Molina-Par?s C., Lythe G. (editors) Mathematical Models and Immune Cell Biology
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8 Diversity Maintenance of Na¨ıve T Cells 183
and
m
D
1
C
m1
X
kD1
1
2
:::
k
1
2
:::
k
C1
X
nDkC1
1
2
:::
n1
1
2
:::
n
1
C
m1
X
kD1
1
2
:::
k
1
2
:::
k
C1
X
nDkC1
1
'nŠ
'
n
1
C
m1
X
kD1
C1
X
nDkC1
kŠ
'nŠ
'
nk
D
1
C
m1
X
kD1
kŠ
'
kC1
"
e
'
k
X
nD0
'
n
1
nŠ
#
for m 2; (8.41)
so the mean extinction times in the most general case are bounded by (8.38)–(8.39).
Therefore, the hard niche case provides an upper bound on the mean extinction
times. This is shown in Fig. 8.3 where
1
(calculated numerically) is plotted as a
function of for fixed ', and hni.For 1,
1
is independent of , in agreement
with (8.38). Clones with 1 have the longest lifetimes in the repertoire. These
are the clonotypes that are most different from other T cells in the repertoire in terms
of the APPs they are able to receive survival signals from and so their long residence
times in the repertoire ensure that diversity is maintained.
1
100000
1e+10
1e+15
1e+20
1e+25
0.0001 0.01 1 100 10000
τ
1
ν
ϕ =10
ϕ =50
♦♦♦♦♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦
♦♦♦♦♦♦♦
♦
Fig. 8.3
1
as a function of for ' D 10; 50, hniD10 and D 1
184 C. Molina-Par´ıs et al.
Homeostasis and the Limiting Conditional Distribution
Since extinction is guaranteed with probability one, the limiting distribution of the
process is given by .1;0;0;:::/
T
. To investigate the behaviour of the system before
extinction takes place, we consider the limiting conditional probability distribution
(LCD) of the process, which represents the homeostatic distribution of T cells.
Case 1
In the hard niche case, the birth and death rates are given by
0
D 0,
n
D ',for
n 1 and
n
D n,forn 0.Then
C1
X
nD2
1
2
:::
n1
2
3
:::
n
D
C1
X
nD2
'
n1
1
nŠ
D
'
C1
X
nD2
'
n
1
nŠ
D
'
e
'
1
'
:
The first approximation (8.20) to the LCD is given by
.1/
1
D
1
1 C
P
C1
nD2
1
2
:::
n1
2
3
:::
n
D
'
.e
'
1/
;
.1/
n
D
1
2
:::
n1
2
3
:::
n
.1/
1
D
'
n
1
nŠ.e
'
1/
for n 2:
This distribution has mean
C1
X
nD1
n
.1/
n
D
'
1 e
'
;
and variance
C1
X
nD1
n
2
.1/
n
C1
X
nD1
n
.1/
n
!
2
D
1
1 e
'
'
2
1 C
'
1
1 e
'
:
8 Diversity Maintenance of Na¨ıve T Cells 185
The second approximating distribution (8.22)totheLCDis:
C1
X
nD2
1
2
:::
n1
1
2
:::
n1
D
C1
X
nD2
'
n1
1
.n 1/Š
D e
'
1:
This means that
.2/
1
D
1
1 C
P
C1
nD2
1
2
:::
n1
1
2
:::
n1
D e
'
;
.2/
n
D
1
2
:::
n1
1
2
:::
n1
.2/
1
D
'
n1
1
.n 1/Š
e
'
; for n 2:
This distribution has mean
C1
X
nD1
n
.2/
n
D
'
C 1;
and variance
C1
X
nD1
n
2
.2/
n
C1
X
nD1
n
.2/
n
!
2
D
'
:
The mean time to extinction increases as ' increases. Hence,
.2/
is the best
approximation when the mean time to extinction is short, while the accuracy of
approximation
.1/
improves with increasing mean time to extinction.
Case 1
In the soft niche case, the birth rates are given by (8.34) and the first approximating
distribution to the LCD is given by
.1/
1
D
1
1 C
P
C1
nD2
1
n
'
n1
Q
n1
kD1
1
kChni
;
.1/
n
D
1
n
'
n1
.1/
1
n1
Y
kD1
1
k C hni
for n 2;
while the second approximating distribution is given by
.2/
1
D
1
1 C
P
C1
nD2
'
n1
Q
n1
kD1
1
kChni
;
.2/
n
D
'
n1
.2/
1
n1
Y
kD1
1
k C hni
for n 2:
186 C. Molina-Par´ıs et al.
If ' is large enough, the clonotype behaves as a hard niche clone even though 1,
and
.1/
is a good approximation to the LCD for long mean extinction times, while
.2/
is a better approximation than
.1/
when the mean time to extinction is short,
as for the case 1.
References
1. Starr TK, Jameson SC, Hogquist KA (2003) Positive and negative selection of T cells. Annu
Rev Immunol 21:139–176
2. Goronzy JJ, Lee WW, Weyand CM (2007) Aging and T-cell diversity. Exp Gerontol 42:
400–406
3. Arstila TP, Casrouge A, Baron V, Even J, Kanellopoulos J, Kourilsky P (1999) A direct estimate
of the human ˛ˇ T cell receptor diversity. Science 286:958–961
4. Davis MM, Bjorkman PJ (1988) T cell antigen receptor genes and T cell recognition. Nature
334:395–402
5. Mahajan VS, Leskov IB, Chen J (2005) Homeostasis of T cell diversity. Cell Mol Immunol
2:1–10
6. van den Berg HA, Burroughs NJ, Rand DA (2002) Quantifying the strength of ligand antago-
nism in TCR triggering. Bull Math Biol 64:781–808
7. Mason D (1998) A very high level of crossreactivity is an essential feature of the T cell receptor.
Immunol Today 19:395–404
8. Davenport MP, Price DA, McMichael AJ (2007) The T cell repertoire in infection and vacci-
nation: implications for control of persistent viruses. Curr Opin Immunol 19:294–300
9. Stirk E, Molina-Par´ıs C, van den Berg H (2008) Stochastic niche structure and diversity
maintenance in the T cell repertoire. J Theor Biol 255:237–249
10. Tanchot C, Rosado MM, Agenes F, Freitas AA, Rocha B (1997) Lymphocyte homeostasis.
Semin Immunol 9:331–337
11. Goldrath AW, Bevan MJ (1999) Selecting and maintaining a diverse T cell repertoire. Nature
402:255–262
12. Allen LJS (2003) An introduction to stochastic processes with applications to biology. Pearson
Education, New Jersey, USA
13. Taylor HM, Karlin S (1998) An introduction to stochastic modeling, 3rd edn. Academic,
San Diego
14. Van Doorn EA (1991) Quasi-stationary distributions and convergence to quasi-stationarity of
birth-death processes. Adv Appl Probab 23:683–700
15. Nasell I (2001) Extinction and quasi-stationarity in the Verhulst logistic model. J Theor Biol
211:11–27
Chapter 9
Multivariate Competition Processes: A Model
for Two Competing T Cell Clonotypes
Carmen Molina-Par´ıs, Grant Lythe, and Emily Stirk
Abstract Diversity in the na¨ıve T cell repertoire is maintained throughout the
majority of an individual’s lifetime. The homeostatic mechanisms involved include
competition for survival stimuli furnished by antigen-presenting cells, dependent on
weak recognition of arrays of self-peptides by the T cell antigen receptor. We study
the dynamics of this process from the point of view of stochastic competitive exclu-
sion between a pair of T cell clonotypes which are similar in terms of the specific
survival stimuli which they are able to receive. This dynamics is formulated as a bi-
variate continuous-time Markov process for the number of T cells belonging to the
pair of clonotypes. We prove that the ultimate fate of both clonotypes is extinction
and provide a bound on mean extinction times. We concentrate mainly on the case
where the two clonotypes exhibit little competition with other T cell clonotypes in
the repertoire, since this case provides an upper bound on the mean extinction times.
As the two clonotypes become more similar in terms of the proportion of their re-
sources which are shared between them, one clonotype quickly becomes extinct in
a process resembling the ecological principle of classical competitive exclusion. We
consider the limiting probability distribution for the bivariate process, conditioned
on non-extinction of both clonotypes.
Introduction: Bivariate Competition Processes
A healthy adult human possesses approximately 10
11
na¨ıve T cells [1], which
belong to around 10
7
–10
8
different clonotypes (all T cells with identical T cell
antigen receptor) [2]. The number of na¨ıve T cells is controlled homeostatically,
meaning that it remains approximately constant throughout much of an individ-
ual’s lifetime [3]. Experimental evidence suggests that T cell homeostasis is driven
by interactions with self-antigens (antigens derived from the body’s own proteins)
G. Lythe (
)
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
e-mail: grant@maths.leeds.ac.uk
C. Molina-Par´ıs and G. Lythe (eds.), Mathematical Models and Immune Cell Biology,
DOI 10.1007/978-1-4419-7725-0
9,
c
Springer Science+Business Media, LLC 2011
187
188 C. Molina-Par´ıs et al.
displayed on the surface of antigen-presenting cells [4–6]. Na¨ıve T cells undergo
infrequent cell divisions after receiving a survival signal from an antigen-presenting
cell (APC). Whether or not a particular T cell can receive a survival signal from a
given APC depends on the TCR it expresses and the array of self-antigens presented
on the APC surface [7]. Competition amongst T cells for these interactions regulates
the diversity of the T cell repertoire [8].
In Chap. 9, we constructed a univariate birth and death process to model the num-
ber of T cells belonging to a particular clonotype. The model relied upon a mean
field approximation of the competition between different clonotypes. It was assumed
that, even though a given T cell may compete with T cells of many other clonotypes
for access to survival signals from an antigen-presentation profile (APP), individual
competitive interactions between pairs of clonotypes are weak and do not have a
significant impact on the fate of either clonotype. In terms of the sets defined in the
previous chapter, this means that jQ
i
\ Q
j
jjQ
i
j for i ¤ j (see Fig. 9.1). In this
chapter, we introduce a model for the case when this condition does not hold. This
means that the two clonotypes i and j overlap significantly in terms of the APPs
from which they can receive survival signals and so the number of T cells of one
clonotype affects the level of survival signals available to T cells of the other clono-
type. We will assume that jQ
i
\ Q
k
jjQ
i
j, jQ
j
\ Q
k
jjQ
j
j for k ¤ i; j and
so competition between pairs of clonotypes (other than the pair i and j ) is small.
Therefore, the birth rates for the two competing clonotypes are coupled and we re-
quire a bivariate continuous-time Markov process to model the number of T cells
belonging to both clonotypes [9,10].
In the next section, a type of bivariate continuous-time Markov chain called a
competition process is defined. This type of process is then used to model the num-
ber of T cells belonging to two closely competing clonotypes. It is then shown
Fig. 9.1 The diagram on the left represents pairs of clonotypes for which the mean field approx-
imation, described in the previous chapter, is reasonable. In this chapter, we consider a pair of
clonotypes such as that represented on the right
9 Competing T Cell Clonotypes 189
that both clonotypes eventually become extinct with probability one. The model
is extended to include any number of clonotypes for which the mean field approx-
imation introduced in the previous chapter does not hold. Finally, we conclude the
chapter with a numerical simulation of the full model using the Gillespie algorithm,
where it is not necessary to make any mean field assumptions regarding the compe-
tition between different clonotypes.
We will construct a mathematical model for the number of T cells belonging
to two different clonotypes, which we refer to as clonotypes 1 and 2,forwhich
jQ
1
\ Q
2
jjQ
1
j. This means that the two clonotypes overlap significantly in
terms of the APPs from which they are able to receive survival signals. The random
variable X.t/ describes the number of T cells of clonotype 1 at time t, while the
random variable Y.t/ describes the number of T cells of clonotype 2 at time t.At
time t D
Q
t
1
, Qn
1
cells of clonotype 1 are released from the thymus and at time t D
Q
t
2
,
Qn
2
cells of clonotype 2 are released from the thymus. It is assumed that after
Q
t
1
no
T cells of clonotype 1 are produced from the thymus and similarly, after
Q
t
2
no T cells
of clonotype 2 are produced from the thymus. Without loss of generality we may
assume that
Q
t
1
Q
t
2
def
D t
0
.Fort<t
0
, only one of the clonotypes is present in the
T cell repertoire and so the univariate model introduced in the previous chapter may
be used. After this time we model the number of T cells belonging to clonotype 1
and 2 as a bivariate competition process.
Competition Processes
Let f.X.t/; Y.t// W t t
0
g be a continuous-time bivariate Markov process on the
state-space S Df.n
1
;n
2
/ W n
1
;n
2
D 0; 1; : : :g. The transition probabilities are
defined by
p
nm
.t/ D PfX.t C t/ D m
1
; Y.t C t/ D m
2
jX.t/ D n
1
; Y.t/ D n
2
g (9.1)
for n D .n
1
;n
2
/ 2 S and m D .m
1
;m
2
/ 2 S, where it is assumed that the tran-
sition probabilities are stationary (they only depend on the time interval between
transitions and not the time at which transitions occur), as in the previous chapter.
A bivariate competition process is a continuous-time Markov process where tran-
sitions are only allowed to adjacent states. Then the transition probabilities satisfy
p
nn
.t/ D 1 .
.1/
n
1
;n
2
C
.2/
n
1
;n
2
C
.1/
n
1
;n
2
C
.2/
n
1
;n
2
/t C o.t/ and
p
nm
.t/ D
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
.1/
n
1
;n
2
t C o.t/ m D .n
1
C 1; n
2
/
.2/
n
1
;n
2
t C o.t/ m D .n
1
;n
2
C 1/
.1/
n
1
;n
2
t C o.t/ m D .n
1
1; n
2
/
.2/
n
1
;n
2
t C o.t/ m D .n
1
;n
2
1/
o.t/ otherwise,
(9.2)
where f.t/ D o.t/ if lim
t!0
C
f.t/
t
D 0.
190 C. Molina-Par´ıs et al.
Fig. 9.2 A schematic representation of the bivariate competition process and the transitions
between different states
A schematic representation of the process is given in Fig. 9.2. The quantity
.1/
n
1
;n
2
is the birth rate of T cells of clonotype 1 and is the rate of transition from state
.n
1
;n
2
/ to .n
1
C 1; n
2
/. Similarly, the birth rate of T cells of clonotype 2, denoted
.2/
n
1
;n
2
, is the rate of transition from state .n
1
;n
2
/ to .n
1
;n
2
C 1/. The death rate
for T cells of clonotype 1 is given by
.1/
n
1
;n
2
and this is the rate of transition from
state .n
1
;n
2
/ to .n
1
1; n
2
/. The death rate of T cells of clonotype 2,
.2/
n
1
;n
2
,isthe
rate of transition from state .n
1
;n
2
/ to .n
1
;n
2
1/.Weset
.1/
0;j
D
.2/
j;0
D 0 for all
j 0 so that transitions outside of the state space S cannot occur.
Let p
n
1
;n
2
be the probability that the process is in state .n
1
;n
2
/ at time t.These
probabilities satisfy the forward Kolmogorov equations, which are given by
9 Competing T Cell Clonotypes 191
dp
n
1
;n
2
.t/
dt
D
.1/
n
1
1;n
2
p
n
1
1;n
2
.t/ C
.2/
n
1
;n
2
1
p
n
1
;n
2
1
.t/ C
.1/
n
1
C1;n
2
p
n
1
C1;n
2
.t/
C
.2/
n
1
;n
2
C1
p
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
p
n
1
;n
2
.t/; (9.3)
for n
1
1; n
2
1,plus
dp
0;0
.t/
dt
D
.1/
1;0
p
1;0
.t/ C
.2/
0;1
p
0;1
.t/
.1/
0;0
C
.2/
0;0
p
0;0
.t/;
dp
0;n
2
.t/
dt
D
.2/
0;n
2
1
p
0;n
2
1
.t/ C
.1/
1;n
2
p
1;n
2
.t/ C
.2/
0;n
2
C1
p
0;n
2
C1
.t/
.1/
0;n
2
C
.2/
0;n
2
C
.2/
0;n
2
p
0;n
2
.t/; n
2
1;
dp
n
1
;0
.t/
dt
D
.1/
n
1
1;0
p
n
1
1;0
.t/ C
.1/
n
1
C1;0
p
n
1
C1;0
.t/ C
.2/
n
1
;1
p
n
1
;1
.t/
.1/
n
1
;0
C
.2/
n
1
;0
C
.1/
n
1
;0
p
n
1
;0
.t/; n
1
1: (9.4)
If the state-space of the process is finite, i.e., S Df.n
1
;n
2
/ W n
1
;n
2
D 0;1;:::;Ng
and
.1/
N;n
2
D
.2/
n
1
;N
D 0 for n
1
;n
2
D 0;1;:::;N so that transitions outside of
the state-space cannot occur, there are additional equations for the states on the
boundary which are given by
dp
N;0
.t/
dt
D
.1/
N 1;0
p
N 1;0
.t/ C
.2/
N;1
p
N;1
.t/
.2/
N;0
C
.1/
N;0
p
N;0
.t/;
dp
0;N
.t/
dt
D
.2/
0;N 1
p
0;N 1
.t/ C
.1/
1;N
p
1;N
.t/
.1/
0;N
C
.2/
0;N
p
0;N
.t/;
dp
N;n
2
.t/
dt
D
.1/
N 1;n
2
p
N 1;n
2
.t/ C
.2/
N;n
2
1
p
N;n
2
1
.t/ C
.2/
N;n
2
C1
p
N;n
2
C1
.t/
.2/
N;n
2
C
.1/
N;n
2
C
.2/
N;n
2
p
N;n
2
.t/; 1 n
2
N 1;
dp
n
1
;N
.t/
dt
D
.1/
n
1
1;N
p
n
1
1;N
.t/ C
.2/
n
1
;N 1
p
n
1
;N 1
.t/ C
.1/
n
1
C1;N
p
n
1
C1;N
.1/
n
1
;N
C
.1/
n
1
;N
C
.2/
n
1
;N
p
n
1
;N
.t/; 1 n
1
N 1;
dp
N;N
.t/
dt
D
.1/
N 1;N
p
N 1;N
.t/ C
.2/
N;N 1
p
N;N 1
.t/
.1/
N;N
C
.2/
N;N
p
N;N
.t/: (9.5)
192 C. Molina-Par´ıs et al.
Mathematical Model for Two Competing Clonotypes
The transition probabilities defining the competition process take the form of those
in (9.2) and are now derived. Recall that C is the set of all T cells in the na¨ıve
repertoire and C
q
is the set of all T cells that are capable of receiving a survival
signal from APP q.Also,Q is the set of all APPs which may occur in the periphery
and Q
i
is the subset of APPs from which T cells of clonotype i can receive a survival
signal. We define n
q
DjC
q
j, which is the total number of T cells that can receive a
survival signal from APP q.Let
q
be the rate of survival signals from all the APCs
presenting APP q,and
.1/
be the per-cell birth rate of T cells of clonotype 1. Then,
assuming that
q
D and that the survival signals from APP q are shared equally
among all the T cells capable of receiving them, we have that
.1/
D
X
q2Q
1
jC
q
j
D
X
q2Q
1
n
q
; (9.6)
as for the univariate case. Let n
1
be the number of T cells belonging to clonotype 1,
and n
1q
be the number of T cells not belonging to clonotype 1 that receive a survival
signal from an APP q 2 Q
1
. Hence, n
q
D n
1
C n
1q
.
We next divide the set Q
1
into two disjoint subsets by defining
Q
12
D Q
1
\ Q
2
; (9.7)
which is the set of APPs from which T cells of both clonotype 1 and clonotype 2
can receive a survival signal, and
Q
1=2
D Q
1
\
N
Q
2
; (9.8)
which is the set of APPs from which T cells of clonotype 1 receive a survival signal,
but T cells of clonotype 2 do not. Here,
N
Q
2
denotes the complement of the set Q
2
.
Then, Q
1
D Q
12
[ Q
1=2
and Q
12
\ Q
1=2
D;. Therefore,
.1/
D
X
q2Q
12
n
1
C n
2
C n
12q
C
X
q2Q
1=2
n
1
C n
1q
;
where n
2
denotes the number of T cells belonging to clonotype 2 and n
12q
D n
q
n
1
n
2
.
Now, we partition the sets Q
12
and Q
1=2
into disjoint subsets as follows:
Q
12
D
C1
[
rD0
Q
12r
and Q
1=2
D
C1
[
rD0
Q
1r=2
;
where Q
12r
is the set of APPs which provide survival signals to T cells of clono-
type 1 and clonotype 2 and to r other distinct clonotypes in the repertoire, and Q
1r=2