Jean-Charles Walter

Stochastic Self-Assembly of ParB Proteins Builds the Bacterial DNA Segregation Apparatus

Many canonical processes in molecular biology rely on the dynamic assembly of higher-order nucleoprotein complexes. In bacteria, the assembly mechanism of ParABS, the nucleoprotein super-complex that actively segregates the bacterial chromosome and many plasmids, remains elusive. We combined super-resolution microscopy, quantitative genome-wide surveys, biochemistry, and mathematical modeling to investigate the assembly of ParB at the centromere-like sequences parS. We found that nearly all ParB molecules are actively confined around parS by a network of synergistic protein-protein and protein-DNA interactions. Interrogation of the empirically determined, high-resolution ParB genomic distribution with modeling suggests that instead of binding only to specific sequences and subsequently spreading, ParB binds stochastically around parS over long distances. We propose a new model for the formation of the ParABS partition complex based on nucleation and caging: ParB forms a dynamic lattice with the DNA around parS. This assembly model and approach to characterizing large-scale, dynamic interactions between macromolecules may be generalizable to many unrelated machineries that self-assemble in superstructures.

An introduction to Monte Carlo methods

Monte Carlo simulations are methods for simulating statistical systems. The aim is to generate a representative ensemble of configurations to access thermodynamical quantities without the need to solve the system analytically or to perform an exact enumeration. The main principles of Monte Carlo simulations are ergodicity and detailed balance. The Ising model is a lattice spin system with nearest neighbor interactions that is appropriate to illustrate different examples of Monte Carlo simulations. It displays a second order phase transition between disordered (high temperature) and ordered (low temperature) phases, leading to different strategies of simulations. The Metropolis algorithm and the Glauber dynamics are efficient at high temperature. Close to the critical temperature, where the spins display long range correlations, cluster algorithms are more efficient. We introduce the rejection free (or continuous time) algorithm and describe in details an interesting alternative representation of the Ising model using graphs instead of spins with the so-called Worm algorithm. We conclude with an important discussion of the dynamical effects such as thermalization and correlation time.

Hyperscaling above the upper critical dimension

Abstract Above the upper critical dimension, the breakdown of hyperscaling is associated with dangerous irrelevant variables in the renormalization group formalism at least for systems with periodic boundary conditions. While these have been extensively studied, there have been only a few analyses of finite-size scaling with free boundary conditions. The conventional expectation there is that, in contrast to periodic geometries, finite-size scaling is Gaussian, governed by a correlation length commensurate with the lattice extent. Here, detailed numerical studies of the five-dimensional Ising model indicate that this expectation is unsupported, both at the infinite-volume critical point and at the pseudocritical point where the finite-size susceptibility peaks. Instead the evidence indicates that finite-size scaling at the pseudocritical point is similar to that in the periodic case. An analytic explanation is offered which allows hyperscaling to be extended beyond the upper critical dimension.

Unwinding Relaxation Dynamics of Polymers

The relaxation dynamics of a polymer wound around a fixed obstacle constitutes a fundamental instance of polymer with twist and torque, and it is also of relevance for DNA denaturation dynamics. We investigate it by simulations and Langevin equation analysis. The latter predicts a relaxation time scaling as a power of the polymer length times a logarithmic correction related to the equilibrium fluctuations of the winding angle. The numerical data support this result and show that at short times the winding angle decreases as a power law. This is also in agreement with the Langevin equation provided a winding-dependent friction is used, suggesting that such reduced description of the system captures the basic features of the problem.

Unwinding Dynamics of a Helically Wrapped Polymer

We study the rotational dynamics of a flexible polymer initially wrapped around a rigid rod and unwinding from it. This dynamics is of interest in several problems in biology and constitutes a fundamental instance of polymer relaxation from a state of minimal entropy. We investigate the dynamics of several quantities such as the total and local winding angles and metric quantities. The results of simulations performed in two and three dimensions, with and without self-avoidance, are explained by a theory based on scaling arguments and on a balance between frictional and entropic forces. The early stage of the dynamics is particularly rich, being characterized by three coexisting phases.

The equilibrium winding angle of a polymer around a bar

The probability distribution of the winding angle ? of a planar self-avoiding walk has been known exactly for a long time: it has a Gaussian shape with a variance growing as ?2 ~ lnL. For the three-dimensional case of a walk winding around a bar, the same scaling is suggested, based on a first-order epsilon-expansion. We tested this three-dimensional case by means of Monte Carlo simulations up to length L?25?000 and using exact enumeration data for sizes L ? 20. We find that the variance of the winding angle scales as ?2 ~ (lnL)2?, with ? = 0.75(1). The ratio ? = ?4/?22 = 3.74(5) is incompatible with the Gaussian value ? = 3, but consistent with the observation that the tail of the probability distribution function p(?) is found to decrease more slowly than a Gaussian function. These findings are at odds with the existing first-order ?-expansion results.

Fractional Brownian motion and the critical dynamics of zipping polymers

We consider two complementary polymer strands of length L attached by a common-end monomer. The two strands bind through complementary monomers and at low temperatures form a double-stranded conformation (zipping), while at high temperature they dissociate (unzipping). This is a simple model of DNA (or RNA) hairpin formation. Here we investigate the dynamics of the strands at the equilibrium critical temperature T=T(c) using Monte Carlo Rouse dynamics. We find that the dynamics is anomalous, with a characteristic time scaling as τ∼L(2.26(2)), exceeding the Rouse time ∼L(2.18). We investigate the probability distribution function, velocity autocorrelation function, survival probability, and boundary behavior of the underlying stochastic process. These quantities scale as expected from a fractional Brownian motion with a Hurst exponent H=0.44(1). We discuss similarities to and differences from unbiased polymer translocation.

Numerical investigation of the ageing of the fully frustrated XY model

We study the out-of-equilibrium dynamics of the fully frustrated XY model. At equilibrium, this model undergoes two phase transitions at two very close temperatures: a Kosterlitz–Thouless topological transition and a second-order phase transition between a paramagnetic phase and a low-temperature phase where the chiralities of the lattice plaquettes are antiferromagnetically ordered. We compute by means of Monte Carlo simulations two-time spin–spin and chirality–chirality autocorrelation and response functions. From the dynamics of the spin waves in the low-temperature phase, we extract the temperature-dependent exponent η. We provide evidence for logarithmic corrections above the Kosterlitz–Thouless temperature and interpret them as a manifestation of free topological defects. Our estimates of the autocorrelation exponent and the fluctuation-dissipation ratio differ from the XY values, while η(TKT) lies at the boundary of the error bar. Indications for logarithmic corrections at the second-order critical temperature are presented. However, the coupling between angles and chiralities is still strong and explains why the autocorrelation exponent and fluctuation-dissipation ratio are far from the Ising values and seem stable.

Logarithmic corrections in the ageing of the fully frustrated Ising model

We study the dynamics of the critical two-dimensional fully frustrated Ising model by means of Monte Carlo simulations. The dynamical exponent is estimated at equilibrium and is shown to be compatible with the value zc = 2. In a second step, the system is prepared in the paramagnetic phase and then quenched at its critical temperature Tc = 0. Numerical evidence for the existence of logarithmic corrections in the ageing regime is presented. These corrections may be related to the topological defects observed in other fully frustrated models. The autocorrelation exponent is estimated to be λ = d as for the Ising chain quenched at Tc = 0.

Rotational dynamics of entangled polymers

Some recent results on the rotational dynamics of polymers are reviewed and extended. We focus here on the relaxation of a polymer, either flexible or semiflexible, initially wrapped around a rigid rod. We also study the steady polymer rotation generated by a constant torque on the rod. The interplay of frictional and entropic forces leads to a complex dynamical behavior characterized by non-trivial universal exponents. The results are based on extensive simulations of polymers undergoing Rouse dynamics and on an analytical approach using force balance and scaling arguments. The analytical results are in general in good agreement with the simulations, showing how a simplified approach can correctly capture the complex dynamical behavior of rotating polymers.