Dieter W. Heermann

Dynamics of polymer solutions and melts. Reptation predictions and scaling of relaxation times

The bond fluctuation model on the simple cubic lattice is studied by Monte Carlo simulations on a multitransputer array, for polymer volume fractions φ in the range 0.025≤φ≤0.500 and chain lengths N in the range 20≤N≤200. Extensive data are presented on the dynamics of monomer displacements, center‐of‐gravity displacements, and relaxation times. This study is complementary to previous work, in which the crossover scaling properties of the chain linear dimensions, structure factor, and self‐diffusion constant were tested for the same athermal model. The simulation technique takes both excluded volume interactions and entanglement constraints into account, but ignores hydrodynamic forces. Our results describe the crossover from Rouse behavior of swollen chains (τ∼N1+2ν, ν being the exponent describing the radius R of the chains, R∼Nν ) to reptation, τ∼N3. Since the excluded volume screening length is found to be smaller than the tube diameter by a factor of about 3, the rescaled times Wτ/N1+2ν decrease firs...

Spatially confined folding of chromatin in the interphase nucleus

Genome function in higher eukaryotes involves major changes in the spatial organization of the chromatin fiber. Nevertheless, our understanding of chromatin folding is remarkably limited. Polymer models have been used to describe chromatin folding. However, none of the proposed models gives a satisfactory explanation of experimental data. In particularly, they ignore that each chromosome occupies a confined space, i.e., the chromosome territory. Here, we present a polymer model that is able to describe key properties of chromatin over length scales ranging from 0.5 to 75 Mb. This random loop (RL) model assumes a self-avoiding random walk folding of the polymer backbone and defines a probability P for 2 monomers to interact, creating loops of a broad size range. Model predictions are compared with systematic measurements of chromatin folding of the q-arms of chromosomes 1 and 11. The RL model can explain our observed data and suggests that on the tens-of-megabases length scale P is small, i.e., 10–30 loops per 100 Mb. This is sufficient to enforce folding inside the confined space of a chromosome territory. On the 0.5- to 3-Mb length scale chromatin compaction differs in different subchromosomal domains. This aspect of chromatin structure is incorporated in the RL model by introducing heterogeneity along the fiber contour length due to different local looping probabilities. The RL model creates a quantitative and predictive framework for the identification of nuclear components that are responsible for chromatin–chromatin interactions and determine the 3-dimensional organization of the chromatin fiber.

Computer-Simulation Methods

In this chapter we give a condensed introduction to the basic features of computer simulation methods. We particularly review the Monte Carlo method of importance sampling and the principles of molecular dynamics simulations. Some of the systematic effects associated with computer simulations, such as finite-size effects, statistical errors and so on, are reviewed. The Ising model is introduced in order to provide a concrete illustration of the Monte Carlo method, which will be useful also in later chapters as a simple context in which to present some of the algorithms we will discuss. It is assumed that the reader has at least some prior understanding or experience of the material covered in this chapter, since it is meant mainly as a reference base for the material addressed in subsequent chapters. For a more complete description of the material presented here we recommend referring to one of the number of books that are now available [2.1–5]. The reader embarking upon computer simulations for the first time is encouraged to consult one of these.

Diffusion-driven looping provides a consistent framework for chromatin organization.

Chromatin folding inside the interphase nucleus of eukaryotic cells is done on multiple scales of length and time. Despite recent progress in understanding the folding motifs of chromatin, the higher-order structure still remains elusive. Various experimental studies reveal a tight connection between genome folding and function. Chromosomes fold into a confined subspace of the nucleus and form distinct territories. Chromatin looping seems to play a dominant role both in transcriptional regulation as well as in chromatin organization and has been assumed to be mediated by long-range interactions in many polymer models. However, it remains a crucial question which mechanisms are necessary to make two chromatin regions become co-located, i.e. have them in spatial proximity. We demonstrate that the formation of loops can be accomplished solely on the basis of diffusional motion. The probabilistic nature of temporary contacts mimics the effects of proteins, e.g. transcription factors, in the solvent. We establish testable quantitative predictions by deriving scale-independent measures for comparison to experimental data. In this Dynamic Loop (DL) model, the co-localization probability of distant elements is strongly increased compared to linear non-looping chains. The model correctly describes folding into a confined space as well as the experimentally observed cell-to-cell variation. Most importantly, at biological densities, model chromosomes occupy distinct territories showing less inter-chromosomal contacts than linear chains. Thus, dynamic diffusion-based looping, i.e. gene co-localization, provides a consistent framework for chromatin organization in eukaryotic interphase nuclei.

Chromatin folding - from biology to polymer models and back

There is rapidly growing evidence that folding of the chromatin fibre inside the interphase nucleus has an important role in the regulation of gene expression. In particular, the formation of loops mediated by the interaction between specific regulatory elements, for instance enhancers and promoters, is crucial in gene control. Biochemical studies that were based on the chromosome conformation capture (3C) technology have confirmed that eukaryotic genomes are highly looped. Insight into the underlying principles comes from polymer models that explore the properties of the chromatin fibre inside the nucleus. Recent models indicate that chromatin looping can explain various properties of interphase chromatin, including chromatin compaction and compartmentalisation of chromosomes. Entropic effects have a key role in these models. In this Commentary, we give an overview of the recent conjunction of ideas regarding chromatin looping in the fields of biology and polymer physics. Starting from simple linear polymer models, we explain how specific folding properties emerge upon introducing loops and how this explains a variety of experimental observations. We also discuss different polymer models that describe chromatin folding and compare them to experimental data. Experimentally testing the predictions of such polymer models and their subsequent improvement on the basis of measurements provides a solid framework to begin to understand how our genome is folded and how folding relates to function.

Random loop model for long polymers.

Remarkably little is known about the higher-order folding motifs of the chromatin fiber inside the cell nucleus. Folding depends among others on local gene density and transcriptional activity and plays an important role in gene regulation. Strikingly, at fiber lengths above 5 to 10 Mb the measured mean square distance between any two points on the chromatin fiber is independent of polymer length. We propose a polymer model that can explain this leveling-off by means of random looping. We derive an analytical expression for the mean square displacement between two arbitrary beads. Here the average is taken over the thermal ensemble with a fixed but random loop configuration, while quenched averaging over the ensemble of different loop configurations--which turns out to be equivalent to averaging over an ensemble of random matrices--is performed numerically. A detailed investigation of this model shows that loops on all scales are necessary to fit experimental data.

Adsorption on stepped surfaces: a Monte-Carlo simulation

Within the lattice gas model for adsorption on cubic (100) surfaces, the effect of surface steps running along lattice directions is modelled by considering adsorption on terraces L lattice spacings wide, with various types of boundary energies on the right and left edges of the terrace. Both the cases of attractive and of repulsive nearest-neighbor interaction between the adparticles are considered. While adsorption isotherms are not much affected by boundary energies in the repulsive case, a drastic influence of various choices of boundary energies is identified for the case of attractive interactions, where the system separates in phases of low and high coverage, respectively. Then adsorption will occur typically near one of the terrace boundaries, and a “domain wall” separating the high and low coverage phases will run parallel to the steps. A wetting transition is identified between a phase where the domain wall is bound to one of the steps and a phase where it is “unbound”, delocalized in the bulk of the terrace. The phase diagram found in the simulation for this wetting transition agrees with a theoretical prediction due to Abraham. In contrast, for repulsive interactions where the system orders in the c(2 × 2) structure, antiphase boundaries occur which typically run perpendicular to the steps. The generalization of these results to other models as well as the application to experiments is briefly discussed.

Localization Microscopy Reveals Expression-Dependent Parameters of Chromatin Nanostructure

A combined approach of 2D high-resolution localization light microscopy and statistical methods is presented to infer structural features and density fluctuations at the nuclear nanoscale. Hallmarks of nuclear nanostructure are found on the scale below 100 nm for both human fibroblast and HeLa cells. Mechanical measures were extracted as a quantitative tool from the histone density fluctuations inside the cell to obtain structural fluctuations on the scale of several micrometers. Results show that different mechanisms of expression of the same nuclear protein type lead to significantly different patterns on the nanoscale and to pronounced differences in the detected compressibility of chromatin. The observed fluctuations, including the experimental evidence for dynamic looping, are consistent with a recently proposed chromatin model.

A model for Escherichia coli chromosome packaging supports transcription factor-induced DNA domain formation

What physical mechanism leads to organization of a highly condensed and confined circular chromosome? Computational modeling shows that confinement-induced organization is able to overcome the chromosomes propensity to mix by the formation of topological domains. The experimentally observed high precision of separate subcellular positioning of loci (located on different chromosomal domains) in Escherichia coli naturally emerges as a result of entropic demixing of such chromosomal loops. We propose one possible mechanism for organizing these domains: regulatory control defined by the underlying E. coli gene regulatory network requires the colocalization of transcription factor genes and target genes. Investigating this assumption, we find the DNA chain to self-organize into several topologically distinguishable domains where the interplay between the entropic repulsion of chromosomal loops and their compression due to the confining geometry induces an effective nucleoid filament-type of structure. Thus, we propose that the physical structure of the chromosome is a direct result of regulatory interactions. To reproduce the observed precise ordering of the chromosome, we estimate that the domain sizes are distributed between 10 and 700 kb, in agreement with the size of topological domains identified in the context of DNA supercoiling.

Fluctuations and lack of self-averaging in the kinetics of domain growth

The fluctuations occurring when an initially disordered system is quenched at timet=0 to a state, where in equilibrium it is ordered, are studied with a scaling theory. Both the mean-sizel(t)d of thed-dimensional ordered domains and their fluctuations in size are found to increase with the same power of the time; their relative size fluctuations are independent of the total volumeLd of the system. This lack of self-averaging is tested for both the Ising model and the φ4 model on the square lattice. Both models exhibit the same lawl(t)=(Rt)x withx=1/2, although the φ4 model has “soft walls”. However, spurious results withx≷1/2 are obtained if “bad” pseudorandom numbers are used, and if the numbern of independent runs is too small (n itself should be of the order of 103). We also predict a critical singularity of the rateR∝(1−T/Tc)v(z−1/x),v being the correlation length exponent,z the dynamic exponent.Also quenches to the critical temperatureTc itself are considered, and a related lack of self-averaging in equilibrium computer simulations is pointed out for quantities sampled from thermodynamic fluctuation relations.