## 5 The Specific Case of DnaA in the _E. coli_ Cell Cycle
### DARSs  
DARS1 and DARS2 [52], in the chromosome are directly responsible for part of the conversion of ADP-DnaA into ATP-DnaA needed for initiation. DARS1 (located between _bioD_ and _uvrB_) and DARS2 (located between _ygpD_ and _mutH_) contain three similarly organised DnaA-binding sites and, in the case of assembly of DnaA proteins into homomultimers on DARS1, the resultant interactions between them reduce the affinity of DnaA for ADP. Since the reactivation of DnaA could be observed _in vitro_ with a fragment DARS1, supercoiling is not needed. This excludes one way in which DARSs could form part of an intensity sensing, non-equilibrium hyperstructure. There may, however, be something else related to a quantity sensing, equilibrium hyperstructure. To follow the reasoning, a reminder about ion condensation may be useful here. The condensation of positive ions such as potassium, magnesium, calcium and polyamines onto negatively charged linear polymers (such as DNA, RNA and certain protein filaments) and onto anionic membrane domains leads to these counterions being delocalised and diffusing in the near region in intimate contact with the polymers and membranes [121, 122, 123]. Ion condensation onto linear polymers occurs when the dimensionless charge parameter, \(\xi\), reaches a critical value where \(\xi=\beta\)e/(4\(\pi\varepsilon_{0}\)ekT), \(\beta\) is the linear charge density (defined as \(\beta=\) z\({}_{\rm p}\)e/b with z\({}_{\rm p}\) the valence of the charges on the polymer, b the distance in nm between these fixed charges and e the elementary charge), \(\varepsilon_{0}\) is the permittivity of vacuum, \(\varepsilon\) is the relative dielectric constant of the medium, k the Boltzmann constant and T the absolute temperature. In water at 25 \({}^{\circ}\)C, \(\xi=0.71\)/b for monovalent fixed charges on the polymer. When \(\xi\) exceeds a critical value of 1/z, counterions of valence z condense onto the polymer. As counterions condense onto such polymers they lower the linear charge density. Counterions condense onto the polymer in the order of their valence, for example, first all the tetravalent ions condense (providing the charge parameter remains high enough), then the trivalent, then the divalent and finally the monovalent ions condense. The dielectric constant of water, on which the charge parameter depends, must vary with water structure, which remains an important area of uncertainty. Ion condensation onto charged polymers, such as DNA or protein filaments, is related to temperature so that it might occur at 30 \({}^{\circ}\)C but not at 0 \({}^{\circ}\)C [7, 8]. It may therefore be significant that a PCR-generated fragment containing DARS1 promoted dissociation _in vitro_ of ADP-DnaA at 30 \({}^{\circ}\)C but not at 0 \({}^{\circ}\)C [52]; the binding of Dna itself to DARS1 and DARS2 is not affected by these temperatures [52]. As mentioned above, such condensation is related to both the charge of the counterion and the density of charge along the polymer: at the critical charge parameter of the polymer, for example, all the quadrivalent ions such as ATP\({}^{4-}\) would condense onto the polymer before the trivalent ones such as ADP\({}^{3-}\); moreover, the linear structure of the polymer is important. It may therefore also be significant that ATP can recharge DARS-bound DnaA and that, for this to happen, several copies of DnaA must bind to the DARS [52]; this might be explained by the condensation of ATP\({}^{4-}\) onto a DARS-bound DnaA structure. In the Dualism approach, it is speculated that quantity sensing may take the form of positively charged counterions condensing onto equilibrium hyperstructures until the threshold is attained at which counterions decondense from the origin region and the consequent repulsion of the phosphates leads to strand separation and initiation of replication [28, 124, 125].