or removed relative to the total number of turns in the relaxed molecule/plasmid, indicating the level of supercoiling. The Gibbs free energy associated with the coiling is given by the equation below The difference in Gibbs free energy between the supercoiled circular DNA and uncoiled circular DNA with N > 2000 bp is approximated by: or, 16 cal/bp. Since the linking number L of supercoiled DNA is the number of times the two strands are intertwined (and both strands remain covalently intact), L cannot change. The reference state (or parameter) L0 of a circular DNA duplex is its relaxed state. In this state, its writhe W = 0. Since L = T + W, in a relaxed state T = L. Thus, if we have a 400 bp relaxed circular DNA duplex, L ~ 40 (assuming ~10 bp per turn in B-DNA). Then T ~ 40. Positively supercoiling: T = 0, W = 0, then L = 0 T = +3, W = 0, then L = +3 T = +2, W = +1, then L = +3 Negatively supercoiling: T = 0, W = 0, then L = 0 T = -3, W = 0, then L = -3 T = -2, W = -1, then L = -3 Negative supercoils favor local unwinding of the DNA, allowing processes such as transcription, DNA replication, and recombination. Negative supercoiling is also thought to favour the transition between B-DNA and Z-DNA, and moderate the interactions of DNA binding proteins involved in gene regulation. Stochastic models Some stochastic models have been proposed to account for the effects of positive supercoiling buildup (PSB) in gene expression dynamics (e.g. in bacterial gene expression), differing in, e.g., the level of detail. In general, the detail increases when adding processes affected by and affecting supercoiling. As this addition occurs, the complexity of the model increases. For example, in two models of different complexity are proposed. In the most detailed one, events were modeled at the nucleotide level, while in the other the events were modeled at the promoter region alone, and thus required much less events to be accounted for. Examples of stochastic models that focus on the effects of PSB on a promoter’s activity can be found in: . In general, such models include a promoter, Pro, which is the region of DNA controlling transcription and, thus, whose activity/locking is affected by PSB. Also included