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How genotype is transformed into phenotype is a fundamental question of systems biology. We will use mathematical models in an attempt to address at least a part of this question.
We use networks to model how genes regulate: the nodes are the genes and the edges indicate interactions. In turn the interactions (edges) provide structure to differential equations that govern the activation and inactivation of genes and hence the quantities of proteins in the organism.
These different levels of proteins correspond to different functions and physiological expression, i.e. phenotypes of the organism. The simplest dynamical expression to identify is that of a steady state or fixed point, and thus in our mathematical model different fixed points correspond to different phenotypes.
There are two classical approaches to this problem. First, assume an explicit differential equation and solve for the equilibria. The challenge in this setting is that for large networks finding all the equilibria is computationally expensive. To a large extent this is due to the fact that these biological systems involve many unknown parameters and the associated dynamics depends on the parameters. Second, assume that the genes have a boolean behavior, either on or off. This allows for extremely fast computations. The disadvantage is that the associated parameter space is extremely restrictive.
A new modeling approach based on combinatorics and algebraic topology combines the speed of boolean computations with the richness of parameter space associated with differential equations. The goal of this project is twofold. First, to use this new approach and the associated software DSGRN to identify fixed points (phenotypes) for complex networks and understand how robust these fixed points are, e.g. are the fixed points preserved over large regions of parameter space. Second, to relate these findings back to classical differential equation models.
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