  The immune response to a pathogen has two basic features. The first is the
expansion of a few pathogen-specific cells to form a population large enough to
control the pathogen. The second is the process of differentiation of cells
from an initial naive phenotype to an effector phenotype which controls the
pathogen, and subsequently to a memory phenotype that is maintained and
responsible for long-term protection. The expansion and the differentiation
have been considered largely independently. Changes in cell populations are
typically described using ecologically based ordinary differential equation
models. In contrast, differentiation of single cells is studied within systems
biology and is frequently modeled by considering changes in gene and protein
expression in individual cells. Recent advances in experimental systems biology
make available for the first time data to allow the coupling of population and
high dimensional expression data of immune cells during infections. Here we
describe and develop population-expression models which integrate these two
processes into systems biology on the multicellular level. When translated into
mathematical equations, these models result in non-conservative, non-local
advection-diffusion equations. We describe situations where the
population-expression approach can make correct inference from data while
previous modeling approaches based on common simplifying assumptions would
fail. We also explore how model reduction techniques can be used to build
population-expression models, minimizing the complexity of the model while
keeping the essential features of the system. While we consider problems in
immunology in this paper, we expect population-expression models to be more
broadly applicable.
