In this paper, we describe block matrix algorithms for the iterative solution of a large-scale linear-quadratic optimal control problem involving a parabolic partial differential equation over a finite control horizon. We consider an “all at once” discretization of the problem and formulate three iterative algorithms. The first algorithm is based on preconditioning a symmetric positive definite reduced linear system involving only the unknown control variables; however inner-outer iterations are required. The second algorithm modifies the first algorithm to avoid inner-outer iterations by introducing an auxiliary variable. It yields a symmetric indefinite system with a positive definite block preconditioner. The third algorithm is the central focus of this paper. It modifies the preconditioner in the second algorithm by a parallel-in-time preconditioner based on the parareal algorithm. Theoretical results show that the preconditioned algorithms have optimal convergence properties and parallel scalability. Numerical experiments confirm the theoretical results.

The dynamics of DNA topology during replication is still poorly understood. Bacterial plasmids are negatively supercoiled. This underwinding facilitates strand separation of the DNA duplex during replication. Leading the replisome, a DNA helicase separates the parental strands that are to be used as templates. This strand separation causes overwinding of the duplex ahead. If this overwinding persists it would eventually impede fork progression. In bacteria DNA gyrase and topoisomerase IV act ahead of the fork to keep DNA underwound. However, the processivity of the DNA helicase might overcome DNA gyrase and topoisomerase IV. It was proposed the overwinding that builds-up ahead of the fork could force it to swivel and diffuse this positive supercoiling behind the fork where topoisomerase IV would also act to maintain replicating DNA underwound. Putative intertwines of sister duplexes in the replicated region are called precatenanes. Fork swiveling and the formation of precatenanes, though, are still questioned. Here we used classical genetics and high-resolution two-dimensional agarose gel electrophoresis to examine the torsional tension of replication intermediates of three bacterial plasmids with the fork stalled at different sites before termination. The results obtained indicated that precatenanes do form as replication progresses before termination.

We systematically varied conditions of two-dimensional (2D) agarose gel electrophoresis to optimize separation of DNA topoisomers that differ either by the extent of knotting, the extent of catenation or the extent of supercoiling. To this aim we compared electrophoretic behavior of three different families of DNA topoisomers: (i) supercoiled DNA molecules, where supercoiling covered the range extending from covalently closed relaxed up to naturally supercoiled DNA molecules; (ii) postreplicative catenanes with catenation number increasing from 1 to ∼15, where both catenated rings were nicked; (iii) knotted but nicked DNA molecules with a naturally arising spectrum of knots. For better comparison, we studied topoisomer families where each member had the same total molecular mass. For knotted and supercoiled molecules, we analyzed dimeric plasmids whereas catenanes were composed of monomeric forms of the same plasmid. We observed that catenated, knotted and supercoiled families of topoisomers showed different reactions to changes of agarose concentration and voltage during electrophoresis. These differences permitted us to optimize conditions for their separation and shed light on physical characteristics of these different types of DNA topoisomers during electrophoresis.