A new class of methods for the solution of stiff initial value problems is introduced that is parallel by design. It has a two-step character and propagates s different "peer" solution variables with essentially identical characteristics from step to step. The main work lies in the solution of s independent linear stage equations which may be solved in parallel. Convergence of order s-1 and stability for general stepsize sequences are proved. Conditions for order s and stronger stability criteria are addressed as well. Promising methods up to order 7 are identified by numerical tests with some widely used stiff test problems. Some of these are competitive with existing software even in sequential computations.

DnaA is a conserved key regulator of replication initiation in bacteria, and is homologous to ORC proteins in archaea and in eukaryotic cells. The ATPase binds to several high affinity binding sites at the origin region and upon an unknown molecular trigger, spreads to several adjacent sites, inducing the formation of a helical super structure leading to initiation of replication. Using FRAP analysis of a functional YFP-DnaA allele in Bacillus subtilis, we show that DnaA is bound to oriC with a half-time of 2.5 seconds. DnaA shows similarly high turnover at the replication machinery, where DnaA is bound to DNA polymerase via YabA. The absence of YabA increases the half time binding of DnaA at oriC, showing that YabA plays a dual role in the regulation of DnaA, as a tether at the replication forks, and as a chaser at origin regions. Likewise, a deletion of soj (encoding a ParA protein) leads to an increase in residence time and to overinitiation, while a mutation in DnaA that leads to lowered initiation frequency, due to a reduced ATPase activity, shows a decreased residence time on binding sites. Finally, our single molecule tracking experiments show that DnaA rapidly moves between chromosomal binding sites, and does not arrest for more than few hundreds of milliseconds. In Escherichia coli, DnaA also shows low residence times in the range of 200 ms and oscillates between spatially opposite chromosome regions in a time frame of one to two seconds, independently of ongoing transcription. Thus, DnaA shows extremely rapid binding turnover on the chromosome including oriC regions in two bacterial species, which is influenced by Soj and YabA proteins in B. subtilis, and is crucial for balanced initiation control, likely preventing fatal premature multimerization and strand opening of DnaA at oriC. Initiation of replication is a key event in the cell cycle of all living cells, and is mediated by the ATPase DnaA in bacteria, and by ORC proteins in eukaryotic cells. DnaA binds to several high affinity binding sites at the origin region of replication (oriC) on the bacterial chromosome, triggers the unwinding of the DNA duplex nearby, and additionally supports loading of the DNA helicase, which in turn leads to the establishment of the DNA replication machinery. How the binding of DnaA to oriC and the triggering of duplex opening are regulated is under extensive investigation. Using two different fluorescence microscopy techniques, we show that DnaA binding and unbinding to oriC is very rapid in two bacterial species and occurs in the range of few seconds. Moreover, DnaA binds to several additional sites on the chromosome, but with an even shorter binding half-time than at oriC: average residence time throughout the chromosome is about 200 ms, as determined by single molecule microscopy. In the absence of two negative regulators, YabA and Soj, DnaA in Bacillus subtilis binds longer to oriC and to other sites on the chromosome, accompanied by a higher frequency of initiation per cell cycle, whereas the expression of a DnaA mutant protein that shows even faster exchange rates results in decreased initiation frequency. Our data reveal that DnaA exchanges rapidly at oriC, and that tight regulation of turnover is important for proper initiation control. We also show that YabA has a dual role, a) in tethering DnaA to the replication machinery and restricting its mobility within the cell and b) in increasing DnaA turnover at oriC, both of which activities reduce the risk of reinitiation during later stages in the cell cycle.