A model of sequence-dependent protein diffusion along DNA

The Interplay between Phase Separation and Gene-Enhancer Communication: A Theoretical Study

The phase separation occurring in a system of mutually interacting proteins that can bind on specific sites of a chromatin fiber is investigated here. This is achieved by means of extensive molecular dynamics simulations of a simple polymer model that includes regulatory proteins as interacting spherical particles. Our interest is particularly focused on the role played by phase separation in the formation of molecule aggregates that can join distant regulatory elements, such as gene promoters and enhancers, along the DNA. We find that the overall equilibrium state of the system resulting from the mutual interplay between binding molecules and chromatin can lead, under suitable conditions that depend on molecules concentration, molecule-molecule, and molecule-DNA interactions, to the formation of phase-separated molecular clusters, allowing robust contacts between regulatory sites. Vice versa, the presence of regulatory sites can promote the phase-separation process. Different dynamical regimes can generate the enhancer-promoter contact, either by cluster nucleation at binding sites or by bulk spontaneous formation of the mediating cluster to which binding sites are successively attracted. The possibility that such processes can explain experimental live-cell imaging data measuring distances between regulatory sites during time is also discussed.

Diffusion of DNA-Binding Species in the Nucleus: A Transient Anomalous Subdiffusion Model

Single-particle tracking experiments have measured escape times of DNA-binding species diffusing in living cells: CRISPR-Cas9, TetR, and LacI. The observed distribution is a truncated power law. Working backward from the experimental results, the observed distribution appears inconsistent with a Gaussian distribution of binding energies. Working forward, the observed distribution leads to transient anomalous subdiffusion, in which diffusion is anomalous at short times and normal at long times, here only mildly anomalous. Monte Carlo simulations are used to characterize the time-dependent diffusion coefficient D(t) in terms of the anomalous exponent α, the crossover time tcross, and the limits D(0) and D(∞) and to relate these quantities to the escape time distribution. The simplest interpretations identify the escape time as the actual binding time to DNA or the period of one-dimensional diffusion on DNA in the standard model combining one-dimensional and three-dimensional search, but a more complicated interpretation may be required. The model has several implications for cell biophysics. 1) The initial anomalous regime represents the search of the DNA-binding species for its target DNA sequence. 2) Non-target DNA sites have a significant effect on search kinetics. False positives in bioinformatic searches of the genome are potentially rate-determining in vivo. For simple binding, the search would be speeded if false-positive sequences were eliminated from the genome. 3) Both binding and obstruction affect diffusion. Obstruction ought to be measured directly, using as the primary probe the DNA-binding species with the binding site inactivated and eGFP as a calibration standard among laboratories and cell types. 4) Overexpression of the DNA-binding species reduces anomalous subdiffusion because the deepest binding sites are occupied and unavailable. 5) The model provides a coarse-grained phenomenological description of diffusion of a DNA-binding species, useful in larger-scale modeling of kinetics, FCS, and FRAP.

Acceleration of bursty multiprotein target search kinetics on DNA by colocalisation

Proteins are capable of locating specific targets on DNA by employing a facilitated diffusion process with intermittent 1D and 3D search steps. We here uncover the implications of colocalisation of protein production and DNA binding sites via computer simulations.

Rad4 recognition-at-a-distance: Physical basis of conformation-specific anomalous diffusion of DNA repair proteins

Since Robert Brown's first observations of random walks by pollen particles suspended in solution, the concept of diffusion has been subject to countless theoretical and experimental studies in diverse fields from finance and social sciences, to physics and biology. Diffusive transport of macromolecules in cells is intimately linked to essential cellular functions including nutrient uptake, signal transduction, gene expression, as well as DNA replication and repair. Advancement in experimental techniques has allowed precise measurements of these diffusion processes. Mathematical and physical descriptions and computer simulations have been applied to model complicated biological systems in which anomalous diffusion, in addition to simple Brownian motion, was observed. The purpose of this review is to provide an overview of the major physical models of anomalous diffusion and corresponding experimental evidence on the target search problem faced by DNA-binding proteins, with an emphasis on DNA repair proteins and the role of anomalous diffusion in DNA target recognition.

Sequence heterogeneity accelerates protein search for targets on DNA

The process of protein search for specific binding sites on DNA is fundamentally important since it marks the beginning of all major biological processes. We present a theoretical investigation that probes the role of DNA sequence symmetry, heterogeneity, and chemical composition in the protein search dynamics. Using a discrete-state stochastic approach with a first-passage events analysis, which takes into account the most relevant physical-chemical processes, a full analytical description of the search dynamics is obtained. It is found that, contrary to existing views, the protein search is generally faster on DNA with more heterogeneous sequences. In addition, the search dynamics might be affected by the chemical composition near the target site. The physical origins of these phenomena are discussed. Our results suggest that biological processes might be effectively regulated by modifying chemical composition, symmetry, and heterogeneity of a genome.

Real sequence effects on the search dynamics of transcription factors on DNA

Recent experiments show that transcription factors (TFs) indeed use the facilitated diffusion mechanism to locate their target sequences on DNA in living bacteria cells: TFs alternate between sliding motion along DNA and relocation events through the cytoplasm. From simulations and theoretical analysis we study the TF-sliding motion for a large section of the DNA-sequence of a common E. coli strain, based on the two-state TF-model with a fast-sliding search state and a recognition state enabling target detection. For the probability to detect the target before dissociating from DNA the TF-search times self-consistently depend heavily on whether or not an auxiliary operator (an accessible sequence similar to the main operator) is present in the genome section. Importantly, within our model the extent to which the interconversion rates between search and recognition states depend on the underlying nucleotide sequence is varied. A moderate dependence maximises the capability to distinguish between the main operator and similar sequences. Moreover, these auxiliary operators serve as starting points for DNA looping with the main operator, yielding a spectrum of target detection times spanning several orders of magnitude. Auxiliary operators are shown to act as funnels facilitating target detection by TFs.

Proteins searching for their target on DNA by one-dimensional diffusion: overcoming the "speed-stability" paradox

The sequence dependence of DNA-protein interactions that allows proteins to find the correct reaction site also slows down the 1D diffusion of the protein along the DNA molecule, leading to the so-called "speed-stability paradox," wherein fast diffusion along the DNA molecule is seemingly incompatible with stable targeting of the reaction site. Here, we develop diffusion-reaction models that use discrete and continuous Gaussian random 1D diffusion landscapes with or without a high-energy cut-off, and two-state models with a transition to and from a "searching" mode in which the protein diffuses rapidly without recognizing the target. We show the conditions under which such considerations lead to a predicted speed-up of the targeting process, and under which the presence of a "searching" mode in a two-state model is nearly equivalent to the existence of a high-energy cut-off in a one-state model. We also determine the conditions under which the search is either diffusion-limited or reaction-limited, and develop quantitative expressions for the rate of successful targeting as a function of the site-specific reaction rate, the roughness of the DNA-protein interaction potential, and the presence of a "searching" mode. In general, we find that a rough landscape is compatible with a fast search if the highest energy barriers can be avoided by "hopping" or by the protein transitioning to a lower-energy "searching" mode. We validate these predictions with the results of Brownian dynamics, kinetic Metropolis, and kinetic Monte Carlo simulations of the diffusion and targeting process, and apply these concepts to the case of T7 RNA polymerase searching for its target site on T7 DNA.

Computational models for large-scale simulations of facilitated diffusion

The binding of site-specific transcription factors to their genomic target sites is a key step in gene regulation. While the genome is huge, transcription factors belong to the least abundant protein classes in the cell. It is therefore fascinating how short the time frame is that they require to home in on their target sites. The underlying search mechanism is called facilitated diffusion and assumes a combination of three-dimensional diffusion in the space around the DNA combined with one-dimensional random walk on it. In this review, we present the current understanding of the facilitated diffusion mechanism and identify questions that lack a clear or detailed answer. One way to investigate these questions is through stochastic simulation and, in this manuscript, we support the idea that such simulations are able to address them. Finally, we review which biological parameters need to be included in such computational models in order to obtain a detailed representation of the actual process.

A comprehensive computational model of facilitated diffusion in prokaryotes

MOTIVATION

Gene activity is mediated by site-specific transcription factors (TFs). Their binding to defined regions in the genome determines the rate at which their target genes are transcribed.

RESULTS

We present a comprehensive computational model of the search process of TF for their genomic target site(s). The computational model considers: the DNA sequence, various TF species and the interaction of the individual molecules with the DNA or between themselves. We also demonstrate a systematic approach how to parametrize the system using available experimental data.

Single Qdot-labeled glycosylase molecules use a wedge amino acid to probe for lesions while scanning along DNA

Within the base excision repair (BER) pathway, the DNA N-glycosylases are responsible for locating and removing the majority of oxidative base damages. Endonuclease III (Nth), formamidopyrimidine DNA glycosylase (Fpg) and endonuclease VIII (Nei) are members of two glycosylase families: the helix–hairpin–helix (HhH) superfamily and the Fpg/Nei family. The search mechanisms employed by these two families of glycosylases were examined using a single molecule assay to image quantum dot (Qdot)-labeled glycosylases interacting with YOYO-1 stained λ-DNA molecules suspended between 5 µm silica beads. The HhH and Fpg/Nei families were found to have a similar diffusive search mechanism described as a continuum of motion, in keeping with rotational diffusion along the DNA molecule ranging from slow, sub-diffusive to faster, unrestricted diffusion. The search mechanism for an Fpg variant, F111A, lacking a phenylalanine wedge residue no longer displayed slow, sub-diffusive motion compared to wild type, suggesting that Fpg base interrogation may be accomplished by Phe¹¹¹ insertion.

Continuous-time random walk: crossover from anomalous regime to normal regime

We consider decoupled continuous time random walk with finite characteristic waiting time and jump length variance. We take approximate jump length probability distribution and waiting time probability distribution given by a product of power-law and exponential function. Using this waiting time probability distribution we study diffusion behaviors for all the time. Due to the finite characteristic waiting time and jump length variance the model presents normal diffusive behavior in the long-time limit. However, the model can describe anomalous behavior at the short and intermediate times. In particular, the model can describe subdiffusive, normal, and superdiffusive behaviors at the short times. Moreover, exact solution for probability distribution of the system is also investigated.

Model of gene transcription including the return of a RNA polymerase to the beginning of a transcriptional cycle

The gene transcription occurs via the RNA polymerase (RNAP) recruitment on the DNA promoter sequence, formation of a locally open DNA chain, promoter escape, steps of the RNA synthesis, and RNA and RNAP release after reading the final DNA base. Just after the end of the RNA synthesis, RNAP surrounds the closed DNA chain and may diffuse along DNA, desorb, or reach the promoter and start the RNA-synthesis cycle again. We present a generic kinetic model taking the latter steps into account and show analytically and by Monte Carlo simulations that it predicts transcriptional bursts even in the absence of explicit regulation of the transcription by master proteins.

Nonspecific DNA-protein interaction: why proteins can diffuse along DNA

Recent single molecule experiments have reported that DNA binding proteins (DNA-BPs) can diffuse along DNA. This suggests that interactions between proteins and DNA play a role during the target search even far from their specific site on DNA. Here we show by means of Monte Carlo simulations and analytical calculations that there is a counterintuitive repulsion between the two oppositely charged macromolecules at a nanometer range. For the concave shape of DNA-BPs, and for realistic protein charge densities, we find that the DNA-protein interaction free energy has a minimum at a finite surface-to-surface separation, in which proteins can easily slide. When a protein encounters its target, the free energy barrier is completely counterbalanced by the H-bond interaction, thus enabling the sequence recognition.

Modeling promoter search by E. coli RNA polymerase: one-dimensional diffusion in a sequence-dependent energy landscape

We present a biophysical model of promoter search by Escherichia coli RNA polymerase. We use an unconventional weight matrix derived from promoter strength data to extract the energy landscape common to a large set of known promoters. This exhibits a continuous strengthening of the binding energy when approaching the transcription start site from either side. During promoter search, the RNA polymerase slides along the DNA double helix (one-dimensional diffusion) after randomly binding to it. We discuss the possibility that the sliding has a sequence-dependent component, which implies that the energy landscape influences the movement with respect to speed, direction and efficiency. Based on this assumption, we relate the obtained energy landscape around the promoters to the one-dimensional diffusion of the RNA polymerase. Our analytical results suggest that the sequence-dependent random walk slows down and gets directed upon entering a region of 500 bp around the transcription start site, which significantly increases the efficiency of promoter search. These results may explain how the RNA polymerase is able to find the promoter in biologically relevant times out of a vast excess of non-target sites. Moreover, they provide evidence for a sequence-dependent component of one-dimensional diffusion.

Description of nonspecific DNA-protein interaction and facilitated diffusion with a dynamical model

We propose a dynamical model for nonspecific DNA-protein interaction, which is based on the "bead-spring" model previously developed by other groups, and investigate its properties using Brownian dynamics simulations. We show that the model successfully reproduces some of the observed properties of real systems and predictions of kinetic models. For example, sampling of the DNA sequence by the protein proceeds via a succession of three-dimensional motion in the solvent, one-dimensional sliding along the sequence, short hops between neighboring sites, and intersegmental transfers. Moreover, facilitated diffusion takes place in a certain range of values of the protein effective charge, that is, the combination of one-dimensional sliding and three-dimensional motion leads to faster DNA sampling than pure three-dimensional motion. At last, the number of base pairs visited during a sliding event is comparable to the values deduced from single-molecule experiments. We also point out and discuss some discrepancies between the predictions of this model and some recent experimental results as well as some hypotheses and predictions of kinetic models.

The role of chromatin conformations in diffusional transport of chromatin-binding proteins: Cartesian lattice simulations

In this paper, a lattice model for the diffusional transport of chromatin-binding particles in the interphase cell nucleus is proposed. Sliding effects are studied in dense networks of chromatin fibers created by three different methods: Randomly distributed, noninterconnected obstacles, a random walk chain model with an attractive step potential, and a self-avoiding random walk chain model with a hard repulsive core and attractive surroundings. By comparing a discrete and continuous version of the random walk chain model, we demonstrate that lattice discretization does not alter the diffusion of chromatin-binding particles. The influence of conformational properties of the fiber network on the particle sliding is investigated in detail while varying occupation volume, sliding probability, chain length, and persistence length. It is observed that adjacency of the monomers, the excluded volume effect incorporated in the self-avoiding random walk model, and the persistence length affect the chromatin-binding particle diffusion. It is demonstrated that sliding particles sense local chain structures. When plotting the diffusion coefficient as a function of the accessible volume for diffusing particles, the data fall onto master curves depending on the persistence length. However, once intersegment transfer is involved, chromatin-binding proteins no longer perceive local chain structures.

A biological interpretation of transient anomalous subdiffusion. I. Qualitative model

Anomalous subdiffusion has been reported for two-dimensional diffusion in the plasma membrane and three-dimensional diffusion in the nucleus and cytoplasm. If a particle diffuses in a suitable infinite hierarchy of binding sites, diffusion is well known to be anomalous at all times. But if the hierarchy is finite, diffusion is anomalous at short times and normal at long times. For a prescribed set of binding sites, Monte Carlo calculations yield the anomalous diffusion exponent and the average time over which diffusion is anomalous. If even a single binding site is present, there is a very short, almost artifactual, period of anomalous subdiffusion, but a hierarchy of binding sites extends the anomalous regime considerably. As is well known, an essential requirement for anomalous subdiffusion due to binding is that the diffusing particle cannot be in thermal equilibrium with the binding sites; an equilibrated particle diffuses normally at all times. Anomalous subdiffusion due to barriers, however, still occurs at thermal equilibrium, and anomalous subdiffusion due to a combination of binding sites and barriers is reduced but not eliminated on equilibration. This physical model is translated directly into a plausible biological model testable by single-particle tracking.

How does a protein search for the specific site on DNA: The role of disorder

Proteins can locate their specific targets on DNA up to two orders of magnitude faster than the Smoluchowski three-dimensional diffusion rate. This happens due to nonspecific adsorption of proteins to DNA and subsequent one-dimensional sliding along DNA. We call such a one-dimensional route towards the target an "antenna." We studied the role of the dispersion of nonspecific binding energies within the antenna due to a quasirandom sequence of natural DNA. A random energy profile for sliding proteins slows the searching rate for the target. We show that this slowdown is different for macroscopic and mesoscopic antennas.

Salerno's model of DNA re-analysed: could breather solitons have biological significance?

We investigate the sequence-dependent behaviour of localised excitations in a toy, nonlinear model of DNA base-pair opening originally proposed by Salerno. Specifically we ask whether "breather" solitons could play a role in the facilitated location of promoters by RNA polymerase (RNAP). In an effective potential formalism, we find excellent correlation between potential minima and Escherichia coli promoter recognition sites in the T7 bacteriophage genome. Evidence for a similar relationship between phage promoters and downstream coding regions is found and alternative reasons for links between AT richness and transcriptionally-significant sites are discussed. Consideration of the soliton energy of translocation provides a novel dynamical picture of sliding: steep potential gradients correspond to deterministic motion, while "flat" regions, corresponding to homogeneous AT or GC content, are governed by random, thermal motion. Finally we demonstrate an interesting equivalence between planar, breather solitons and the helical motion of a sliding protein "particle" about a bent DNA axis.