Asymptotic behaviour and boundedness of linear systems with time varying coefficients

Motivated by developments in spacecraft dynamics, the asymptotic behaviour and boundedness of solution of a special class of time varying systems in which each term appears as the sum of a constant and a time varying part, are analysed in this paper. It is not possible to apply standard textbook results to such systems, which are originally in second order. Some of the existing results are reformulated. Four theorems which explore the relations between the asymptotic behaviour/boundedness of the constant coefficient system, obtained by equating the time varying terms to zero, to the corresponding behaviour of the time varying system, are developed. The results show the behaviour of the two systems to be intimately related, provided the solutions of the constant coefficient system approach zero are bounded for large values of time, and the time varying terms are suitably restrained. Two problems are tackled using these theorems.

Evolution of bacterial transcription factors: how proteins take on new tasks, but do not always stop doing the old ones

Many bacterial transcription factors do not behave as per the textbook operon model. We draw on whole genome work, as well as reported diversity across different bacteria, to argue that transcription factors may have evolved from nucleoid-associated proteins. This view would explain a large amount of recent data gleaned from high-throughput sequencing and bioinformatic analyses.