Amplification of high affinity cells in the memory population is a logarithmic function of their selection coefficient

Having described the basic model, I will now sketch the derivation of a measure of germinal center efficiency, which I call amplification. This is defined as the ratio between the average affinity of the memory cell pool and the average affinity of the dark zone cells. The first ones constitute the output, the latter the input to the selective filter of the germinal centers.

The antigen dynamics is trivial and simply given by the exponential decay of equation (). The B-cell dynamics is given by equation (). To solve these equations, I will assume that antigen decay is slow compared to the influx, death, and selection dynamics, which amounts to f being constant in equations (). This means that during short periods of time over which the antigen concentration f is roughly constant, the B cells equilibrate to the above values and that these values slowly shift under the change of f as determined by equations (). Under this assumption the centrocytes will reach a quasi-steady state in which

$$B_i = \frac{I_i}{s_i f + \mu}.$$ (5.4)

Let us now solve for the number of cells that have entered the memory pool as a function of time for each affinity class, and for the average affinity of the output cells over time. The output flux into the memory pool for cells in affinity class i, O_i(t), is given by

$$O_i(t) \equiv \frac{d N_i}{dt} = s_i B_i f = \frac{s_i f I_i}{s_i f + \mu}.$$ (5.5)

Substituting f from equation () we obtain the output flux explicitly as a function of time

$$O_i(t) = \frac{s_i e^{-\gamma t}}{s_i e^{-\gamma t} + \mu} I_i.$$ (5.6)

The above expressions demonstrate the main qualitative features of the model. First, the output flux is at most as high as the input flux at any time. Obviously, when there is no recycling or division of centrocytes, the number of cells in class i entering the memory pool cannot be larger than the influx I_i of cells in that class. This means that if mutation only creates a small number of cells in high affinity classes, only a small number of high affinity cells can enter the memory pool. Second, the output flux in each class is maximal at the start of the germinal center reaction and decays to zero at late times as antigen decays. Third, the behavior of the output flux O_i(t) of class i is completely independent of the affinities s_j and input fluxes I_j of cells in the other affinity classes. That is, the output fluxes are not the result of competition between cells. Rather, it is a "competition" between rescue and death that determines the output flux. I will briefly elaborate on this issue, as it seems somewhat controversial at a first reading. Specifically, the notion of competition implicitly assumes some limiting resource, which in this case would be the antigen. Preliminary simulations of this model showed that, for biologically reasonable choices of the parameter values, the cells will rapidly equilibrate with the free antigenic sites on follicular dendritic cells. If at the end of this period there will be free antigenic sites left, then the equilibrium will slowly shift under the independent dynamics of the antigen. If all the antigen is quickly bound by B cells at the beginning of the germinal center reaction, I would expect that the number of high affinity cells that will be generated will be even smaller. In this case, it would not be guaranteed that the few high affinity variants generated during the germinal center reaction will all get to bind the antigen to get rescued. It still seems possible though that the amplification factor will be higher. As I will show below, the number of cells that are generated in a one-pass selection model is already too low to account for the experimental data. Further decreasing these numbers, even with a better amplification of high affinity cells, does not change the basic conclusion that multiple rounds of division, mutation and selection must take place in the germinal centers. However, in the context of a recycling model, the limiting antigen hypothesis would clearly merit consideration.

At all times, the output flux is proportional to the input flux I_i. As long as $s_i e^{-\gamma t} > \mu$, most input cells in class i get rescued. As soon as $s_i e^{-\gamma t} < \mu$, most input cells in class i die, and, as time goes on, the output flux O_i starts decreasing exponentially at the same rate as the antigen. This behavior is illustrated in Figure for two affinity classes, class 1 being a high affinity class and class 0 a low affinity class. The output of the zero class drops exponentially from the start, while the output of class 1 cells remains roughly constant for a while and then starts dropping exponentially.

 

Note that the time interval over which $s_i \exp(-\gamma t) > \mu$ and the output flux is roughly constant increases only logarithmically with s_i. This feature has important consequences for the efficiency of this type of selection dynamics as will be discussed below. Another thing to note from Fig. is that the output per day can be on the order of 1 cell or less, which makes it clear that stochastic finite size effects should be important (for a more detailed discussion see Radmacher et al. (1998)). Therefore, the above results should be thought to represent the average output per day, and we expect a stochastic variant of our model to exhibit considerable fluctuations in these numbers. These fluctuations do not, however, alter the conclusions that we can draw from this model.

The total output N_i(t) into the memory pool at time t can be obtained by integrating O_i over time. We find

$$N_i(t) = \frac{I_i}{\gamma} \log\left[ \frac{s_i + \mu}{s_i e^{-\gamma t}+ \mu}\right].$$ (5.7)

The asymptotic outputs N_i in the limit of $t \rightarrow \infty$ are given by

$$N_i = \frac{I_i}{\gamma} \log\left[ 1 + \frac{s_i}{\mu}\right].$$ (5.8)

Again, note that the total output of a certain class i is proportional to its input, I_i, and is independent of the affinities and inputs of the other classes, showing that there is no competition between classes. Let us now consider the differential "amplification" of cells in different affinity classes as produced by equation (). Consider an affinity class i for which $s_i \ll \mu$. Most cells in this class will die, so the total output of cells in affinity class i is small. However, since $\log(1+\epsilon) \approx \epsilon$ for small $\epsilon$ the output in class i is roughly proportional to the affinity, s_i. That is, for affinity classes that have an initial rescue rate smaller than the death rate, the output is proportional to the affinity. Next, consider an affinity class j for which $s_j \gg \mu$. For this class, the output is roughly proportional to the logarithm of its initial rescue rate s_j. In short, affinity classes with rescue rates below the death rate undergo approximately affinity proportional selection, but affinity classes with affinities above the death rate undergo selection that is only proportional to the logarithms of their affinity. In this way, the affinity maturation that is achieved is largely set by the death rate $\mu$. If most classes have rescue rates above $\mu$ the selection will be very weak. The strongest selection occurs when all affinity classes have rescue rates well below the death rate, in which case selection is approximately proportional to affinity. In those cases the total outputs into the memory pool will be small since most cells die. This behavior is shown by all of the one-pass selection scenarios that I mentioned at the beginning of this section.

Let us formally determine the efficiency of the germinal center reaction. I defined this as the ratio of the average affinity $\langle s(t) \rangle_m$ of the memory pool at time t and the average affinity $\langle s \rangle$ of the cells entering the selective compartment. The latter is given by

$$\langle s \rangle = \sum_i s_i \frac{I_i}{I},$$ (5.9)

where I is the total input into the light zone per unit time. The average affinity $\langle s(t) \rangle_m$ of the memory pool at time t is given by

$$\langle s(t)\rangle_{m} = \sum_i s_i \frac{N_i(t)}{N(t)},$$ (5.10)

where N(t) is the total output into the memory pool at time t. The amplification factor due to selection, A_s(t), will then be

$$A_s(t) = \frac{\langle s(t)\rangle_{m}}{\langle s \rangle}.$$ (5.11)

The asymptotic amplification A_s is given by the limit of the above expression as $t \rightarrow \infty$. Since we know that the most stringent selection occurs when all $s_i \ll \mu,$ we can immediately derive an upper bound for the asymptotic amplification. This will also be an upper bound on the amplification at any time during the germinal center reaction. Relation could be used in specific cases, when the selection coefficients of different mutants are known, to calculate the affinity amplification at any time t of the germinal center reaction. When all $s_i \ll \mu$, we have for the output in class i

$$N_i = \frac{s_i I_i}{\gamma \mu},$$ (5.12)

from which we can easily derive that

$$A_s = 1 + \frac{{\rm var}(s)}{\langle s \rangle^2}.$$ (5.13)

This form is typical of affinity proportional selection. The amplification is roughly proportional to var(s), the variance of the input affinity distribution. In this limit case, the total output into the memory pool N, is given by

$$N = \frac{\langle s \rangle I}{\mu \gamma}.$$ (5.14)