New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Facilitated diffusion with DNA coiling

Communicated by Joshua Jortner, Tel Aviv University, Tel Aviv, Israel, March 27, 2009 (received for review January 26, 2009)
Abstract
When DNAbinding proteins search for their specific binding site on a DNA molecule they alternate between linear 1dimensional diffusion along the DNA molecule, mediated by nonspecific binding, and 3dimensional 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 3dimensional 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 socalled 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 3dimensional 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 DNAbinding proteins.
At any given instant of time a biological cell only uses part of its genes for production of other molecules (RNA, proteins). During the development of the cell, or the organism the cell belongs to, different genes may be turned on. A classical example is the lactose metabolism of bacteria: in absence of lactose the Lac repressor is specifically bound at the lacZ gene and prevents unnecessary production of the lactose enzyme used to digest the milk sugar; when only lactose and no glucose is present the lacZ gene is activated and lactosedigesting enzymes are produced. Equally famed is the λ switch in Escherichia coli bacteria that are infected by bacteriophage λ. There, the λ switch decides between the dormant lysogenic state or the state of lysis that leads to the production of new phages and ultimate death of the E.coli host cell. Molecularly, the regulation of a gene relies on the presence of certain DNAbinding proteins, socalled transcription factors, that bind to a specific binding site close to the starting sequence of the related gene. The transcription factor then promotes or inhibits binding of RNA polymerase and thus regulates the transcription of this gene. This strategy allows eukaryotic and prokaryotic cells as well as viruses to respond to different internal or external signals (1, 2).
The binding of the protein to its specific binding site (cognate site, “target”) is necessarily preceded by a search process of the protein for this target site, this search being of a purely stochastic nature. Yet, some active binding proteins can locate their target orders of magnitude faster (3) than estimated for pure 3dimensional diffusion limited search according to the celebrated Smoluchowski result for the search rate k_{on}. Here, the target is assumed spherical and of radius b, whereas D_{3d} denotes the 3dimensional diffusion coefficient D_{3d} (4). An explanation for this discrepancy is based on the linear topology of the DNA chain in which the target site is embedded. In fact, in their seminal work Berg and von Hippel analyzed the search by DNAbinding proteins and demonstrated that the combination of different search mechanisms explains the observed binding rates. According to the Berg–von Hippelfacilitated diffusion model (5) DNAbinding proteins combine 3dimensional diffusion in the bulk and 1dimensional diffusion along the DNA chain (6–9). This linear diffusion is mediated by nonspecific binding, the protein's binding affinity to noncognate DNA made possible by the heteropolymeric nature of DNA (10–12). An additional search mechanism already proposed in the works of Berg and von Hippel are intersegmental transfers: some proteins with two DNAbinding sites can intermittently bind nonspecifically to 2 DNAsegments that are far apart along the DNA contour but are brought in close contact in the embedding (real) space by DNA looping. Such proteins thereby transfer across a point of close contact without actually leaving the DNA (5).
Facilitated diffusion has recently received renewed interest, following a variety of new conceptual viewpoints of gene regulation (13–18). Fresh impetus has also been given by the possibility to monitor the target search by DNAbinding proteins on the level of single DNA molecules. Various aspects of the search mechanisms of DNAbinding proteins have been studied either by fluorescence methods or indirectly by probing the response to external manipulation by optical tweezers (6–8, 19, 20). In particular, it was demonstrated that the local DNA density indeed significantly influences the target search rate (20). This observation is consistent with the Berg–von Hippel idea of intersegmental transfers mentioned above. However, in contrast to the Lac repressor protein employed in the original work by Berg and von Hippel, in the optical tweezers assay reported in ref. 20 the target search was performed by EcoRV restriction enzymes that only have one DNAbinding site. The idea of intersegmental transfers therefore has to be extended, such that the protein dissociates from one DNA segment, shortly diffuses threedimensionally, and then immediately rebinds to another DNA segment that is close by because of DNA looping. We refer to this extended mechanism as intersegmental jumps. In what follows we develop a theoretical framework to include intersegmental jumps. This extension of the facilitated diffusion model allows us to study the influence of the local DNA density on the rate of search of DNAbinding proteins for their specific binding site. Our generalized model therefore accounts for different degrees of DNA coiling. We show that this systematic extension of the Berg–von Hippel model is quantitative and consistent with the results from our experimental study in ref. 20.
Model
Previous approaches to facilitated protein diffusion consider bulk exchange for a straight cylindrical chain (21); they invoke scaling properties in different chain configurations (22) without quantitative comparison of the search rates between these; or they reformulate the problem with uncorrelated bulk excursions (23–26). Here, we relate a multiparticle picture to single protein jumps along the DNA contour allowing us to consider DNA as a fluctuating chain.
In our description of the target search process, we use the density per length n(x,t) of proteins on the DNA as the relevant dynamical quantity (x is the distance along the DNA contour). We include 1dimensional diffusion along the DNA with diffusivity D_{1d}, protein dissociation with rate k_{off}^{ns} and (re)adsorption after diffusion through the bulk fluid with diffusion constant D_{3d}. The dynamics of n(x,t) is thus governed by Here, j(t) is the flux into the target site located at x = 0, W_{bulk}(x − x′,t − t′) is the joint probability that a protein returns to the point x at time t after leaving the DNA at x′, t′ for a bulk excursion. G(x,t) is the flux onto the DNA of proteins that have not previously been bound to the DNA (virgin flux). j(t) is determined by imposing the boundary condition n(x = 0,t) = 0 at the target. To consider the transfer kernel W_{bulk} as homogeneous in space and time we made 2 assumptions that we found to be valid for the system studied in ref. 20: First, that the DNA can be treated as being infinitely long when considering the search rate. In practice this requires that the DNA should be longer than the effective sliding length l_{sl}^{eff} introduced in Eq. 14. Note that even in bacteria the DNA length measures several millimeters, whereas DNA's persistence length is of the order of 50 nm. The second assumption is that the coiled DNA conformation fluctuates sufficiently quickly such that subsequent excursion distances x − x′ can be treated as independent.
To proceed, we Laplace and Fourier transform Eq. 2: where we denote the Fourier and Laplace transform of a function by explicit dependence on the transformed variable. Thus n(q,u) = L[n(q,t)], n(q,t) = F[n(x,t)] and n_{0}(x) = n(x,t = 0). The Fourier–Laplace transform of W(x,t) is here defined through We rewrite these equations in the form where the density is the solution of the diffusion problem in absence of the target, i.e., when only nonspecific binding between proteins and the DNA occurs. In position space at x = 0 one obtains j(u) = n^{nsx}(x = 0,u)/W(x = 0,u).
In this study we focus on the long time behavior of j(t) for which the density n^{nsx}(x,t) will reach an equilibrium value n_{eq}^{ns}. Since, by Tauberian theorems, large t correspond to small u we study the u → 0 behavior of j(u), with n^{ns}(x,u) ∼ n_{eq}^{ns}/u. For W_{0}(x = 0,u) we assume that the limit W(x = 0,u = 0) (time average) is finite and nonzero; this is true when the distance distribution of bulk excursions is sufficiently long tailed, as fulfilled for the transfer kernel W_{bulk} encountered below. In practice, being interested in the first arrival at the target, we assume that the protein concentration is sufficiently dilute such that this arrival happens after reaching n_{eq}^{ns} (“rapid equilibrium”) but still with many searching proteins. Thus j(u) ∼ k_{1d}n_{eq}^{ns}/u, and therefore at long times the stationary current j_{stat} ∼ k_{1d}n_{eq}^{ns} into the target is reached. Here, we introduce the constant k_{1d}^{−1} = W(x = 0,u = 0): of dimension sec over cm. We also recognize that λ_{bulk}(x) = W_{bulk}(x,u = 0) is the distribution of relocation lengths along the DNA after a single 3dimensional excursion.
We express our results in terms of the association rate to the target where n_{bulk} is the density of unbound proteins in the bulk (at nonspecific equilibrium). The nonspecific binding constant per length of DNA is K_{ns} = n_{eq}^{ns}/n_{bulk}. In terms of the total volume concentration of proteins n_{total} = n_{bulk} + l_{DNA}^{total}n_{eq}^{ns}, where l_{DNA}^{total} is the total length of DNA divided by the volume of the entire system, we have The rate k_{on} is related to the mean first arrival time for the steadystate association. Namely, k_{on}n_{bulk} is the probability per time for protein association with the target. Thus, the (survival) probability that no protein has arrived at the target yet is P_{surv}(t) = exp(−k_{on}n_{bulk}t), and the average target search time is T = 1/(k_{on}n_{bulk}). The flux j(t) being linear in the protein concentration n_{total} implies that for sufficiently low n_{total} the steady state is indeed reached well before the first protein actually binds to the specific target (note that in vivo protein concentrations can be as low as nanomolar).
Results
Straight Rod Configuration.
Consider first stretched DNA (see Fig. 1) corresponding to a cylinder of radius r_{int} (the range of nonspecific interaction), with nonspecific reaction rate k_{on}^{ns} at the boundary: the flux of proteins, per length of DNA, onto the DNA is k_{on}^{ns} times the bulk concentration next to the DNA. This implies that at equilibrium of nonspecific binding, n_{eq}^{ns}k_{off}^{ns} = n_{bulk}k_{on}^{ns} such that K_{ns} = k_{on}^{ns}/k_{off}^{ns}. (We deviate here from the conventions of ref. 20 by only letting n_{bulk} count “active enzymes.”)
To find the distribution W_{bulk}(x,t) for return to the DNA of a protein released at the DNA at x = 0 and t = 0, we solve the cylindrical diffusion equation for the probability density P(x,r,t) of the protein's position
where r measures the distance to the cylinder axis. The reactive boundary condition for the probability flux out of the cylinder boundary is
The initial distribution outside the cylinder is P(r,t)_{t=0} = 0. From the solution of this problem we obtain W_{bulk} as k_{on}^{ns} P_{r=rint} in terms of modified Bessel functions K_{ν},
where we abbreviate
With these ingredients one can evaluate numerically the search rate to reach the specific target on a straight DNA, as given by Eq. 7. However, some limits allow for an analytic approximation for the integral in Eq. 7: for k_{on}^{ns} ≪ D_{3d} one may take 1 − λ_{bulk}(q) ≈ 1, and thus
In the opposite limit k_{on}^{ns} ≫ D_{3d} one may approximate 1 − λ_{bulk}(q) ≈ 2πD_{3d}/[k_{on}^{ns}/ln(l_{sl}^{eff}/r_{int})] where the length This produces the result In this limit the protein can rebind quickly after unbinding. This leads to oversampling and a lowered value of k_{on} compared to Eq. 13. l_{sl}^{eff} can be interpreted as an effective sliding length including immediate rebindings (21). Expression 15 generalizes Smoluchowski's result Eq. 1, replacing the naked target size b by the “antenna length” entering Eq. 15. Results 7 and 12 are identical to the classical result for straight, infinitely long DNA (21). However, the approach here allows us to explicitly consider coiled DNA.
Coil with Persistence Length.
To obtain the transfer kernel W_{bulk} for a random coil DNA configuration (see Fig. 2) we treat the DNA as consisting locally of straight segments. We assume that the local density l_{DNA} of DNA (length per volume) around a segment is dilute such that the typical distance
Discussion
A single stochastic search mechanism is usually not efficient at optimizing a search process. Thus, Brownian motion, whose fractal dimension is d_{f} = 2, in 1dimensional and 2dimensional geometries leads to significant oversampling in the sense that a given site is revisited many times (27, 28). Lévy motion with a jump length distribution λ(x) ≃ x^{−1−α} (0 < α < 2) has fractal dimension d_{f} = α and thus a comparatively reduced degree of oversampling. However, the first arrival of Lévy motion is hampered by leapovers, i.e., frequent overshoots of the target (29). In 3dimensional space both Brownian and Lévy motions are inefficient at locating a small target. Whereas typical chemical reactants in a liquid environment have no choice but to passively diffuse until mutual encounter (4) intermittent models combine different search mechanisms (28, 30). An important example of intermittent search models is the facilitated diffusion of transcription factors in gene regulation.
Here, we have developed a theoretical framework analyzing facilitated diffusion in the presence of intersegmental jumps whose relevance depends on the degree of DNA coiling. These results together with the experimental evidence from our single DNA experiments reported in ref. 20 demonstrate that the binding rate of DNAbinding proteins can be significantly enhanced when DNA is allowed to coil freely. This effect would be further amplified when the local density of DNA is increased beyond that of a relaxed coil, for instance, by packaging in a living cell. This is consistent with observations of increased cleavage rate of restriction enzymes in supercoiled over relaxed DNA plasmids (31).
In our derivation of the probability P_{surv}^{foreign} we assumed that the density of foreign segments is uniform in space. Although this is not true for a freely fluctuating coil (the density decreases on a scale of the persistence length of the DNA) we argue that this nonuniformity is not important for evaluating the search rate: in fact, if the protein diffuses far away from the original segment it will most likely perform a long relocation measured along the DNA contour. Again it does not matter for the search rate how we bookkeep these long relocations. To estimate the density l_{DNA} for a random relaxed coil we employ the Worm Like Chain model. A good approximation for the probability density to loop back, for a point on the chain a contour distance s away from another point on the chain, is (32)
For a target in the middle of a chain of length L, this leads to the DNA density
The theory presented here quantitatively describes recent experimental data from an optical tweezers setup reported in ref. 20. In that assay the search rate k_{on} was measured at various degrees of DNA coiling for a DNA of length L = 6,538 base pairs (approximately 2.24 μ m corresponding to 42 persistence lengths). The relative increase R of k_{on} in the maximally relaxed DNA configuration compared with k_{on} in the stretched state is shown in Fig. 3 as a function of the NaCl concentration. R was measured to be ≈ 1.1 – 1.3 in buffers with salt concentrations of 5 mM MgCl_{2}, and 0–25 mM for NaCl, respectively. This relative increase R compares well with the theory presented here by using estimates of l_{DNA} based on Eq. 23 giving
We presented a model for facilitated diffusion that is useful to calculate the search rate of DNAbinding proteins for their specific binding site on a fluctuating DNA coil with variable DNA density. This model represents a convenient way to rephrase the Berg–von Hippel theory with the additional advantage that it interpolates between stretched and coiled DNA configurations. In the development of this theory a number of assumptions were made, in particular, that of sufficiently fast local relaxation of the DNA configuration. This assumption was found to be satisfied for the protein EcoRV used in ref. 20, but it will not hold universally. It would therefore be beneficial to generalize the theory to slowly fluctuating or static (quenched) DNA configurations, potentially leading to effects similar to those found in polymer models (36). Another interesting question for future work concerns the potential effect on in vivo gene regulation of the slow diffusion caused by molecular crowding (37, 38), including interesting consequences for the exchange dynamics of proteins with the DNA (39).
Acknowledgments
We thank Tobias Ambjörnsson, Alexander Grosberg, Mehran Kardar, Leonid Mirny, and Roland Winkler for discussions. This work was partially funded by the Villum Kann Rasmussen Foundation, Deutsche Forschungsgemeinschaft, and by a Fundamenteel Onderzoek der Materie Projectruimte Grant, a Netherlands Organization for Scientific ResearchVernieuwings Impuls grant.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: metz{at}ph.tum.de

Author contributions: M.A.L., B.v.d.B., S.M.J.K., G.J.L.W., and R.M. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

↵* Cases where this assumption is justified are given by systems in which (i) the persistence length of the DNA is large such that the chemical distance along DNA loops will necessarily be larger than the effective sliding length l_{sl}^{eff}, or (ii) the DNA attains a globular configuration, for instance, through mutual attraction bringing many chemically distant DNA segments together.
References
 ↵
 Alberts B,
 et al.
 ↵
 Ptashne M
 ↵
 ↵
 von Smoluchowski M
 ↵
 von Hippel PH,
 Berg OG
 ↵
 ↵
 Elf J,
 Li GW,
 Xie XS
 ↵
 ↵
 Gowers DM,
 Wilson GG,
 Halford SE
 ↵
 ↵
 ↵
 ↵
 ↵
 Walczak AM,
 Onuchic JN,
 Wolynes PG
 ↵
 ↵
 Kolesov G,
 Wunderlich Z,
 Laikova ON,
 Gelfand MS,
 Mirny LA
 ↵
 ↵
 ↵
 Bonnet I,
 et al.
 ↵
 van den Broek B,
 Lomholt MA,
 Kalisch SMJ,
 Metzler R,
 Wuite GJL
 ↵
 ↵
 ↵
 Halford SE,
 Marko JF
 ↵
 ↵
 ↵
 Lomholt MA,
 Ambjörnsson T,
 Metzler R
 ↵
 Hughes BD
 ↵
 Lomholt MA,
 Koren T,
 Metzler R,
 Klafter J
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
Citation Manager Formats
Sign up for Article Alerts
Jump to section
You May Also be Interested in
More Articles of This Classification
Biological Sciences
Related Content
 No related articles found.