Target search on a dynamic DNA molecule

Dynamics of chromosomal target search by a membrane-integrated one-component receptor

Membrane proteins account for about one third of the cellular proteome, but it is still unclear how dynamic they are and how they establish functional contacts with cytoplasmic interaction partners. Here, we consider a membrane-integrated one-component receptor that also acts as a transcriptional activator, and analyze how it kinetically locates its specific binding site on the genome. We focus on the case of CadC, the pH receptor of the acid stress response Cad system in E. coli. CadC is a prime example of a one-component signaling protein that directly binds to its cognate target site on the chromosome to regulate transcription. We combined fluorescence microscopy experiments, mathematical analysis, and kinetic Monte Carlo simulations to probe this target search process. Using fluorescently labeled CadC, we measured the time from activation of the receptor until successful binding to the DNA in single cells, exploiting that stable receptor-DNA complexes are visible as fluorescent spots. Our experimental data indicate that CadC is highly mobile in the membrane and finds its target by a 2D diffusion and capture mechanism. DNA mobility is constrained due to the overall chromosome organization, but a labeled DNA locus in the vicinity of the target site appears sufficiently mobile to randomly come close to the membrane. Relocation of the DNA target site to a distant position on the chromosome had almost no effect on the mean search time, which was between four and five minutes in either case. However, a mutant strain with two binding sites displayed a mean search time that was reduced by about a factor of two. This behavior is consistent with simulations of a coarse-grained lattice model for the coupled dynamics of DNA within a cell volume and proteins on its surface. The model also rationalizes the experimentally determined distribution of search times. Overall our findings reveal that DNA target search does not present a much bigger kinetic challenge for membrane-integrated proteins than for cytoplasmic proteins. More generally, diffusion and capture mechanisms may be sufficient for bacterial membrane proteins to establish functional contacts with cytoplasmic targets.

Adaptation to changing environments is vital to bacteria and is enabled by sophisticated signal transduction systems. While signal transduction by two-component systems is well studied, the signal transduction of membrane-integrated one-component systems, where one protein performs both sensing and response regulation, are insufficiently understood. How can a membrane-integrated protein bind to specific sites on the genome to regulate transcription? Here, we study the kinetics of this process, which involves both protein diffusion within the membrane and conformational fluctuations of the genomic DNA. A well-suited model system for this question is CadC, the signaling protein of the E. coli Cad system involved in pH stress response. Fluorescently labeled CadC forms visible spots in single cells upon stable DNA-binding, marking the end of the protein-DNA search process. Moreover, the start of the search is triggered by a medium shift exposing cells to pH stress. We probe the underlying mechanism by varying the number and position of DNA target sites. We combine these experiments with mathematical analysis and kinetic Monte Carlo simulations of lattice models for the search process. Our results suggest that CadC diffusion in the membrane is pivotal for this search, while the DNA target site is just mobile enough to reach the membrane.

Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces

Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of “white noise” to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α<2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation–dissipation theorem derived for weak external forcing.

Switches in a genetic regulatory system under multiplicative non-Gaussian noise

The non-Gaussian noise is multiplicatively introduced to model the universal fluctuation in the gene regulation of the bacteriophage λ. To investigate the key effect of non-Gaussian noise on the genetic on/off switch dynamics from the viewpoint of quantitative analysis, we employ the high-order perturbation expansion to deduce the stationary probability density of repressor concentration and the mean first passage time from low concentration to high concentration and vice versa. The occupation probability of different concentration states can be estimated from the height and shape of the peaks of the stationary probability density, which could be used to determine the overall expression level. A further concern is the mean first passage time, also referred to as the mean switching time, which can be adopted as an important measure to characterize the adaptability of gene expression to the environmental variation. Through our investigation, it is observed that the non-Gaussian heavy-tailed noise can better induce the switches between distinct genetic expression states and additionally, it accelerates the switching process more evidently compared to the Gaussian noise and the bounded noise.

How nonspecifically DNA-binding proteins search for the target in crowded environments

We investigate how a tracer particle searches a target located in DNA modeled by a stiff chain in crowded environments using theoretical analysis and Langevin dynamics simulations. First, we show that the three-dimensional (3D) diffusion coefficient of the tracer only depends on the density of crowders ϕ, while its one-dimensional (1D) diffusion coefficient is affected by not only ϕ but also the nonspecific binding energy ε. With increasing ϕ and ε, no obvious change in the average 3D diffusion time is observed, while the average 1D sliding time apparently increases. We propose theoretically that the 1D sliding of the tracer along the chain could be well captured by the Kramers' law of escaping rather than the Arrhenius law, which is verified directly by the simulations. Finally, the average search time increases monotonously with an increase in ϕ while it has a minimum as a function of ε, which could be understood from the different behaviors of the average number of search rounds with the increasing ϕ or ε. These results provide a deeper understanding of the role of facilitated diffusion in target search of proteins on DNA in vivo.

Transitions in a genetic transcriptional regulatory system under Lévy motion

Based on a stochastic differential equation model for a single genetic regulatory system, we examine the dynamical effects of noisy fluctuations, arising in the synthesis reaction, on the evolution of the transcription factor activator in terms of its concentration. The fluctuations are modeled by Brownian motion and α-stable Lévy motion. Two deterministic quantities, the mean first exit time (MFET) and the first escape probability (FEP), are used to analyse the transitions from the low to high concentration states. A shorter MFET or higher FEP in the low concentration region facilitates such a transition. We have observed that higher noise intensities and larger jumps of the Lévy motion shortens the MFET and thus benefits transitions. The Lévy motion activates a transition from the low concentration region to the non-adjacent high concentration region, while Brownian motion can not induce this phenomenon. There are optimal proportions of Gaussian and non-Gaussian noises, which maximise the quantities MFET and FEP for each concentration, when the total sum of noise intensities are kept constant. Because a weaker stability indicates a higher transition probability, a new geometric concept is introduced to quantify the basin stability of the low concentration region, characterised by the escaping behaviour.

Effect of DNA conformation on facilitated diffusion

Within a living cell, site-specific DNA-binding proteins need to search the whole genome to find a target of ~10-20 bp. That they find the target, and do so quickly, is vital for the correct functioning of the DNA, and of the cell as a whole. The current understanding is that this search is performed via facilitated diffusion, i.e. by combining three-dimensional bulk diffusion within the cytoplasm or nucleoplasm, with one-dimensional diffusion along the DNA backbone, to which the protein binds non-specifically. After reviewing the standard theory of facilitated diffusion, we discuss in the present article the still rather rare direct computer simulations of this process, focusing on the three-dimensional part of the search, and the effect of DNA looping and the general DNA conformation on its efficiency. We close by highlighting some open questions in this field.

Lévy noise induced switch in the gene transcriptional regulatory system

The study of random fluctuations in a gene transcriptional regulatory system is extended to the case of non-Gaussian Lévy noise, which can describe unpredictable jump changes of the random environment. The stationary probability densities are given to explore the key roles of Lévy noise in a gene transcriptional regulatory system. The results demonstrate that the parameters of Lévy noise, including noise intensity, stability index, and skewness parameter, can induce switches between distinct gene-expression states. A further concern is the switching time (from the high concentration state to the low concentration one or from the low concentration state to the high concentration one), which is a random variable and often referred to as the mean first passage time. The effects of Lévy noise on expression and degradation time are studied by computing the mean first passage time in two directions and a number of different peculiarities of non-Gaussian Lévy noise compared with Gaussian noise are observed.

Facilitated diffusion on mobile DNA: configurational traps and sequence heterogeneity

We present Brownian dynamics simulations of the facilitated diffusion of a protein, modeled as a sphere with a binding site on its surface, along DNA, modeled as a semiflexible polymer. We consider both the effect of DNA organization in three dimensions and of sequence heterogeneity. We find that in a network of DNA loops, which are thought to be present in bacterial DNA, the search process is very sensitive to the spatial location of the target within such loops. Therefore, specific genes might be repressed or promoted by changing the local topology of the genome. On the other hand, sequence heterogeneity creates traps which normally slow down facilitated diffusion. When suitably positioned, though, these traps can, surprisingly, render the search process much more efficient.