! n = 1, we have the CK equation
∂ r0
=
∂t
∂ r1
=
∂t
∂ [F0 (P)r0 ]
+ w r1 w+ r0
∂P
∂ [F1 (P)r1 ]
w r1 + w+ r0 .
∂P
(5B.33a)
(5B.33b)
From the specific form of the functions Fn , we see that P is always contained in the
interval [P0 , P0 (1 + s0 )].
Under the adiabatic approximation that the phosphotase dynamics occurs on a
much faster time-scale than changes in the density of complexes C, we can analyze
the quasi-steady-state solution along similar lines to the Dogterom-Leibler model of
microtubule catastrophes in Sect. 4.1.2 and the simple model of bacterial chemotaxis
in Sect 5.3.3. Setting time derivatives to zero and summing equations (5B.33a,b)
yields the conservation condition F0 (P)r0 (P) + F1 (P)r1 (P) = H. Setting P = P0
shows that H > 0, whereas setting P = P0 (1 + s0 ) shows that H < 0 (since rn is
positive). Hence, H = 0 and we obtain the following equation for r1 :
∂ [F1 r1 ]
= w r1
∂P
w+
F1
r1 ,
F0
which can be rewritten as
∂ r1
=
∂P
This has the solution

w
w+ ∂ ln F1
+
+
r1 .
F1
F0
∂P
r1 (P) = N eF(P) ,
where N is a normalization factor and
13
T cell density T
2500
2000
1500
1000
500
0
0
40
20
60
80 100
adaptation capacity P0/K0
Fig. 5B.6: Bifurcation diagram for steady-state T cell densities as a function of the adaptation
capacity P0 /K0 . Stable and unstable fixed points are indicated by solid and dashed curves respectively. Parameter values are s = 1000, rP = 0.027 day 1 , A = 8 cells, d = 6 day 1 , c =
0.06 cell 1 day 1 , d = 0.02 day 1 .
Z 
w
w+
∂ ln F1 (P)
+
+
dP
F1 (P) F0 (P)
∂P
w
w+
=
ln |P0 (1 + s0 ) P| +
ln |P0 P|
rP
rP
F(P) =
ln |P0 (1 + s0 )
P|.
Hence, the steady-state probability density for finding a phosphotase concentration
P in the bound population of T cells is
r1 (P) = N |P0 (1 + s0 )
P|d/rP
1
|P0
P|c(A
C)/rP
(5B.34)
with P0  P  P0 (1 + s0 ). Finally, the fraction a of T cells that is activated and
divides when dissociation from an APC occurs is given by the fraction of T cells
that have K > P when releasing from an APC. It follows that
a=
Z K0 (1+s0 )
P0
r1 (P)dP.
(5B.35)
The value of a tracks the density C of T cell-APC complexes, which itself depends on the total density T of T cells according to equation (5B.30). At low T cell
densities, the density of complexes C is small but the frequency at which an individual T cell interacts with an APC is high. This means that the level of phosphotase
activity within each T cells tends towards P0 (1 + s0 ), resulting in a low value of a.
This tends to stabilize a state in which T = 0 (extinction). On the other hand, if T is
high, then the density C ⇡ A (saturation of APCs) so that the encounter rate is small
and P tends towards P0 . Now a is large and promotes cell division, which further
14
increases the T cell density. This positive feedback loop when combined with saturation can stabilize a high density T cell state. One thus finds that over a range of
values of P0 /K0 , the system exhibits bistability between an extinct state and a high
density state, whereas for sufficiently large P0 /K0 only extinction occurs. This is
illustrated in Fig. 5B.7. Note that without adaptation (P0 = 0, K0 > 0 and a fixed),
the extinct state is unstable.
Regulatory T cells
The above activation threshold model is not able to fully capture the suppressive
effects of regulatory T cells on autoreactive T cells transferred to an animal without
T cells [13]. Therefore, it is necessary to model the interactions between both types
of T cell explicitly. Here we review a mathematical model due to Leon et al [9],
which posits that the persistence and expansion of regulatory T cells (denoted by
TR ) depend on the interactions with autoreactive T cells (denoted by TE ), which
are mediated by the presence of antigen presenting cells (APCs). For the sake of
illustration, we will consider a simplified version of this crossregulation model, in
which each APC has two binding sites for T cells [3]. (A more general model has
multiple binding sites and multiple interactions between T cells). The basic reactions
of the model are illustrated in Fig. 5B.7, in which TR and TE cells compete for the
APC binding sites. If a TE cell is bound to an APC then it can divide provided that the
other site is not occupied by a TR cell. On the other, a bound TR cell can only divide
if the other binding site is occupied by a TE cell, whose own activation is inhibited
by the TR cell. Following along similar lines to the previous model, we assume that
both cell types degrade at a rate d , bind to an APC at a rate c, and dissociate at a
rate d. Finally, in situations where a T cell can be activated, the probability of cell
division following dissociation is given by aE and aR , respectively. However, in
contrast to the previous model, the rates aE,R are not adaptable.
Let E and R be the density of TE and TR receptors, and let EF , EB and RF , RB
be the corresponding densities of free and bound receptors. The density of APCs
with no sites occupied is denoted by A0 , the densities of APCs with one bound site
of a given type are given by AE and AR , whereas the densities of APCs with two
bound sites are taken to be AEE , ARR and AER , with the subscripts indicating which
combination of T cell types is involved. Finally, let A be the total density of APCs,
F the density of free binding sites, T the total density of T cells, and TB = EB + RB
the total density of bound T cells. We thus have the conservation relations
T = E + R = EF + RF + EB + RB ,
EB = AE + 2AEE + AER ,
2A = F + EB + RB ,
RB = AR + 2ARR + AER
A = A0 + AE + AR + AEE + ARR + AER .
(5B.36a)
(5B.36b)
(5B.36c)
From the law mass action, we have the following kinetic equations for E and R:
15
antigen presenting cell
TE cell
TR cell
δ
(c)
(b)
(a)
c
d(1-α)
c
δ
dα
d(1-α)
c
d(1-α)
c
d
dα
c
dα
d
Fig. 5B.7: Reaction diagram of the minimal crossregulation model describing the interactions between (a) one or two TE cells with an APC, (b) one or two TR cells with an APC, (c) one TE cell and
one TR cell with an APC. For simplicity, both TR and TE cells degrade at a rate d , bind to APCs at
a rate c, and dissociate at a rate d. If only one site is bound by a T cell then cell division can occur
with probability aE if it is a TE cell but not if it is a TR cell. The same holds if the other site is bound
by a T cell of the same type. However, if both a TR cell and a TE cell are simultaneously bound
then the TE cell promotes activation of the TR cell at a rate aR , which in turn inhibits activation of
the TE cell.
dE
= (1 + aE )d[AE + 2AEE ]
dt
dR
= (1 + aR )dAER d R.
dt
d E,
(5B.37a)
(5B.37b)
Similarly, the kinetic equations for the various APC subpopulations are
dA0
= dAE + dAR 2cA0 [E EB ] 2cA0 [R RB ],
dt
dAE
= 2cA0 [E EB ] + dAER + 2dAEE dAE cAE [R
dt
dAR
= 2cA0 [R
dt
dAEE
= cAE [E
dt
dARR
= cAR [R
dt
dAER
= cAR [E
dt
RB ] + dAER + 2dARR
dAR
cAR [E
(5B.38a)
RB ]
cAE [E
EB ],
(5B.38b)
EB ]
cAR [R
RB ]
(5B.38c)
EB ]
2dAEE
(5B.38d)
RB ]
2dARR
(5B.38e)
EB ] + cAE [R
RB ]
2dAER
16
(5B.38f)
The first equation is linearly dependent on the other four due to conservation of
APCs.
As in the previous model, we make the adiabatic assumption that the dynamics
of E and R is much slower than the dynamics of binding/unbinding. We can then
assume that the various APC subpopulations are in quasi-equilibrium. This now
generates algebraic equations for E EB and R RB in terms of the APC densities. First, seting all time time derivatives to zero in equations (5B.38), consider the
combinations (b)+(d)+(f) and (c)+(e)+(f). This yields the pair of equations
(2A0 + AE + AR )[E
(2A0 + AE + AR )[R
EB ]
RB ]
d
[AE + 2AEE + AER ] = 0,
c
d
[AR + 2ARR + AER ] = 0.
c
Using the conservation relations, these reduce to
KF[E
EB ] = EB ,
KF[R
RB ] = RB
(5B.39)
with equilibrium constant K = c/d, which can be rearranged to yield
EB =
FK
E,
1 + FK
RB =
FK
R
1 + FK
(5B.40)
Finally, substituting these equations into the conservation relation 2A = F + EB + RB
results in a quadratic for F, which has the unique positive solution
p
[1 + K(T 2A)] + [1 + K(T 2A)]2 + 8AT K 2
F=
.
(5B.41)
2K
(If the binding/unbinding rates for the TR and TE receptors were different, then F
would satisfy a cubic.) Once F = F(T, A) has been determined, we can calculate
EB = EB (E, T, A) and RB = RB (R, T, A), and thus solve the steady-state equations
(5B.38b)- (5B.38f) to determine the densities AE , AR , ARR , AEE , ARE as functions of
E, R, A. More specifically,
AE =
FEB
,
2A
AR =
FRB
,
2A
AEE =
EB2
,
4A
ARR =
R2B
,
4A
AER =
EB RB
,
2A
A0 =
F2
.
4A
In summary, using the quasi-steady-state solutions, we can write equations
(5B.38) as the closed planar system
dE
= F1 (E, R; A),
dt
dR
= F2 (E, R; A),
dt
(5B.42)
with the total density of APCs A acting as a parameter. We can now determine the
fixed points for the receptor densities (E, R) and how they depend on A. An example
bifurcation diagram is shown in Fig. 5B.8. It can be seen that there is a critical APC
density A = aE that is required in order to sustain a population of TE cells. Beyond
17
the persistence and
ecurrent interactions
growth factors are
/2007
A
Total T cell density, E+R
1000
100
E
R
10
1
R
E
R
0.1
0.001
E
0.01
aE
0.1
aR
1
10
100
B
Fig. 5B.8: Bifurcation diagram of the minimal crossregulation model, showing the equilibria for
the total receptor density T = R + E as a function of the APC density A. [Adapted from Carniero
et al [3].] Solid (dashed) lines indicate stable (unstable) equilibria. The pie charts show the relative
1000
proportions of TR and TE cells at particular points on the stable branches. Parameter values are
K = 1, d = 0.01, pE ⌘ (1 + aE )d = 1.1 and pR ⌘ (1 + aR )d = 1.0.
Total T cell density, E+R
in vitro cultures of TR
can reproduce patterns
in real in vitro cultures
enriched cells (labeled
hed cells (labeled with
ell-depleted, APCed with DDAO blue)
s with RPMI media,
CD3. Aggregates were
y. (B) Individual cellgregation parameters
n rules for the cell life
ults adapted from
produced by TE cells in their vicinity. These simple principles
imply that there is a critical APC density, aE, that is necessary and
sufficient to sustain a TE cell population (Fig. 6, top). The
persistence of a TR cell population requires a higher density of
100
this critical density there is a stable branch of equilibria in which the population of
TE cells is dominant. However, at a second critical density aR there is a saddle-node
10resulting in the emergence of a second stable branch along which TR
bifurcation
cells dominate, thus suppressing the autoimmune response of the TE cells.
1
Supplementary references
0.1
1. Carneiro, J., Paixao, T., Milutinovic, D., Sousa, J., Leon, K., Gardner, R., Faro, J.: Immunolog0.001 Lessons0.01
0.1 modeling. J.1Comput. Appl.10
100
ical self-tolerance:
from mathematical
Math. 184, 77-100
(2005).
Cognate APC density
2. Coombs, D., Goldstein, B.: T cell activation: Kinetic proofreading, serial engagement and cell
adhesion. J. Comput. Appl. Math. 184, 121-139 (2005).
Fig.
6. Equilibrium
densities
of den
specific
populations
R andV.TOliveira,
E cell M.
3. Carneiro,
J., K. Leon, I. Caramalho,
C. Van
Dool, R.TGardner,
Bergman, N.
Sepulveda,
T.
Paixao,
J.
Faro,
and
J.
Demengeot.
When
three
is
not
a
crowd:
a
cross-regulation
as a function of the density of their cognate APCs. (A) Bifurcation
model of the dynamics and repertoire selection of regulatory CD4 T cells. Immunol. Rev. 216,
diagram
of the model representing all possible equilibria. The lines
48-68 (2007).
4.
De
Boer,
R. J.,total
Perelson,
A. S.: T cell repertoires
exclusion,
Theor.Biol.
169
indicate the
equilibrium
density and
of competitive
T cells (sum
of J.the
variables
375-390 (1994).
E þ R) as a function of the cognate APC density (parameter A). Solid
lines indicate stable equilibria, and the dashed line indicates unstable
18
equilibria. The pies indicate the relative proportions of TE and TR cells at
equilibrium [respectively E/(E þ R) and R/(E þ R)] for the indicated
values of A. (B) The lines indicate the equilibria that are actually
reached by solving the system with fixed initial conditions (solid line:
E ¼ 2/3 and R ¼ 1/3; dashed line: E ¼ 1/3 and R ¼ 2/3), as a function
of the density of cognate APCs. Remaining parameter values as in Fig. 2.
5. Dembo, A., Zeitouni, O.: Large Deviations: Techniques and Applications, 2nd edn. Springer,
NewYork (2004).
6. Goldstein, B., Faeder, J. R., Hlavacek, W. S. Mathematical and computational models of
immune-receptor signaling. Nat. Rev. Immun. 4. 445-456 (2004)
7. Grossman, Z., Paul, W. E.: Autoreactivity, dynamic tuning and selectivity, Curr. Opin. Immunol.13, 687-698 (2001).
8. Kim, P. S., Lee, P. P., and Levy, D. Modeling regulation mechanisms of the immune system. J.
Theor. Biol. 246, 33-69 (2007).
9. Leon, K., R. Perez, A. Lage, and J. Carneiro. 2000. Modelling T-cell-mediated suppression
dependent on interactions in multicellular conjugates. J. Theor. Biol. 207, 231-254 (2000).
10. Lever, M., Maini, P. K., van der Merwe, P. A., Dushek, O. Phenotypic models of T cell activation. Nat. Rev. Immun. 14 619-629 (2014)
11. McKeithan, K.: Kinetic proofreading in T-cell receptor signal transduction. Proc. Natl. Acad.
Sci. USA 92, 5042-5046 (1995).
12. Pereira, P., Forni, L. et al., Autonomous activation of B and T cells in antigen-free mice. Eur.
J. Immunol. 16, 685-688 (1986).
13. Sakaguchi, S., Sakaguchi, N. et al.: Immunologic tolerance maintained by CD25+ CD4+ regulatory T cells: their common role in controlling autoimmunity, tumor immunity, and transplantation tolerance, Immunol.Re 182, 18-32 (2001).
14. Touchette, H. The large deviation approach to statistical mechanics. Phys. Rep. 478, 1-69
(2009).
15. Valitutti, S., Muller, S., Cella, M., Padovan, E., Lanzavecchia, A.: Serial triggering of many T
cell receptors by a few peptide-MHC complexes. Nature 375, 148-151 (1995)
16. Valitutti, S., Lanzavecchia, A.: Serial triggering of TCRs: a basis for the sensitivity and specificity of antigen recognition. Immunol. Today 18, 299-304 (1997)
17. van den Berg, H.A., Rand, D.A., Burroughs, N.J.: A reliable and safe T cell repertoire based
on low-affinity T cell receptors. J. Theor. Biol. 209, 465486 (2001)
18. van den Berg, H.A., Burroughs, N.J., Rand, D.A.: Quantifying the strength of ligand antagonism in TCR triggering. Bull. Math. Biol. 64, 781808 (2002)
19. Zint, N., Baake, E., Hollander, F.: How T cells use large deviations to recognize foreign antigens. J. Math. Biol. 57, 841-861 (2008).
19