Molina-Par?s C., Lythe G. (editors) Mathematical Models and Immune Cell Biology

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5 The Cyton Model 113

First and Subsequent Divisions and Progressor Fraction

Although the PDFs for death and division may, in principle, vary for each division,

we ﬁnd that for most cases it is sufﬁcient to distinguish between undivided cells,

and those in subsequent generations. Detailed analysis of a CpG-stimulated B cell

approach in vitro shows that the PDFs for death and division times vary only slightly

with generation after initial division [8].

We introduced the concept of the progressor fraction and discussed its division

dependence. Here we apply a functional form to this division dependence, thus re-

ducing the number of free parameters that are necessary to specify the progressor

fraction in the model. The form used in [4], is based on the observation that the

ultimate division number achieved by a cohort of cells is approximately normally

distributed over the generations [5]. As is the case with death and division times, the

undivided cells behave in a way that is different from cells in subsequent genera-

tions, and for this reason we let the progressor fraction for undivided cells remain a

free parameter unconstrained by the functional form used for the subsequent gener-

ations. This implies that the fraction of cells in generation i that are division capable

is given by



if i D 0,

iC1

N .n; 

dest

;

dest

/dn

N .n; 

dest

;

dest

/dn

if i>0,

(5.5)

where 

dest

;

dest

are the mean and standard deviation of the Gaussian PDF that

describes division destiny:

N .n;;/ D

2

exp



.n  /

2



Comparison with Experiment

This gives a model with 11 free parameters, listed in Table 5.1, which are obtained

empirically by ﬁtting to experimental data in the form of CFSE timecourses. The

values of these parameters will be dependent on experimental conditions such as the

strength and type of stimulus, as well as the presence of different cytokines and their

concentrations. In Fig. 5.4 we show a comparison between the best-ﬁt model pre-

diction for the population of cells in each division compared with experimental time

course. The data was obtained from CFSE analysis of a system of LPS stimulated

B cells. The best-ﬁt values for the free parameters were obtained by minimising the

sum of the squared residuals between predicted and measured values.

114 C. Wellard et al.

Table 5.1 A list of free parameters used in the cyton model of lymphocyte proliferation

Parameter Description Parameter Description



Mean time to ﬁrst division 

Mean time to die for cells

in subsequent divisions



Standard deviation of the time

to ﬁrst division



Standard deviation of the time

to die in subsequent division



Mean time to die for undivided

cells



Progressor fraction for undivided

cells



Standard deviation of the time

to die for undivided cells



dist

Mean generation number for the

PDF characterising the division

destiny



Mean time to subsequent division 

dist

Standard deviation of the generation

number for the PDF

characterising the division

destiny



Standard deviation of the time

to subsequent division

Fig. 5.4 A ﬁt to time course data obtained from CFSE analysis of LPS stimulated B cells. Here

we show the best-ﬁt prediction for the populations of each generation as a function of time (line),

compared to the measured values (points)

5 The Cyton Model 115

Stochastic Variation in Population Number

Although the cyton model is based on the idea that the fundamental cellular

processes of death and division are inherently stochastic, we have only presented

a solution for the average behaviour of the system. The solution of these equa-

tions gives an estimate for the number of cells that would be expected over many

replicates of an identical experiment, but does not give any information about ﬂuctu-

ations in this result due to the stochastic nature of the underlying processes. In large

systems the variation between replicates due to these ﬂuctuations will usually be

small, generally much lower than the variation due to experimental considerations,

however in small systems these stochastic ﬂuctuations can be observed. In [8], the

authors present multiple time course experiments obtained from the time-lapse mi-

croscopy of CpG-stimulated B cells. In each case the clone produced from a single

progenitor was followed throughout the duration of the response, and the number of

cells in each generation recorded as the response progressed. In this experiment sig-

niﬁcant variation was observed in the population sizes of the different clones. While

the model as formulated in (5.4) is sufﬁcient to reproduce the average behaviour of

these clones, it is not sufﬁcient to encapsulate the stochastic variation. To do this

requires a new approach.

Branching Process Models

The theory of branching process [9] has been used extensively in mathematical

biology to simulate cell population dynamics, from discrete time models [10]to

more complex continuous time models [11–13]. Here we use a generalisation of the

Bellman–Harris model to obtain a set of recursion relations for the probability gen-

erating function (PGF) for the number of cells in each generation as a function of

time. This PGF can be used to calculate the mean population dynamics, as well as

the expected stochastic ﬂuctuations around this mean.

We begin by deﬁning the random variable Z

.t/ to denote the number of live

cells in generation i at time t. The PGF for the number of cells in each generation,

given a single cell in generation i at time t D 0 is deﬁned as

.s;t/ D E

.t/

.0/ D 1; Z

.0/ D 0 8j ¤ i

; (5.6)

where EŒx denotes the expectation value of the random variable x. Following the

prescription for a generalised Bellman–Harris model [12,13] gives a recursion rela-

tion for this PGF

.s;t/ D .1Q

.t//.1Q

.t//s

.t/C

iC1

.s;t/

./d: (5.7)

116 C. Wellard et al.

Here the q

.t/; q

.t/ are the effective probability distributions deﬁned in (5.1),

and Q

.t/; Q

.t/ are the corresponding cumulative functions. The ﬁrst term in

the above equation takes into account the case in which the cell has neither died

nor divided, the contribution for this case is s

. The second term takes into account

the case in which the cell has died, here the contribution is one. Finally the third

term takes into account the cases in which the cell divided at a time t  ,andthe

contribution here is equal to the product of two PGFs for cells in the subsequent

generation,born at t  .

Mean Population Dynamics

The mean dynamics of the model can be obtained using 

.t/ D

.s;t/

sD0

,where



.t/ gives the expected number of cells of generation j at time t given a single

progenitor of generation i at t D 0. When applied to (5.7)thisgives



.t/ D .1  Q

.t//.1 Q

.t//ı

i;j

C 2



iC1

.t  /q

./d: (5.8)

This recursion relation can be evaluated by noting that 

.t/ D0; 8j<i, and by

assuming some limit on the number of divisions that a cell has undergone. This limit

may be imposed either by a progressor fraction that effectively vanishes beyond a

certain generation, or by assuming a minimum division time for cells.

Stochastic Fluctuations

The advantage of the branching process approach is that, in principle, it is possible

to calculate the entire probability distribution for the populations size at any point

in time. In practice this is quite difﬁcult, and usually unnecessary. A much easier

prospect, and one that offers great utility, is to calculate the variance in population

size. This variance will give a measure of the uncertainty in the population size

resulting from the stochastic nature of the underlying processes. To calculate the

variance, we begin with the second derivative of the PGF 

j;k

.t/ D

.s;t/

sD0

When applied to (5.7)thisgives



j;k

.t/ D 2





j;k

iC1

.t  / C

iC1

.t  /

iC1

.t  /



./d: (5.9)

From this we can use the relations



j;k

.t/ D

(



j;k

.t/ 

.t/

.t/ j ¤ k



j;j

.t/ C

.t/ 

.t/

j D k

(5.10)

5 The Cyton Model 117

to calculate the covariances of the population of cells in generations j; k at time t

resulting from an initial cell of generation i.

Differentiation

Branching process models have been used to model differentiation in neuronal

cells [11]. Here we extend our model to include possible differentiation processes

for lymphocytes. The cyton model can be generalised to treat general differentiation

processes in the same way as it treats death and division. To motivate this generali-

sation we note that various differentiation events in activated B cells, such as isotype

switching and commitment to antibody-secreting cells, have been shown to be divi-

sion linked [14,15]. Similarly we assume that the death and division processes are

dependent on the differentiation state, or type of the cell, and that the probability of

differentiating to a particular state is dependent on the current state of the cell.

Thus we hypothesise the existence of a molecular mechanism responsible for

each possible differentiation process, that acts independently of all other process,

and acts in a stochastic manner. In analogy to death and division these processes

can be characterised by a PDF for the differentiation time, which is dependent on

the state of the cell, as well as its generation number. We deﬁne the PDF for time-

to-differentiation from type j to type k for a cell in division i as p

i;j

.t/. In analogy

with progressor fraction, we deﬁne a division dependent differentiation fraction 

i;j

which gives the fraction of cells of type j that are capable of differentiation to type

k in division i. In this case the effective distributions, which give the probability

density at time t,aregivenby

i;j

.t/ D 

i;j

.t/



1  P

i;j

.t/





1  

i;j

.t/



;

i;j

.t/ D p

i;j

.t/



1  

i;j

.t/





1  

i;j

.t/



;

i;j

.t/ D 

i;j

.t/.1  P

i;j

.t//



1  

i;j

.t/



¤k;j



1  

i;j

.t/



Here P.t/ denotes the cumulative distribution of the relevant PDF.

Using the same methodology as previously we can derive a recursion relation for

the PGF

i;j

.s;t/ D



1  Q

i;j

.t/



1  Q

i;j

.t/



k¤j



1  Q

i;j

.t/



i;j

C Q

i;j

.t/

iC1;j

.s;t  /

i;j

./d C

k¤j

i;k

.s;t  /q

i;j

./d:

(5.11)

118 C. Wellard et al.

As previously, we can use this relation to derive an expression for the expected

population of cells of type j

in generation i

given an initial cell of type j in

generation i



i;j

.t/ D .1 Q

i;j

.t//.1  Q

i;j

.t//

k¤j

.1  Q

i;j

.t//ı

i;i

j;j



iC1;j

.t  /q

i;j

./d C

k¤j



i;k

.; t  /q

i;j

./d:

Similarly we can use



i;j

.t/ D 2





iC1;j

.t  /

iC1;j

.t  / C

iC1;j

.t  /



i;j

./d

k¤j



i;k

.t  /q

i;j

./d;

to calculate values for the covariances.

Simulation

The methods mentioned so far require numerical solution of a set of deterministic

integral equations for the mean number of cells or the variance. Higher order mo-

ments can also be calculated however each comes at further computational expense

to the previous. An alternative approach is to simulate the cells directly. Although

this approach is generally slower to calculate the mean value, there is no further cost

in calculating the variance and higher moments. Further, it is more ﬂexible and can

be quickly adapted to different sorts of models. Also it can be readily extended to

provide readouts of common experimentally observed quantities.

Modelling Cell Properties

Each cell being simulated carries with it a set of numbers which represent its physi-

cal state. In some cases these numbers are decided when the cell is created, in others

they vary over (simulated) time. The following is a non-exhaustive list of what might

be stored:

– When the cell was created. This allows age related properties of the cell to be

calculated.

5 The Cyton Model 119

– A unique ID. This way relationships between cells can be kept so that correlations

and clonal effects can be explored.

– Cell fate. In the case of the Cyton model, whether a cell dies or divides and when

this happens can be decided at the time a cell is created.

– Cell type. Whether the cell has committed to becoming antibody secreting cell

and if isotype switching has occurred.

– Stage of cell cycle and cell size. By storing where a cell is in the cell cycle

the amount of DNA can be calculated. This allows comparison to be made with

ﬂow cytometry techniques that can measure DNA content such as DAPI and

Hoechst stains which bind to the DNA in a stoichiometric manner. Estimating

cell size allows connection to be made with forward scatter measurements from

ﬂow cytometry.

– Fluorophore content. If the stage of the cell cycle and DNA content are known

then it is also possible to make connection with Bromodeoxyuridine (BrdU)

experiments. BrdU is a molecule that can replace thymidine during DNA repli-

cation. In BrdU experiments, cells’ exposure to BrdU is limited to certain times.

Its incorporation can be measured using immunohistochemistry techniques. Nor-

mally it is used to provide conﬁrmation that cells are dividing during a particular

time (when BrdU is present) but potentially it can be used to provide more infor-

mation about the parameters for a model such as Cyton [16–18].

– Afﬁnity for antigen. This enables the models to be implemented which link a

cell’s ability to proliferate to its afﬁnity, thereby addressing repertoire issues.

Implementation

In practice the simulation of cells is done using a discrete event simulator. This

requires the maintenance of a list of cells and their associated states and a time-

ordered list of events that can occur to them. Simulation is done asynchronously.

That is, instead of stepping through time, the programme processes a sequence of

queued events. In this way, time resolution can be made arbitrarily small (within

machine precision) without incurring computational expense. Examples of events

that need to be queued are

– Cell division and death

– Changing of some cell property such as a cell type and phase

– Making measurement corresponding to those made at experimental time points

Assuming that there are no interactions between cells then the computational cost

of this method scales no worse than O.N/ times the cost of insertion into the event

list where N is the number of cells.

120 C. Wellard et al.

References

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sis of lymphocyte differentiation and proliferation in vitro using carboxyﬂuorescein diacetate

succinimidyl ester. Immunol Cell Biol 77:516–522

3. Gett A, Hodgkin P (2000) A cellular calculus for signal integration by T cells. Nat Immunol

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5. Turner M, Hawkins E, Hodgkin P (2008) Quantitative regulation of B cell division destiny by

signal strength. J Immunol 181:374–382

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division and death rates from CFSE data. Bull Math Biol 68:1011–1031

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alternative stochastic lymphocyte fates. Proc Natl Acad Sci USA 106:13457

9. Kimmel M, Axelrod DE (2001) Branching processes in biology. Springer, New York

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tion in tissue culture. J Math Biol 37:49

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clonal data on multi-type cell populations. Biometrics 61:199–207

13. Subramanian V, Duffy K, Turner M, Hodgkin P (2008) Determining the expected variability

of immune responses using the cyton model. J Math Biol 56:861–892

14. Hodgkin P, Lee J, Lyons A (1996) B cell differentiation and isotype switching is related to

division cycle number. J Exp Med 184:277–281

15. Hasbold J, Corcoran L, Tarlinton D, Tangye S, Hodgkin P (2004) Evidence from the gen-

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Monitoring T cell proliferation. In: Nagorsen D, Marincola FM (eds) Analyzing T Cell

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Springer, chapter 6, pp 123–142

Chapter 6

Modelling Intravital Two-Photon Data

of Lymphocyte Migration and Interaction

Marc Thilo Figge and Michael Meyer-Hermann

Abstract Multi-photon microscopy is a powerful tool for imaging lymphocyte

migration and interaction in intact biological organs. These experiments gener-

ate quantitative data on the cell motility, shape dynamics, and contact duration of

cellular interactions. In this chapter we review mathematical models that have suc-

cessfully contributed to the interpretation of these data with regard to lymphocyte

migration and interaction. The examples involve different modelling approaches and

range from T cell priming in secondary lymphoid tissue to B cell afﬁnity maturation

in germinal centres.

Introduction

Adaptive immunity implies the diligent communication of speciﬁc information that

is facilitated by lymphocyte migration in lymphoid tissue and their interaction with

other components of the immune system. For example, T cells scan antigen present-

ing cells and may become activated if the T cell receptors recognize speciﬁc foreign

peptides embedded in major histocompatibility molecules. T cell priming by den-

dritic cells in lymph nodes occurs in subsequent phases that clearly differ in the T

cell behaviour with regard to migration and interaction [1]. Differences in the mi-

gration behaviour of the distinct types of T cells in diverse lymphoid tissues remain

to be explained [2].

The salient feature of B cells is that the B cell receptors undergo afﬁnity matu-

ration in order to optimize the immune response against pathogenic antigens. This

process takes place in follicular structures of lymphoid organs, referred to as ger-

minal centres [3, 4], and involves intercellular interactions in the selection process

for high-afﬁnity B cells. Germinal centres have a peculiar morphology consisting of

M.T. Figge (



)

Applied Systems Biology, Friedrich Schiller University Jena, Leibniz Institute

for Natural Product Research and Infection Biology – Hans Kn¨oll Institute,

Beutenbergstrasse 11a, 07745 Jena, Germany

e-mail: thilo.ﬁgge@hki-jena.de

C. Molina-Par´ıs and G. Lythe (eds.), Mathematical Models and Immune Cell Biology,

DOI 10.1007/978-1-4419-7725-0

 Springer Science+Business Media, LLC 2011

121

122 M.T. Figge and M. Meyer-Hermann

two distinct zones, termed the dark and the light zone. The dark zone is the region

of B cell proliferation, while in the light zone B cells undergo selection. How B cell

migration between the zones is realized is still a matter of debate today [5–8].

For the last few years, lymphocyte migration and interaction have been exten-

sively explored by non-invasive intravital microscopy [9]. The main achievement

of this technique is that lymphocytes can be followed in real time and within their

natural cellular environment. The generated data include records of cell tracks, i.e.

the time-dependent position of single cells, as well as cellular contact times of in-

teractions. The working principle of imaging by multi-photon microscopy is brieﬂy

summarized, together with an overview of dynamical cell properties that can be ex-

plored by this experimental technique. The rapidly growing body of experimental

data is calling for mathematical methods that are suitable for analyzing these data

in order to extract meaningful conclusions initiating new insights and experiments.

The art of mathematical modelling is to choose an adequate modelling approach.

This process starts by specifying the questions to be answered and continues by

identifying all variables and connections between them that are relevant for answer-

ing these questions. Each modelling approach has its assets and drawbacks. As a

rule of thumb, computationally cheap methods often only give a very rough repre-

sentation of reality and, therefore, they often only provide answers to questions of a

very general kind. Questions with regard to speciﬁc system properties typically re-

quire detailed calculations involving sophisticated computational methods. In many

cases, however, a detailed modelling approach also demands knowledge about pro-

cesses and parameters that are not yet accessible experimentally. Estimating the

quantitative impact of these processes ultimately leaves the questioner with the task

to decide whether or not the obtained answers are reasonable. Ideally, different mod-

elling approaches can be combined where answers to questions obtained at a certain

level of resolution serve as input for modelling at a level of higher resolution.

We consider three modelling approaches of lymphocyte migration and interac-

tion that differ in the level of resolution with respect to the functional properties of

the cells. The ﬁrst is a statistical modelling approach for the analysis of cell tracks as

observed in two-photon experiments. Single cells are viewed as independent point

particles and the only functional aspect that is taken into account is their ability to

migrate. Next, we consider cellular systems in which different types of cells migrate

and interact with each other. This is captured by an agent-based modelling approach,

where each cell represents a discrete agent that interacts with other cells and is mon-

itored during its whole lifetime. Finally, the agent-based modelling approach is used

with a sufﬁciently high spatial resolution to go beyond the point particle represen-

tation of cells in order to study the dynamics of cellular shape deformations under

migration and interaction.