  The zebrafish (\emph{Danio rerio}) is one of the model animals for study of
immunology because the dynamics in the adaptive immune system of zebrafish are
similar to that in higher animals. In this work, we built a multi-scale model
to simulate the dynamics of B cells in the primary and secondary immune
responses of zebrafish. We use this model to explain the reported correlation
between VDJ usage of B cell repertoires in individual zebrafish. We use a delay
ordinary differential equation (ODE) system to model the immune responses in
the 6-month lifespan of a zebrafish. This mean field theory gives the number of
high affinity B cells as a function of time during an infection. The sequences
of those B cells are then taken from a distribution calculated by a
"microscopic" random energy model. This generalized $NK$ model shows that
mature B cells specific to one antigen largely possess a single VDJ
recombination. The model allows first-principles calculation of the
probability, $p$, that two zebrafish responding to the same antigen will select
the same VDJ recombination. This probability $p$ increases with the B cell
population size and the B cell selection intensity. The probability $p$
decreases with the B cell hypermutation rate. The multi-scale model predicts
correlations in the immune system of the zebrafish that are highly similar to
that from experiment.
