  This article presents an algorithm that allows modeling of biological
networks in a qualitative framework with continuous time. Mathematical modeling
is used as a systems biology tool to answer biological questions, and more
precisely, to validate a network that describes biological observations and to
predict the effect of perturbations.
  We propose a modeling approach that is intrinsically continuous in time. The
algorithm presented here fills the gap between qualitative and quantitative
modeling. It is based on continuous time Markov process applied on a Boolean
state space. In order to describe the temporal evolution, we explicitly specify
the transition rates for each node. For that purpose, we built a language that
can be seen as a generalization of Boolean equations. The values of transition
rates have a natural interpretation: it is the inverse of the time for the
transition to occur. Mathematically, this approach can be translated in a set
of ordinary differential equations on probability distributions; therefore, it
can be seen as an approach in between quantitative and qualitative.
  We developed a C++ software, MaBoSS, that is able to simulate such a system
by applying Kinetic Monte-Carlo (or Gillespie algorithm) in the Boolean state
space. This software, parallelized and optimized, computes temporal evolution
of probability distributions and can also estimate stationary distributions.
Applications of Boolean Kinetic Monte-Carlo have been demonstrated for two
qualitative models: a toy model and a published p53/Mdm2 model. Our approach
allows to describe kinetic phenomena which were difficult to handle in the
original models. In particular, transient effects are represented by time
dependent probability distributions, interpretable in terms of cell
populations.
