Trade-offs in antibody repertoires to complex antigens

Pathogens vary in their antigenic complexity. While some pathogens such as measles present a few relatively invariant targets to the immune system, others such as malaria display considerable antigenic diversity. How the immune response copes in the presence of multiple antigens, and whether a trade-off exists between the breadth and efficacy of antibody (Ab)-mediated immune responses, are unsolved problems. We present a theoretical model of affinity maturation of B-cell receptors (BCRs) during a primary infection and examine how variation in the number of accessible antigenic sites alters the Ab repertoire. Naive B cells with randomly generated receptor sequences initiate the germinal centre (GC) reaction. The binding affinity of a BCR to an antigen is quantified via a genotype–phenotype map, based on a random energy landscape, that combines local and distant interactions between residues. In the presence of numerous antigens or epitopes, B-cell clones with different specificities compete for stimulation during rounds of mutation within GCs. We find that the availability of many epitopes reduces the affinity and relative breadth of the Ab repertoire. Despite the stochasticity of somatic hypermutation, patterns of immunodominance are strongly shaped by chance selection of naive B cells with specificities for particular epitopes. Our model provides a mechanistic basis for the diversity of Ab repertoires and the evolutionary advantage of antigenically complex pathogens.


Supplemental Material
Detailed Methods Calculation of GC dissolution threshold, F T As the GC reaction proceeds, epitope masking by antibodies lowers the mean effective affinities of B cells. When no B cells have effective affinity above a threshold, F T , the GC reaction ends. We set F T = 10 4.125 . This is determined by competition of antibodies with affinity much above that of the B cell receptor as follows: where α governs the competition between the masking antibody and the B cell receptor; and F Abj is the affinity of the masking antibody. The masking antibody is chosen by maximizing over all plasma cells i, and the affinity is normalized over all epitopes k:

Simulation Description
This purpose of this section is to provide a description of the model of B-cell affinity maturation, with sufficient detail to implement the simulation in code. Context and justification for the steps can be found in in the Methods. Although the software supports multiple infections, for simplicity this description is limited to a single course of infection in a naive host. At the beginning of the simulation, an antigen is initialized with q epitopes. N G,0 germinal centers are seeded with founder cells. Affinity maturation proceeds through R rounds of growth and selection. At the end of each round, memory and plasma cells are copied and exported from the cells.

GC Creation
1. Choose a random epitope j.

Calculate the energy threshold
where Φ is the cumulative distribution function of the standard normal distribution. The probability that a randomly generated cell will have binding energy lower than U f to a randomly generated epitope is a priori equal to f .

Repeat:
(a) Create a cell with sequence x drawn uniformly randomly from all possible sequences of length L with alphabet size A. (b) Calculate the binding energy U j (x) of the cell to the epitope (Energy Computation). (c) If U j (x) < U f , then choose this cell to seed the GC and exit the loop. 4. Initialize a list of cells for the GC with the seed cell.

Affinity Maturation
Affinity maturation in round r proceeds as follows for GC g:

Energy Computation
The energy U j (x) is equal to the sum of energies at each site: The individual energies U ij are normally distributed via a pseudo-random mapping from the energy seed s j , the sequence position i, and the neighbor sequence ν i (x), using the following computation: 1. Form a byte sequence b ij := s j |i|ν i (x), using four bytes for s j , two bytes for i, and K + 1 bytes for the neighbor sequence ν i (x), one for each amino acid.
3. Set u = (U + 1)/2 32 , v = (V + 1)/2 32 , where U is the first four bytes of sha1 ij and V is the next four bytes, and thus u, v are to a discrete approximation uniformly distributed on (0, 1]. 4. Set U ij := √ −2 ln u cos(2πv), so that U ij is normally distributed according to the Box-Muller transform [2].

Affinity Computation
The intrinsic affinity F j (x) of a cell with sequence x to epitope j is equal to The effective affinity E j (x) is equal to where α governs the competition between the masking antibody and the B cell receptor, and F Abj is the affinity of the masking antibody. The masking antibody is chosen by maximizing over all plasma cells i, and the affinity is normalized over all epitopes k:

Cell Export
Memory and plasma cells are chosen for export from GC g in round r as follows: 1. For each cell i in GC g: , add a copy of cell i to the list of memory cells.
with probability p add a copy of cell i to the list of memory cells with initial concentration C.

Concentration
The concentration C i (t) of plasma cell i at time t is given by where t i is the time at which the plasma cell was exported and δ is the decay of the antibody concentration, which we assume is 0 during a primary infection.  Figure S1. Effective affinity. The effective affinity function increases sharply as the affinity of the B cell receptor, F , approaches the affinity of the masking antibody, F Ab . The relative effectiveness of the masking antibody is controlled by the parameter α.   Figure S12. Occurrence of cross-reactive antibodies. The frequency of memory (A) or plasma (B) B cells that target one or more epitopes. Targeting is determined by affinity greater than 10 6.5 , which is the threshold for plasma cells to be exported from the GC.