Kinetics of target site localization of a protein on DNA: a stochastic approach

The effects of intersegmental transfers on target location by proteins

We study a model of protein searching for a target, using facilitated diffusion, on a DNA molecule confined in a finite volume. The model includes three distinct pathways for facilitated diffusion: (a) sliding--in which the protein diffuses along the contour of the DNA, (b) jumping--where the protein travels between two sites along the DNA by three-dimensional diffusion and finally (c) intersegmental transfer--which allows the protein to move from one site to another by transiently binding both at the same time. The typical search time is calculated using scaling arguments which are verified numerically. Our results suggest that the inclusion of intersegmental transfer (i) decreases the search time considerably, (ii) makes the search time much more robust to variations in the parameters of the model and (iii) that the optimal search time occurs in a regime very different than that found for models which ignore intersegmental transfers. The behavior we find is rich and shows surprising dependences, for example on the DNA length.

Optimizing intermittent reaction paths

Various examples of biochemical reactions in cells, such as DNA/protein interactions, reveal that in extremely diluted regimes reaction paths are not always simple brownian trajectories. They can rather be qualified as intermittent, since they combine slow diffusion phases on one hand and a second mode of faster transport on the other hand, which can be either a faster diffusion mode, as in the case of DNA-binding proteins, or a ballistic mode powered by molecular motors in the case of intracellular transport. In this article, we introduce simple theoretical models which permit to calculate explicitly the reaction rates for reactions limited by intermittent transport. This approach shows quantitatively that intermittent reaction pathways are actually very efficient, since they permit to significantly increase the reaction rates, which could explain why they are observed so often. Moreover, we give theoretical arguments which suggest that intermittent transport could also be useful for in vitro chemistry. Indeed, we show that intermittent transport naturally pops up in the context of reaction at interfaces, where reactants combine surface diffusion phases and bulk excursions, and could permit to enhance reactivity. In this case, adjusting chemically the affinity of reactants with the interface makes possible to optimize the reaction rate.

How DNA coiling enhances target localization by proteins

Many genetic processes depend on proteins interacting with specific sequences on DNA. Despite the large excess of nonspecific DNA in the cell, proteins can locate their targets rapidly. After initial nonspecific binding, they are believed to find the target site by 1D diffusion ("sliding") interspersed by 3D dissociation/reassociation, a process usually referred to as facilitated diffusion. The 3D events combine short intrasegmental "hops" along the DNA contour, intersegmental "jumps" between nearby DNA segments, and longer volume "excursions." The impact of DNA conformation on the search pathway is, however, still unknown. Here, we show direct evidence that DNA coiling influences the specific association rate of EcoRV restriction enzymes. Using optical tweezers together with a fast buffer exchange system, we obtained association times of EcoRV on single DNA molecules as a function of DNA extension, separating intersegmental jumping from other search pathways. Depending on salt concentration, targeting rates almost double when the DNA conformation is changed from fully extended to a coiled configuration. Quantitative analysis by an extended facilitated diffusion model reveals that only a fraction of enzymes are ready to bind to DNA. Generalizing our results to the crowded environment of the cell we predict a major impact of intersegmental jumps on target localization speed on DNA.

Zero constant formula for first-passage observables in bounded domains

In this Letter, we develop an analytical approach which provides an explicit determination of mean first-passage times (MFPTs) for random walks in bounded domains for a wide class of transport processes. In particular, we derive for the first time explicit expressions of MFPTs for emblematic models of transport in complex media, such as diffusion on deterministic and random fractals. This approach relies on a scale-invariance hypothesis and a large volume expansion of the MFPT, which actually proves to be very accurate even for small system sizes as shown by numerical simulations. This explicit determination of MFPTs can be straightforwardly generalized to further useful first-passage observables such as occupation times and splitting probabilities.

Intermittent Brownian dynamics over a rigid strand: heavily tailed relocation statistics in a simple geometry

We analyze the intermittent Brownian dynamics (a succession of adsorption and bulk relocation steps) of a test particle over a single strand. We propose an analytic expression of the relocation time distribution at all times. We show that this distribution has a nontrivial heavily tailed statistics at long time with a diverging average relocation time. In order to experimentally probe this first passage statistics, we follow the intermittent Brownian dynamics of water molecules over long and stiff imogolite mineral strands, using a field cycling NMR dispersion technique. Our analytic derivation is found to be in good agreement with experimental data on a large domain of observation. Implications for the efficiency of a search strategy on a single filament are then discussed and the importance of the confinement and/or the finite size effect is emphasized.

Sliding and jumping of single EcoRV restriction enzymes on non-cognate DNA

The restriction endonuclease EcoRV can rapidly locate a short recognition site within long non-cognate DNA using 'facilitated diffusion'. This process has long been attributed to a sliding mechanism, in which the enzyme first binds to the DNA via nonspecific interaction and then moves along the DNA by 1D diffusion. Recent studies, however, provided evidence that 3D translocations (hopping/jumping) also help EcoRV to locate its target site. Here we report the first direct observation of sliding and jumping of individual EcoRV molecules along nonspecific DNA. Using fluorescence microscopy, we could distinguish between a slow 1D diffusion of the enzyme and a fast translocation mechanism that was demonstrated to stem from 3D jumps. Salt effects on both sliding and jumping were investigated, and we developed numerical simulations to account for both the jump frequency and the jump length distribution. We deduced from our study the 1D diffusion coefficient of EcoRV, and we estimated the number of jumps occurring during an interaction event with nonspecific DNA. Our results substantiate that sliding alternates with hopping/jumping during the facilitated diffusion of EcoRV and, furthermore, set up a framework for the investigation of target site location by other DNA-binding proteins.

Spatial effects on the speed and reliability of protein-DNA search

Strong experimental and theoretical evidence shows that transcription factors (TFs) and other specific DNA-binding proteins find their sites using a two-mode search: alternating between three-dimensional (3D) diffusion through the cell and one-dimensional (1D) sliding along the DNA. We show that, due to the 1D component of the search process, the search time of a TF can depend on the initial position of the TF. We formalize this effect by discriminating between two types of searches: global and local. Using analytical calculations and simulations, we estimate how close a TF and binding site need to be to make a local search likely. We then use our model to interpret the wide range of experimental measurements of this parameter. We also show that local and global searches differ significantly in average search time and the variability of search time. These results lead to a number of biological implications, including suggestions of how prokaryotes achieve rapid gene regulation and the relationship between the search mechanism and noise in gene expression. Lastly, we propose a number of experiments to verify the existence and quantify the extent of spatial effects on the TF search process in prokaryotes.

Narrow-escape time problem: time needed for a particle to exit a confining domain through a small window

The time needed for a particle to exit a confining domain through a small window, called the narrow-escape time (NET), is a limiting factor of various processes, such as some biochemical reactions in cells. Obtaining an estimate of the mean NET for a given geometric environment is therefore a requisite step to quantify the reaction rate constant of such processes, which has raised a growing interest in the past few years. In this Letter, we determine explicitly the scaling dependence of the mean NET on both the volume of the confining domain and the starting point to aperture distance. We show that this analytical approach is applicable to a very wide range of stochastic processes, including anomalous diffusion or diffusion in the presence of an external force field, which cover situations of biological relevance.

Probing microscopic origins of confined subdiffusion by first-passage observables

Subdiffusive motion of tracer particles in complex crowded environments, such as biological cells, has been shown to be widespread. This deviation from Brownian motion is usually characterized by a sublinear time dependence of the mean square displacement (MSD). However, subdiffusive behavior can stem from different microscopic scenarios that cannot be identified solely by the MSD data. In this article we present a theoretical framework that permits the analytical calculation of first-passage observables (mean first-passage times, splitting probabilities, and occupation times distributions) in disordered media in any dimensions. This analysis is applied to two representative microscopic models of subdiffusion: continuous-time random walks with heavy tailed waiting times and diffusion on fractals. Our results show that first-passage observables provide tools to unambiguously discriminate between the two possible microscopic scenarios of subdiffusion. Moreover, we suggest experiments based on first-passage observables that could help in determining the origin of subdiffusion in complex media, such as living cells, and discuss the implications of anomalous transport to reaction kinetics in cells.

Efficiency of trapping processes in regular and disordered networks

We use a Markov method to study the efficiency of trapping processes involving both a random walker and a deep trap in regular and disordered networks. The efficiency is gauged by the mean absorption time (average of the mean number of steps performed by the random walker before being absorbed by the trap). We compute this quantity in terms of different control parameters, namely, the length of the walker jumps, the mobility of the trap, and the degree of spatial disorder of the network. For a proper choice of the system size, we find in all cases a nonmonotonic behavior of the efficiency in terms of the corresponding control parameter. We thus arrive at the conclusion that, despite the decrease of the effective system size underlying the increase of the control parameter, the efficiency is reduced as a result of an increase of the escape probability of the walker once it finds itself in the interaction zone of the trap. This somewhat anti-intuitive effect is very robust in the sense that it is observed regardless of the specific choice of the control parameter. For the case of a ring lattice, results for the mean absorption time in systems of arbitrary size are given in terms of a two-parameter scaling function. For the case of a mobile trap, we deal with both trapping via a single channel (walker-trap overlap) and via two channels (walker-trap overlap and walker-trap crossing), thereby generalizing previous work. As for the disordered case, our analysis concerns small world networks, for which we see several crossovers of the absorption time as a function of the control parameter and the system size. The methodology used may be well suited to exploring characteristic time scales of encounter-controlled phenomena in networks with a few interacting elements and the effect of geometric constraints in nanoscale systems with a very small number of particles.

First-passage times in complex scale-invariant media

How long does it take a random walker to reach a given target point? This quantity, known as a first-passage time (FPT), has led to a growing number of theoretical investigations over the past decade. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media, neuron firing dynamics, spreading of diseases or target search processes. Most methods of determining FPT properties in confining domains have been limited to effectively one-dimensional geometries, or to higher spatial dimensions only in homogeneous media. Here we develop a general theory that allows accurate evaluation of the mean FPT in complex media. Our analytical approach provides a universal scaling dependence of the mean FPT on both the volume of the confining domain and the source-target distance. The analysis is applicable to a broad range of stochastic processes characterized by length-scale-invariant properties. Our theoretical predictions are confirmed by numerical simulations for several representative models of disordered media, fractals, anomalous diffusion and scale-free networks.

Modeling transcription factor binding events to DNA using a random walker/jumper representation on a 1D/2D lattice with different affinity sites

Surviving in a diverse environment requires corresponding organism responses. At the cellular level, such adjustment relies on the transcription factors (TFs) which must rapidly find their target sequences amidst a vast amount of non-relevant sequences on DNA molecules. Whether these transcription factors locate their target sites through a 1D or 3D pathway is still a matter of speculation. It has been suggested that the optimum search time is when the protein equally shares its search time between 1D and 3D diffusions. In this paper, we study the above problem using Monte Carlo simulations by considering a simple physical model. A 1D strip, representing a DNA, with a number of low affinity sites, corresponding to non-target sites, and high affinity sites, corresponding to target sites, is considered and later extended to a 2D strip. We study the 1D and 3D exploration pathways, and combinations thereof by considering three different types of molecules: a walker that randomly walks along the strip with no dissociation; a jumper that represents dissociation and then re-association of a TF with the strip at later time at a distant site; and a hopper that is similar to the jumper but it dissociates and then re-associates at a faster rate than the jumper. We analyze the final probability distribution of molecules for each case and find that TFs can locate their targets on the experimental time scale even if they spend only 15% of their search time diffusing freely in the solution. This agrees with recent experimental results obtained by Elf et al (2007 Science 316 1191) and is in contrast to previously reported theoretical predictions. Our results also agree with the experimental evidence for the role of chaperons and proteasomes in stabilizing and destabilizing TFs binding, respectively, during the regulation process. Therefore, the results of our manuscript can provide a refined theoretical framework for the process.

Optimal length distribution of termite tunnel branches for efficient food search and resource transportation

Subterranean termites excavate branching tunnels for searching and transporting food in soil. Experimentally, the length distribution of the branch tunnels, P(L), was characterized by the exponentially decaying function, P(L) approximately exp(-alphaL) with a branch length exponent of alpha=0.15. To evaluate the significance of this alpha value, we used a lattice model to simulate tunnels of the Formosan subterranean termite, Coptotermes formosanus Shiraki in featureless soil and computed the ratio of energy gain for obtained food to loss for transporting food for a given time, gamma for various simulated tunnel patterns with the different values of alpha. In simulation, the gamma was maximized at 0.15<alpha<0.20 for the number of primary tunnels N=6, 8, and 10. Our results indicate that tunnels with branch length distributions similar to those derived from empirical tunnel patterns result in tunnels made up of highly efficient paths to search and transport resources.

Modeling diffusional transport in the interphase cell nucleus

In this paper a lattice model for the diffusional transport of particles in the interphase cell nucleus is proposed. Dense networks of chromatin fibers are created by three different methods: Randomly distributed, noninterconnected obstacles, a random walk chain model, and a self-avoiding random walk chain model with persistence length. By comparing a discrete and a continuous version of the random walk chain model, we demonstrate that lattice discretization does not alter particle diffusion. The influence of the three dimensional geometry of the fiber network on the particle diffusion is investigated in detail while varying the occupation volume, chain length, persistence length, and walker size. It is shown that adjacency of the monomers, the excluded volume effect incorporated in the self-avoiding random walk model, and, to a lesser extent, the persistence length affect particle diffusion. It is demonstrated how the introduction of the effective chain occupancy, which is a convolution of the geometric chain volume with the walker size, eliminates the conformational effects of the network on the diffusion, i.e., when plotting the diffusion coefficient as a function of the effective chain volume, the data fall onto a master curve.

How gene order is influenced by the biophysics of transcription regulation

What are the forces that shape the structure of prokaryotic genomes: the order of genes, their proximity, and their orientation? Coregulation and coordinated horizontal gene transfer are believed to promote the proximity of functionally related genes and the formation of operons. However, forces that influence the structure of the genome beyond the level of a single operon remain unknown. Here, we show that the biophysical mechanism by which regulatory proteins search for their sites on DNA can impose constraints on genome structure. Using simulations, we demonstrate that rapid and reliable gene regulation requires that the transcription factor (TF) gene be close to the site on DNA the TF has to bind, thus promoting the colocalization of TF genes and their targets on the genome. We use parameters that have been measured in recent experiments to estimate the relevant length and times scales of this process and demonstrate that the search for a cognate site may be prohibitively slow if a TF has a low copy number and is not colocalized. We also analyze TFs and their sites in a number of bacterial genomes, confirm that they are colocalized significantly more often than expected, and show that this observation cannot be attributed to the pressure for coregulation or formation of selfish gene clusters, thus supporting the role of the biophysical constraint in shaping the structure of prokaryotic genomes. Our results demonstrate how spatial organization can influence timing and noise in gene expression.

Intermittent search process and teleportation

The authors study an intermittent search process combining diffusion and "teleportation" phases in a d-dimensional spherical continuous system and in a regular lattice. The searcher alternates diffusive phases, during which targets can be discovered, and fast phases (teleportation) which randomly relocate the searcher, but do not allow for target detection. The authors show that this alternation can be favorable for minimizing the time of first discovery, and that this time can be optimized by a convenient choice of the mean waiting times of each motion phase. The optimal search strategy is explicitly derived in the continuous case and in the lattice case. Arguments are given to show that much more general intermittent motions do provide optimal search strategies in d dimensions. These results can be useful in the context of heterogeneous catalysis or in various biological examples of transport through membrane pores.

Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary

We derive the boundary condition for a subdiffusive particle interacting with a reactive boundary with a finite reaction rate. Molecular crowding conditions, that are found to cause subdiffusion of larger molecules in biological cells, are shown to effect long-tailed distributions with an identical exponent for both the unbinding times from the boundary to the bulk and the rebinding times from the bulk. This causes a weak ergodicity breaking: typically, an individual particle either stays bound or remains in the bulk for very long times. We discuss why this may be beneficial for in vivo gene regulation by DNA-binding proteins, whose typical concentrations are nanomolar.

A quantitative study of lambda-phage SWITCH and its components

We propose what we believe is a new model to quantitatively describe the lambda-phage SWITCH system. The model incorporates facilitated transfer mechanism of transcription factor, which can be simplified into a two-step reaction. We first sequentially obtain two indispensable parameters by fitting our model to experimental data of two simple systems, and then apply them to study the natural lambda-SWITCH system. By incorporating the facilitated transfer mechanism, we find that in RecA(-) host Escherichia coli, the wild-type lambda-lysogenic state is in a monostable regime rather than in a bistable regime. Furthermore, the model explains the weak role of Cro protein and probably sheds light on the evolution of lambda-Cro protein, which is known to be structurally distinct from the other Cros in lambdoid family members.

Diffusion of transcription factors can drastically enhance the noise in gene expression

We study by Green's Function Reaction Dynamics the effect of the diffusive motion of repressor molecules on the noise in mRNA and protein levels for a gene that is under the control of a repressor. We find that spatial fluctuations due to diffusion can drastically enhance the noise in gene expression. After dissociation from the operator, a repressor can rapidly rebind to the DNA. Our results show that the rebinding trajectories are so short that, on this timescale, the RNA polymerase (RNAP) cannot effectively compete with the repressor for binding to the promoter. As a result, a dissociated repressor molecule will on average rebind many times, before it eventually diffuses away. These rebindings thus lower the effective dissociation rate, and this increases the noise in gene expression. Another consequence of the timescale separation between repressor rebinding and RNAP association is that the effect of spatial fluctuations can be described by a well-stirred, zero-dimensional, model by renormalizing the reaction rates for repressor-DNA (un) binding. Our results thus support the use of well-stirred, zero-dimensional models for describing noise in gene expression. We also show that for a fixed repressor strength, the noise due to diffusion can be minimized by increasing the number of repressors or by decreasing the rate of the open complex formation. Lastly, our results emphasize that power spectra are a highly useful tool for studying the propagation of noise through the different stages of gene expression.

Two-dimensional intermittent search processes: An alternative to Lévy flight strategies

Lévy flights are known to be optimal search strategies in the particular case of revisitable targets. In the relevant situation of nonrevisitable targets, we propose an alternative model of two-dimensional (2D) search processes, which explicitly relies on the widely observed intermittent behavior of foraging animals. We show analytically that intermittent strategies can minimize the search time, and therefore do constitute real optimal strategies. We study two representative modes of target detection and determine which features of the search time are robust and do not depend on the specific characteristics of detection mechanisms. In particular, both modes lead to a global minimum of the search time as a function of the typical times spent in each state, for the same optimal duration of the ballistic phase. This last quantity could be a universal feature of 2D intermittent search strategies.

How proteins search for their specific sites on DNA: the role of DNA conformation

It is known since the early days of molecular biology that proteins locate their specific targets on DNA up to two orders-of-magnitude faster than the Smoluchowski three-dimensional diffusion rate. An accepted explanation of this fact is that proteins are nonspecifically adsorbed on DNA, and sliding along DNA provides for the faster one-dimensional search. Surprisingly, the role of DNA conformation was never considered in this context. In this article, we explicitly address the relative role of three-dimensional diffusion and one-dimensional sliding along coiled or globular DNA and the possibility of correlated readsorption of desorbed proteins. We have identified a wealth of new different scaling regimes. We also found the maximal possible acceleration of the reaction due to sliding. We found that the maximum on the rate-versus-ionic strength curve is asymmetric, and that sliding can lead not only to acceleration, but also in some regimes to dramatic deceleration of the reaction.

Optimal target search on a fast-folding polymer chain with volume exchange

We study the search process of a target on a rapidly folding polymer ("DNA") by an ensemble of particles ("proteins"), whose search combines 1D diffusion along the chain, Lévy type diffusion mediated by chain looping, and volume exchange. A rich behavior of the search process is obtained with respect to the physical parameters, in particular, for the optimal search.

Measurement of the contributions of 1D and 3D pathways to the translocation of a protein along DNA

Proteins that act at specific DNA sequences bind DNA randomly and then translocate to the target site. The translocation is often ascribed to the protein sliding along the DNA while maintaining continuous contact with it. Proteins also can move on DNA by multiple cycles of dissociation/reassociation within the same chain. To distinguish these pathways, a strategy was developed to analyze protein motion between DNA sites. The strategy reveals whether the protein maintains contact with the DNA as it transfers from one site to another by sliding or whether it loses contact by a dissociation/reassociation step. In reactions at low salt, the test protein stayed on the DNA as it traveled between sites, but only when the sites were <50 bp apart. Transfers of >30 bp at in vivo salt, and over distances of >50 bp at any salt, always included at least one dissociation step. Hence, for this enzyme, 1D sliding operates only over short distances at low salt, and 3D dissociation/reassociation is its main mode of translocation.

First passage time of N excluded-volume particles on a line

Motivated by recent single-molecule studies of proteins sliding on a DNA molecule, we explore the targeting dynamics of particles ("proteins") sliding diffusively along a line ("DNA") in search of their target site (specific target sequence). At lower particle densities, one observes an expected reduction of the mean first passage time proportional to N(-2), with corrections at higher concentrations. We explicitly take adsorption and desorption effects, to and from the DNA, into account. For this general case, we also consider finite-size effects when the continuum approximation based on the number density of particles breaks down. Moreover, we address the first-passage-time problem of a tagged particle diffusing among other particles.

First-passage method for the study of the efficiency of a two-channel reaction on a lattice

We study the efficiency of a two-channel reaction between two walkers on a finite one-dimensional periodic lattice. The walkers perform a combination of synchronous and asynchronous jumps on the lattice and react instantaneously when they meet at the same site (first channel) or upon position exchange (second channel). We develop a method based on a conditional first-passage problem to obtain exact results for the mean number of time steps needed for the reaction to take place as well as for higher order moments. Previous results obtained in the framework of a difference equation approach are fully confirmed, including the existence of a parity effect. For even lattices the maximum efficiency corresponds to a mixture of synchronous events and a small amount of asynchronous events, while for odd lattices the reaction time is minimized by a purely synchronous process. We provide an intuitive explanation for this behavior. In addition, we give explicit expressions for the variance of the reaction time. The latter displays a similar even-odd behavior, suggesting that the parity effect extends to higher order moments.

Target search of N sliding proteins on a DNA

At low to moderate ambient salt concentrations, DNA-binding proteins bind relatively tightly to DNA, and only very rarely detach. Intersegmental transfer due to DNA-looping can be excluded by applying an external pulling force to the DNA molecule. Under such conditions, we explore the targeting dynamics of N proteins sliding diffusively along DNA in search of their specific target sequence. At lower densities of binding proteins, we find a reduction of the characteristic search time proportional to N(-2), with corrections at higher concentrations. Rates for detachment and attachment of binding proteins are incorporated in the model. Our findings are in agreement with recent single molecule studies in the presence of bacteriophage T4 gene 32 protein for which the unbinding rate is much lower than the specific binding rate.

Optimal search strategies for hidden targets

What is the fastest way of finding a randomly hidden target? Experimental observations reveal that the search behavior of foraging animals is generally intermittent: active search phases randomly alternate with phases of fast ballistic motion. Here, we study the efficiency of this two state search strategy by calculating analytically the mean first passage time at the target. We model the perception mechanism involved in the active search phase by a diffusive process. We show that the search strategy is optimal when the average duration of "motion phases" varies like the power either 3/5 or 2/3 of the average duration of "search phases" depending on the regime. This scaling accounts for experimental data over a wide range of species, which suggests that the kinetics of search trajectories is a determining factor optimized by foragers and that the perception activity is adequately described by a diffusion process.

A model for the mediation of processivity of DNA-targeting proteins by nonspecific binding: dependence on DNA length and presence of obstacles

A physical and mathematical model is presented to explain processivity of proteins on DNA. In this model, a DNA-targeting protein such as a restriction enzyme can diffuse to the DNA surface and nonspecifically bind to it. Once on the DNA surface it will either move along the DNA or equilibrate with the surrounding region. Owing to the nonspecific binding, the search for a specific site on the DNA occurs in a reduced dimensionality, and the protein appears processive when moving from one specific site to another. The simplest version of this nonspecific-binding-facilitated diffusion model is solved and the results quantitatively explain experimentally observed dependence of the processivity ratio on the intervening DNA length between two specific sites.