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

Physical constraints determine the logic of bacterial promoter architectures

Site-specific transcription factors (TFs) bind to their target sites on the DNA, where they regulate the rate at which genes are transcribed. Bacterial TFs undergo facilitated diffusion (a combination of 3D diffusion around and 1D random walk on the DNA) when searching for their target sites. Using computer simulations of this search process, we show that the organization of the binding sites, in conjunction with TF copy number and binding site affinity, plays an important role in determining not only the steady state of promoter occupancy, but also the order at which TFs bind. These effects can be captured by facilitated diffusion-based models, but not by standard thermodynamics. We show that the spacing of binding sites encodes complex logic, which can be derived from combinations of three basic building blocks: switches, barriers and clusters, whose response alone and in higher orders of organization we characterize in detail. Effective promoter organizations are commonly found in the E. coli genome and are highly conserved between strains. This will allow studies of gene regulation at a previously unprecedented level of detail, where our framework can create testable hypothesis of promoter logic.

FOUNDATIONS FOR MODELING THE DYNAMICS OF GENE REGULATORY NETWORKS: A MULTILEVEL-PERSPECTIVE REVIEW

A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom–up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.

A mathematical model for scanning and catalysis on single-stranded DNA, illustrated with activation-induced deoxycytidine deaminase

We formulated a master equation-based mathematical model to analyze random scanning and catalysis for enzymes that act on single-stranded DNA (ssDNA) substrates. Catalytic efficiencies and intrinsic scanning distances are deduced from the distribution of positions and gap lengths between a series of catalytic events occurring over time, which are detected as point mutations in a lacZα-based reporter sequence containing enzyme target motifs. Mathematical analysis of the model shows how scanning motions become separable from the catalysis when the proper statistical properties of the mutation pattern are used to interpret the readouts. Two-point correlations between all catalytic events determine intrinsic scanning distances, whereas gap statistics between mutations determine their catalytic efficiencies. Applying this model to activation-induced deoxycytidine deaminase (AID), which catalyzes C→U deaminations processively on ssDNA, we have established that deaminations of AGC hot motifs occur at a low rate, ∼0.03 s(-1), and low efficiency, ∼3%. AID performs random bidirectional movements for an average distance of 6.2 motifs, at a rate of about 15 nucleotides per second, and "dwells" at a motif site for 2.7 s while bound >4 min to the same DNA molecule. These results provide new and important insights on how AID may be optimized for generating mutational diversity in Ig genes, and we discuss how the properties of AID acting freely on a "naked" ssDNA relate to the constrained action of AID during transcription-dependent somatic hypermutation and class-switch recombination.

Complex tracer diffusion dynamics in polymer solutions

We employ fluorescence correlation spectroscopy (FCS) and coarse-grained molecular dynamics simulations to study the mobility of tracers in polymer solutions. Excluded volume interactions result in crowding-induced slowdown, depending only on the polymer concentration. With specific tracer-polymer attractions, the tracer is slowed down at much lower concentrations, and a second diffusion component appears that is sensitive to the polymer chain length. The two components can be resolved by FCS, only if the distance traveled by the tracer in the polymer-bound state is greater than the FCS focal spot size. The tracer dynamics can be used as a sensitive probe of the nature and strength of interactions, which-despite their local character-emphasize the role of chain connectivity.

Encounter dynamics of a small target by a polymer diffusing in a confined domain

We study the first passage time for a polymer, that we call the narrow encounter time (NETP), to reach a small target located on the surface of a microdomain. The polymer is modeled as a freely joint chain (beads connected by springs with a resting non zero length) and we use Brownian simulations to study two cases: when (i) any of the monomer or (ii) only one can be absorbed at the target window. Interestingly, we find that in the first case, the NETP is an increasing function of the polymer length until a critical length, after which it decreases. Moreover, in the long polymer regime, we identified an exponential scaling law for the NETP as a function of the polymer length. In the second case, the position of the absorbed monomer along the polymer chain strongly influences the NETP. Our analysis can be applied to estimate the mean first time of a DNA fragment to a small target in the chromatin structure or for mRNA to find a small target.

Distance matters: the impact of gene proximity in bacterial gene regulation

Following recent discoveries of colocalization of downstream-regulating genes in living cells, the impact of the spatial distance between such genes on the kinetics of gene product formation is increasingly recognized. We here show from analytical and numerical analysis that the distance between a transcription factor (TF) gene and its target gene drastically affects the speed and reliability of transcriptional regulation in bacterial cells. For an explicit model system, we develop a general theory for the interactions between a TF and a transcription unit. The observed variations in regulation efficiency are linked to the magnitude of the variation of the TF concentration peaks as a function of the binding site distance from the signal source. Our results support the role of rapid binding site search for gene colocalization and emphasize the role of local concentration differences.

Speed-selectivity paradox in the protein search for targets on DNA: is it real or not?

Protein search for targets on DNA starts all major biological processes. Although significant experimental and theoretical efforts have been devoted to investigation of these phenomena, mechanisms of protein-DNA interactions during the search remain not fully understood. One of the most surprising observations is known as a speed-selectivity paradox. It suggests that experimentally observed fast findings of targets require smooth protein-DNA binding potentials, while the stability of the specific protein-DNA complex imposes a large energy gap which should significantly slow down the protein molecule. We developed a discrete-state stochastic approach that allowed us to investigate explicitly target search phenomena and to analyze the speed-selectivity paradox. A general dynamic phase diagram for different search regimes is constructed. The effect of the target position on search dynamics is investigated. Using experimentally observed parameters, it is found that slow protein diffusion on DNA does not lead to an increase in the search times. Thus, our theory resolves the speed-selectivity paradox by arguing that it does not exist. It is just an artifact of using approximate continuum theoretical models for analyzing protein search in the region of the parameter space beyond the range of validity of these models. In addition, the presented method, for the first time, provides an explanation for fast target search at the level of single protein molecules. Our theoretical predictions agree with all available experimental observations, and extensive Monte Carlo computer simulations are performed to support analytical calculations.

In vivo facilitated diffusion model

Under dilute in vitro conditions transcription factors rapidly locate their target sequence on DNA by using the facilitated diffusion mechanism. However, whether this strategy of alternating between three-dimensional bulk diffusion and one-dimensional sliding along the DNA contour is still beneficial in the crowded interior of cells is highly disputed. Here we use a simple model for the bacterial genome inside the cell and present a semi-analytical model for the in vivo target search of transcription factors within the facilitated diffusion framework. Without having to resort to extensive simulations we determine the mean search time of a lac repressor in a living E. coli cell by including parameters deduced from experimental measurements. The results agree very well with experimental findings, and thus the facilitated diffusion picture emerges as a quantitative approach to gene regulation in living bacteria cells. Furthermore we see that the search time is not very sensitive to the parameters characterizing the DNA configuration and that the cell seems to operate very close to optimal conditions for target localization. Local searches as implied by the colocalization mechanism are only found to mildly accelerate the mean search time within our model.

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.

Narrow-escape-time problem: the imperfect trapping case

We present a master equation approach to the narrow escape time (NET) problem, i.e., the time needed for a particle contained in a confining domain with a single narrow opening to exit the domain for the first time. We introduce a finite transition probability, ν, at the narrow escape window, allowing the study of the imperfect trapping case. Ranging from 0 to ∞, ν allowed the study of both extremes of the trapping process: that of a highly deficient capture and situations where escape is certain ("perfect trapping" case). We have obtained analytic results for the basic quantity studied in the NET problem, the mean escape time, and we have studied its dependence in terms of the transition (desorption) probability over (from) the surface boundary, the confining domain dimensions, and the finite transition probability at the escape window. Particularly we show that the existence of a global minimum in the NET depends on the "imperfection" of the trapping process. In addition to our analytical approach, we have implemented Monte Carlo simulations, finding excellent agreement between the theoretical results and simulations.

How to accelerate protein search on DNA: location and dissociation

One of the most important features of biological systems that controls their functioning is the ability of protein molecules to find and recognize quickly specific target sites on DNA. Although these phenomena have been studied extensively, detailed mechanisms of protein-DNA interactions during the search are still not well understood. Experiments suggest that proteins typically find their targets fast by combining three-dimensional and one-dimensional motions, and most of the searching time proteins are non-specifically bound to DNA. However these observations are surprising since proteins diffuse very slowly on DNA, and it seems that the observed fast search cannot be achieved under these conditions for single proteins. Here we propose two simple mechanisms that might explain some of these controversial observations. Using first-passage time analysis, it is shown explicitly that the search can be accelerated by changing the location of the target and by effectively irreversible dissociations of proteins. Our theoretical predictions are supported by Monte Carlo computer simulations.

Generalized facilitated diffusion model for DNA-binding proteins with search and recognition states

Transcription factors (TFs) such as the lac repressor find their target sequence on DNA at remarkably high rates. In the established Berg-von Hippel model for this search process, the TF alternates between three-dimensional diffusion in the bulk solution and one-dimensional sliding along the DNA chain. To overcome the so-called speed-stability paradox, in similar models the TF was considered as being present in two conformations (search state and recognition state) between which it switches stochastically. Combining both the facilitated diffusion model and alternating states, we obtain a generalized model. We explicitly treat bulk excursions for rodlike chains arranged in parallel and consider a simplified model for coiled DNA. Compared to previously considered facilitated diffusion models, corresponding to limiting cases of our generalized model, we surprisingly find a reduced target search rate. Moreover, at optimal conditions there is no longer an equipartition between the time spent by the protein on and off the DNA chain.

Geometry-induced bursting dynamics in gene expression

In prokaryotes and eukaryotes, genes are transcribed stochastically according to various temporal patterns that range from simple first-order kinetics to marked bursts, resulting in temporal and cell-to-cell variations of mRNA and protein levels. Here, we consider the effect of the transport of regulatory molecules on the noise in gene expression by taking into account explicitly the dynamics of a finite number of transcription factors confined in the cell. We calculate analytically time-dependent correlation functions of mRNA levels for a wide range of transport mechanisms and find that in the limit of small-transcription-factor copy number, the results differ significantly from standard approaches, which ignore confinement. It is shown how such dynamical quantities, which can now be obtained experimentally, can be used to identify the underlying mechanisms of transcription. Of particular importance, it is demonstrated that the geometry of transcription-factor trajectories in the cellular environment plays a key role in transcription kinetics, and can intrinsically generate the observed various transcription patterns ranging from simple first-order kinetics to bursts.

Interfacial territory covered by surface-mediated diffusion

We consider a minimal model of heterogeneous catalysis in which a molecule performs surface-mediated diffusion inside a confining domain whose boundary contains catalytic sites. We explicitly take into account the combination of surface and bulk diffusion, and we obtain exact results for the mean and variance of the territory covered on the boundary by the particle before its exit in the case of a two-dimensional spherical domain. Depending on the relative positions of the entrance and exit points, very different behaviors with respect to the mean adsorption time of the molecule on the surface are found. We also determine both exact lower and upper bounds and an approximate expression of the probability of reacting with catalytic sites before exiting the domain. These results provide a quantitative measure of the efficiency of an idealized catalyst.

Filling of a Poisson trap by a population of random intermittent searchers

We extend the continuum theory of random intermittent search processes to the case of N independent searchers looking to deliver cargo to a single hidden target located somewhere on a semi-infinite track. Each searcher randomly switches between a stationary state and either a leftward or rightward constant velocity state. We assume that all of the particles start at one end of the track and realize sample trajectories independently generated from the same underlying stochastic process. The hidden target is treated as a partially absorbing trap in which a particle can only detect the target and deliver its cargo if it is stationary and within range of the target; the particle is removed from the system after delivering its cargo. As a further generalization of previous models, we assume that up to n successive particles can find the target and deliver its cargo. Assuming that the rate of target detection scales as 1/N, we show that there exists a well-defined mean-field limit N→∞, in which the stochastic model reduces to a deterministic system of linear reaction-hyperbolic equations for the concentrations of particles in each of the internal states. These equations decouple from the stochastic process associated with filling the target with cargo. The latter can be modeled as a Poisson process in which the time-dependent rate of filling λ(t) depends on the concentration of stationary particles within the target domain. Hence, we refer to the target as a Poisson trap. We analyze the efficiency of filling the Poisson trap with n particles in terms of the waiting time density f(n)(t). The latter is determined by the integrated Poisson rate μ(t)=∫(0)(t)λ(s)ds, which in turn depends on the solution to the reaction-hyperbolic equations. We obtain an approximate solution for the particle concentrations by reducing the system of reaction-hyperbolic equations to a scalar advection-diffusion equation using a quasisteady-state analysis. We compare our analytical results for the mean-field model with Monte Carlo simulations for finite N. We thus determine how the mean first passage time (MFPT) for filling the target depends on N and n.

Intra-nuclear mobility and target search mechanisms of transcription factors: a single-molecule perspective on gene expression

Precise expression of specific genes in time and space is at the basis of cellular viability as well as correct development of organisms. Understanding the mechanisms of gene regulation is fundamental and still one of the great challenges for biology. Gene expression is regulated also by specific transcription factors that recognize and bind to specific DNA sequences. Transcription factors dynamics, and especially the way they sample the nucleoplasmic space during the search for their specific target in the genome, are a key aspect for regulation and it has been puzzling researchers for forty years. The scope of this review is to give a state-of-the-art perspective over the intra-nuclear mobility and the target search mechanisms of specific transcription factors at the molecular level. Going through the seminal biochemical experiments that have raised the first questions about target localization and the theoretical grounds concerning target search processes, we describe the most recent experimental achievements and current challenges in understanding transcription factors dynamics and interactions with DNA using in vitro assays as well as in live prokaryotic and eukaryotic cells. This article is part of a Special Issue entitled: Nuclear Transport and RNA Processing.

Classes of fast and specific search mechanisms for proteins on DNA

Problems of search and recognition appear over different scales in biological systems. In this review we focus on the challenges posed by interactions between proteins, in particular transcription factors, and DNA and possible mechanisms which allow for fast and selective target location. Initially we argue that DNA-binding proteins can be classified, broadly, into three distinct classes which we illustrate using experimental data. Each class calls for a different search process and we discuss the possible application of different search mechanisms proposed over the years to each class. The main thrust of this review is a new mechanism which is based on barrier discrimination. We introduce the model and analyze in detail its consequences. It is shown that this mechanism applies to all classes of transcription factors and can lead to a fast and specific search. Moreover, it is shown that the mechanism has interesting transient features which allow for stability at the target despite rapid binding and unbinding of the transcription factor from the target.

Physical limits on cooperative protein-DNA binding and the kinetics of combinatorial transcription regulation

Much of the complexity observed in gene regulation originates from cooperative protein-DNA binding. Although studies of the target search of proteins for their specific binding sites on the DNA have revealed design principles for the quantitative characteristics of protein-DNA interactions, no such principles are known for the cooperative interactions between DNA-binding proteins. We consider a simple theoretical model for two interacting transcription factor (TF) species, searching for and binding to two adjacent target sites hidden in the genomic background. We study the kinetic competition of a dimer search pathway and a monomer search pathway, as well as the steady-state regulation function mediated by the two TFs over a broad range of TF-TF interaction strengths. Using a transcriptional AND-logic as exemplary functional context, we identify the functionally desirable regime for the interaction. We find that both weak and very strong TF-TF interactions are favorable, albeit with different characteristics. However, there is also an unfavorable regime of intermediate interactions where the genetic response is prohibitively slow.

Target search on a dynamic DNA molecule

We study a protein-DNA target search model with explicit DNA dynamics applicable to in vitro experiments. We show that the DNA dynamics plays a crucial role for the effectiveness of protein "jumps" between sites distant along the DNA contour but close in three-dimensional space. A strongly binding protein that searches by one-dimensional sliding and jumping alone explores the search space less redundantly when the DNA dynamics is fast on the time scale of protein jumps than in the opposite "frozen DNA" limit. We characterize the crossover between these limits using simulations and scaling theory. We also rationalize the slow exploration in the frozen limit as a subtle interplay between long jumps and long trapping times of the protein in "islands" within random DNA configurations in solution.

Mathematical descriptions of biochemical networks: stability, stochasticity, evolution

In this paper, we review some fundamental aspects, as well as some new developments, in the emerging field of network biology. The focus of attention is placed on mathematical approaches to conceptual modeling of biomolecular networks with special emphasis on dynamic stability, stochasticity and evolution.

Transcription factor search for a DNA promoter in a three-state model

To ensure fast gene activation, transcription factors (TFs) use a mechanism known as facilitated diffusion to find their DNA promoter site. Here we analyze such a process where a TF alternates between three- and one-dimensional diffusion. In the latter (TF bound to the DNA), the TF further switches between a fast translocation state dominated by interaction with the DNA backbone, and a slow examination state where interaction with DNA base pairs (bp) is predominant. We derive a formula for the mean search time, and show that it is faster and less sensitive to the binding-energy fluctuations as compared to the case with a single sliding state. We find that for an optimal search, the time spent bound to the DNA is larger compared to the three-dimensional time, in agreement with recent experimental data.

The elastic network model reveals a consistent picture on intrinsic functional dynamics of type II restriction endonucleases

The vibrational dynamics of various type II restriction endonucleases, in complex with cognate/non-cognate DNA and in the apo form, are investigated with the elastic network model in order to reveal common functional mechanisms in this enzyme family. Scissor-like and tong-like motions observed in the slowest modes of all enzymes and their complexes point to common DNA recognition and cleavage mechanisms. Normal mode analysis further points out that the scissor-like motion has an important role in differentiating between cognate and non-cognate sequences at the recognition site, thus implying its catalytic relevance. Flexible regions observed around the DNA-binding site of the enzyme usually concentrate on the highly conserved β-strands, especially after DNA binding. These β-strands may have a structurally stabilizing role in functional dynamics for target site recognition and cleavage. In addition, hot spot residues based on high-frequency modes reveal possible communication pathways between the two distant cleavage sites in the enzyme family. Some of these hot spots also exist on the shortest path between the catalytic sites and are highly conserved.

Quasi-steady-state analysis of two-dimensional random intermittent search processes

We use perturbation methods to analyze a two-dimensional random intermittent search process, in which a searcher alternates between a diffusive search phase and a ballistic movement phase whose velocity direction is random. A hidden target is introduced within a rectangular domain with reflecting boundaries. If the searcher moves within range of the target and is in the search phase, it has a chance of detecting the target. A quasi-steady-state analysis is applied to the corresponding Chapman-Kolmogorov equation. This generates a reduced Fokker-Planck description of the search process involving a nonzero drift term and an anisotropic diffusion tensor. In the case of a uniform direction distribution, for which there is zero drift, and isotropic diffusion, we use the method of matched asymptotics to compute the mean first passage time (MFPT) to the target, under the assumption that the detection range of the target is much smaller than the size of the domain. We show that an optimal search strategy exists, consistent with previous studies of intermittent search in a radially symmetric domain that were based on a decoupling or moment closure approximation. We also show how the decoupling approximation can break down in the case of biased search processes. Finally, we analyze the MFPT in the case of anisotropic diffusion and find that anisotropy can be useful when the searcher starts from a fixed location.

Theory on the dynamic memory in the transcription-factor-mediated transcription activation

We develop a theory to explain the origin of the static and dynamical memory effects in transcription-factor-mediated transcription activation. Our results suggest that the following inequality conditions should be satisfied to observe such memory effects: (a) τ(L)≫max(τ(R),τ(E)), (b) τ(LT)≫τ(T), and (c) τ(I)≥(τ(EL)+τ(TR)) where τ(L) is the average time required for the looping-mediated spatial interactions of enhancer-transcription-factor complex with the corresponding promoter--RNA-polymerase or eukaryotic RNA polymerase type II (PolII in eukaryotes) complex that is located L base pairs away from the cis-acting element, (τ(R),τ(E)) are respectively the search times required for the site-specific binding of the RNA polymerase and the transcription factor with the respective promoter and the cis-regulatory module, τ(LT) is the time associated with the relaxation of the looped-out segment of DNA that connects the cis-acting site and promoter, τ(T) is the time required to generate a complete transcript, τ(I) is the transcription initiation time, τ(EL) is the elongation time, and τ(TR) is the termination time. We have theoretically derived the expressions for the various searching, looping, and loop-relaxation time components. Using the experimentally determined values of various time components we further show that the dynamical memory effects cannot be experimentally observed whenever the segment of DNA that connects the cis-regulatory element with the promoter is not loaded with bulky histone bodies. Our analysis suggests that the presence of histone-mediated compaction of the connecting segment of DNA can result in higher values of looping and loop-relaxation times, which is the origin of the static memory in the transcription activation that is mediated by the memory gene loops in eukaryotes.

Dancing on DNA: kinetic aspects of search processes on DNA

Recognition and binding of specific sites on DNA by proteins is central for many cellular functions such as transcription, replication, and recombination. In the search for its target site, the DNA-associated protein is facing both thermodynamic and kinetic difficulties. The thermodynamic challenge lies in recognizing and tightly binding a cognate (specific) site among the billions of other (non-specific) sequences on the DNA. The kinetic difficulty lies in finding a cognate site in mere seconds amidst the crowded cellular environment that is filled with other DNA sequences and proteins. Herein, we discuss the history of the DNA search problem, the theoretical background and the various experimental methods used to study the kinetics of proteins searching for target sites on DNA.

Rapid search for specific sites on DNA through conformational switch of nonspecifically bound proteins

We develop a theory for the rapid search of specific sites on DNA, via a mechanism in which a nonspecifically-bound protein can switch between two conformations. In the "inactive" conformation, the bound protein has favorable, nonspecific interactions with the DNA, but cannot be recognized by the target site. In the "active" conformation, the protein is recognized by the target site but has a very rugged energy surface elsewhere on the DNA. The rate constant for protein binding to the specific site is calculated by an approach in which the protein, after reaching the DNA surface via 3D diffusion, searches for the target site via 1D diffusion while being allowed to escape to the bulk solution. Mindful of the pitfalls of many previous approximate treatments, we validate our approach against a rigorous solution of the problem when the protein has a fixed conformation. In the 1D diffusion toward the specific site, a conformationally switchable protein predominantly adopts the inactive conformation due to the favorable nonspecific interactions with the DNA, thus maximizing the 1D diffusion constant and minimizing the chance of escape to the bulk solution. Once at the target site, a transition to the active conformation allows the protein to be captured. This induced-switch mechanism provides robust speedup of protein-DNA binding rates, and appears to be adopted by many transcription factors and DNA-modifying enzymes.

Universality classes of first-passage-time distribution in confined media

We study the first-passage time (FPT) distribution to a target site for a random walker evolving in a bounded domain. We show that in the limit of large volume of the confining domain, this distribution falls into universality classes indexed by the walk dimension d(w) and the fractal dimension d(f) of the medium, which have been recently identified previously [Bénichou et al., Nat. Chem. 2, 472 (2010)]. We present in this paper a complete derivation of these universal distributions, discuss extensively the range of applicability of the results, and extend the method to continuous-time random walks. This analysis puts forward the importance of the geometry, and in particular the position of the starting point, in first-passage statistics. Analytical results are validated by numerical simulations, applied to various models of transport in disordered media, which illustrate the universality classes of the FPT distribution.

The diffusion constant of a labeled protein sliding along DNA

Long ago inferred by biochemists, the linear diffusion of proteins along DNA has recently been observed at a single-molecule level using fluorescence microscopy. This imaging technique requires labeling the protein of interest with a fluorophore, usually an organic nanosized dye that is not supposed to impact the dynamics of the protein. Yet individual proteins can also be tracked using much larger labels, like quantum dots or beads. We investigate here the impact of such a large label on the protein diffusion along DNA. Solving a Fokker-Planck equation, we estimate the diffusion constant of a protein-label complex diffusing in a periodic potential that mimics the DNA-protein interaction, the link between the protein and the label being modeled as a Hookean spring. Our results indicate that the diffusion constant can generally be calculated by considering that the motion of the protein in the DNA potential is decoupled from the Brownian motion of the label. Our conclusions are in good agreement with the experimental results we obtained with the restriction enzyme EcoRV, assuming a rotation-coupled diffusion of the enzyme along DNA.

Facilitated diffusion of proteins on chromatin

We present a theoretical model of facilitated diffusion of proteins in the cell nucleus. This model, which takes into account the successive binding and unbinding events of proteins to DNA, relies on a fractal description of the chromatin which has been recently evidenced experimentally. Facilitated diffusion is shown quantitatively to be favorable for a fast localization of a target locus by a transcription factor and even to enable the minimization of the search time by tuning the affinity of the transcription factor with DNA. This study shows the robustness of the facilitated diffusion mechanism, invoked so far only for linear conformations of DNA.

Physics of protein-DNA interactions: mechanisms of facilitated target search

One of the most critical aspects of protein-DNA interactions is the ability of protein molecules to quickly find and recognize specific target sequences on DNA. Experimental measurements indicate that the corresponding association rates to few specific sites among large number of non-specific sites are typically large. For some proteins they might be even larger than maximal allowed three-dimensional diffusion rates. Although significant progress in understanding protein search and recognition of targets on DNA has been achieved, detailed mechanisms of these processes are still strongly debated. Here we present a critical review of current theoretical approaches and some experimental observations in this area. Specifically, the role of lowering dimensionality, non-specific interactions, diffusion along the DNA molecules, protein and target sites concentrations, and electrostatic effects are critically analyzed. Possible future directions and outstanding problems are also presented and discussed.

Theory of site-specific DNA-protein interactions in the presence of conformational fluctuations of DNA binding domains

We develop a theory that explains how the thermally driven conformational fluctuations in the DNA binding domains (DBDs) of the DNA binding proteins (DBPs) are effectively coupled to the one-dimensional searching dynamics of DBPs for their cognate sites on DNA. We show that the rate gammaopt, associated with the flipping of conformational states of DBDs beyond which the maximum search efficiency of DBPs is achieved, varies with the one-dimensional sliding length L as gammaopt proportional, L(-2) and with the number of roadblock protein molecules present on the same DNA m as gammaopt proportional, m2. The required free energy barrier ERTO associated with this flipping transition seems to be varying with L as ERTO proportional, variant ln L2. When the barrier height associated with the conformational flipping of DBDs is comparable with that of the thermal free energy, then the possible value of L under in vivo conditions seems to be L<or=70 bps.

Dynamic strategies for target-site search by DNA-binding proteins

Gene regulatory proteins find their target sites on DNA remarkably quickly; the experimental binding rate for lac repressor is orders-of-magnitude higher than predicted by free diffusion alone. It has been proposed that nonspecific binding aids the search by allowing proteins to slide and hop along DNA. We develop a reaction-diffusion theory of protein translocation that accounts for transport both on and off the strand and incorporates the physical conformation of DNA. For linear DNA modeled as a wormlike chain, the distribution of hops available to a protein exhibits long, power-law tails that make the long-time displacement along the strand superdiffusive. Our analysis predicts effective superdiffusion coefficients for given nonspecific binding and unbinding rate parameters. Translocation rate exhibits a maximum at intermediate values of the binding rate constant, while search efficiency is optimized at larger binding rate constant values. Thus, our theory predicts a region of values of the nonspecific binding and unbinding rate parameters that balance the protein translocation rate and the efficiency of the search. Published data for several proteins falls within this predicted region of parameter values.

Facilitated search of proteins on DNA: correlations are important

A starting point of many biological processes is protein binding to specific regions on DNA. Although typical concentrations of DNA-binding proteins are low, and target sites are typically buried among huge number of non-specific sites, the search process is frequently achieved at a remarkably fast rate. For some proteins it has been confirmed that association rates might be even larger than the maximal allowed three-dimensional diffusion rates. The current theoretical view of this phenomenon is based on the idea of lowering dimensionality, i.e., the overall search process is viewed as a combination of uncorrelated three-dimensional excursions in the solution and one-dimensional hoppings on DNA. However, some predictions of this theoretical picture contradict recent single-molecule measurements of protein diffusion processes. An alternative theoretical approach points out the importance of correlations during the search process that appear due to non-specific interactions between protein and DNA molecules. To test different theoretical ideas we performed extensive lattice Monte Carlo computer simulations of the facilitated diffusion. Our results revealed that correlations are important, and the acceleration in the search could only be achieved at some intermediate non-specific binding energies and protein concentrations. Physico-chemical aspects and the origins of these correlations are discussed.

Survival probability of a subdiffusive particle in a d-dimensional sea of mobile traps

We investigate the long-time behavior of the survival probability P(t) of a mobile particle in d-dimensional continuous Euclidean media doped with noninteracting mobile traps. The particle is strictly subdiffusive, implying that its mean-square displacement grows as tgamma' with 0<gamma'<1. Initially, the traps are scattered randomly and their subsequent mean-square displacement grows as tgamma with 0<gamma<or=1. Instantaneous annihilation of the particle takes place upon contact with any of the traps. The solution to this problem is obtained by deriving lower and upper asymptotic bounds of the survival probability and showing that they converge to one another for long times, thereby unambiguously determining the long-time decay of P(t). For d>or=2 we find that at late times the survival probability is that of the pure target problem (the problem where the particle remains immobile) in agreement with previous studies for the d=1 case. These decay laws remain invariant over the whole gamma range as opposed to the dynamical crossover observed for the case of a purely diffusive particle (gamma'=1) where, for gamma<2/(2+d) , the survival probability becomes that of the so-called trapping problem (the problem where the particle moves in a sea of static traps). This behavior implies that for sufficiently low values of gamma(gamma<2/(2+d)) the survival probability becomes singular in the limit gamma'-->1: trappinglike for gamma'=1 and targetlike for any gamma'<1.

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.

Diffusion of the restriction nuclease EcoRI along DNA

Many specific sequence DNA binding proteins locate their target sequence by first binding to DNA nonspecifically, then by linearly diffusing or hopping along DNA until either the protein dissociates from the DNA or it finds the recognition sequence. We have devised a method for measuring one-dimensional diffusion along DNA based on the ratio of the dissociation rate of protein from DNA fragments containing one specific binding site to the dissociation rate from DNA fragments containing two specific binding sites. Our extensive measurements of dissociation rates and specific-nonspecific relative binding constants of the restriction nuclease EcoRI enable us to determine the diffusion rate of nonspecifically bound protein along the DNA. By varying the distance between the two binding sites, we confirm a linear diffusion mechanism. The sliding rate is relatively insensitive to salt concentration and osmotic pressure, indicating that the protein moves smoothly along the DNA probably following the helical phosphate-sugar backbone of DNA. We calculate a diffusion coefficient for EcoRI of 3 x 10(4) bp(2) s(-)(1) EcoRI is able to diffuse approximately 150 bp, on average, along the DNA in 1 s. This diffusion rate is about 2000-fold slower than the diffusion of free protein in solution. A factor of 40-50 can be accounted for by rotational friction resulting from following the helical path of the DNA backbone. Two possibilities could account for the remaining activation energy: salt bridges between the DNA and the protein are transiently broken, or the water structure at the protein-DNA interface is disrupted as the two surfaces move past each other.

Characteristics of transcriptional activity in nonlinear dynamics of genetic regulatory networks

Microarray measurements of mRNA abundances is a standard tool for evaluation of transcriptional activity in functional genomics. The methodology underlying these measurements assumes existence of a direct link between transcription levels, that is, gene-specific mRNA copy numbers present in the cell, and transcription rates, that is, the numbers of gene-specific mRNA molecules synthesized per unit of time. In this paper, the question of whether or not such a tight interdependence may exist is examined in the context of nonlinear dynamics of genetic regulatory networks. Using the equations of chemical kinetics, a model has been constructed that is capable of explicitly taking into consideration nonlinear interactions between the genes through the teamwork of transcription factors. Jacobian analysis of stability has shown that steady state equilibrium is impossible in such systems. However, phase space compressibility is found to be negative, thus suggesting that asymptotic stability may exist and assume either the form of limit cycle or of a chaotic attractor. It is argued that in rapidly fluctuating or chaotic systems, direct evaluation of transcription rates through transcription levels is highly problematic. It is also noted that even if a hypothetical steady state did exist, the knowledge of transcription levels alone would not be sufficient for the evaluation of transcription rates; an additional set of parameters, namely the mRNA decay rates, would be required. An overall conclusion of the work is that the measurements of mRNA abundances are not truly representative of the functionality of genes and structural fidelity of the genetic codes.

Searching fast for a target on DNA without falling to traps

Genomic expression depends critically on both the ability of regulatory proteins to locate specific target sites on DNA within seconds and on the formation of long-lived (many minutes) complexes between these proteins and the DNA. Equilibrium experiments show that indeed regulatory proteins bind tightly to their target site. However, they also find strong binding to other nonspecific sites which act as traps that can dramatically increase the time needed to locate the target. This gives rise to a conflict between the speed and stability requirements. Here we suggest a simple mechanism which can resolve this long-standing paradox.

Robustness of optimal intermittent search strategies in one, two, and three dimensions

Search problems at various scales involve a searcher, be it a molecule before reaction or a foraging animal, which performs an intermittent motion. Here we analyze a generic model based on such type of intermittent motion, in which the searcher alternates phases of slow motion allowing detection and phases of fast motion without detection. We present full and systematic results for different modeling hypotheses of the detection mechanism in space in one, two, and three dimensions. Our study completes and extends the results of our recent letter [Loverdo, Nat. Phys. 4, 134 (2008)] and gives the necessary calculation details. In addition, another modeling of the detection case is presented. We show that the mean target detection time can be minimized as a function of the mean duration of each phase in one, two, and three dimensions. Importantly, this optimal strategy does not depend on the details of the modeling of the slow detection phase, which shows the robustness of our results. We believe that this systematic analysis can be used as a basis to study quantitatively various real search problems involving intermittent behaviors.

Quantifying hopping and jumping in facilitated diffusion of DNA-binding proteins

Facilitated diffusion of DNA-binding proteins is known to speed up target site location by combining three dimensional excursions and linear diffusion along the DNA. Here we explicitly calculate the distribution of the relocation lengths of such 3D excursions, and we quantify the short-range correlated excursions, also called hops, and the long-range uncorrelated jumps. Our results substantiate recent single-molecule experiments that reported sliding and 3D excursions of the restriction enzyme EcoRV on elongated DNA molecules. We extend our analysis to the case of anomalous 3D diffusion, likely to occur in a crowded cellular medium.

Facilitated diffusion with DNA coiling

When DNA-binding proteins search for their specific binding site on a DNA molecule they alternate between linear 1-dimensional diffusion along the DNA molecule, mediated by nonspecific binding, and 3-dimensional volume excursion events between successive dissociation from and rebinding to DNA. If the DNA molecule is kept in a straight configuration, for instance, by optical tweezers, these 3-dimensional excursions may be divided into long volume excursions and short hops along the DNA. These short hops correspond to immediate rebindings after dissociation such that a rebinding event to the DNA occurs at a site that is close to the site of the preceding dissociation. When the DNA molecule is allowed to coil up, immediate rebinding may also lead to so-called intersegmental jumps, i.e., immediate rebindings to a DNA segment that is far away from the unbinding site when measured in the chemical distance along the DNA, but close by in the embedding 3-dimensional space. This effect is made possible by DNA looping. The significance of intersegmental jumps was recently demonstrated in a single DNA optical tweezers setup. Here we present a theoretical approach in which we explicitly take the effect of DNA coiling into account. By including the spatial correlations of the short hops we demonstrate how the facilitated diffusion model can be extended to account for intersegmental jumping at varying DNA densities. It is also shown that our approach provides a quantitative interpretation of the experimentally measured enhancement of the target location by DNA-binding proteins.

Diffusion, dimensionality, and noise in transcriptional regulation

The precision of biochemical signaling is limited by randomness in the diffusive arrival of molecules at their targets. For proteins binding to specific sites on DNA and regulating transcription, the ability of the proteins to diffuse in one dimension by sliding along the length of the DNA, in addition to their diffusion in bulk solution, would seem to generate a larger target for DNA binding, consequently reducing the noise in the occupancy of the regulatory site. Here we show that this effect is largely canceled by the enhanced temporal correlations in one-dimensional diffusion. With realistic parameters, sliding along DNA has surprisingly little effect on the physical limits to the precision of transcriptional regulation.