Mathematical approaches to differentiation and gene regulation

We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. Once such a graph has been established, there remains the difficult task to decide which dynamical properties of the gene network can be inferred from it, in the absence of precise quantitative data about their regulation. In this paper we discuss a rule proposed by R.Thomas according to which the possibility for the network to have several stationary states implies the existence of a positive circuit in the corresponding interaction graph. We prove that, when properly formulated in rigorous terms, this rule becomes a theorem valid for several different types of formal models of gene networks. This result is already known for models of differential or boolean type. We show here that a stronger version of it holds in the differential setup when the decay of protein concentrations is taken into account. This allows us to verify also the validity of Thomas' rule in the context of piecewise-linear models and the corresponding discrete models. We discuss open problems as well.Comment: To appear in Notes Comptes-Rendus Acad. Sc. Paris, Biologi

On the multistationarity of chemical reaction networks

We present a new conjecture about a necessary condition that a (bio)chemical network has to satisfy for it to exhibit multistationarity. According to a Theorem of Feliu and Wiuf [27, 12], the conjecture is known for strictly monotonic kinetics. We give several examples illustrating our conjecture

Upper bounds for regularized determinants

Let $E$ be a holomorphic vector bundle on a compact K\"ahler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $\Delta$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one defines its regularized determinant $det'(\Delta)$. We conjectured elsewhere that, when $h$ varies, this determinant $det'(\Delta)$ remains bounded from above. In this paper we prove this in two special cases. The first case is when $X$ is a Riemann surface, $E$ is a line bundle and $dim(H^0 (X,E)) + dim(H^1 (X,E)) \leq 2$, and the second case is when $X$ is the projective line, $E$ is a line bundle, and all metrics under consideration are invariant under rotation around a fixed axis.Comment: 22 pages, plain Te

The Prosecution’s Choice: Admitting a Non-testifying Domestic Violence Victim’s Statements Under Crawford v. Washington

This Comment explores these available options in light of Crawford\u27s holding and reasoning. In Part II this Comment provides an overview of the background of the Confrontation Clause from its development as a constitutional amendment through its application in Ohio v. Roberts and the admissibility of hearsay. Part III examines Crawford and its holding. Part IV discusses the policy concerns leading to the use of police reports and officer testimony in the prosecution of domestic violence cases and the problems that arise in the prosecution of domestic violence cases. Part V examines excited utterance in Texas, the Texas Court of Criminal Appeals\u27s recent opinion defining testimonial, and how Texas courts may apply this definition in domestic violence cases. Finally, Part VI discusses the applicability of the forfeiture-by-wrongdoing doctrine