Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation

We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam, and later improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove that ||u||_2 < C L^1.5. This result is slightly weaker than that recently announced by Giacomelli and Otto, but applies in the presence of an additional linear destabilizing term. We further show that for a large class of Lyapunov functions \phi the exponent 1.5 is the best possible from this line of argument. Further, this result together with a result of Molinet gives an improved estimate for L_2 boundedness of the Kuramoto-Sivashinsky equation in thin rectangular domains in two spatial dimensions.Comment: 17 pages, 1 figure; typos corrected, references added; figure modifie

Posttranslational Regulation of the Scaffold for Fe-S Cluster Biogenesis, Isu

Isu, the scaffold protein on which Fe-S clusters are built in the mitochondrial matrix, plays a central role in the biogenesis of Fe-S cluster proteins. We report that the reduction in the activity of several components of the cluster biogenesis system, including the specialized Hsp70 Ssq1, causes a 15–20-fold up-regulation of Isu. This up-regulation results from changes at both the transcriptional and posttranslational level: an increase in ISU mRNA levels and in stability of ISU protein. Its biological importance is demonstrated by the fact that cells lacking Ssq1 grow poorly when Isu levels are prevented from rising above those found in wild-type cells. Of the biogenesis factors tested, Nfs1, the sulfur donor, was unique. Little increase in Isu levels occurred when Nfs1 was depleted. However, its presence was required for the up-regulation caused by reduction in activity of other components. Our results are consistent with the existence of a mechanism to increase the stability of Isu, and thus its level, that is dependent on the presence of the cysteine desulfurase Nfs1