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 173

Continuous-Time Markov Chains

A stochastic process fX

W t 2 T g is a collection of random variables indexed by a

set T , the elements of which usually correspond to time values. The state-space S

of the process is the range of all possible values that the random variables X

can

take. In this chapter, the values of the random variables will represent the number

of T cells belonging to a particular clonotype and so the state-space will be discrete.

Production of a new cell or death of an existing cell may occur at any time and so

the set T will be continuous.

A stochastic process fX

W t  t

g on the state-space S Df0; 1; : : :g is called a

continuous-time Markov chain if it satisﬁes the following condition. For any 0 

< :::< t

j C1

P.X

j C1

D n

j C1

D n

;:::;X

D n

/ D P.X

j C1

D n

j C1

D n

This is the Markov property. It says that, given the current state of the process, the

probability of future behaviour is not inﬂuenced by any additional knowledge of the

past history of the process.

The transition probabilities are denoted by

/ D P.X

D mjX

D n/

for t

and n, m 2 S.Thatis,p

/ is the probability that the process

is in state m at time t

given that it was in state n at time t

. We will say that the

transition probabilities are stationary (homogeneous) if the transition probabilities

depend, not on both t

and t

, but only on t

 t

. Then the notation becomes

 t

/ D P.X

D mjX

D n/:

The transition probabilities have the property

mD0

.t/ D 1 for t  t

;n2 S;

and satisfy the equations

.t C s/ D

kD0

.t/p

.s/; (8.1)

for all s, t 2 Œt

; C1/ and all n, m 2 S, which are known as the Chapman–

Kolmogorov equations.

174 C. Molina-Par´ıs et al.

Birth and Death Processes

A birth and death process is a continuous-time Markov chain where transitions are

only allowed to adjacent states. The state-space of the process may be ﬁnite i.e.,

S Df0;1;:::;Ng or inﬁnite i.e., S Df0; 1; : : :g. A birth and death process has the

following transition probabilities as t ! 0

.t/ D P.X

tCt

D mjX

D n/



t C o.t/ m D n C 1



t C o.t/ m D n  1

1  .

C 

/t C o.t/ m D n

o.t/ otherwise,

(8.2)

where f.t/ D o.t/ as t ! 0

if lim

t!0

f.t/

t

D 0. The birth rate, 

,is

the rate of transition from state n to n C 1 while the death rate, 

, is the rate of

transition from state n to n  1. The birth and death rates satisfy 

 0, 

 0

for n D 0; 1; 2; : : :,but

D 0 so that transitions outside of the state-space cannot

occur. The process can be represented as

0 1 2 3

n+1

···

n−1

n+1

Let

.t/ D P.X

D njX

D n

/; n 2 S; (8.3)

which is the probability that the birth and death process is in state n at time t,given

that the initial state of the process is n

. If the state-space of the process is ﬁnite and



D 0 so that transitions outside of the state-space cannot occur, the equation for

the endpoint, n D N ,is

.t C t/ D 

N 1

tp

N 1

.t/ C.1  

t/p

.t/ Co.t/: (8.4)

Taking the limit t ! 0

, results in

.t/

D 

n1

.t/ C

nC1

.t/ .

C 

.t/; 1  n  N  1:

(8.5)

Similarly,

.t/

D 

.t/ 

.t/; (8.6)

.t/

D 

N 1

.t/ 

.t/: (8.7)

The differential equations (8.5)–(8.7) are known as the forward Kolmogorov

equations.

8 Diversity Maintenance of Na¨ıve T Cells 175

Stationary Probability Distribution

A stationary probability distribution  D .

;

;:::/,where

 0 for n 2 S,



D 0 for n … S and

nD0



D 1, satisﬁes the equations

0 D 



 



;

0 D 

n1



n1

C 

nC1



nC1

 .

C 

/

;1 n  N 1;

0 D 

N 1



N 1

 



;

which are the forward Kolmogorov equations with the time derivatives set to zero.

If the state-space of the birth and death process fX

W t  t

g is inﬁnite i.e., S D

f0; 1; : : :g, a unique positive stationary probability distribution, , exists if and only

if [12]



n1

>0 and 

>0 for n D 1;2;::: (8.8)

and

nD1



:::

n1



:::

< C1: (8.9)

If these conditions are satisﬁed, the stationary probability distribution is given by



1 C

nD1



:::

n1



:::

; (8.10)





:::

n1



:::



for n D 1;2;:::: (8.11)

If the state space of the birth and death process is ﬁnite i.e., S Df0; 1; 2; : : : ; N g,

then a unique positive stationary probability distribution  exists if and only if



n1

>0 and 

>0 for n D 1;2;:::;N:

Then the stationary probability distribution is given by (8.10)–(8.11), where the

index n and the summation on n extend from 1 to N .

Continuous-Time Birth and Death Process with Absorbing States

If 

D 

D 0 then n D 0 is an absorbing state, meaning that once the process

reaches this state, it remains there forever. The process can be represented as

0 1 2 3

··· ···

n+1

n−1

n+1

176 C. Molina-Par´ıs et al.

The stationary probability distribution of the process has all its mass at the absorbing

state, i.e.,  D .1;0;0;:::/

. The probability of the process reaching the absorbing

state is given by lim

t!C1

.t/. If the state-space of the process is inﬁnite i.e., S D

f0; 1; : : : g and

nD1



:::



:::

DC1; (8.12)

then lim

t!C1

.t/ D 1;which means that eventual absorption at n D 0 is guaran-

teed [13]. On the other hand, if (8.12) does not hold, then

lim

t!C1

.t/ D

nDm



:::



:::

1 C

nD1



:::



:::

; (8.13)

where m is the initial state of the process.

The Limiting Conditional Probability Distribution

Prior to extinction at n D 0 occurring, the probability distribution of a birth and

death process may adopt a stationary shape for a long period of time, especially

if the expected time until absorption is relatively large. In order to study the be-

haviour of the process before extinction occurs, we deﬁne the following conditional

probabilities:

.t/ D P.X

D n j X

¤ 0/ D

.t/

1  p

.t/

for n D 1;2;:::; (8.14)

which is the probability that the process is in state n at time t, given that absorption

has not yet occurred. These conditional probabilities satisfy:

.t/

1  p

.t/

1  p

.t/

1  p

.t/

D 

n1

.t/ .

C 

.t/

C

nC1

.t/ C

.t/q

.t/; (8.15)

for n  2 and

.t/

1  p

.t/

1  p

.t/

1  p

.t/

D 

.t/ .

C 

.t/ C

.t/q

.t/: (8.16)

8 Diversity Maintenance of Na¨ıve T Cells 177

If the state space is ﬁnite, we have

.t/

1  p

.t/

1  p

.t/

1  p

.t/

D 

N 1

.t/ 

.t/ C

.t/q

.t/: (8.17)

A distribution Nq is called a quasi-stationary probability distribution (QSD) if it is

a solution of (8.15)–(8.17) where the time derivatives are set to zero. The limiting

conditional probability distribution (LCD) of the process is deﬁned as

lim

t!C1

.t/; n  1: (8.18)

Since the LCD is independent of time, it is also a QSD. If the state-space of the

process is ﬁnite, there is a unique QSD, which is also the LCD of the process.

However, if the process has an inﬁnite state-space, there may be no QSD and, if a

QSD does exist, it is not necessarily unique [14]. In most cases, it is not possible

to ﬁnd an explicit solution for the QSD or the LCD, but numerical methods can be

used (see [15] for an iterative procedure).

The LCD can be approximated analytically by setting 

D 0 [15]. This results

in a new birth and death process which has no absorbing state and can be repre-

sented as

··· ···

01 2 3

n+1

n−1

n+1

If 

n1

>0 and 

>0 for n D 2;3;:::;and

nD2



:::

n1



:::

< C1 (8.19)

the new process has a unique positive stationary probability distribution which is

given by



.1/

1 C

nD2



:::

n1



:::

;



.1/



:::

n1



:::



.1/

for n  2: (8.20)

This distribution is an approximation to the true LCD of the process.

The LCD may also be approximated by the stationary distribution of a birth and

death process which has the same birth rates as the original process, but where the

death rates 

are replaced by 

n1

to allow for one immortal individual [15]. This

can be represented as

178 C. Molina-Par´ıs et al.

··· ···

n−1

n−1 n+1

If 

>0 and 

>0 for n D 1;2;:::;and

nD2



:::

n1



:::

n1

< C1 (8.21)

this process has a unique positive stationary probability distribution which is

given by



.2/

1 C

nD2



:::

n1



:::

n1

;



.2/



:::

n1



:::

n1



.2/

for n  2: (8.22)

In general, 

.1/

is a better approximation to the LCD when the mean time until

extinction is long, whereas 

.2/

provides a better approximation when the mean

extinction time is short [15].

Mathematical Model of Peripheral Maintenance

We will use the index i to label T cell clonotypes and the index q to label APPs. Let

C be the set of all T cells in the na¨ıve repertoire. Whether or not a T cell of clonotype

i can receive a survival signal from a given APP q depends on the particular TCR

it expresses on its surface. The subset C

is deﬁned to be 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

is the subset of APPs from which T cells

of clonotype i can receive a survival signal. We deﬁne n

DjC

j, which is the

total number of T cells that can receive a survival signal from APP q. These sets are

illustrated in Fig. 8.2.

Let 

be the rate of survival signals from all the APCs presenting APP q,which

we assume remains constant in time. We will also assume that the survival signals

from any APP q are shared equally among all the T cells capable of receiving them.

Now deﬁne 

.i/

to be the per cell birth rate for T cells of clonotype i .Then



.i/

q2Q



: (8.23)

8 Diversity Maintenance of Na¨ıve T Cells 179

APPs T cell clonotypes

Fig. 8.2 The sets of APPs and T cell clonotypes. Each circle on the left represents an APP, while

each circle on the right represents a T cell clonotype (containing all the T cells with an identical

TCR). A line between the two circles indicates that T cells of this clonotype receive a survival

signal from the APP

Let n

be the number of T cells belonging to clonotype i and n

be the number of

T cells not of clonotype i that receive a survival signal from an APP q 2 Q

.Then

D n

C n

so that



.i/

q2Q



C n

; (8.24)

wherewehaveassumedthat

D  for simplicity. Next, we partition the set Q

into disjoint subsets as follows:

[

rD0

; (8.25)

where Q

is the set of APPs which provide survival signals to T cells of clonotype i

and to r other distinct clonotypes in the repertoire. Then Q

D;for r ¤ r

and hence,



.i/

D 

rD0

q2Q

C n

: (8.26)

The previous equation implies that the birth rate per T cell of clonotype i depends

not only on the number of T cells of clonotype i, but also on the number of T cells

of any other clonotype that receives a survival signal from an APP q 2 Q

through

the term n

We proceed to develop a mean ﬁeld approximation to decouple the birth rates so

that the expression for 

.i/

depends only on the number of T cells of clonotype i.

180 C. Molina-Par´ıs et al.

Treating the mean number of T cells per clonotype as a parameter, hni, the approx-

imation is [9]



.i/

D 

rD0

rhniCn

: (8.27)

Let 

be the number of clonotypes that compete with T cells of clonotype i for

survival signals from an APP, with the average taken over all the APPs belonging to

the set Q

. Assuming that

jDjQ





rŠ

(8.28)

completes the model.

Summary of the Model

Using the mean ﬁeld approximation, we have modelled the number of T cells

belonging to a given clonotype, i, as a continuous-time birth and death process

W t  t

g on the state-space S Df0; 1; 2; : : :g with the birth and death rates



D 0; (8.29)



D 'ne



rD0



rŠ

rhniCn

;n 1; (8.30)



D n ; n  0; (8.31)

where the clonotype label i has been dropped for notational convenience. The model

has four parameters:

(1) ' is a parameter proportional to the number of APPs which can provide survival

stimuli to T cells of the ﬁxed clonotype i.Then'

1

is proportional to the mean

time until a T cell of this clonotype receives a survival signal from an APP in

the absence of competition with T cells of other clonotypes.

(2)  is the “mean niche overlap” and encodes competition for survival stimuli be-

tween T cells of the ﬁxed clonotype, i, and T cells of other clonotypes.

(3) hni is the average clonotype size over the na¨ıve T cell repertoire.

(4)  is the death rate per T cell of clonotype i.

Special Cases

We introduce two special cases of the model which are deﬁned by the value of the

mean niche overlap parameter , which encodes competition. If   1 we say that

the clone occupies a “hard niche”, while if   1 we say that it occupies a “soft

niche”. Biologically, a clonotype with   1 possesses a TCR that is very different

8 Diversity Maintenance of Na¨ıve T Cells 181

from the other clonotypes in the repertoire, while a clonotype with   1 has a

TCR that is similar to the TCRs of many other clonotypes in terms of the APPs

from which it is able to receive survival signals. We now derive expressions for the

forms of the birth rates in these special cases.

For   1, the ﬁrst term in the sum in (8.30) dominates so that



 'ne





0Š

 '; for n  1;

which means that the birth rate is approximately constant in this case. On the other

hand, for   1 we have



D 'n

rD0





rŠ

rhniCn

D 'nE



YhniCn



; (8.32)

where Y is Poisson distributed with mean . Carrying out a Taylor expansion about

the mean gives



D 'n



hniCn

hni



.hniCn/

C :::



: (8.33)

Since   1, the second and subsequent terms in the above expansion may be

neglected, which results in



hniCn

: (8.34)

Clonal Extinction and Mean Extinction Times

Since 

D 

D 0, the birth and death process has an absorbing state at n D 0.

Clonal extinction occurs if the process reaches this state. We now calculate the prob-

ability of extinction using condition (8.12).

Firstly, note that the birth rate is bounded from above as follows:



D 'ne



rD0



rŠ

rhniCn

 'ne



rD0



rŠ

D 'e





D ': (8.35)

182 C. Molina-Par´ıs et al.

Therefore

nD1



:::



:::



nD1



nŠ

; (8.36)

using the upper bound (8.35). Let a



nŠ

.Then

nC1

.n C 1/

!C1as n !C1;

and so

nD1

diverges by the ratio test. Hence, condition (8.12) holds by the

comparison test and we conclude that absorption at n D 0 is certain for all values

of the parameters. This means that the eventual fate of any clone is extinction and

disappearance from the repertoire.

We now compute the mean times to extinction. The mean time to extinction from

initial state m isgivenby[12]





nD2



:::

n1



:::

;mD 1;



m1

kD1



:::



:::

nDkC1



:::

n1



:::

;mD 2;3;::::

(8.37)

We ﬁrst calculate the mean time to extinction for the hard niche case (  1). Sub-

stituting the birth and death rates 

D ' and 

D n into the above expressions

results in







 1



; (8.38)



D 

m1

kD1

kŠ





kC1





nD0





nŠ

for m  2: (8.39)

We now consider the most general form of the birth and death rates given by (8.30)–

(8.31). Then



nD1



:::

n1



:::



nD1

n1



nŠ

using the bound 

 '





 1



; (8.40)