Abstract. We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction-diffusion problems. We first investigate the physical origin of universality in these systems, before comparing RG methods to other available analytic techniques, including exact solutions and Smoluchowski-type approximations. Starting from the microscopic reaction-diffusion master equation, we then pedagogically detail the mapping to a field theory for the single-species reaction kA → ℓA (ℓ < k). We employ this particularly simple but non-trivial system to introduce the field-theoretic RG tools, including the diagrammatic perturbation expansion, renormalization, and Callan-Symanzik RG flow equation. We demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative ǫ = d c − d expansions with respect to the upper critical dimension d c . With these basics established, we then provide an overview of more sophisticated applications to multiple species reactions, disorder effects, Lévy flights, persistence problems, and the influence of spatial boundaries. We also analyze field-theoretic approaches to nonequilibrium phase transitions separating active from absorbing states. We focus particularly on the generic directed percolation universality class, as well as on the most prominent exception to this class: evenoffspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.

The conserved Polycomb repressive complex 2 (PRC2) generates trimethylation of histone 3 lysine 27 (H3K27me3), a modification associated with stable epigenetic silencing. Much is known about PRC2-induced silencing but key questions remain concerning its nucleation and stability. Vernalization, the perception and memory of winter in plants, is a classic epigenetic process that, in Arabidopsis, involves PRC2-based silencing of the floral repressor FLC. The slow dynamics of vernalization, taking place over weeks in the cold, generate a level of stable silencing of FLC in the subsequent warm that depends quantitatively on the length of the prior cold. These features make vernalization an ideal experimental system to investigate both the maintenance of epigenetic states and the switching between them. Here, using mathematical modelling, chromatin immunoprecipitation and an FLC:GUS reporter assay, we show that the quantitative nature of vernalization is generated by H3K27me3-mediated FLC silencing in the warm in a subpopulation of cells whose number depends on the length of the prior cold. During the cold, H3K27me3 levels progressively increase at a tightly localized nucleation region within FLC. At the end of the cold, numerical simulations predict that such a nucleation region is capable of switching the bistable epigenetic state of an individual locus, with the probability of overall FLC coverage by silencing H3K27me3 marks depending on the length of cold exposure. Thus, the model predicts a bistable pattern of FLC gene expression in individual cells, a prediction we verify using the FLC:GUS reporter system. Our proposed switching mechanism, involving the local nucleation of an opposing histone modification, is likely to be widely relevant in epigenetic reprogramming.

In prokaryotes, DNA can be segregated by three different types of cytoskeletal filaments. The best-understood type of partitioning (par) locus encodes an actin homolog called ParM, which forms dynamically unstable filaments that push plasmids apart in a process reminiscent of mitosis. However, the most common type of par locus, which is present on many plasmids and most bacterial chromosomes, encodes a P loop ATPase (ParA) that distributes plasmids equidistant from one another on the bacterial nucleoid. A third type of par locus encodes a tubulin homolog (TubZ) that forms cytoskeletal filaments that move rapidly with treadmill dynamics.

Bacterial plasmids encode partitioning (par) loci that confer stable plasmid inheritance. We showed previously that, in the presence of ParB and parC encoded by the par2 locus of plasmid pB171, ParA formed cytoskeletal-like structures that dynamically relocated over the nucleoid. Simultaneously, the par2 locus distributed plasmids regularly over the nucleoid. We show here that the dynamic ParA patterns are not simple oscillations. Rather, ParA nucleates and polymerizes in between plasmids. When a ParA assembly reaches a plasmid, the assembly reaction reverses into disassembly. Strikingly, plasmids consistently migrate behind disassembling ParA cytoskeletal structures, suggesting that ParA filaments pull plasmids by depolymerization. The perpetual cycles of ParA assembly and disassembly result in continuous relocation of plasmids, which, on time averaging, results in equidistribution of the plasmids. Mathematical modeling of ParA and plasmid dynamics support these interpretations. Mutational analysis supports a molecular mechanism in which the ParB/parC complex controls ParA filament depolymerization.cytoskeleton ͉ DNA segregation ͉ mathematical modeling ͉ ParA ParB ͉ pulling I n bacteria, it has been difficult to analyze how chromosomes are segregated. To gain insight into the problem, partitioning (par) loci encoded by plasmids have been used extensively as model systems. Type I par loci encode 3 components: a Walker Box ATPase (ParA), a DNA binding protein (ParB), and one or more cis-acting DNA regions where the proteins act (parC). The ParB proteins bind site-specifically to their cognate parC sites to form a ''partition complex.'' ParB also interacts with the cognate ParA protein and thereby functions as an adaptor between ParA and parC DNA. Thus, the parC region at which the segregation apparatus congregates is functionally equivalent of a eukaryotic centromere. Interestingly, ParA ATPases form helical structures that dynamically relocate over the nucleoid (1-6). ParA relocation but not the formation of filamentous structures depends on the presence of ParB bound to parC (1, 2, 4, 6). The presence of helical ParA structures in living cells is consistent with the ability of the proteins to polymerize in vitro (4, 6-13).Purified ParAs of Thermus thermophilus and plasmid pSM19035 both dimerize in the presence of ATP (6, 14), whereas ParA of P1 dimerizes also without nucleotide (13). The ParA-ATP dimers bind cooperatively and nonspecifically to DNA. Thus, the in vitro DNA binding activity of ParA proteins is consistent with the nucleoid association seen in vivo (1,8). In all cases investigated, ParB stimulates ParA ATPase activity, either on its own or in the presence of its cognate centromere site (6,9,11,15).We showed previously that the type I par2 locus of pB171, on average, distributes plasmids regularly over the bacterial nucleoid (7). Our observations raised the possibility that the dynamic ParA filaments generate the mechanical force that move and position plasmids within the cell.Here we analyze the re...