Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Access to Ribosomal Protein Rpl25p by the Signal Recognition Particle Is Required for Efficient Cotranslational Translocation

Targeting of proteins to the endoplasmic reticulum (ER) occurs cotranslationally necessitating the interaction of the signal recognition particle (SRP) and the translocon with the ribosome. Biochemical and structural studies implicate ribosomal protein Rpl25p as a major ribosome interaction site for both these factors. Here we characterize an RPL25GFP fusion, which behaves as a dominant mutant leading to defects in co- but not posttranslational translocation in vivo. In these cells, ribosomes still interact with ER membrane and the translocon, but are defective in binding SRP. Overexpression of SRP can restore ribosome binding of SRP, but only partially rescues growth and translocation defects. Our results indicate that Rpl25p plays a critical role in the recruitment of SRP to the ribosome