Basic model

We considered germinal center cells falling into two phenotypic classes, centroblasts and centrocytes, and referred to these two pools of cells as the "proliferative compartment" and the "selective compartment". After mutation generates B cell variants, the interactions among B cells, antigen and T cells, can then be thought of as a filter, selecting for the high affinity cells, and letting the low affinity cells die. The selected cells move into the memory pool and do not re-enter the proliferative compartment of the GC. If there is no feedback from the selective compartment to the proliferative compartment, we can neglect the internal dynamics of the proliferative compartment, and only consider its input into the selective compartment. Thus, we assumed that, due to proliferation, there is a constant flow of B cells into the selective compartment, where antigen is held on the surface of follicular dendritic cells. This assumption is supported by the observation that once the distinguishable light and dark zones could be observed on GC sections, the dark zone did not seem to undergo further expansion (), suggesting that as cells divide half of the progeny, on average, leave the dark zone.

We group the B cells entering the selective compartment into a small number of affinity classes (1993), where all cells in class i are assumed to have similar affinity for the antigen. The number of cells of class i in the selective compartment is denoted by B_i.

The concentration of antigen is denoted by F, and I will only consider here the case of a non-replicating antigen. Antigen within the GC is allowed to decay, the initial concentration being F₀. At any particular time in the germinal center reaction, f=F/F₀ is the fraction of antigen remaining in the GC. While the rate of decay of antigen trapped on follicular dendritic cells (FDC) is not known precisely, measurements by Tew and coworkers () and Tew & Perelson (unpublished results) using radioactive protein antigens show approximately exponential decay with half-lives of 1 to 2 months.

We further assumed that the survival of the B cells is the result of their interaction with the antigen, and that the rate of rescue of B cells in class i is proportional to a single factor, s_i. This factor determines the quality of interaction of B cells in class i with FDC-associated antigen when the antigen concentration is maximal. If the antigen decays, we assumed that the rescue rate is proportional to f, the fraction of remaining antigen. One could imagine this factor s_i being proportional to the affinity or the binding rate constant between cells of class i and the antigen, or alternatively, it could denote the amount of antigen that cells of class i manage to present to the T cells. The amount of presented antigen should depend on the affinity of the B cell for antigen, since B cells need to strip the antigen off the surface of FDCs. The model is robust against the specific implementation of the rescue dynamics, as long as rescue of cells in class i is proportional to both the amount of antigen with which they can interact and a single affinity class-specific factor s_i. What we are essentially implementing through this assumption is the view that centrocytes are programmed to die unless "rescued" by the interaction with antigen-loaded dendritic cells, this interaction being affinity-dependent. From now on, the factors s_i will be referred to as affinities, keeping in mind that there need not be a simple mapping between these factors and the affinity of the B cell receptors for the antigen. However, it seems reasonable to assume that the factor s_i is a monotonically increasing function of the affinity of cells in class i.

Let us denote the number of cells of type i that have entered the memory pool by N_i. I will assume that these cells are long-lived on the time scale of the germinal center reaction, such that no significant loss from this pool occurs during this time period. As I will focus on the efficiency of the germinal center reaction itself, I will not discuss possible dynamics of the memory cell compartment. Affinity selection, and even affinity maturation seem to occur at post-germinal center stages (1998). They do not, however, affect our conclusions on the efficiency of the germinal center reaction.

Under these assumptions, the rate at which rescued cells of class i enter the memory pool is given by

$$\frac{d N_i}{dt} = s_i B_i f.$$ (5.1)

If the antigen decays at a constant rate, gamma, from an initial amount F₀ to F at time t, the fraction of antigen present in the GC as a function of time is simply

$$f(t) = e^{-\gamma t}.$$ (5.2)

As I mentioned above, I will only present here the case of cells mutating when they replicate in the dark zone. Provided that the mutation dynamics is fast with respect to the rate at which FDC-associated antigen is depleted, and that the total influx of B cells into the selective compartment is constant over the duration of the germinal center reaction (1990), we can assume that there is a constant input flux, I, of B cells into the light zone, and that this influx has constant numbers, I_i, of cells in the different affinity classes. Thus, if mutation asymptotically produces a proportion $\rho$ of high affinity cells, then we assume that this proportion occurs in the input to the light zone from the start of the germinal center reaction. Note that this is an upper bound on the average affinity of the cells entering the light zone since at the start of the GC reaction fewer high affinity cells might be produced.

Once in the selective compartment, centrocytes get rescued and move into the memory pool, or die at rate $\mu$. Thus, the dynamics of centrocytes of affinity class i is described by

$$\frac{d B_i}{dt} = I_i - \mu B_i - s_i B_i f.$$ (5.3)

The first term on the right hand side denotes the constant influx of cells of affinity class i of centroblasts into the selective compartment, the second term accounts for cell death, and the third term for depletion of centrocytes due to their being rescued and converted into memory cells that exit the GC. We can also take into account the possibility of lethal mutants occurring in B cells at a rate q_l. This would not change the above formulae but would effectively increase the death rate $\mu \rightarrow \mu + q_l$.