A possible role of chemotaxis in germinal center formation

Tilo Beyer1, Michael Meyer-Hermann1 and Gerhard Soff1

1 Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany

Correspondence to: T. Beyer; E-mail: tilbey{at}theory.phy.tu-dresden.de
Transmitting editor: T. Tedder


    Abstract
 Top
 Abstract
 Introduction
 The model
 Results
 Discussion
 References
 
During the germinal center (GC) reaction a characteristic morphology is developed. In the framework of a recently developed space-time model for the GC, a mechanism for the formation of dark and light zones has been proposed. There, the mechanism is based on a diffusing differentiation signal which is distinguished by follicular dendritic cells (FDC). Here, we investigate a possible influence of recently found chemoattractants on GC formation in the framework of a single cell-based stochastic and discrete three-dimensional model. This necessitates a more detailed spatial description. The model is enlarged by a detailed prescription of cell motility and it is introduced as a consistent volume concept. We consider various possible chemotactic pathways that may play a role for the development of both zones. Our results suggest that the centrocyte motility resulting from a FDC-derived chemoattractant has to exceed a lower limit to allow the separation of centroblasts and centrocytes. In contrast to light microscopy, the dark zone is ring shaped. This suggests that FDC-derived chemoattractants alone cannot explain the typical GC morphology.

Keywords: B cells, centroblasts, centrocytes, follicular dendritic cells, spatial model


    Introduction
 Top
 Abstract
 Introduction
 The model
 Results
 Discussion
 References
 
Germinal centers (GC) play an important role during secondary immune response [for an overview, see (1)]. The reaction starts with the activation of B cells by antigen during a T cell-dependent or -independent immune response (2). Within secondary lymphoid organs such as the spleen, tonsils or lymph nodes the major part differentiates into plasma cells while a small number of activated B cells—the seeder cells—migrate into the primary follicle which now develops to a secondary follicle. There, the seeder cells develop to centroblasts (2,3) which start to proliferate rapidly and replace the naive B cells inside a dense network of follicular dendritic cells (FDC) within 3–4 days (2,4). At about the same time a yet unknown signal provides the centroblasts to start somatic hypermutation (1,5) leading to a diversification of antibodies amongst the monoclonally expanded cells (2). Several experiments suggest that the expression of antibodies is suppressed on the centroblast surface (6). The centroblasts differentiate into smaller apoptotic centrocytes which express antibodies to a higher degree (7,8). To rescue centrocytes from programmed cell death they first have to interact with antigen-presenting FDC (911). In order to do so the centrocyte has to present an antibody with high affinity for the antigen. If the interaction is successful then centrocytes are positively selected. Furthermore, they have to pass a second checkpoint where T cells test for autoimmune reaction and provide signals for further differentiation either into plasma cells or memory cells (11,12). The process of somatic hypermutation together with those two stages of selection leads to affinity maturation, i.e. optimization of antibodies (13).

During the GC reaction a characteristic morphology occurs. When the processes of somatic hypermutation and differentiation of centroblasts to centrocytes have started, basically two specific areas can be seen in the GC around day 5: a centroblast-rich dark zone and a light zone primarily filled with centrocytes (1,2,4,14). Until now it has not been resolved which mechanisms lead to the separation of both cell types. In the framework of a recent model for GC morphology it has been shown that a slowly diffusing signal which is distinguished by FDC and consumed by centroblasts, indeed, leads to the intermediate development of dark and light zones as observed experimentally (15).

The importance of an intermediately appearing dark zone for affinity maturation has been emphasized. The two zones remain for at least a few days before the light zone enriches with centroblasts and the GC is homogeneously filled with centroblasts and centrocytes (4). The total cell number then decreases until the whole GC reaction stops ~3 weeks after initiation and only a few centroblasts remain (1,2).

Today the model mentioned above (15) is the only one considering spatial aspects of the GC reaction. Many other models deal exclusively with kinetic aspects or affinity maturation (1620). In this article we will present the first three-dimensional model of the GC reaction, which is a generalization of the model introduced in (15). Again, the model is lattice based. However, the new model includes a detailed description of cell motility allowing for the consideration of chemotaxis in GC reactions. To this end it is also necessary to introduce a consistent volume concept to account for differences in cell diameters of centroblasts and centrocytes important for their motility. On the basis of this generalized model we discuss several known chemotactic pathways which may play an essential role in GC formation (2128).


    The model
 Top
 Abstract
 Introduction
 The model
 Results
 Discussion
 References
 
To represent a GC in a spatial model we have taken into account three important concepts which are mirrored in the following three subsections: antibodies and antigen, the dynamics of cell–cell interactions, and the spatial distribution of cells. In all cases we want to stay as close as possible to experimental results. Of course there are still unknown parameters. One part is the physiological parameters which are not yet measured or which are not determined consistently in experiments. A second part consists of model parameters that have no direct physiological correspondence but have to be motivated by physiological properties. All parameter values are summarized in Table 1. The model incorporates individual cells on a three-dimensional lattice. Each cell has its unique set of properties described in the following sections.


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Table 1. Summary of model parameters
 
Representation of antibody and antigen
The representation of the antibody phenotype of each cell follows the model of Perelson and Oster (29). In this model a shape space, i.e. a N-dimensional finite size lattice, is defined in which each point corresponds to an antibody phenotype {Phi}. The antigen is represented by the complementary antibody position {Phi}0, the shape space. Since non-optimal antibodies may also bind an antigen, the interaction between both is described by an affinity distribution function:


where ||.||1 denotes the 1-norm (sum over the absolute values of the coordinates) in the shape space, {Gamma} = 2.8 is the affinity width and {eta} = 2 is the exponential weight (20). The affinity function is interpreted as probability of an antibody {Phi} to bind the antigen {Phi}0 ranging from 1 for a perfect matching antibody to 0 for a non-binding antibody. In this scheme hypermutation is described as a jump from an antibody position {Phi} to one of 2N possible next-neighbor antibody positions {Phi} + {Delta}{Phi}. Here we do not consider key mutations which would require a description closer to the details of somatic hypermutation. This may be provided by a genetic model including hot spots and their implication for antibodies. For a discussion of the influence of key mutations, see (20,30,31). Further we do not consider effects such as antigen consumption or decay (16,17).

GC dynamics
There exists a great variety of different models for the dynamics and morphological properties of GC reactions. In our model we want to consider a rather general and simplified perspective on a representative GC. We will concentrate on B cells and FDC, and do not model T cells, monocytes or tingible body macrophages which may also influence the morphology. It is known that GC can develop their typical structure in the absence of T cells (3234). The GC reaction is initiated by activated B cells which migrate into the FDC network. It has been shown that the seeder cells have at least a low affinity for the antigen which is presented by the FDC (35). In the model this is taken into account by using an antibody position {Phi} ~5–10 mutation steps away from the antigen position {Phi}0 in the shape space. This is in agreement with experimental observations (36,37).

Proliferation phase
When the seeder cells have entered the FDC network they receive a signal that induces a fast proliferation phase with an average cell division time of 6–7 h (1,2,8,38,39). During the first 3 days of the GC reaction ~3 seeder cells (2,40) increase their number to ~104 cells, replacing the naive B cells inside the FDC network. The naive B cells now form the mantle zone (2).

Since our model is single-cell based we will not simulate the proliferation in terms of a linear differential equation with a constant cell division rate. Instead, let us consider an ensemble of equal cells in the same stage of the cell cycle and measure the time they need to complete the process of mitosis. Qualitatively this leads to a peak centered at the mean cell cycle time TP = 6 h with width {sigma} = 0.6 h. As a simple approach we use a Gauss function which leads to a proliferation rate:


where ti denotes the eigentime of cell i. The eigentime describes an internal clock of the cell indicating its cell cycle status and is an individual property of each cell. We will make use of this concept throughout the model to describe time-dependent processes such as proliferation, differentiation or centrocyte–FDC interaction (see below).

Differentiation to centrocytes
Some experiments show that centroblasts express low levels of antibodies on their surface (6). Thus selection is very unlikely to take place in terms of antibody affinity. After 3 days of monoclonal expansion centroblasts differentiate into centrocytes which express higher levels of antibody. Now B cells can be selected according to their affinity for the antigen presented by the FDC (12). The first step of B cell selection within GC is a close cell–cell interaction between centrocytes and FDC which takes ~1–4 h (41,42).

It has been shown that about the same time when the differentiation to centrocytes starts, the process of somatic hypermutation is initiated in centroblasts (1,5).

The capability of positively selected centrocytes to recycle back to centroblasts has been frequently discussed (20,4345). This is incorporated to the model using the recycling probability r = 0.8 (16,20). This implies that 20% of all positively selected centrocytes become either plasma or memory cells. The model does not distinguish between plasma and memory cells which are denoted as output cells in the following. In fact, not all centrocytes that survive the first selection step become output cells due to the last selection step taking place in interaction with T cells. Thus only a certain fraction s = 0.9 of these cells will survive (1,12,20).

We also include a time delay {Delta}{tau} = 48 h between the onset of centroblast differentiation and the start of output cell production. All positively selected centrocytes are recycled during this optimization phase (20,46,47), which lasts until the onset of output cell production.

Spatial modeling of the GC
At first we consider the spatial resolution of the model. We do not need to consider the details of the cell shape since we are not primarily interested in surface properties. However, we cannot neglect the volume of the cells because centroblasts are >10 times larger than centrocytes [centroblast diameter dCB = 15 µm (48), centrocyte diameter dCC = 6.5 µm (8,49)]. If we allow only one cell per lattice point we have essentially two possibilities to choose the lattice constant. On the one hand, the lattice constant can be chosen to be of the order of the centroblast diameter dCB. This provides a well-defined volume concept in the sense that one lattice point embeds a volume larger than the volume of one cell. However, this implies that a lattice area filled with centrocytes would mainly consist of empty space. On the other hand, if we choose a lattice constant equal to the centrocyte diameter dCC we do not have problems with empty spaces but one lattice point provides not enough space for a centroblast. As a consequence, we are led to a subcellular description that would contain more information than necessary for the present purposes.

To solve this conflict we proceed as follows. The lattice constant a is defined as the average of the diameters of centroblasts and centrocytes. In the following the lattice constant a denotes the distance between nearest-neighbor lattice points. This is a natural unit for the discretization of parameters depending on the lattice constant (such as cell velocities). To minimize effects of lattice anisotropy we choose a face-centered cubic (f.c.c.) lattice which has the highest symmetry amongst regular lattices. Second, we consider polyhedrons at each lattice point formed by nearest-neighbor points. Within this polyhedron the volume of all cells that intersect with the polyhedron are summed up. We assume spherical cells and the lattice constant to be large enough to exclude an intersection of cells on second-next-neighbor points with the polyhedron. Then, only nearest neighbors have to be considered. A cell is allowed to move to a next-neighbor point if three conditions are fulfilled: the point is not occupied by another cell, there is enough volume remaining in the corresponding polyhedron to take up the volume of the new cell and the polyhedrons belonging to the next neighbors of the considered target point can take up the intersecting volume of the new cell.

We consider two origins for cell movement: undirectioned random movement and chemotaxis. This is incorporated into the model by separating the cell movement into an isotropic random part and a cell type-dependent deterministic part. In each timestep a randomly chosen cell is highlighted to perform a motion. All next-neighbor points are checked if the cell is allowed to move to this point in the sense explained above. Then the deterministic part is calculated as a velocity vector based on chemoattractant concentration gradients to free next-neighbor points (the calculation of the concentrations is described below). If the cell responds to multiple chemoattractants, the corresponding velocity vectors are summed up to give the resulting velocity vector v. This vector is projected onto the lattice via unit vectors en, where n denotes all 12 possible directions of movement. The result is multiplied with the factor {Delta}t/a in order to convert it into a probability for moving to a lattice point via channel n in the time interval {Delta}t:


In an analogous way we calculate a probability for undirected random movement:


where vrandom is the cell velocity for undirectioned movement and Nnn = 12 is the number of next neighbors in the f.c.c. lattice. The total probability for the movement of a cell to a neighbor point using the channel n is then given by:


where {delta}n,free is the Kronecker symbol for channel n which is equal to 1 if the channel is free and 0 otherwise.

In order to interpret wn as probability we demand wn >= 0 and set wn = 0 otherwise. As only movements to next neighbor points are allowed we have to choose the time step {Delta}t small enough to keep {sum}nwn < 0. This condition depends on the highest possible cell velocity.

Knowing the probabilities of cell movement for each free channel, one of the available channels is selected by using a pseudo random number. Note, that this procedure alters the effective values for the random velocity and the chemotactic response, e.g. in the case that all next-neighbor points are occupied by other cells both values are 0.

Representation of B cells
B cells are modeled as individual cells with several properties: position, volume Vi, antibody phenotype {Phi}i and eigentime ti. Each cell has a certain probability to proliferate, if it is a centroblast, or to die by apoptosis, if it is a centrocyte, which grows according to eq. (2) when the eigentime ti reaches the average cell division time TP or lifetime TL, respectively. The eigentime is always set to 0 when the cell has proliferated, i.e. both new cells start with eigentime ti= 0. The same holds true if the cell has differentiated.

Growth and differentiation of cells. Since we included the cell volume in the model we have to explain how to differentiate from large centroblasts to small centrocytes and back. If centroblasts differentiate to centrocytes an additional volume around the cell becomes available which is not a real problem. However, the other way round, an additional volume is suddenly required. In order to provide a more realistic description of cell volume we include cell growth in the model. We use an equation of the type:


where {alpha} and ß determine the time course of the growth process.

When a centrocyte recycles back into a centroblast it starts to grow. To ensure the volume restriction the cell may only grow if the polyhedron associated to the occupied lattice point and the polyhedrons associated to the next-neighbor points can hold the additional volume ßVi{alpha}{Delta}t. If this is not possible the cell stops growing until the required space becomes available. Note that this process depends on the movement and growth of cells in the vicinity. Therefore, it is important to calculate cell growth by choosing cells in a random sequence in each time step—as is done for cell movement.

Additionally, the growth process of recycled cells takes longer compared to a new cell emerging from mitosis because the initial volume is substantially smaller. Proliferation requires the cell to have an above threshold volume. If this is not required for the process of differentiation as well, recycled cells would preferentially differentiate and proliferation would become a rare event. However, the multiplication of high-affinity B cells through recycling is an important feature of GC dynamics. Therefore, we forbid differentiation of a recycled centroblast until it reaches the volume of a freshly proliferated cell.

The differentiation of centroblasts to centrocytes is assumed to occur with a constant rate after activation by a quantum of signal molecules (50,51). The influence of spatial inhomogeneous differentiation signals on GC morphology has been studied before (15). We want to exclude a spatial inhomogeneity of signal molecules in order to avoid an interference with the chemotactic response of GC B cells. Therefore we introduce a homogeneous density of quanta of signal molecules {rho}. A centroblast consumes such quanta with a rate u = {rho}/h proportional to this density {rho}. One quantum of signal molecules is assumed to contain enough molecules to induce the differentiation process. The centroblast then differentiates with rate 1/TD = 1/(3 h). The production of signal quanta itself is assumed to be initiated by an unknown signal at day 3 that causes the secretion of these signal molecules with constant rate q during the whole GC reaction. The molecules are not completely removed from the GC by consumption by B cells or diffusion and therefore accumulate during the reaction. This results in an enhanced activation of differentiation of centroblasts to centrocytes in the later stages of the reaction which finally terminates the GC reaction.

In order to perform the differentiation process the cell ‘shrinks’ to the proper centrocyte volume and acquires all the characteristic properties, such as the ability to interact with FDC and the initiation of the process of apoptosis. The cell is assumed to be already able to respond to chemoattractants before the typical centrocyte volume has been reached.

The constants {alpha} and ß in eq. (6) have yet to be specified. The exponent {alpha} describes the volume dependence of the growth process and is determined to be {alpha} = 3/4 according to observations made for many different organisms (52). The coefficient ß is chosen for the process of shrinking and growing separately. In the case of cell shrinking ßs is determined by the time TS a centroblast needs to fully differentiate into a centrocyte. Unfortunately this time is not known, but seems to be substantially smaller than 7 h (2,4). Therefore, we assume TS = 3 h. In the case of cell growth we chose ßg in respect to the proliferation of centroblasts. The centroblast volume is doubled during the G1 phase of the cell cycle. We assume the duration of the G1 phase to be of the order of TG1 = 5.4 h. It results in:


for the process of growing [V0 = ({pi}/6)dCB3 denotes the maximum centroblast volume] and:


in the case of shrinking. The centroblast diameter dCB is a little more than 2 times larger than the centrocyte diameter dCC, resulting in a volume ratio of ~12:1.

Proliferation of centroblasts. The proliferation of the centroblasts is described as the replacement of a large mother cell by two daughter cells with half volume, thus respecting the volume conservation but violating the condition that only one cell can occupy one lattice point. The latter condition is restored automatically when a lattice point in the close proximity becomes available for one of the cells to move to. An eigentime of 0 is attributed to each new cell and it restarts to grow. If the growth process stops for a short period of time because there is not enough space available the eigentime is frozen as well. This mirrors the influence of the growth process—the G1 phase of the cell cycle—on the proliferation time. The other phases of the cell cycle are more or less of constant duration and do not depend on space restrictions since the volume does not change.

Both daughter cells have to maintain a certain distance in order to have sufficient space to grow. In the case of very low mobility the B cells may not reach this minimum distance at every moment and therefore proliferation is inhibited, i.e. the expected 104 cells after 3 days of clonal expansion will not be seen. This implies a lower bound for the mobility of the B cells.

We do not consider apoptosis of centroblasts. Experiments show that centroblasts can enter apoptosis when left without proper surviving signals (50,5355). Thus we neglect two possible effects: a time dependence and a spatial dependence of an effective proliferation rate. Also, we do not include proliferation signals (51). Thus the proliferation probability p(ti) depends neither on the position of the cell nor on the time course of the GC reaction. Only volume restrictions modify the proliferation rate. The results will be discussed in this context.

Hypermutation. For the selection process each B cell has its own antibody phenotype {Phi}i. Before the onset of somatic hypermutation all centroblasts have the same antibody phenotype. Hypermutation is started at the same time point as differentiation. A cell jumps to a next-neighbor point in the shape space with hypermutation probability m = 0.5 (15,56). Hypermutation is allowed during cell division only and both new cells mutate with the same probability.

FDC representation
A FDC is represented as a soma with dendrites attached to it. The soma is assumed to be immobile and to have a volume comparable to B cells. The volume of the dendrites is neglected so that B cells and dendrites may occupy the same lattice points. Each FDC has six dendrites with a length of four lattice constants a resulting in an overall length of 254 µm, which is in good agreement with morphological studies (57). The total number of FDC is NFDC = 104 (58). This provides enough sites for centrocytes to interact with FDC without saturation effects.

Interaction between centrocytes and FDC
The interaction between centrocytes and FDC takes place when the centrocyte is a next-neighbor of a dendrite or the soma of a FDC, or shares a lattice point with a dendrite. We assume that the process of selection takes ~3 h (41,42). During that time the centrocyte is in close contact with the FDC and therefore the centrocyte is assumed not to move. Thereafter the affinity between antigen and the antibody phenotype of the centrocyte is calculated according to eq. (1). The affinity is interpreted as probability of positive selection. If the cell is positively selected it may become an output cell or a centroblast again. During the optimization phase, i.e. before the production of output cells is started [day ~5 (20,46,47)], all positively selected B cells are recycled.

Chemotaxis
Every FDC soma generates a chemotactic field by secreting a chemoattractant into the GC. We assume that the decay and the uptake of signal molecules by B cells is small enough to neglect the feedback on the chemotactic field. Thus the concentration of the chemoattractant remains in equilibrium during the whole GC reaction. The concentration of the chemoattractant is proportional to one over distance for each FDC and is calculated once at the beginning of the GC reaction. We further assume that the response of cells to a chemotactic gradient {nabla}c is constant: v = {gamma}a,b {nabla}c/||{nabla}c|| ({gamma}a,b denoting the velocity of the cell type b in response to a chemoattractant stemming from cell type a), i.e. the cell detects the gradient and actively moves into that direction. At every lattice point we calculate all concentration differences to free next-neighbor points resulting from all FDC and use a vector sum to compute the corresponding velocity vector v which enters eq. (3). For our purposes {gamma}a,b is the key parameter and is varied over several magnitudes to investigate its influence on the GC morphology.

Boundary and initial conditions
We restrict the whole GC reaction to a sphere with diameter dGC = 345 µm (59) of a fully developed GC. This ensures sufficient space for slightly more than 12,000 centroblasts on the lattice. No cells except output cells can leave this volume. We do not consider adhesion or the pressure of surrounding naive B cells. This assumption, or more generally the boundary conditions, will have to be further discussed.

The starting point of our model is based on are three activated B cells that have already entered the network of FDC which are homogeneously distributed on one half of the sphere mentioned above. Some experiments indicate that the FDC network is not homogeneous. There may exist regions where FDC present antigen and other regions where FDC present less or almost no antigen. Centrocytes are primarily found In the first region, while centroblasts dominate in the second region (60). In our model only the antigen-containing region is taken into account.

In all of our simulations we will use the same configuration of the initial antibody phenotype of the seeder cells and the antigen held by the FDC for reasons of comparability. The initial affinity of the antibody is a({Phi}seeder cell, {Phi}0) = 0.04.


    Results
 Top
 Abstract
 Introduction
 The model
 Results
 Discussion
 References
 
GC population kinetics
The inclusion of a differentiation signal allows various scenarios for GC population kinetics. Therefore, at first, we have to determine realistic time courses of the GC reaction.

The GC kinetics is strongly dependent on the production rate q of the signal molecule quanta and the boundary conditions. High production rate q results in an exponentially declining cell population. Too small production rate q implies a domination of proliferation over differentiation and the cells fill the whole lattice until the proliferation is inhibited by space limitations. This leads to a durable dynamic equilibrium with a stable B cell population (data not shown). For production rates between these two extreme scenarios the time course of the B cell population becomes realistic. At the beginning B cells fill the whole lattice. After a certain time (depending on the signal production rate q) the signal molecule density increases and more centroblasts start to differentiate. This results in a controlled reduction of the centroblast population (Fig. 1).



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Fig. 1. Different cell populations during the GC reaction. Recycled and output cells are shown as time-integrated cell numbers. Centroblast and centrocyte cell numbers at given time points are plotted. After 3 days the differentiation into centrocytes starts and the centrocyte population increases rapidly. The total B cell number (centroblast and centrocytes) has a peak at day ~4 followed by a plateau. After 10 days the populations are declining.

 
The typical GC reaction in the model shows a nearly exponential increase of the centroblast population during the first 3 days (Fig. 1). Then the differentiation process starts and the centrocyte population increases. The peak of the reaction is reached at about day 4, in good agreement with experiment (34). The total population then stabilizes on a plateau for some days before it declines rapidly, entering a long-lasting, low-level reaction with a very slowly decreasing B cell population. The duration of the plateau depends on the production rate q of the differentiation signal. A reduction of the signal production of 10% leads to a prolongation of the plateau phase of 6 days, while an increase of >10% leads to the disappearance of the plateau.

The time course of such a GC reaction is only slightly dependent on the differentiation rate 1/TD, describing the rate with which activated centroblasts differentiate into centrocytes. The differentiation rate basically has to be large enough to guarantee a declining GC population in the late stages of the reaction (20). Note, that observable differentiation rates are effective rates which already include the effect of signal molecules.

FDC-derived chemoattractant for centrocytes
For our purpose we define the appearance of a light zone as a cluster of cells in which centrocytes dominate. The dark zone is analogously defined. To achieve a separation of centrocytes and centroblasts in a GC the FDC may selectively attract centrocytes but not centroblasts (22). To test this hypothesis we simulate the GC reaction with a chemotactic response of centrocytes only.

A light zone is observed if the centrocyte velocity resulting from the chemoattractant allows the cells to leave the centroblast population before they die by apoptosis. A dark zone develops if this velocity is high enough to allow centrocytes to leave the centroblast population before new centrocytes arise from centroblast differentiation, thus generating a centroblast-dominated area. For very small chemotactic coefficients {gamma}FDC,CC no zones occur at all (data not shown). Large chemotactic coefficients {gamma}FDC,CC result in the formation of a light and dark zone. While the light zone consists of a dense cluster of centrocytes with only few centroblasts, the dark zone is formed in an asymmetric ring-like structure of less dense packed centroblasts (Fig. 2). This can be understood by assuming a quasi-stationary situation in which cell numbers do not significantly change. Then the centrocytes get entrapped in the local minima of the chemotactic field reproducing the equipotential lines of the signal molecule concentration (Fig. 3). Note that not all cells can achieve local minima since volume restrictions have to be respected. The unphysiological ring structure of the dark zone is caused by the fact that the centroblasts have no attractor and behave like a gas inside the GC sphere. With intermediate chemotactic coefficients the zones appear less clearly and later (Fig. 4, left column). The resulting deterministic velocity critical for the formation of light zones is of the order of several µm/min.



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Fig. 2. A 2a-thick slice of the germinal center (centrocytes, white; FDC, gray; centroblasts, black). Only centrocytes respond to the chemoattractant distinguished by the FDC with {gamma}FDC,CC = 13 µm/min (for all other parameters see Table 1). From day 5 on a light zone can be observed. It remains stable for at least 5 days but never exclusively consists of centrocytes. There are always centroblasts present which are mostly recycled centroblasts (not distinguished). The dark zone has a ring-like shape. The last panel shows the GC after 3 weeks with only small numbers of centroblasts and centrocytes remaining.

 


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Fig. 3. The equipotential lines of the chemotactic signal concentration. From outside to inside every line represents an increase of 0.5 of the chemoattractant concentration in arbitrary units.

 


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Fig. 4. A 2a-thick slice of the GC (centrocytes, white; FDC, gray; centroblasts, black). The left column shows a GC reaction with an intermediate chemotactic response of centrocytes to a FDC-derived chemoattractant {gamma}FDC,CC = 1 µm/min. The light zone is less clearly formed compared to Fig. 2 (days 5–10). Only when the density of the cells is reduced at day ~7 can a more pronounced light zone be observed. The right column shows a GC with centroblasts responding to a FDC-derived chemoattractant {gamma}FDC,CB = 2 µm/min. Almost no separation of centrocytes and centroblasts can be seen. Only a small dark zone appears. All other parameters remain as in Table 1.

 
Required motility of centroblasts
The reaction also requires that the undirected movement of the centroblasts is fast enough (with a random velocity vrandom ~ 2 µm/min) in order to allow the centrocytes to escape the centroblast-dominated areas. For random velocities which are significantly below this threshold no light zone develops as long as the centroblast population forms a densely packed cluster. Only when the density of centroblasts declines does a light zone occur for a short period of time (data not shown). Remarkably, the necessary random velocity shows no significant dependence on the chemotactic coefficient {gamma}FDC,CC. The undirected movement of centroblasts is necessary to offer the possibility for centrocytes to use small gaps for movement. Higher centrocyte velocities basically have no effect if the space for movement is lacking. On the other hand, a faster undirected movement of centroblasts may offer enough dynamically produced gaps to ensure the mobility of centrocytes. However, the directed centrocyte movement may be too slow to use the available space. We conclude that a reasonable relation of centroblasts and centrocytes mobility is necessary to allow the separation of both cell types.

In the cases where dark and light zones appear it can be observed that the centrocytes can leave the centroblast-dominated areas. This results from the high mobility and small diameter of centrocytes which use every available space around the large centroblasts to perform movement in the direction of the FDC network. However, they do not replace all of the centroblasts inside the FDC network. This is in part related to the process of recycling which acts as an additional source of centroblasts in the FDC network. In addition, centroblasts rarely leave the FDC network due to their relatively weak mobility. In conclusion, dark zones with very low numbers of centrocytes and light zones with relatively high fractions of centroblasts are generated.

Chemoattractant for centroblasts
In a second step we let the centroblasts also respond to the FDC chemoattractant with equal, higher and weaker responses compared to the centrocytes. In all three cases there exists neither a dark nor a light zone during the whole GC reaction (Fig. 4, right column) except if the chemotactic response of the centroblasts is weak enough so that the undirected movement dominates. Then, the centroblasts behave similar to a free gas as in the scenario where only centrocytes respond to the chemoattractant. In the case of weak chemotactic response of centroblasts ({gamma}FDC,CB) and strong chemotactic response of centrocytes ({gamma}FDC,CC) a very small, almost symmetric ring of centroblasts occurs (Fig. 4, right column, day 5).

Mantle zone-derived chemotactic signals
In order to avoid the ring structure we investigate the mantle zone as an alternative source for a chemoattractant that may act on centrocytes. We include a preformed mantle zone of a GC which is polarized, i.e. the mantle zone is thicker on the side of the FDC network. Similar to the FDC we let the cells in the mantle zone segregate to a chemoattractant. As before, this is reflected in a stationary configuration. This signal alone is sufficient to form light and dark zones (Fig. 5). The dark zone is even more physiological than with FDC-derived chemoattractant, but begins to penetrate the FDC network from day 8 on, while the light zone moves towards the boundary of the FDC network around day 9 (Fig. 5). This is due to the new location of the local minima of the chemotactic field situated at the outer boundary of the FDC network. Centrocytes are densely packed in these local minima. When their number begins to decline, the centrocytes no longer extend to the FDC network. The centroblasts behave like a gas and are dispersed over the free space including the part of the FDC network opposite the mantle zone. This also affects the late stage of the GC reaction. The B cell population declines much faster than with FDC-derived chemoattractant and therefore generates less output cells (data not shown).



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Fig. 5. A 2a-thick slice of the GC (centrocytes, white; FDC, gray; centroblasts, black). A GC reaction with mantle zone-derived chemoattractants only ({gamma}MZ,CC = 13 µm/min.). A dark and a light zone can be distinguished. The dark zone is sickle shaped instead of ring shaped (Fig. 2, days 5–10), but begins to penetrate the FDC network from day 8 on. The light zone is shifted towards the boundary of the FDC network after day 9 and the GC reaction stops very soon because positive selection and recycling are suppressed (not shown).

 
Combined chemotactic signals
When the centrocytes respond to signals from FDC and mantle zone cells, a light zone and a dark zone can be observed (Fig. 6). The light zone stays within the FDC network during the whole reaction. The dark zone is neither ring shaped nor penetrates the FDC network. The GC morphology is closer to observed morphologies than with FDC-derived chemoattractant alone. Also the GC kinetics stays in good agreement with experiment (data not shown). Only the differentiation rate 1/TD has to be adjusted to restore the late stage of the reaction. Otherwise, the B cell population would decline faster than with FDC-derived chemoattractant only, because the FDC–centrocyte interaction is slightly reduced when some of the centrocytes are entrapped at the boundary of the FDC network.



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Fig. 6. A 2a-thick slice of the GC (centrocytes, white; FDC, gray; centroblasts, black). The centrocytes respond to chemoattractants from FDC {gamma}FDC,CC = 7 µm/min and mantle zone cells {gamma}MZ,CC = 13 µm/min. We can observe a sickle-shaped dark zone and a light zone. The ring structure of the dark zone is clearly reduced (cf. Fig. 2, days 5–10). The dark zone does not penetrate the FDC network and the centrocytes stay within the light zone (cf. Fig. 5, days 8 and 9). The differentiation rate was adjusted to TD = 3.5/h (all other parameters remain as in Table 1).

 
The relative strength of the chemotactic response must be tightly balanced. The velocity in response to the mantle zone-derived chemoattractant has to be twice the strength compared to the response to FDC-derived chemoattractants. Higher values lead to structures like in Fig. 5 and smaller values to structures like in Fig. 2. To explain this behavior we plot the equipotential lines of the resulting chemotactic field (Fig. 7). The shift of the isolines towards the mantle zone compared to Fig. 3 can clearly be seen. We want to emphasize that the B cells in the mantle zone are not necessarily the source of the chemoattractant. Any other cell type with similar spatial distribution leads to the same result.



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Fig. 7. The equipotential lines of the effective chemoattractant concentration. Isolines of the FDC-derived plus the mantle zone-derived chemoattractant concentrations are shown. The relative strength of the signals is 1:2. From outside to inside every line represents an increase of 0.5 of the chemoattractant concentration in arbitrary units (as in Fig. 3). The global minimum is shifted to the mantle zone but remains within the FDC network.

 
Affinity maturation
If we investigate the affinity of the output cells we recognize that we have ~60% output cells with high-affinity ([a({Phi},{Phi}0) >= 0.8] and ~30% medium-affinity cells [0.4 < a({Phi},{Phi}0) < 0.8] (Fig. 8). The plot shows the fraction of corresponding numbers of output cells integrated over time intervals of 2 h and reflects the quality of produced output cells. The small numbers of produced output cells during those short time intervals lead to huge statistical fluctuations. Remarkably, the ratio between medium- and high-affinity cells is reversed after 2 weeks. At the beginning of the output production most cells still have low affinity for the antigen. Then, the fraction of medium- and high-affinity cells increases. After 2 weeks the fraction of high-affinity cells increases further, while the fraction of medium-affinity cells is declining. This behavior is in good quantitative agreement with experiment (61).



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Fig. 8. The fraction of medium- and high-affinity output cells in a small time windows of 2 h. The corresponding bold lines show the average over 24 h. Medium affinity is defined as 0.4 < a({Phi}*,{Phi}0) < 0.8 and high affinity as [0.8 < a({Phi}*,{Phi}0) < 1.0]. {Phi}* denotes all {Phi} in the given intervals and {Phi}0 the position of the antigen. The output production begins after 5 days.

 
Interestingly, this behavior seems to be independent of the GC morphology as long as the general GC kinetics remains unaffected. In contrast, the number of output cells changes for different morphologies. The output quantity depends on the probability for centrocytes to interact with FDC. Some morphologies of the GC reaction induce volume constellations that inhibit the movement of centrocytes and in this way reduce this interaction probability. However, the fact that the quality remains almost the same (data not shown), at first sight, seems to be in contradiction to results of Meyer-Hermann (15). However, we use another indicator for the quality of the GC reaction. Considering the time course of the time integrated output quality instead, the GC morphology, indeed, influences the total output quality (data not shown). This value memorizes the total output production and is therefore sensitive to the effectivity of affinity maturation in the early phase (day ~6) of the GC reaction which is inhibited by the absence of zones (15).

Robustness of the model and the results
The model results are stable against small variations of most of the experimentally unknown parameters. This applies especially to the parameters of cell movement (vrandom, {gamma}a,b). The most sensitive parameter is the production rate of the signal molecule quanta q. Its value determines the duration of the plateau phase in the GC kinetics and variations of >10% cause exponentially declining reactions or persistent plateau phases respectively. Also the total size of the GC represented by dGC influences the GC kinetics. The plateau phase of the reaction is reached when the proliferation is inhibited due to limited space. This results in an increased number of centroblasts. When the signal production rate q is adjusted the GC kinetics in Fig. 1 is restored on a higher level of cell numbers. In order to fit the cell numbers to experiment (3), the value for the diameter of the GC is fixed to dGC = 345 µm.

The recycling hypothesis is still more a prediction from theory than a measured fact. The recycling probability r of positively selected centrocytes has been estimated on the basis of experiments providing indirect evidence for the process of recycling to be of the order of 80% (20,37). We tested how changes in the recycling probability r change the results. To achieve reasonable GC kinetics for smaller r we have to raise the differentiation time TD in order to prevent too fast declining B cell populations (up to TD = 5.3 h for r = 0). With decreasing recycling probabilities r the population kinetics becomes more and more exponential and the plateau phase diminishes. The morphology remains unaffected apart from a small reduction of the necessary random movement of centroblasts. This is due to smaller cell densities that occur when the plateau is missing. The major change resulting from the reduced recycling probability r is an impaired affinity maturation when r < 60%. In particular, there is no increase of the output cell affinity after day 10 (data not shown).

Despite the stochastical nature of the model, the results are reproducible without significant statistical variations. The only exceptions are the details of the affinity maturation (Fig. 8) due to small cell numbers.

Also the morphology is independent of the details of the FDC distribution. An expansion of the FDC network slightly beyond the boundary of the GC does not alter the results in general.


    Discussion
 Top
 Abstract
 Introduction
 The model
 Results
 Discussion
 References
 
In the present study we enlarged our previous model for the GC morphology in order to investigate the influence of chemotaxis on GC formation. To this end it was necessary to develop a more detailed description of cell mobility. In addition to a minimization of lattice-anisotropy effects, this, especially, includes a self-consistent cell volume concept on a discrete regular lattice without going into the details on a subcellular level. We included cell growth and shrinking, as well as cell differentiation. The resulting model is the first three-dimensional attempt to simulate GC cell population dynamics and GC morphology. It should, therefore, provide a more realistic description of processes involved in GC development.

Within our model the different cell volumes of centroblasts and centrocytes strongly influence their mobility. We observe in the simulations that small centrocytes are able to move in the environment of large centroblasts even when the latter ones are densely packed. The centrocytes find small gaps and slip through the centroblasts. The mobility of the centroblasts is inhibited by their larger volume. The restrictions on the total GC volume inhibit centroblast growth and thus centroblast proliferation. All of these observations in the GC simulations are quite realistic effects which have an equivalent in real GC and which seem to be important on the way towards producing a realistic picture of moving cells during the GC reaction. Therefore, we consider our model to be suitable for studying possible effects of chemotactic signals.

First, we investigated the influence of a FDC-derived chemoattractant (21,22) on the GC morphology. This signal is sufficient to separate centroblasts and centrocytes, but cannot explain the formation of the characteristic GC structures. If the chemotactic response of the centrocytes and the random movement of the centroblasts exceeds given values ({gamma}FDC,CC = 4 µm/min for centrocytes and vrandom = 2 µm/min for centroblasts respectively) a light and a dark zone develop. However, in contrast to experiment, the centroblasts form an asymmetric ring-like structure (Fig. 2). For smaller values of cell mobility (~1 order of magnitude) the zones are developed later in the GC reaction when the total density of cells becomes small enough (Fig. 4). For significantly reduced cell mobility no separation into light and dark zones occurs at all. The corresponding velocities of both cell types are within physiological relevant values and are comparable to relatively slowly moving cells (6267).

Interestingly, chicken GC seem to have such a ring structure suggesting that FDC-derived chemotaxis acting on centrocytes is adequate to describe their morphology (68). In contrast, in mammalian GC other or additional mechanisms cause the formation of dark and light zones. This difference possibly is related to the absence of a mantle zone in chicken GC (68). To test this hypothesis the mantle zone was taken into consideration as a source of a chemoattractant acting on centrocytes. Again a separation into light and dark zones can be observed. The dark zone is sickle shaped and the light zone is shifted within the FDC network towards the source of the chemoattractant. This causes an inhibited FDC–centrocyte interaction resulting in reduced numbers of recycled centroblasts and output cells. We conclude that the mantle zone may influence the GC morphology in mammalians. Indeed, there exists evidence for a chemotactic activity of the mantle zone. B cells and to some extent T cells in the mantle zone of mammalians GC secrete IL-16 (69), a chemoattractant for T cells (70). However, it seems that the mantle zone is not responsible for typical GC morphologies on its own.

In addition, we investigated centroblasts responding to FDC-derived chemoattractants. Only a small ring-shaped dark zone can be achieved for small values of the chemotactic coefficient {gamma}FDC,CB = 2 µm/min and a more intense chemotactic response of centrocytes {gamma}FDC,CC = 10 µm/min (Fig. 4). The ring of centroblasts is now symmetric in contrast to the results when the centroblasts do not respond to the FDC-derived chemoattractant. The asymmetric ring structure is restored for significantly reduced chemotactic response when the random motility dominates (data not shown). It is known that the seeder B cells enter the primary follicle via chemotaxis provided by the B lymphocyte chemoattractant (BLC)–CXC receptor 5 pathway (24,25,27,71). Our results suggest that centroblasts only weakly respond to FDC-derived chemoattractants during the GC reaction. In agreement with this, GC B cells showed no response to BLC (72), stromal cell-derived factor-1{alpha} (23,72), secondary lymphoid tissue chemoattractant and macrophage inflammatory protein-3{alpha} (72), and a weak response to rC5a (73).

We also analyzed alternative sources for the chemoattractants. If the mantle zone B cells or cells with a similar distribution secrete a chemoattractant acting as an attractant for centrocytes, dark and light zones are formed (Fig. 5). The separation of dark and light zone is more physiological but results in a shorter GC reaction because the light zone shifts to the boundary of the FDC network while the dark zone enters it. This may be related to the model assumptions. We did not include the dynamics of the mantle zone, thus neglecting a possible movement of the minima of the chemotactic field relative to the FDC network. The duration of the GC reaction can only be prolonged if unrecycled centroblasts are attributed with a longer lifetime. However, the quality of the output cells still remains too low (data not shown).

When centrocytes respond to both chemoattractants—the mantle zone-derived and FDC-derived chemoattractants—a longer-lasting light zone can be observed which remains inside the FDC network. The dark zone is sickle shaped and does not expand into the FDC network (Fig. 6). This is the most realistic scenario we could generate. The GC kinetics and the affinity maturation are similar to the scenario with FDC chemoattractant alone (Figs 1 and 8), while the unphysiological ring structure of the dark zone turns into a more realistic sickle-shaped dark zone. This result still does not reproduce the observations of light microscopy (2,7476), suggesting that other mechanisms are necessary to generate the typical GC morphology. One possible mechanism is a diffusing differentiation signal distinguished by the FDC (1).

In general we achieved physiologically realistic GC population kinetics for the different scenarios of the GC morphology (Fig. 1). In the late stages of the reaction when low numbers of GC B cells remain differences occur affecting the number of output cells. This results in different total average affinities of output cells depending on the various GC morphologies. However, the qualities of output cells produced at the end of the reaction are comparable provided that the number of output cells is large enough. The distribution between high-, medium- and low-affinity cells (Fig. 8) quantitatively mirrors experimental results very well (61). We can conclude that the morphology of the GC mainly influences the quantity and not the quality of output cells.

One assumption of the model is that centrocytes respond to a chemoattractant with constant velocity only detecting the direction of the chemotactic gradient. We addressed the question if a linear dependency of the velocity on the chemoattractant concentration gradient (v ~ {nabla}c), would change the results. Indeed, the results cannot be restored within reasonable parameter values, i.e. velocities. Centrocytes far away from the FDC network have the smallest velocities and at the same time have the longest way to go. However, increasing the overall velocity in response to the chemotactic gradient means, especially, also increasing the velocity of cells close to the FDC network, reaching unphysiological high values. If the chemotactic response was linear to the concentration itself one would get almost the same result with a slightly flatter velocity distribution (1/distance compared to 1/distance2). Taken together, this suggests that centrocytes use the chemoattractant concentration mainly to detect the direction of movement and have a more or less constant velocity in response to the chemoattractant.

One may think about a possible influence of cell–cell adhesion. It is known that GC B cells and FDC form clusters in vitro (77). The cooperation of chemotaxis and differential adhesion for the separation of cells has been studied in other systems (78). The major difference to this study is that centroblasts differentiate into centrocytes and vice versa, while in the model of Jiang et al. (78) the cell fractions remain constant. In addition, the cells have no gaps between them, while in our model small gaps are necessary to allow the centrocytes to move towards the FDC. It would be interesting to investigate if adhesion could substitute the boundary condition, i.e. the restriction of cell movement to a sphere, and thus circumvent the gas behavior of the centroblasts.

We conclude that FDC-derived chemoattractants acting on centrocytes lead to a separation of centroblasts and centrocytes in the GC reaction. However, GC morphology as observed in mammalians is not correctly reproduced. This problem persists even using a combined centrocyte response to mantle zone-derived and FDC-derived chemoattractants. These results suggests that chemotaxis on its own is not sufficient to induce a realistic development of dark and light zones. The present model points towards an additional role either of cell–cell adhesion or cells surrounding the GC.


    Abbreviations
 
BLC—B lymphocyte chemoattractant

CB—centroblast

CC—centrocyte

FDC—follicular dendritic cell

GC—germinal center


    References
 Top
 Abstract
 Introduction
 The model
 Results
 Discussion
 References
 

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