Optimality of mutation and selection in germinal centers

Abstract

The population dynamics theory of B cells in a typical germinal center could play an important role in revealing how affinity maturation is achieved. However, the existing models encountered some conflicts with experiments. To resolve these conflicts, we present a coarse-grained model to calculate the B cell population development in affinity maturation, which allows a comprehensive analysis of its parameter space to look for optimal values of mutation rate, selection strength, and initial antibody-antigen binding level that maximize the affinity improvement. With these optimized parameters, the model is compatible with the experimental observations such as the approximately 100-fold affinity improvements, the number of mutations, the hypermutation rate, and the "all or none" phenomenon. Moreover, we study the reasons behind the optimal parameters. The optimal mutation rate, in agreement with the hypermutation rate in vivo, results from a tradeoff between accumulating enough beneficial mutations and avoiding too many deleterious or lethal mutations. The optimal selection strength evolves as a balance between the need for affinity improvement and the requirement to pass the population bottleneck. These findings point to the conclusion that germinal centers have been optimized by evolution to generate strong affinity antibodies effectively and rapidly. In addition, we study the enhancement of affinity improvement due to B cell migration between germinal centers. These results could enhance our understanding of the functions of germinal centers.

Figure 1. Histogram of affinity improvement upon single mutations derived from the PINT database.

The silent or lethal mutation is not included in the figure. Here the bin size is h = 0.5 kcal/mol (equivalent to 2.3-fold change in Ka), and the unit of W is . Only 4.9% of the affinity affecting mutations could improve affinity.

Figure 2. Total population as a function of time for various initial population sizes, starting from germline (initial) Ig-antigen binding level or , with and .

Solid line: exact analytical result for infinite population size, Eq. (14). Dotted lines: numerical results for initial population sizes (green), (yellow), and (red) respectively. The population in red goes extinct at the bottleneck.

Figure 3. The improvement of affinity for the whole spleen including many GCs in the limit of rapid B cell migration between GCs.

The improvement of A (sum of Ka over all B cells) is shown in color code, where A(0) is the initial value. AM is assumed to terminate (a) in 14 days or when B cell population reaches the initial size, whichever comes first or (b) when the population recovers the initial size (B cells) after going through the bottleneck, no matter how long it takes. (a): The optimal improvement of A occurs when about 50% daughter cells are mutated at divisions. (b): The grey scale shows the probability for the whole population to survive through the bottleneck. The black region at high mutation rate indicates lethal mutagenesis where there are too many lethal mutations for B cell population to increase.

Figure 4. The improvement of affinity for an isolated GC, i.e., in the limit of no B cell migration between GCs.

The improvement of A (sum of Ka) is shown in the same color code as in Figure 3a, assuming AM is terminated when the population recovers the initial size (3000 B cells). The optimal improvement of affinity occurs when about 60% daughter cells are mutated at divisions, and takes 16 days. The grey scale shows the probability for a GC to survive through the bottleneck.

Figure 5. The distribution of F, fraction of strong affinity B cells, for many GCs which follow similar development patterns but each starts from a random time, is consistent with the “all or none” phenomenon.

Every GC has initial binding level , 50% mutated daughter cells, and selection strength . The calculation is terminated when the population in the GC recovers the initial size (3000 B cells), and F reaches 85% at the termination moment.