Molina-Par?s C., Lythe G. (editors) Mathematical Models and Immune Cell Biology
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9 Competing T Cell Clonotypes 193
is the set of APPs which provide survival signals to T cells of clonotype 1 and to r
other distinct clonotypes, none of which is clonotype 2. Hence,
.1/
D
C1
X
rD0
0
@
X
q2Q
12r
1
n
1
C n
2
C n
12q
C
X
q2Q
1r=2
1
n
1
C n
1q
1
A
: (9.9)
Note that the birth rate of T cells of clonotype 1 depends on the number of T cells
of any other clonotype that competes with clonotype 1 for access to survival signals
from an APP, through the terms n
1q
and n
12q
.
We now introduce a mean field approximation so that this birth rate depends only
on the number of T cells belonging to clonotypes 1 and 2. We denote by
– p
1
the probability that a randomly chosen APP provides a survival stimulus to
T cells of clonotype 2, given that it provides a survival stimulus to T cells of
clonotype 1
– p
2
the probability that a randomly chosen APP provides a survival stimulus to
T cells of clonotype 1, given that it provides a survival stimulus to T cells of
clonotype 2
– p
ji=j
the probability that an APP chosen at random from Q
i=j
will belong to the
set Q
k
of a different clonotype k selected at random
With the mean field approximation we write
.1/
D
C1
X
rD0
jQ
12r
j
n
1
C n
2
C rhni
C
jQ
1r=2
j
n
1
C rhni
; (9.10)
where
jQ
12r
jDp
1
jQ
1
j
r
12
e
12
rŠ
and jQ
1r=2
jD.1 p
1
/jQ
1
j
r
1
e
1
rŠ
:
The parameter
12
is the mean niche overlap for APPs in the set Q
12
, which is the
average number of clonotypes that are competing with clonotype 1 and clonotype
2 for an APP in the set Q
12
, whereas
1
is the mean niche overlap for APPs in
the set Q
1=2
, which is the average number of clonotypes that are competing with
clonotype 1 for an APP in the set Q
1=2
. Hence, these parameters represent the
strength of competition between T cells of clonotype 1 and T cells of other clono-
types in the repertoire (other than clonotype 2) for APPs in these sets, Q
12
and
Q
1=2
, respectively. These parameters are not the same as the mean niche overlap
parameter for all APPs in the set Q
1
, , which is a parameter of the corresponding
univariate model described in the previous chapter. However, these parameters are
related by
D
12
p
1
C
1
.1 p
1
/: (9.11)
194 C. Molina-Par´ıs et al.
Hence, the per-cell birth rate of T cells of clonotype 1 is given by
.1/
D'
1
p
1
e
12
C1
X
rD0
r
12
rŠ
1
n
1
C n
2
C rhni
C.1 p
1
/e
1
C1
X
rD0
r
1
rŠ
1
n
1
C rhni
!
;
(9.12)
where '
1
D jQ
1
j is a parameter proportional to the number of APPs from which
T cells of clonotype 1 receive a survival signal. By a similar derivation, the per-cell
birth rate of T cells of clonotype 2 is given by
.2/
D'
2
p
2
e
12
C1
X
rD0
r
12
rŠ
1
n
1
Cn
2
Crhni
C.1 p
2
/e
2
C1
X
rD0
r
2
rŠ
1
n
2
C rhni
!
;
(9.13)
where '
2
D jQ
2
j and p
2
D P.q 2 Q
1
jq 2 Q
2
/. The per-cell death rate of a given
clonotype is assumed to be constant and is given by
1
for T cells of clonotype 1
and
2
for T cells of clonotype 2. Therefore, the birth and death rates of the bivariate
competition process are given by
.1/
n
1
;n
2
D
.1/
n
1
;
.2/
n
1
;n
2
D
.2/
n
2
;
.1/
D
1
n
1
and
.2/
D
2
n
2
:
Summary of the Model
We have constructed a bivariate competition process fX.t/; Y.t/ W t t
0
g on the
state-space f.n
1
;n
2
/ W n
1
;n
2
D 0; 1; : : :g which models the number of T cells
belonging to clonotypes 1 and 2 at time t. The birth rates are given by (9.12), (9.13)
plus
.1/
0;n
2
D 0 for n
2
0 and
.2/
n
1
;0
D 0 for n
1
0. The death rates are given
by
.1/
n
1
;n
2
D
1
n
1
and
.2/
n
1
;n
2
D
2
n
2
for n
1
;n
2
0. By definition, Q
1
\ Q
2
D
Q
12
D Q
21
D Q
2
\ Q
1
.WealsohavethatjQ
12
jDp
1
jQ
1
jD'
1
p
1
and,
similarly, jQ
21
jDp
2
jQ
2
jD'
2
p
2
. This leads to the constraint on the parameters
'
1
p
1
D '
2
p
2
; (9.14)
where 0 p
1
1 and 0 p
2
1, as the parameters p
1
and p
2
represent
probabilities. Hence, the model has nine independent parameters,
'
1
;'
2
;p
1
;
12
;
1
;
2
;
1
;
2
; hni;
compared to the four parameters of the univariate model described in the previous
chapter.
9 Competing T Cell Clonotypes 195
Clonal Extinction and Mean Extinction Times
Sincewehavethat
.1/
0;j
D
.2/
j;0
D
.1/
0;j
D
.2/
j;0
D 0 for all j 0,thesetof
states A Df.n
1
;n
2
/ W n
1
D 0 or n
2
D 0g forms an absorbing set. (Once the
process enters this set, it will never leave.) This corresponds to extinction of one of
the clonotypes from the T cell repertoire. The state .n
1
;n
2
/ D .0; 0/ is an absorbing
state corresponding to extinction of both clonotypes. In this section, we prove that
the probability of the process eventually reaching this absorbing state is one and
show that the mean time until the absorbing state is reached is finite. In order to do
this, we require the following upper bound on the birth rates:
.1/
n
1
;n
2
'
1
n
1
p
1
e
12
C1
X
rD0
r
12
rŠ
1
n
1
C .1 p
1
/e
1
C1
X
rD0
r
1
rŠ
1
n
1
!
D '
1
n
1
p
1
e
12
e
12
n
1
C .1 p
1
/e
1
e
1
n
1
D '
1
: (9.15)
Similarly, we can show that
.2/
n
1
;n
2
'
2
.
Both Clonotypes Become Extinct with Probability One
In order to prove that the absorbing state at .n
1
;n
2
/ D .0; 0/ is reached with prob-
ability one, we use the method of Iglehart [11] and bound the bivariate competition
process with a univariate birth and death process which moves towards the absorb-
ing state at the origin at a slower rate than the bivariate process. We then show that
the univariate process reaches the absorbing state with certainty, and hence we may
conclude that the bivariate competition process also reaches state .0; 0/ with proba-
bility one. We first divide the state-space S of the process into the following disjoint
subsets:
S
k
Df.n
1
;n
2
/ W n
1
C n
2
D kg for k 0:
A schematic diagram of these sets is given in Fig. 9.3. The birth and death rates for
the univariate birth and death process are now defined in such a way that this new
process moves towards the origin at a slower rate than the bivariate process. Let
0
k
D max
.n
1
;n
2
/2S
k
n
.1/
n
1
;n
2
C
.2/
n
1
;n
2
o
and
0
k
D min
.n
1
;n
2
/2S
k
n
.1/
n
1
;n
2
C
.2/
n
1
;n
2
o
;
with
0
k
D
0
k
D 0 when k D 0. If the process is in the set S
k
, at the next transition
it moves to either S
k1
or S
kC1
.Therate
0
k
is the maximum rate at which the
bivariate process moves upwards from the set S
k
to S
kC1
and the rate
0
k
is the
196 C. Molina-Par´ıs et al.
S
0
S
1
S
2
S
3
S
4
n
2
n
1
Fig. 9.3 Diagram of the sets S
k
for k 0
minimum rate at which the bivariate process moves downwards from the set S
k
to
S
k1
. Hence, these rates define a univariate birth and death process on the state
space fS
0
; S
1
; S
2
;:::g where S
0
is an absorbing state and S
k
is now treated as an
element rather than a set. This process moves towards the absorbing state at a slower
rate than the bivariate process and can be represented by the following diagram:
A sufficient condition for this process to reach the absorbing state with probability
one is that the series
C1
X
kD1
0
1
0
2
:::
0
k
0
1
0
2
:::
0
k
(9.16)
diverges (see previous chapter). In order to prove this, we first observe that
0
k
D max
.n
1
;n
2
/2S
k
n
.1/
n
1
;n
2
C
.2/
n
1
;n
2
o
'
1
C '
2
; (9.17)
from (9.15)and
0
k
D min
.n
1
;n
2
/2S
k
f
1
n
1
C
2
n
2
g
D k min.
1
;
2
/: (9.18)
9 Competing T Cell Clonotypes 197
Therefore,
C1
X
kD1
0
1
0
2
:::
0
k
0
1
0
2
:::
0
k
C1
X
kD1
kŠŒmin.
1
;
2
/
k
.'
1
C '
2
/
k
: (9.19)
Let
a
k
D
kŠŒmin.
1
;
2
/
k
.'
1
C '
2
/
k
; (9.20)
so that
a
kC1
a
k
D
.k C 1/ min.
1
;
2
/
.'
1
C '
2
/
!C1as k !C1: (9.21)
Hence, the series
P
C1
kD1
a
k
diverges by the ratio test and so the series (9.16)diverges
by comparison. Therefore, the univariate process reaches the origin with certainty
and so we conclude that the bivariate competition process ultimately reaches the ab-
sorbing state .n
1
;n
2
/ D .0; 0/ with probability one for all values of the parameters,
corresponding to extinction of both clonotypes from the repertoire.
Finite Mean Extinction Times
The mean time until both clonotypes become extinct, when the initial state of the
process is .n
1
;n
2
/, is denoted by
n
1
;n
2
. This quantity is finite for all .n
1
;n
2
/ 2
S nf.0; 0/g if the series
C1
X
kD1
0
1
0
2
:::
0
k1
0
1
0
2
:::
0
k
(9.22)
converges [11]. For the rates (9.17)–(9.18),
C1
X
kD1
0
1
0
2
:::
0
k1
0
1
0
2
:::
0
k
C1
X
kD1
.'
1
C '
2
/
k1
kŠŒmin.
1
;
2
/
k
: (9.23)
Let
b
k
D
.'
1
C '
2
/
k1
kŠŒmin.
1
;
2
/
k
: (9.24)
Then,
b
kC1
b
k
D
.'
1
C '
2
/
.k C1/ min.
1
;
2
/
! 0 as k !C1; (9.25)
which means that the series
P
C1
kD1
b
k
converges by the ratio test. Hence, the series
(9.22) converges by comparison and therefore the mean time to reach the absorb-
ing state from all initial states .n
1
;n
2
/ 2 S nf.0; 0/g is finite for all values of
198 C. Molina-Par´ıs et al.
the parameters. This is because the convergence of the series (9.22) is sufficient
to prove that the univariate process with the birth and death rates
0
k
and
0
k
reaches the absorbing state in finite time from all initial states, and this process
moves towards the absorbing state at a slower rate than the bivariate competition
process.
The Limiting Conditional Probability Distribution
The limiting conditional probability distribution of the process represents the late-
time behaviour of the system before extinction occurs. Since extinction of both
clonotypes occurs with certainty in finite time, the limiting probability distribution
of the process, lim
t!C1
p
n
1
;n
2
.t/, has all its mass at the state .n
1
;n
2
/ D .0; 0/.In
order to study the behaviour of the process before extinction takes place, we condi-
tion on the event that the process has not yet reached the absorbing set A.Letp
A
.t/
denote the probability that the process is not in set A at time t.Wedefine
q
n
1
;n
2
.t/ D
p
n
1
;n
2
.t/
p
A
.t/
; (9.26)
which is the probability that the process is in state .n
1
;n
2
/ at time t, given that
extinction of either clonotype has not yet occurred. Then
dq
n
1
;n
2
.t/
dt
D
1
p
A
.t/
dp
n
1
;n
2
.t/
dt
p
n
1
;n
2
.t/
.p
A
.t//
2
dp
A
.t/
dt
: (9.27)
Using the fact that
P
C1
n
1
D0
P
C1
n
2
D0
p
n
1
;n
2
D 1,wehave
p
A
.t/ D 1
C1
X
n
2
D0
p
0;n
2
.t/
C1
X
n
1
D0
p
n
1
;0
.t/ C p
0;0
.t/; (9.28)
and hence
dp
A
.t/
dt
D
d
dt
C1
X
n
2
D0
p
0;n
2
.t/
d
dt
C1
X
n
1
D0
p
n
1
;0
.t/ C
dp
0;0
.t/
dt
: (9.29)
We now derive expressions for the terms on the right hand side of this equation.
d
dt
C1
X
n
2
D0
p
0;n
2
.t/ D
1
C1
X
n
2
D0
p
1;n
2
.t/ and
d
dt
C1
X
n
1
D0
p
n
1
;0
.t/ D
2
C1
X
n
1
D0
p
n
1
;1
.t/:
9 Competing T Cell Clonotypes 199
Therefore,
dp
A
.t/
dt
D
1
C1
X
n
2
D1
p
1;n
2
.t/
2
C1
X
n
1
D1
p
n
1
;1
.t/;
and
dq
n
1
;n
2
.t/
dt
D
.1/
n
1
1;n
2
q
n
1
1;n
2
.t/ C
.2/
n
1
;n
2
1
q
n
1
;n
2
1
.t/ C
.1/
n
1
C1;n
2
q
n
1
C1;n
2
.t/
C
.2/
n
1
;n
2
C1
q
n
1
;n
2
C1
.t/
.1/
n
1
;n
2
C
.2/
n
1
;n
2
C
.1/
n
1
;n
2
C
.2/
n
1
;n
2
q
n
1
;n
2
.t/
C
1
q
n
1
;n
2
.t/
C1
X
n
2
D1
q
1;n
2
.t/ C
2
q
n
1
;n
2
.t/
C1
X
n
1
D1
q
n
1
;1
.t/; (9.30)
for n
1
1, n
2
1. A distribution, Nq, is called a quasi-stationary probability distri-
bution (QSD) if it is a solution of the equations obtained from (9.30) by setting the
time derivatives to zero. The limiting conditional probability distribution (LCD) is
defined by
lim
t!C1
q
n
1
;n
2
.t/; (9.31)
where n
1
1, n
2
1. This distribution is independent of time and so is a QSD.
For a process where the state-space is finite, the QSD of the process is unique and
is equal to the LCD [12]. This will always be the case when the LCD of the process
is calculated numerically.
A Process with m Competing Clonotypes
In previous sections we have studied a process modelling the number of T cells
belonging to two competing clonotypes. In this section we extend the model to
include m competing clonotypes for which the mean field approximation does not
hold, and prove that all clonotypes eventually become extinct with probability one
and that the mean time until extinction is finite. In this case, we do not make use of
a mean field approximation.
The number of T cell of clonotypes 1;2;:::, m at time t is modelled as a
continuous-time multivariate Markov process fX
1
.t/; X
2
.t/; : : : ; X
m
.t/ W t t
0
g
on the state-space S Df.n
1
;n
2
;:::;n
m
/ W n
1
;n
2
;:::;n
m
D 0; 1; 2; : : :g.Asbe-
fore, transitions are only allowed to adjacent states and so we have an m-dimensional
generalisation of the birth and death process, which we call a multivariate competi-
tion process [13].
200 C. Molina-Par´ıs et al.
For n D .n
1
;n
2
;:::;n
m
/, n
0
D .n
0
1
;n
0
2
;:::;n
0
m
/ we introduce the transition
probabilities
p
nn
0
.t/ D PfX
1
.t C t/ D n
0
1
; X
2
.t C t/ D n
0
2
;:::;
X
m
.t C t/ D n
0
m
jX
1
.t/ D n
1
; X
2
.t/ D n
2
;:::;X
m
.t/ D n
m
g
for n and n
0
2 S.Ast ! 0
C
these probabilities satisfy
p
nn
.t/ D 1
.1/
n
C :::C
.m/
n
C
.1/
n
C :::C
.m/
n
t C o.t/
and
p
nn
0
.t/ D
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
.1/
n
t C o.t/ n
0
D .n
1
C 1; n
2
;:::;n
m
/
.2/
n
t C o.t/ n
0
D .n
1
;n
2
C 1;:::;n
m
/
:
:
:
:
:
:
.m/
n
t C o.t/ n
0
D .n
1
;n
2
;:::;n
m
C 1/
.1/
n
t C o.t/ n
0
D .n
1
1; n
2
;:::;n
m
/
.2/
n
t C o.t/ n
0
D .n
1
;n
2
1;:::;n
m
/
:
:
:
:
:
:
.m/
n
t C o.t/ n
0
D .n
1
;n
2
;:::;n
m
1/
o.t/ otherwise
and p
nn
0
.t/ D 0 for n … S or n
0
… S. Here,
.s/
n
is the birth rate of T cells of
clonotype s and is the rate of transition from state .n
1
;n
2
;:::;n
s
;:::;n
m
/ to state
.n
1
;n
2
;:::;n
s
C 1;:::;n
m
/ for 1 s m. Similarly,
.s/
n
is the death rate of T
cells of clonotype s and is the rate of transition from state .n
1
;n
2
;:::;n
s
;:::;n
m
/
to state .n
1
;n
2
;:::;n
s
1;:::;n
m
/ for 1 s m. Setting
.s/
n
D 0 for n
s
D 0
and 1 s m ensures that transitions outside of the state-space cannot occur.
We will assume that, after the initial time, the thymus does not produce any more
T cells of clonotypes 1;2;:::;m. Hence,
.s/
n
D 0 for n
s
D 0 where 1 s m.
This means that the set of states A Df.n
1
;n
2
;:::;n
m
/ W n
s
D 0 for 1 s mg is
an absorbing set and the state .n
1
;n
2
;:::;n
m
/ D .0;0;:::;0/is an absorbing state.
Rather than explicitly deriving an expression for the birth rate of each clonotype
in the multivariate competition process, we instead formulate a bound on each of
these rates. Let Q
s
be the set of all APPs from which T cells of clonotype s can
receive a survival signal. We define
.s/
to be the per-cell birth rate for T cells of
clonotype s,where1 s m.Then
.s/
D
X
q2Q
s
jC
q
j
D
X
q2Q
s
n
s
C n
sq
; (9.32)
9 Competing T Cell Clonotypes 201
where n
s
is the number of T cells belonging to clonotype s and n
sq
is the number
of T cells in C
q
that are not of clonotype s.Then
.s/
X
q2Q
s
1
n
s
D
jQ
s
j
n
s
D
'
s
n
s
;
where '
s
D jQ
s
jis a parameter representing the strength of the stimulation. There-
fore,
.s/
n
D
.s/
n
s
'
.s/
; (9.33)
for 1 s m. To obtain the death rates, we assume there is a constant per-cell
death rate, which for clonotype s is denoted by
s
. Hence,
.s/
n
D
s
n
s
; (9.34)
for 1 s m.
Extinction Occurs with Probability One
Using the bound on the birth rates derived in the previous section, we now show
that the probability of eventual absorption at state .n
1
;n
2
;:::;n
m
/ D .0;0;:::;0/
is 1 for all values of the parameters. We first introduce the following disjoint subsets
of S:
S
k
Df.n
1
;n
2
;:::;n
m
/ W n
1
C :::C n
m
D kg for k 0:
We now define the birth rates for a univariate birth and death process on the state-
space fS
0
; S
1
;:::g in such a way that it moves towards the absorbing state at the
origin at a slower rate than the multivariate process. Let
0
k
D max
.n
1
;:::;n
m
/2S
k
n
.1/
n
C
.2/
n
C :::C
.m/
n
o
; (9.35)
0
k
D min
.n
1
;:::;n
m
/2S
k
n
.1/
n
C
.2/
n
C :::C
.m/
n
o
: (9.36)
From (9.33)wehavethat
0
k
D max
.n
1
;:::;n
m
/2S
k
n
.1/
n
C
.2/
n
C :::C
.m/
n
o
'
1
C '
2
C :::C '
m
;
0
k
D min
.n
1
;:::;n
m
/2S
k
f
1
n
1
C
2
n
2
C :::C
m
n
m
g
D k min.
1
;
2
;:::;
m
/:
As in the bivariate case,
0
k
is the maximum rate at which the multivariate process
moves from the set S
k
to S
kC1
and
0
k
is the minimum rate at which the multivariate
process moves from the set S
k
to S
k1
. Hence, the univariate birth and death process
with rates
0
k
and
0
k
on the state-space fS
0
; S
1
;:::g moves towards the absorbing
202 C. Molina-Par´ıs et al.
state at S
0
at a slower rate than the multivariate one. The univariate process reaches
the absorbing state with certainty if
C1
X
kD1
0
1
0
2
:::
0
k
0
1
0
2
:::
0
k
(9.37)
diverges. We have
C1
X
kD1
0
1
0
2
:::
0
k
0
1
0
2
:::
0
k
C1
X
kD1
kŠŒmin.
1
;
2
;:::;
m
/
k
.'
1
C '
2
C :::C '
m
/
k
:
Let
c
k
D
kŠŒmin.
1
;
2
;:::;
m
/
k
.'
1
C '
2
C :::C '
m
/
k
;
so that
c
kC1
c
k
D
.k C1/ min.
1
;
2
;:::;
m
/
'
1
C '
2
C :::C '
m
!C1as k !C1;
which means that
P
C1
kD1
c
k
diverges by the ratio test. Hence, the series (9.37)di-
verges by comparison and so the univariate process reaches the absorbing state with
probability one. Since this process moves towards the absorbing state at a slower
rate than the multivariate competition process, we conclude that absorption at the
origin is also certain for this process for all values of the parameters.
Let
n
be the mean extinction time if the initial state of the process is
n D .n
1
;n
2
;:::;n
m
/. This time is finite for all n 2 S nf.0;0;:::;0/g if the
series
C1
X
kD1
0
1
0
2
:::
0
k1
0
1
0
2
:::
0
k
(9.38)
converges [11]. We have
C1
X
kD1
0
1
0
2
:::
0
k1
0
1
0
2
:::
0
k
.'
1
C '
2
C :::C '
m
/
k1
kŠŒmin.
1
;
2
;:::;
m
/
k
:
Let
d
k
D
.'
1
C '
2
C :::C '
m
/
k1
kŠŒmin.
1
;
2
;:::;
m
/
k
:
Then
d
kC1
d
k
D
'
1
C '
2
C :::C '
m
.k C 1/ min.
1
;
2
;:::;
m
/
! 0 as k !C1;