Emergence of high affinity mutants in the germinal centers

Our initial motivation for improving mutation rate estimation techniques was to estimate mutation rates in germinal centers. Compared to the simple computational model that I designed for a bacterial culture, the cell population dynamics in the germinal centers is complicated by a number of factors:

• Germinal centers have an initial phase of exponential expansion of cells, which is followed by an apparent steady-state in cell number. During this phase there is considerable cell death, as well as clonal expansion.
• Cells grow in a selective environment. Certain mutations are advantageous, and of these, some result in preferential expansion of the clone (1998). At the moment we have no quantification of the selective advantage of these clones, and we do not know by what mechanism their rapid expansion occurs.
• Deleterious mutations can also be generated, and they seem to be relatively rapidly followed by the death of the cell.

These constraints make it extremely difficult to attempt an accurate estimate of the mutation rate in germinal centers. An approach that is used experimentally in order to circumvent the selection problem, is to look at passenger genes in B cells that went through a somatic mutation process. A passenger gene is a gene that does not affect the survival probability of the cell in a particular environment. In our case, this environment is the germinal center. The claim is that the association of the passenger gene with a successful or unsuccessful phenotype, that is, with a high or low affinity immunoglobulin receptor is irrelevant. If we focus on one site (nucleotide position) of the passenger gene, the mutant distribution only depends on the relative probability of generating different tree shapes (and, of course, on the probability of mutation per cell replication, assuming that mutations are replication-dependent). This is what we found in the analysis of the L-D distribution. If the association between the passenger gene and a successful or unsuccessful phenotype were relevant, then the relative probabilities of different trees would have to be modified by this association. It is conceivable that successful mutants are selected faster and/or divide at faster rates than unsuccessful ones. Then if the cell harbors a successful mutation, its cell cycle time may have a different distribution than if the cell did not have this mutation. If this were the case, the number of generations that a cell goes through would depend on its selected receptor. Thus, it is not clear that measuring the mutation rate from passenger genes that are carried by cells that have functional, selected receptors, circumvents the selection problem.

However, looking at this experiment differently allows me to design a mutation rate estimation method based on passenger gene mutation. Let us assume that up to the point when the successful mutant appeared in the germinal center, the cells underwent exponential expansion, with cells cycle times being independent, identically-distributed random variables. The consistency of the estimate of the waiting time for a successful mutant (1998) and of the duration of the exponential expansion phase of the germinal center reaction (1991) support this hypothesis. Let us further assume, similar to Radmacher et al. (1998), that the progeny of this successful mutant will take over the germinal center.

Now consider the germinal center cells at the end of germinal center reaction. Sequencing their passenger gene and taking the intersection of the mutation sets in these genes, we should obtain the set of mutations that were present in the founder cell of the clone that stumbled upon the successful mutation. So the set of mutations in the passenger gene of this founder gives us an estimation of the number of mutations in a cell at the end of the exponential expansion phase.

I will now define the quantities that I need for estimating the mutation rate from these data:

• P(g|N) = probability that a cell in a culture of size N is of generation g. I will assume that the generation number of the cell that finds the key mutation when the culture reached size N is the average generation number in the culture at that time. To be accurate, we would have to exclude the cells in the culture whose sister cells divided already. They cannot be recently born, and the mutation can only be in a recently born cell. This density function is completely determined by the cell cycle parameters, and I can determine it by simulation.
• $P_{\mu}(N)$ = probability that the genealogical tree of the culture is of size N when the successful mutation is found. A tree of size N is generated by N-1 divisions. With a constant mutation rate per division, the probability of finding the successful mutant will only be related to the tree size, not to the tree shape. Specifically, let us assume that the successful mutant is only one mutation away from the germline. Let $\mu$ be the probability of a mutation per site per division. Let p be the probability that a mutation at the site of interest produces the appropriate nucleotide (I will assume for the moment that all nucleotides have an equal chance of being produced by a mutation). Then the probability that the mutation occurred at the N-1^st division is
  
  $$P_{\mu}(N) = (1 - \mu p)^{N-2} \mu p.$$
  
  
  Note that I neglect here the deleterious mutations that might have been generated at other sites before the successful mutant was found. We may relax this assumption, and use an effective mutation rate, which would be weighted by the probability that a mutation is lethal. Also, given the results of Radmacher et al. (1998), I would have to weight the mutation rate by the probability that the high-affinity mutation seeds the clone that takes over the germinal center.
• $P_{\mu}(g)$ = probability that the successful mutation is found in a cell of generation g.
  
  $$P_{\mu}(g) = \sum_{N = 1}^{\infty} P_{\mu}(N) P(g\vert N).$$
  
  

• $P_{\mu}(m)$ = probability of m mutations in the passenger gene. Consider one nucleotide position in the passenger gene. Its probability of mutating in one division is $\mu$. Note that here I take the probability of a mutation per site per division, not the probability of a specific substitution, as I did when I determined the probability of producing the successful mutant. $(1 - \mu)^g$ is the probability of no mutation in g generations, and $1 - (1 - \mu)^g$ is the probability of at least one mutation in g generations. Then if the passenger gene is L nucleotides long, the probability of m of them being mutated is given by
  
  $$P_{\mu}(m) = \sum_{g = 1}^{\infty} P_{\mu}(g) \left( \begin{... ...m \end{array}\right) [(1 - \mu)^g]^{L-m} [1 - (1 - \mu)^g]^m.$$
  
  

The estimation procedure would then be as follows. We take passenger sequence data from a number of cells from a number of germinal centers. For each set of sequences that comes from the same germinal center, we take the intersection of the mutation sets of individual sequences. This gives us the set of mutations present in the founder of that particular germinal center. We determine P(g|N) for our experimental system, using a reasonable cell-cycle time distribution. and germinal center size. We know the lengths of the genes, and we can generate a family of curves of mutation frequency distribution as a function of the mutation rate. We can then identify the mutation rate that gives the best fitting curve for the passenger gene mutation frequency.