Modulation of DNA loop lifetimes by the free energy of loop formation

Stretching multistate flexible chains and loops

Polymer loop structure commonly appears in biological phenomena, such as DNA looping and DNA denaturation. When a chain forms a loop, its elastic behavior differs from that of an open chain due to the loss of entropy. In the case of reversible loop formation, interesting behavior emerges related to the multistate nature of the conformations. In this study, we model a multistate reversible loop as a looping Gaussian chain, which can bind (close) reversibly at one or several points to form a loop, or a zipping Gaussian loop, which can zip reversibly to form a double-stranded chain. For each model, we calculate the force-extension relations in the fixed-extension (Helmholtz) and the fixed-force (Gibbs) statistical ensembles. Unlike the single Gaussian chain or loop, the multilevel systems demonstrate qualitatively distinct tensile elasticity and ensemble inequivalence. In addition, we investigate a Gaussian necklace consisting of reversible alternating blocks of the zipped chain and loop and obtain the force-temperature phase diagram. The phase diagram implies a force-induced phase transition from a completely looped (denatured) state to a mixed (bound) state.

Optimizing the random search of a finite-lived target by a Lévy flight

In many random search processes of interest in chemistry, biology, or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk, or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a nontrivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a nonconventional tricritical point. These results pave the way to the optimization of search processes under time constraints.

Robustness of DNA looping across multiple cell divisions in individual bacteria

DNA looping has emerged as a central paradigm of transcriptional regulation, as it is shared across many living systems. One core property of DNA looping-based regulation is its ability to greatly enhance repression or activation of genes with only a few copies of transcriptional regulators. However, this property based on a small number of proteins raises the question of the robustness of such a mechanism with respect to the large intracellular perturbations taking place during growth and division of the cell. Here we address the issue of sensitivity to variations of intracellular parameters of gene regulation by DNA looping. We use the lac system as a prototype to experimentally identify the key features of the robustness of DNA looping in growing Escherichia coli cells. Surprisingly, we observe time intervals of tight repression spanning across division events, which can sometimes exceed 10 generations. Remarkably, the distribution of such long time intervals exhibits memoryless statistics that is mostly insensitive to repressor concentration, cell division events, and the number of distinct loops accessible to the system. By contrast, gene regulation becomes highly sensitive to these perturbations when DNA looping is absent. Using stochastic simulations, we propose that the observed robustness to division emerges from the competition between fast, multiple rebinding events of repressors and slow initiation rate of the RNA polymerase. We argue that fast rebinding events are a direct consequence of DNA looping that ensures robust gene repression across a range of intracellular perturbations.

Coarse-grained modeling reveals the impact of supercoiling and loop length in DNA looping kinetics

Measurements of protein-mediated DNA looping reveal that in vivo conditions favor the formation of loops shorter than those that occur in vitro, yet the precise physical mechanisms underlying this shift remain unclear. To understand the extent to which in vivo supercoiling may explain these shifts, we develop a theoretical model based on coarse-grained molecular simulation and analytical transition state theory, enabling us to map out looping energetics and kinetics as a function of two key biophysical parameters: superhelical density and loop length. We show that loops on the scale of a persistence length respond to supercoiling over a much wider range of superhelical densities and to a larger extent than longer loops. This effect arises from a tendency for loops to be centered on the plectonemic end region, which bends progressively more tightly with superhelical density. This trend reveals a mechanism by which supercoiling favors shorter loop lengths. In addition, our model predicts a complex kinetic response to supercoiling for a given loop length, governed by a competition between an enhanced rate of looping due to torsional buckling and a reduction in looping rate due to chain straightening as the plectoneme tightens at higher superhelical densities. Together, these effects lead to a flattening of the kinetic response to supercoiling within the physiological range for all but the shortest loops. Using experimental estimates for in vivo superhelical densities, we discuss our model's ability to explain available looping data, highlighting both the importance of supercoiling as a regulatory force in genetics and the additional complexities of looping phenomena in vivo.

Determinants of cyclization-decyclization kinetics of short DNA with sticky ends

Cyclization of DNA with sticky ends is commonly used to measure DNA bendability as a function of length and sequence, but how its kinetics depend on the rotational positioning of the sticky ends around the helical axis is less clear. Here, we measured cyclization (looping) and decyclization (unlooping) rates (k_loop and k_unloop) of DNA with sticky ends over three helical periods (100-130 bp) using single-molecule fluorescence resonance energy transfer (FRET). k_loop showed a nontrivial undulation as a function of DNA length whereas k_unloop showed a clear oscillation with a period close to the helical turn of DNA (∼10.5 bp). The oscillation of k_unloop was almost completely suppressed in the presence of gaps around the sticky ends. We explain these findings by modeling double-helical DNA as a twisted wormlike chain with a finite width, intrinsic curvature, and stacking interaction between the end base pairs. We also discuss technical issues for converting the FRET-based cyclization/decyclization rates to an equilibrium quantity known as the J factor that is widely used to characterize DNA bending mechanics.

First Hitting Times to Intermittent Targets

In noisy environments such as the cell, many processes involve target sites that are often hidden or inactive, and thus not always available for reaction with diffusing entities. To understand reaction kinetics in these situations, we study the first hitting time statistics of a one-dimensional Brownian particle searching for a target site that switches stochastically between visible and hidden phases. At high crypticity, an unexpected rate limited power-law regime emerges for the first hitting time density, which markedly differs from the classic t^{-3/2} scaling for steady targets. Our problem admits an asymptotic mapping onto a mixed, or Robin, boundary condition. Similar results are obtained with non-Markov targets and particles diffusing anomalously.

Twist-bend coupling, twist waves, and the shape of DNA loops

By combining analytical and numerical calculations, we investigate the minimal-energy shape of short DNA loops of approximately 100 base pairs (bp). We show that in these loops the excess twist density oscillates as a response to an imposed bending stress, as recently found in DNA minicircles and observed in nucleosomal DNA. These twist oscillations, here referred to as twist waves, are due to the coupling between twist and bending deformations, which in turn originates from the asymmetry between DNA major and minor grooves. We introduce a simple analytical variational shape that reproduces the exact loop energy up to the fourth significant digit and is in very good agreement with shapes obtained from coarse-grained simulations. We, finally, analyze the loop dynamics at room temperature, and show that the twist waves are robust against thermal fluctuations. They perform a normal diffusive motion, whose origin is briefly discussed.

Identifying Physical Causes of Apparent Enhanced Cyclization of Short DNA Molecules with a Coarse-Grained Model

DNA cyclization is a powerful technique to gain insight into the nature of DNA bending. While the wormlike chain model provides a good description of small to moderate bending fluctuations, it is expected to break down for large bending. Recent cyclization experiments on strongly bent shorter molecules indeed suggest enhanced flexibility over and above that expected from the wormlike chain. Here, we use a coarse-grained model of DNA to investigate the subtle thermodynamics of DNA cyclization for molecules ranging from 30 to 210 base pairs. As the molecules get shorter, we find increasing deviations between our computed equilibrium j-factor and the classic wormlike chain predictions of Shimada and Yamakawa for a torsionally aligned looped molecule. These deviations are due to sharp kinking, first at nicks, and only subsequently in the body of the duplex. At the shortest lengths, substantial fraying at the ends of duplex domains is the dominant method of relaxation. We also estimate the dynamic j-factor measured in recent FRET experiments. We find that the dynamic j-factor is systematically larger than its equilibrium counterpart—with the deviation larger for shorter molecules—because not all the stress present in the fully cyclized state is present in the transition state. These observations are important for the interpretation of recent cyclization experiments, suggesting that measured anomalously high j-factors may not necessarily indicate non-WLC behavior in the body of duplexes.

Facilitation of DNA loop formation by protein-DNA non-specific interactions

Complex DNA topological structures, including polymer loops, are frequently observed in biological processes when protein molecules simultaneously bind to several distant sites on DNA. However, the molecular mechanisms of formation of these systems remain not well understood. Existing theoretical studies focus only on specific interactions between protein and DNA molecules at target sequences. However, the electrostatic origin of primary protein-DNA interactions suggests that interactions of proteins with all DNA segments should be considered. Here we theoretically investigate the role of non-specific interactions between protein and DNA molecules on the dynamics of loop formation. Our approach is based on analyzing a discrete-state stochastic model via a method of first-passage probabilities supplemented by Monte Carlo computer simulations. It is found that depending on a protein sliding length during the non-specific binding event three different dynamic regimes of the DNA loop formation might be observed. In addition, the loop formation time might be optimized by varying the protein sliding length, the size of the DNA molecule, and the position of the specific target sequences on DNA. Our results demonstrate the importance of non-specific protein-DNA interactions in the dynamics of DNA loop formations.

Base-Pair Mismatch Can Destabilize Small DNA Loops through Cooperative Kinking

Base-pair mismatch can relieve mechanical stress in highly strained DNA molecules, but how it affects their kinetic stability is not known. Using single-molecule fluorescence resonance energy transfer, we measured the lifetimes of tightly bent DNA loops with and without base-pair mismatch. Surprisingly, for loops captured by stackable sticky ends which leave single-stranded DNA breaks (or nicks) upon annealing, the mismatch decreased the loop lifetime despite reducing the overall bending stress, and the decrease was largest when the mismatch was placed at the DNA midpoint. These findings suggest that base-pair mismatch increases bending stress at the opposite side of the loop through an allosteric mechanism known as cooperative kinking. Based on this mechanism, we present a three-state model that explains the apparent dichotomy between thermodynamic and kinetic stability.

Comparison between Effects of Retroactivity and Resource Competition upon Change in Downstream Reporter Genes of Synthetic Genetic Circuits

Reporter genes have contributed to advancements in molecular biology. Binding of an upstream regulatory protein to a downstream reporter promoter allows quantification of the activity of the upstream protein produced from the corresponding gene. In studies of synthetic biology, analyses of reporter gene activities ensure control of the cell with synthetic genetic circuits, as achieved using a combination of in silico and in vivo experiments. However, unexpected effects of downstream reporter genes on upstream regulatory genes may interfere with in vivo observations. This phenomenon is termed as retroactivity. Using in silico and in vivo experiments, we found that a different copy number of regulatory protein-binding sites in a downstream gene altered the upstream dynamics, suggesting retroactivity of reporters in this synthetic genetic oscillator. Furthermore, by separating the two sources of retroactivity (titration of the component and competition for degradation), we showed that, in the dual-feedback oscillator, the level of the fluorescent protein reporter competing for degradation with the circuits' components is important for the stability of the oscillations. Altogether, our results indicate that the selection of reporter promoters using a combination of in silico and in vivo experiments is essential for the advanced design of genetic circuits.

Measurement of Mesoscale Conformational Dynamics of Freely Diffusing Molecules with Tracking FCS

Few techniques are suited to probe the structure and dynamics of molecular complexes at the mesoscale level (∼100-1000 nm). We have developed a single-molecule technique that uses tracking fluorescence correlation spectroscopy (tFCS) to probe the conformation and dynamics of mesoscale molecular assemblies. tFCS measures the distance fluctuations between two fluorescently labeled sites within an untethered, freely diffusing biomolecule. To achieve subdiffraction spatial resolution, we developed a feedback scheme that allows us to maintain the molecule at an optimal position within the laser intensity gradient for fluorescence correlation spectroscopy. We characterized tFCS spatial sensitivity by measuring the Brownian end-to-end dynamics of DNA molecules as short as 1000 bp. We demonstrate that tFCS detects changes in the compaction of reconstituted nucleosome arrays and can assay transient protein-mediated interactions between distant sites in an individual DNA molecule. Our measurements highlight the applicability of tFCS to a wide variety of biochemical processes involving mesoscale conformational dynamics.

Frequency Modulation of Transcriptional Bursting Enables Sensitive and Rapid Gene Regulation

Gene regulation is a complex non-equilibrium process. Here, we show that quantitating the temporal regulation of key gene states (transcriptionally inactive, active, and refractory) provides a parsimonious framework for analyzing gene regulation. Our theory makes two non-intuitive predictions. First, for transcription factors (TFs) that regulate transcription burst frequency, as opposed to amplitude or duration, weak TF binding is sufficient to elicit strong transcriptional responses. Second, refractoriness of a gene after a transcription burst enables rapid responses to stimuli. We validate both predictions experimentally by exploiting the natural, optogenetic-like responsiveness of the Neurospora GATA-type TF White Collar Complex (WCC) to blue light. Further, we demonstrate that differential regulation of WCC target genes is caused by different gene activation rates, not different TF occupancy, and that these rates are tuned by both the core promoter and the distance between TF-binding site and core promoter. In total, our work demonstrates the relevance of a kinetic, non-equilibrium framework for understanding transcriptional regulation.

Lateral migration of electrospun hydrogel nanofilaments in an oscillatory flow

The recent progress in bioengineering has created great interest in the dynamics and manipulation of long, deformable macromolecules interacting with fluid flow. We report experimental data on the cross-flow migration, bending, and buckling of extremely deformable hydrogel nanofilaments conveyed by an oscillatory flow into a microchannel. The changes in migration velocity and filament orientation are related to the flow velocity and the filament's initial position, deformation, and length. The observed migration dynamics of hydrogel filaments qualitatively confirms the validity of the previously developed worm-like bead-chain hydrodynamic model. The experimental data collected may help to verify the role of hydrodynamic interactions in molecular simulations of long molecular chains dynamics.

The free-energy cost of interaction between DNA loops

From the viewpoint of thermodynamics, the formation of DNA loops and the interaction between them, which are all non-equilibrium processes, result in the change of free energy, affecting gene expression and further cell-to-cell variability as observed experimentally. However, how these processes dissipate free energy remains largely unclear. Here, by analyzing a mechanic model that maps three fundamental topologies of two interacting DNA loops into a 4-state model of gene transcription, we first show that a longer DNA loop needs more mean free energy consumption. Then, independent of the type of interacting two DNA loops (nested, side-by-side or alternating), the promotion between them always consumes less mean free energy whereas the suppression dissipates more mean free energy. More interestingly, we find that in contrast to the mechanism of direct looping between promoter and enhancer, the facilitated-tracking mechanism dissipates less mean free energy but enhances the mean mRNA expression, justifying the facilitated-tracking hypothesis, a long-standing debate in biology. Based on minimal energy principle, we thus speculate that organisms would utilize the mechanisms of loop-loop promotion and facilitated tracking to survive in complex environments. Our studies provide insights into the understanding of gene expression regulation mechanism from the view of energy consumption.

The mechanics of DNA loops bridged by proteins unveiled by single-molecule experiments

Protein-induced DNA bridging and looping is a common mechanism for various and essential processes in bacterial chromosomes. This mechanism is preserved despite the very different bacterial conditions and their expected influence on the thermodynamic and kinetic characteristics of the bridge formation and stability. Over the last two decades, single-molecule techniques carried out on in vitro DNA systems have yielded valuable results which, in combination with theoretical works, have clarified the effects of different parameters of nucleoprotein complexes on the protein-induced DNA bridging and looping process. In this review, I will outline the features that can be measured for such processes with various single-molecule techniques in use in the field. I will then describe both the experimental results and the theoretical models that illuminate the contribution of the DNA molecule itself as well as that of the bridging proteins in the DNA looping mechanism at play in the nucleoid of E. coli.

Buckling a Semiflexible Polymer Chain under Compression

Instability and structural transitions arise in many important problems involving dynamics at molecular length scales. Buckling of an elastic rod under a compressive load offers a useful general picture of such a transition. However, the existing theoretical description of buckling is applicable in the load response of macroscopic structures, only when fluctuations can be neglected, whereas membranes, polymer brushes, filaments, and macromolecular chains undergo considerable Brownian fluctuations. We analyze here the buckling of a fluctuating semiflexible polymer experiencing a compressive load. Previous works rely on approximations to the polymer statistics, resulting in a range of predictions for the buckling transition that disagree on whether fluctuations elevate or depress the critical buckling force. In contrast, our theory exploits exact results for the statistical behavior of the worm-like chain model yielding unambiguous predictions about the buckling conditions and nature of the buckling transition. We find that a fluctuating polymer under compressive load requires a larger force to buckle than an elastic rod in the absence of fluctuations. The nature of the buckling transition exhibits a marked change from being distinctly second order in the absence of fluctuations to being a more gradual, compliant transition in the presence of fluctuations. We analyze the thermodynamic contributions throughout the buckling transition to demonstrate that the chain entropy favors the extended state over the buckled state, providing a thermodynamic justification of the elevated buckling force.

The Role of DNA Looping in the Search for Specific Targets on DNA by Multisite Proteins

Many cellular processes involve simultaneous interactions between DNA and protein molecules at several locations. They are regulated and controlled by special protein-DNA complexes, which are known as synaptic complexes or synaptosomes. Because of the multisite nature of involved proteins, it was suggested that during the formation of synaptic complexes DNA loops might appear, but their role is unclear. We developed a theoretical model that allowed us to evaluate the effect of transient DNA loop formation. It is based on a discrete-state stochastic method that explicitly takes into account the free-energy contributions due to the appearance of DNA loops. The formation of the synaptic complexes is viewed as a search for a specific binding site on DNA by the protein molecule already bound to DNA at another location. It was found that the search might be optimized by varying the position of the target and the total length of DNA. Furthermore, the formation of transient DNA loops leads to faster dynamics if it is associated with favorable enthalpic contributions to nonspecific protein-DNA interactions. It is also shown that DNA looping might reduce stochastic noise in the system.

Aspects of protein-DNA interactions: a review of quantitative thermodynamic theory for modelling synthetic circuits utilising LacI and CI repressors, IPTG and the reporter gene lacZ

Protein-DNA interactions are central to the control of gene expression across all forms of life. The development of approaches to rigorously model such interactions has often been hindered both by a lack of quantitative binding data and by the difficulty in accounting for parameters relevant to the intracellular situation, such as DNA looping and thermodynamic non-ideality. Here, we review these considerations by developing a thermodynamically based mathematical model that attempts to simulate the functioning of an Escherichia coli expression system incorporating two of the best characterised prokaryotic DNA binding proteins, Lac repressor and lambda CI repressor. The key aim was to reproduce experimentally observed reporter gene activities arising from the expression of either wild-type CI repressor or one of three positive-control CI mutants. The model considers the role of several potentially important, but sometimes neglected, biochemical features, including DNA looping, macromolecular crowding and non-specific binding, and allowed us to obtain association constants for the binding of CI and its variants to a specific operator sequence.

Single-molecule fluorescence studies on DNA looping

Structure and dynamics of DNA impact how the genetic code is processed and maintained. In addition to its biological importance, DNA has been utilized as building blocks of various nanomachines and nanostructures. Thus, understanding the physical properties of DNA is of fundamental importance to basic sciences and engineering applications. DNA can undergo various physical changes. Among them, DNA looping is unique in that it can bring two distal sites together, and thus can be used to mediate interactions over long distances. In this paper, we introduce a FRET-based experimental tool to study DNA looping at the single molecule level. We explain the connection between experimental measurables and a theoretical concept known as the J factor with the intent of raising awareness of subtle theoretical details that should be considered when drawing conclusions. We also explore DNA looping-assisted protein diffusion mechanism called intersegmental transfer using protein induced fluorescence enhancement (PIFE). We present some preliminary results and future outlooks.

Interplay of Protein Binding Interactions, DNA Mechanics, and Entropy in DNA Looping Kinetics

DNA looping plays a key role in many fundamental biological processes, including gene regulation, recombination, and chromosomal organization. The looping of DNA is often mediated by proteins whose structural features and physical interactions can alter the length scale at which the looping occurs. Looping and unlooping processes are controlled by thermodynamic contributions associated with mechanical deformation of the DNA strand and entropy arising from thermal fluctuations of the conformation. To determine how these confounding effects influence DNA looping and unlooping kinetics, we present a theoretical model that incorporates the role of the protein interactions, DNA mechanics, and conformational entropy. We show that for shorter DNA strands the interaction distance affects the transition state, resulting in a complex relationship between the looped and unlooped state lifetimes and the physical properties of the looped DNA. We explore the range of behaviors that arise with varying interaction distance and DNA length. These results demonstrate how DNA deformation and entropy dictate the scaling of the looping and unlooping kinetics versus the J-factor, establishing the connection between kinetic and equilibrium behaviors. Our results show how the twist-and-bend elasticity of the DNA chain modulates the kinetics and how the influence of the interaction distance fades away at intermediate to longer chain lengths, in agreement with previous scaling predictions.

Phase diagram of the ground states of DNA condensates

The phase diagram of the ground states of DNA in a bad solvent is studied for a semiflexible polymer model with a generalized local elastic bending potential characterized by a nonlinearity parameter x and effective self-attraction promoting compaction. x=1 corresponds to the wormlike chain model. Surprisingly, the phase diagram as well as the transition lines between the ground states are found to be a function of x. The model provides a simple explanation for the results of prior experimental and computational studies and makes predictions for the specific geometries of the ground states. The results underscore the impact of the form of the microscopic bending energy at macroscopic observable scales.