Hexagonal patterns in finite domains

In many mathematical models for pattern formation, a regular hexagonal pattern is stable in an infinite region. However, laboratory and numerical experiments are carried out in finite domains, and this imposes certain constraints on the possible patterns. In finite rectangular domains, it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational. In practice, it is found experimentally that in a rectangular region, patterns of irregular hexagons are often observed. This work analyses the geometry and dynamics of irregular hexagonal patterns. These patterns occur in two different symmetry types, either with a reflection symmetry, involving two wavenumbers, or without symmetry, involving three different wavenumbers. The relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case. It is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls. Numerical simulations of a model partial differential equation are also presented to illustrate the behaviour.

DnaG interacts with a linker region that joins the N- and C-domains of DnaB and induces the formation of 3-fold symmetric rings

Loading of the replicative ring helicase onto the origin of replication (oriC) is the final outcome of a well coordinated series of events that collectively constitute a primosomal cascade. Once the ring helicase is loaded, it recruits the primase and signals the switch to the polymerization mode. The transient nature of the helicase-primase (DnaB-DnaG) interaction in the Escherichia coli system has hindered our efforts to elucidate its structure and function. Taking advantage of the stable DnaB-DnaG complex in Bacillus stearothermophilus, we have reviewed conflicting mutagenic data from other bacterial systems and shown that DnaG interacts with the flexible linker that connects the N- and C-terminal domains of DnaB. Furthermore, atomic force microscopy (AFM) imaging experiments show that binding of the primase to the helicase induces predominantly a 3-fold symmetric morphology to the hexameric ring. Overall, three DnaG molecules appear to interact with the hexameric ring helicase but a small number of complexes with two and even one DnaG molecule bound to DnaB were also detected. The structural/functional significance of these data is discussed and a speculative structural model for this complex is suggested.