  Stochastic rearrangement of germline DNA by VDJ recombination is at the
origin of immune system diversity. This process is implemented via a series of
stochastic molecular events involving gene choices and random nucleotide
insertions between, and deletions from, genes. We use large sequence
repertoires of the variable CDR3 region of human CD4+ T-cell receptor beta
chains to infer the statistical properties of these basic biochemical events.
Since any given CDR3 sequence can be produced in multiple ways, the probability
distribution of hidden recombination events cannot be inferred directly from
the observed sequences; we therefore develop a maximum likelihood inference
method to achieve this end. To separate the properties of the molecular
rearrangement mechanism from the effects of selection, we focus on
non-productive CDR3 sequences in T-cell DNA. We infer the joint distribution of
the various generative events that occur when a new T-cell receptor gene is
created. We find a rich picture of correlation (and absence thereof), providing
insight into the molecular mechanisms involved. The generative event statistics
are consistent between individuals, suggesting a universal biochemical process.
Our distribution predicts the generation probability of any specific CDR3
sequence by the primitive recombination process, allowing us to quantify the
potential diversity of the T-cell repertoire and to understand why some
sequences are shared between individuals. We argue that the use of formal
statistical inference methods, of the kind presented in this paper, will be
essential for quantitative understanding of the generation and evolution of
diversity in the adaptive immune system.
