We investigate the measurement of DNA supercoiling density ($\sigma$) along chromosomes using interaction frequencies between DNA and DNA-anchored clusters of proteins. Specifically, we show how the physics of DNA supercoiling leads, in bacteria, to the quantitative modeling of binding properties of ParB proteins around their centromere-like site, {\it parS}. Using this framework, we provide an upper bound for $\sigma$ in the {\it Escherichia coli} chromosome, consistent with plasmid values, and offer a proof of concept for a high accuracy measurement. To reach these conclusions, we revisit the problem of the formation of ParB clusters. We predict, in particular, that they result from a non-equilibrium, stationary balance between an influx of produced proteins and an outflux of excess proteins, i.e., they behave like liquid-like protein condensates with unconventional ``leaky'' boundaries.

Translation is one of the main steps in the synthesis of proteins. It consists of ribosomes that translate sequences of nucleotides encoded on mRNA into polypeptide sequences of amino acids. Ribosomes bound to mRNA move unidirectionally, while unbound ribosomes diffuse in the cytoplasm. It has been hypothesized that finite diffusion of ribosomes plays an important role in ribosome recycling and that mRNA circularization enhances the efficiency of translation. In order to estimate the effect of cytoplasmic diffusion on the rate of translation, we consider a Totally Asymmetric Simple Exclusion Process (TASEP) coupled to a finite diffusive reservoir, which we call the Ribosome Transport model with Diffusion (RTD). In this model, we derive an analytical expression for the rate of protein synthesis as a function of the diffusion constant of ribosomes, which is corroborated with results from continuous-time Monte Carlo simulations. Using a wide range of biological relevant parameters, we conclude that diffusion in biological cells is fast enough so that it does not play a role in controlling the rate of translation initiation.

The cellular cytoplasm is organized into compartments. Phase separation is a simple manner to create membraneless compartments in order to confine and localize particles like proteins. In many cases, these particles are bound to fluctuating polymers like DNA or RNA. We propose a general theoretical framework for such polymer-bound particles and derive an effective 1D lattice gas model with both nearest-neighbor and emergent long-range interactions arising from looped configurations of the fluctuating polymer. We argue that 1D phase transitions exist in such systems for both Gaussian and self-avoiding polymers and, using a variational method that goes beyond mean-field theory, we obtain the complete mean occupation-temperature phase diagram. To illustrate this model, we apply it to the biologically relevant case of ParABS, a prevalent bacterial DNA segregation system.

Bacterial ParB partitioning proteins involved in chromosomes and low-copy-number plasmid segregation have recently been shown to belong to a new class of CTP-dependent molecular switches. Strikingly, CTP binding and hydrolysis was shown to induce a conformational change enabling ParB dimers to switch between an open and a closed conformation. This latter conformation clamps ParB dimers on DNA molecules, allowing their diffusion in one dimension along the DNA. It has been proposed that this novel sliding property may explain the spreading capability of ParB over more than 10-Kb from parS centromere sites where ParB is specifically loaded. Here, we modeled such a mechanism as a typical reaction-diffusion system and compared this ‘Clamping & sliding’ model to the ParB DNA binding pattern from high-resolution ChIP-sequencing data. We found that this mechanism can not account for all the in vivo characteristics, especially the long range of ParB binding to DNA. In particular, it predicts a strong effect of the presence of a roadblock on the ParB binding pattern that is not observed in ChIP-seq. Moreover, the rapid assembly kinetics observed in vivo after the duplication of parS sites is not easily explained by this mechanism. We propose that ‘Clamping & sliding’ might explain the ParB spreading pattern at short distances from parS but that another mechanism must apply for ParB recruitment at larger genomic distances.

ABout our activities on physico-mathematical modelling of intracellularmolecular transport

Activities around modelling DNA segregation and positioning in the bacterial genome