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

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6 Modelling Intravital Two-Photon Data 133

The applied agent-based model consists of the following three coupled levels:

(a) The ﬁrst level contains the main lattice corresponding to the three-dimensional

physical space in which cells can migrate and interact. Each lattice point can

carry up to one biological cell that evolves according to reaction rates deﬁning

a probability of action or interaction with neighboring cells on the lattice. In

addition, cells migrate according to a chemokine distribution that is calculated

on a separate lattice (see third level below).

(b) The second level refers to the antibody shape space, which is represented by

a four-dimensional lattice encoding the antibody type of each B cell. Somatic

hypermutation is represented by switching the antibody type to a neighboring

point in the shape space. Each point in the shape space is associated with an

antibody–antigen afﬁnity and determines the binding probability of a B cell to

an antigen-presenting follicular dendritic cell.

the chemokines CXCL12 and CXCL13. The source for CXCL12 are stromal

cells at the border of the follicle in the dark zone, and follicular dendritic cells

in the light zone for CXCL13.

The essential feature of this agent-based model is that it captures the whole ger-

minal centre reaction comprising the population kinetics, cellular interactions, and

afﬁnity maturation. This level of description allows to distinguish between centrob-

lasts, proliferating B cells that undergo somatic hypermutation, and centrocytes,

nondividing B cells with activated apoptosis competing for survival signals. The

included mechanisms follow, to a large extent, the classical model of the germi-

nal centre reaction [3] and the simulations are validated by numerous experimental

facts [7].

The preferred direction of B cell migration is deﬁned by a polarity vector that is

renewed after the persistence time of 1:24 min, which was obtained from the statis-

tical cell track model that was discussed in Section “B Cell Trans-Zone Migration

in Germinal Centres”. The polarity vector consists of two contributions. A random

contribution according to the turning angle distribution as measured in two-photon

experiment [13] and a contribution that depends on the chemokine gradients.

The simulations predicted a novel migration model for B cells in germinal cen-

tres, referred to as “transient chemotaxis model” that reconciles all experimental

and theoretical data [7,8]. According to this model, most of the cells in the dark and

light zone actively perform persistent random walk migration. Centrocytes acquire

additional sensitivity to CXCL13 after or during differentiation from centroblasts

and orient toward the network of follicular dendritic cells in the light zone. In the

network, the centrocytes lose CXCL13 sensitivity. This might be induced by contact

with follicular dendritic cells, by overcritical CXCL13 concentrations and CXCR5

internalization, or simply by down-regulation of CXCR5 after a characteristic time.

Similarly, a return to the dark zone of positively selected centrocytes is facilitated by

a transient chemotactic sensitivity for CXCL12. However, random walk migration

remains the dominant pathway of re-polarization since B cells undergo directed mi-

gration only temporarily. Therefore, transient chemotaxis might well be hidden in

the experimental motility data that seemingly support pure persistent random walk

migration.

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

Furthermore, the functional analysis of germinal centre simulations revealed

insight about the interaction between B cells and follicular dendritic cells. Cen-

trocytes bind antigen on follicular dendritic cells to obtain survival signals and this

binding process is afﬁnity dependent. Contacts observed in vivo have rarely been

found to be longer than 5 min [14]. In the agent-based model it was assumed that,

depending on the afﬁnity of the encoded antibody, the B cells either contact fol-

licular dendritic cells in a static condition of 30 min or in a transient manner that

lasted until the B cell continued to migrate. A distribution of contact durations was

obtained from the simulations and is plotted in Fig. 6.3. It was found that the ma-

jority of B cells exhibit only short contacts with follicular dendritic cells, i.e. less

than 5 min (see Fig. 6.3a), and only 2–4% of the cells are in a static contact dur-

ing the germinal centre reaction (see Fig. 6.3b). These results of the simulations

are in agreement with the two-photon measurements [14]. Therefore, it cannot be

Fig. 6.3 Simulation results

on the overall contact time

and the number of static

contacts between centrocytes

and follicular dendritic cells.

(a) The vast majority of

contacts between centrocytes

and follicular dendritic cells

during a one-hour

measurement is of dynamic

nature and lasts around 3 min.

(b) During the germinal

centre reaction, the fraction of

centrocytes that drive the

afﬁnity maturation and are in

static contact with follicular

dendritic cells for 30 min is

only 2–4%. (This ﬁgure was

modiﬁed after [7])

6 Modelling Intravital Two-Photon Data 135

concluded from the experimental contact data that centrocytesintegrate signals from

short contacts with follicular dendritic cells [14]. Instead, even though signal inte-

gration cannot be ruled out, the simulations were consistent with the idea that rare

static contacts are sufﬁcient to drive afﬁnity maturation.

Agent-Based Models of Cell Shape Dynamics

Two-photon microscopy data reveal that cell motility and cell shape are closely in-

terlinked with each other, since the re-polarization of cells involves the repositioning

of the cellular cytoskeleton. Several modelling approaches exists that capture the

shape dynamics of migrating cells [17, 18, 23]. In this chapter, we concentrate on

the approach that has been applied to analyze the experimentally observed shape

dynamics of migrating cells in lymphoid organs [23].

Two-photon data show a stochastic variation of lymphocyte motility that is

closely interlinked with the cell shape [10]. Therefore, to analyze these data, cells

cannot be treated as point particles but have to be modelled in a spatially resolved

fashion. This is realized by an agent-based model with a sufﬁciently high lattice

resolution that represents each cell by a collection of subunits corresponding to

connected lattice points. In addition, updating rules have to be implemented in the

agent-based model that mimic realistic shape deformations of migrating cells.

Cell Shape Dynamics of Migrating T and B Cells

Following a reductionalist point of view, cells may be represented by the cell vol-

ume, the cell polarity and a list of connected cell subunits that constitute the cell

shape on the lattice. The cell volume determines the number of cell subunits accord-

ing to the chosen space resolution, while velocity states translate into probabilities

of subunit movements in the direction of the cell polarity according to the chosen

time resolution.

Two main contributions of cell dynamics have been identiﬁed [23]:

(1) The rearrangement of subunits with respect to a virtually shifted barycentre of

the cell (active movement). A normalized polarization vector determines the di-

rection of the active movement of a cell, which in reality is a complex function

of the cytoskeleton organization as well as localized signalling pathways. This

vector is considered as an approximation for the direction in which cell protru-

sions are developed and is assumed to change randomly with a probability that

is associated with the persistence time.

(2) The rearrangement of the subunits with respect to the actual barycentre (re-

shaping forces). The subunit movement is realized according to heuristic rules

that are interpretable as physical quantities. This ingredient of the cell motil-

ity model concerns the cell shape stability. During the procedure of active

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

cell movement the details of forces that reshape the cell towards a sphere, i.e.

hydrostatic pressure, reduced actin ﬁlament assembly, actomyosin contraction,

or membrane surface tension, are ignored. Within a phenomenological approx-

imation, all these forces are included in a single reshaping force. This overall

elastic force drives the subunits of an elongated cell back to the current barycen-

tre and promotes a spherical shape.

The simulation of active movement was performed by the following procedure:

The barycentre of the cell is virtually shifted in the direction of the polarization vec-

tor towards the membrane of the cell. Then every subunit representing a membrane

point of the cell is moved in random order towards free lattice points near the virtual

barycentre. Moving a border subunit which is not a direct neighbor of its target point

(e.g. a subunit on the back of the cell) is considered to correspond to the cytosol shift

through the whole cell. Note that in this model all subunits carried the same proper-

ties and if the movement of a membrane subunit would have caused the subunits of

the cell to become disconnected this movement was suppressed. The displacement

of subunits was stopped when either no membrane subunit remained to be moved or

the barycentre of the cell was displaced by one lattice constant. In the new state the

cell had reorganized its membrane and thus changed its shape. Thereby the total cell

volume (or, the total cell area in a two-dimensional implementation) was conserved.

Thus, the movement of the cell barycentre was realized by subunit rearrangements,

and inherently coupled the cell movement to its deformation.

The comparison of the model results with experiment revealed that the two in-

gredients active movement and reshaping forces are sufﬁcient in order to describe

lymphocyte motility data as found by two-photon imaging. Here, we concentrate

on the shape index that was calculated from the data obtained for T cell and B

cell migration in lymph nodes of mice [10]. Projecting the cell volume on a two-

dimensional plane, the area was approximated by an ellipse, which is characterized

by its two axes. The shape index s

.n/ of the i th cell was calculated as the ratio of

the long to the short axis at each time step t

, such that s

.n/ > 1 corresponds to an

elongated cell shape that becomes circular for s

.n/ D 1.

In Fig. 6.4 the shape index distributions are plotted as obtained from the mea-

surement of I D 46 T cells and B cells. The average shape index is around 2:3 for

T cells and 1:6 for B cells, indicating that the cellular elongation is about a factor

1:5 larger for T cells compared to B cells. The differences in the shape index are

in agreement with observations by two-photon microscopy [10]. It was speculated

that this may point to cardinal differences in the cytoskeleton dynamics of migrat-

ing T and B cells [23]. Nevertheless, in the simulations both cell types dynamically

round up and take a symmetric shape during the process of re-polarization. It should

be noted that this result has not been enforced by assuming a pausing time in the

agent-based model, as was done in the statistical cell track model combined with

the assumption of a single speed state for the cells. Rather, in the agent-based model

the shape index is a direct consequence of the reshaping forces that cells experience

while they are continuing to migrate and the shape index is closely connected with

a signiﬁcant width in the underlying speed distributions [23].

6 Modelling Intravital Two-Photon Data 137

Fig. 6.4 Simulation results

on the distribution of the

shape index for 46 T cells and

B cells. The error bars

correspond to one standard

deviation and stars indicate

the mean values of the

distributions. (a) The T cell

shape index distribution has a

mean shape index of 2:3 and

shows that T cells reach

peak elongations between

4 and 5.(b) The B cell shape

index distribution has a mean

shape index of 1:6,which

corresponds to only 70%of

the mean value for T cells.

Peak elongations in B cells

occur between 2:5 and 3:5

Summary

Multi-photon microscopy is a powerful tool for imaging lymphocyte migration and

interaction in intact biological organs. Quantitative information on the cell motility,

shape dynamics and contact durations are derived from these data by mathematical

analysis that shape our interpretation of these data.

It is important to realize that mathematical analysis is not merely a descriptive

approach, rather, its valuable contribution is to constitute a predictive consistency

check of the working hypothesis that is underlying the interpretation of the experi-

mental data. For example, a simple statistical cell track model was used to analyze

the trans-zone migration of B cells in germinal centres. This revealed, on the one

hand, that the measured trans-zone frequency is in agreement with the working

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

hypothesis of B cells performing a persistent random walk migration in germinal

centres. On the other hand, the same analysis disclosed an inconsistency with re-

spect to the observed zonal morphology in germinal centres. This initiated further

analysis using an agent-based model approach that captures the functional aspects

of the germinal centre reaction. It then turned out that, hidden behind the experi-

mental data from which the working hypothesis was derived, transient chemotaxis

restores the zonal morphology in germinal centres. Since transient chemotaxis is

still compatible with the B cell motility data, this implied that the initial working

hypothesis of pure random walk migration with directional persistence time was ac-

tually formulated too narrow. This development of insight is a prime example for

experiment and theory working hand in hand [8].

A ﬁrm experimental data basis is the mandatory prerequisite for realistic models

of cellular dynamics in complex biological systems. Including more detailed aspects

of cell dynamics into mathematical models reveals new system features and is be-

coming more feasible due to the continuously growing computer resources in terms

of memory and processor speed. At the same time experimental techniques are de-

veloping fast and highly accurate data on cellular dynamics that are recorded over

several hours time can be expected in the near future. The real challenge, however,

is for modelers and experimentalists to keep pace in working hand in hand achieving

the highest beneﬁt from the symbiotic effects of theory and experiment.

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Chapter 7

Modelling Lymphocyte Dynamics In Vivo

Becca Asquith and Jos

e A.M. Borghans

Abstract Quantiﬁcation of lymphocyte dynamics contributes greatly to our under-

standing of many fundamental processes in immunology, including homeostasis,

ageing, immunological memory and pathogenesis. Recently developed experimen-

tal techniques to label lymphocytes and mathematical models to interpret the re-

sulting data enable us to accurately measure the rate of T and B lymphocyte

proliferation and disappearance in humans in vivo. Here we describe some of the

experimental and mathematical techniques involved, including a discussion of the

advantages and disadvantages of the most popular methods available, as well as a

practical guide to modelling labelling data, and a discussion of future challenges in

this rapidly-moving area.

Why Quantify Lymphocyte Dynamics?

Lymphocytes are the key cells of our adaptive immune system and their dynamics

determine our health. The correct balance between the production of lymphocytes

(de novo in the thymus and by peripheral proliferation) and lymphocyte loss (by

cell death or differentiation) is essential for immune function. Dysregulation of

lymphocyte dynamics results in a broad spectrum of pathologies including AIDS

(precipitated by CD4

T lymphocyte loss), leukemia (aberrant growth of B or T

lymphocytes) and autoimmune conditions such as multiple sclerosis or arthritis

(inappropriate activity of self-reactive lymphocytes). Even though lymphocyte pro-

liferation and death rates are often regarded as textbook immunology, estimates of

these rates can easily vary by a 100-fold. Nevertheless, researchers are trying to un-

derstand how human diseases like HIV infection, and therapeutic interventions such

B. Asquith (



)

Department of Immunology, Wright-Fleming Institute, Imperial College London,

Norfolk Place, London W2 1PG, UK

e-mail: b.asquith@imperial.ac.uk

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

141

142 B. Asquith and J.A.M. Borghans

as haematopoietic stem cell transplantation, affect lymphocyte kinetics. As long as

there is controversy about lymphocyte kinetics in healthy individuals, such ques-

tions will remain hard to address.

Why Are Mathematical Models Needed?

Thanks to recent experimental advances, including the application of stable isotope

labelling to label and trace cells that undergo proliferation, the time is now ripe to

determine these rates of lymphocyte turnover. Proper interpretation of labelling ex-

perimental data hinges upon the use of mathematical models. Without mathematical

models, labelling curves remain merely descriptive and do not yield the quantitative

turnover parameters that are needed. Moreover, mathematical modelling helps to ob-

tain insights into the complicated dynamical processes underlying the functioning

of the immune system.

Mathematical models are often criticized because of the assumptions that are

made. Indeed, as Lee Segel, a distinguished mathematical biologist himself, once

said “mathematical biologists are people who make oversimpliﬁed models”, but

importantly he added “and do not even feel embarrassed”. It is important to re-

alize, that any interpretation of data incorporates assumptions. The danger with

intuitive, model-free interpretations is that the assumptions are often implicit. Im-

plicit assumptions lack transparency and researchers themselves may be unaware

that they are making these assumptions and so their accuracy is never addressed.

Indeed, as will be repeatedly seen in the following examples, intuitive “assumption-

free” approaches that initially seem persuasive have frequently been demonstrated

to give incorrect results. The fact that mathematical models make their assumptions

explicit, and thereby visible, makes it easier to scrutinise them. Additionally, making

simpliﬁcations and eliminating free parameters has the considerable advantage that

it becomes possible to estimate key parameters such as lymphocyte proliferation and

disappearance rates with limited amounts of data, which would be impossible with

highly complex models (see the Section on Parameter Identiﬁability). Finally, sim-

pliﬁed models can provide real insights into complex and highly dynamical systems

such as the immune system.

Different Methods to Study Lymphocyte Dynamics

In recent decades, a large number of different methods have been applied to estimate

the rates of production and death of different lymphocyte populations. These meth-

ods differ widely and include the interpretation of “experiments of nature”, such as

lymphocyte reconstitution after severe lymphopenia, as well as different markers to

study lymphocyte populations that undergo proliferation and death. Some of those

markers occur naturally, including the expression of the proliferation marker Ki67,