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* Laboratoire de Physique Théorique des Liquides, Université Pierre et Marie Curie, Paris, France; and Laboratoire de Physique de la Matière Condensée, Collège de France, Paris, France
Correspondence: Address reprint requests to Mathieu Coppey, Tel.: 33-1-442-77291; E-mail: coppey{at}lptl.jussieu.fr.
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ABSTRACT |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODELRESULTSCONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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; Riggs et al., 1970). To account for this, it was proposed that the reaction occurs via a facilitated diffusion process (Von Hippel and Berg, 1989). The restriction enzyme first binds to DNA on a nonspecific site, then performs a one-dimensional random walk until it reaches the target DNA sequence. Indeed, it is by scanning the DNA and not by diffusing in a three-dimensional volume that the restriction enzyme reaches its target site sequence. However, results from experiments (Szczelkun and Halford, 1996) using two interlinked rings of DNA (plasmid, each containing a target site for the restriction enzyme EcoRV) rule out this possibility: the mechanism of target site localization does not involve a unique one-dimensional diffusion along DNA. If it were the case, the EcoRV enzyme would cleave the DNA of only one of the two rings, as opposed to what is observed. Moreover, it is expected that molecular crowding of in vivo situations must hinder any long one-dimensional scanning process of the DNA (Wenner and Bloomfield, 1999).
To account for the fast association rate, several strategies have been proposed and modeled from experimental data (Berg et al., 1981; Von Hippel and Berg, 1989; Winter et al., 1981). Four major translocation processes were identified (we recall that translocation is the overall process by which a protein goes from one DNA sequence to another). The first, the sliding process, corresponds to the pure one-dimensional diffusion as discussed above. The second, the intersegmental transfer (Milsom et al., 2001), involves dimer proteins having two binding sites. The restriction enzyme bound on DNA at the first site binds its second site to a remote DNA sequence and then dissociates from the first one. The two other translocation processes are induced by several dissociation-reassociation events. According to the rebinding of the enzyme either near the departure site or to an uncorrelated site, the translocation process is called hopping or jumping (Halford and Szczelkun, 2002). Which of these translocation processes or which combination of them describes the mechanism of target site localization on DNA is still an open question.
Understanding the translocation process is of great importance as it governs the kinetics of genetic events (Misteli, 2001). Several experimental investigations were carried out to elucidate the pathway followed by a restriction enzyme to reach a single target site. Some of them quantify the rate of cleavage reactions, by varying the length of the DNA strand (for a review, see Shimamoto, 1999) or the salt concentration (Winter et al., 1981; Lohman, 1996) which affects the binding properties of DNA-affine proteins on nonspecific sequences. These experimental results allow one to reject the possibility of a unique translocation process, but cannot fully describe the structure of the combined process. Berg et al. (1981) had proposed a theoretical approach to quantify the relevant parameters of the localization of a single target site. Their model describes the overall searching process comprising the primary encounter of the enzyme with a DNA domain and the secondary encounter of the enzyme with the target site. Here we deal with the unvisited case of two competitive target sites to quantitatively analyze the physical properties of the second encounter, i.e., the target site localization of a restriction enzyme initially bounded to the DNA. Only the study of such systems gives access to the detailed pathway of secondary encounter with well-defined initial conditions. Related experimental studies with two differentiable target sites located at well-defined positions on the DNA strand (Langowski et al., 1983; Terry et al., 1985; Stanford et al., 2000) allow one to handle two descriptive quantities: the preference and the processivity of the restriction enzymes. The preference is the ratio of the number of enzymes that react with one target site, over the number of enzymes that react with the other target site. The processivity is the fraction of enzymes that will react successively with the two target sites. To extract from these experiments physical parameters of the enzyme pathway such as the proportion of time spent by the enzyme on the DNA, the average number of dissociation-association events and the average DNA length scanned before the target site localization, it is necessary to build a reliable physical model that can mimic the biological situation.
Here, we propose a simple and general stochastic model to describe the kinetics of target site localization of a restriction enzyme on DNA, which explicitly combines any one-dimensional motion along the DNA and three-dimensional excursions in the solution. In the particular case of one-dimensional diffusing motion, our model allows us to recover the analytic expression for the mean time needed for the enzyme to find a single target site on DNA given by Berg et al. (1981). This mean time presents an optimum, corresponding to the quickest finding strategy that can be discussed in the cases of point-like and extended target sites. The model explicitly gives the mean number of enzyme visits on the DNA and the proportion of the DNA visited until the target site is localized. For two target sites, our model provides theoretical expressions for the preference and the processivity factors. These expressions involve two unknown physical parameters: the one-dimensional and three-dimensional residence frequencies 
and '. We show that is easily evaluated from the confrontation of the theoretical preference to experimental data. The second unknown parameter ', of minor physical relevance, is extracted from the assumption that the searching strategy is optimal which will be justified. The comparison of the theoretical processivity factor to experimental data allows us to predict the value of a dynamic-associated parameter: the probability that after an excursion the enzyme will associate to the same DNA substrate it has left, 
r.
The article is constructed as follows: first we give the general background of such an approach and we present the hypothesis of our model. Then we deduce the mean search time from the study of the density of the first time passage, and for the cases of point-like and extended target sites we discuss the optimal strategy for finding the target site as quickly as possible. We give the condition of existence of this optimal strategy as well as its quantitative characteristics. We discuss the value of the optimal one-dimensional frequency and evaluate finite-size effects. Equation 12 gives the mean target site localization time for an enzyme which starts from a random position on the DNA. The complete distribution of the number of visits of the protein on the DNA is explicitly determined. In particular, its mean value is given by Eq. 18. The average number of distinct basepairs (bp) visited on the DNA is given by Eq. 21. Second, the preference and the processivity factors of the restriction enzyme for two target sites, as functions of the distance between the target sites, are obtained (Eqs. 36 and 39) and compared with experimental results concerning EcoRV (Stanford et al., 2000). The comparison gives us the residence time on the DNA per binding event and other related physical quantities. We then numerically obtain the mean time needed for the enzyme to go from the first target site to the second target site (using Eq. 37), and the mean number of visits on the DNA substrate before the two target sites are cleaved. In conclusion, we discuss the predicted value of r defined previously.
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MODEL |
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. This law relies on a Markovian description of the chemical bond which is commonly used. The probability for the protein to still be bound to DNA at a random time t (knowing that it is bound at t = 0) is then P(T > t) = exp(–t), and the probability that the protein leaves the DNA at a random time T in the interval [t, t + dt] is P(t < T < t + dt) = exp(–t)dt.
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), we assume that the extremities of the DNA chain act on the protein as reflecting boundaries. Thus, a protein when reaching an extremity during a binding event is reflected and continues its one-dimensional motion. The target site sequence is a specific sequence of basepairs (e.g., the restriction enzyme, EcoRV, recognizes the sequence GATATC (Taylor and Halford, 1989). The reaction occurs when the reactive domain of the protein matches the target site sequence. To a first approximation, we model the target site sequence as being a perfect reactive point (Fig. 2). The reaction is assumed to be infinitely fast as soon as the protein meets the target site. Note that in this case the protein can find the target site only by diffusing along DNA. The precise mechanism of this elementary act is still subject to discussion. In particular, the profile of the DNA-protein interaction potential is unknown, and could be attractive over an extended area. It is then reasonable also to treat the case where the target site is a zone of finite extension 2r (Fig. 3). In that case the target site can then be reached either by diffusion along the DNA, or by coming directly from a three-dimensional excursion. This second approach, developed further, gives rise to strongly different behavior of the search time.
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) that, for not-excessively concentrated long molecules in solution, the DNA strands form disjoint domains diluted in the medium. A protein which reaches such a DNA domain will be trapped in it. In this case excursions might be correlated due to the geometric configuration of the DNA. As the configuration of a polymer strand in solution is a random coil, even short three-dimensional excursions can lead to a long effective translocation of the linear position of the protein on DNA. Consequently, a small number of long-range transitions is sufficient to uncorrelate the protein position on DNA.
We now introduce three basic quantities used in this work. The first one, P3D(t), is the probability density that the protein in the solution at time t = 0 will bind DNA at time t at a random position,
 | (1) |
' corresponding to a mean time spent in the surrounding solution 
' = 1/'. Accounting rigorously for the entire law is beyond the scope of this work. Rather we concentrate here on the characteristic time ', which exists and is finite as soon as the system is confined; and the exponential tail of the law, which proves to be valid in most plausible geometries. We will show that this model captures the main relevant characteristics of the problem.
The second quantity, P1D(t|x), is the conditional probability density that the protein, being on the DNA at position x and at time t = 0, will dissociate at time t without any encounter with the target site. Assuming that the dissociation rate is independent of the state of the protein, one has
 | (2) |
The last quantity, 
is the conditional probability density that the protein, being on DNA at position x and at time t = 0, will find the target site for the first time at time t during its one-dimensional diffusion, without leaving the DNA:
 | (3) |
First passage density
By calculating the first passage density, we obtain the mean time needed for the protein to find its specific target site, as well as all associated moments. We assume that the protein starts at t = 0 linked to the DNA at position x. We consider a generic event (Fig. 2) whose bulk number of excursions is n–1, the residence times on DNA t1,...,tn, and the excursion times 1,...,n–1. The probability density of such an event, for which the protein finds the target site for the first time (t = time), 
is
 | (4) |
are averaged over the initial position of the protein as 
and 
We denote by M the DNA length on the "left" side of the target site and by L the length on the "right" side of the target site. The average of a function f over the initial position x is given by 
To obtain the density of first passage at the target site, F(t|x), we sum over all possible numbers of excursions and we integrate over all intervals of time, ensuring that 
The average over the initial position of the protein, 
can be expressed as
 | (5) |
we obtain
 | (6) |
being the Laplace transform of j(t|x). This expression completely solves our problem for any one-dimensional motion. We will see in the next section that the main quantities of physical interest can be extracted from this formula.
Optimal search strategy
The relevant quantity to describe the protein/DNA association reaction is the mean time 
necessary for the protein to find the target site (see above). This mean time is obtained from the derivative of the first passage density by the relation
 | (7) |
 | (8) |
):
 | (9) |
 | (10) |
 | (11) |
 | (12) |
, obtained from a complete description of the three-dimensional motion (Berg and Blomberg, 1976, 1977, 1978). 
or 
), grows linearly with the length of the DNA strand. This mirrors the efficiency of the one-dimensional and three-dimensional combined motion when compared to the quadratic growth obtained in the case of pure sliding. In particular, the boundary effects are negligible for this quantity as soon as the overall length is large enough. 3D corresponding to the three-dimensional motion is finite and independent of the departure and arrival points. The corresponding expression of the mean first passage time is obtained by replacing ' by 3D.
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and recently addressed by Slutsky and Mirny (2004), which concerns the optimum strategy for such a coupled motion. Indeed, it seems reasonable that is large for both very large (in the infinite limit, the protein is never on the DNA), and very small (pure sliding limit). It has been suggested from qualitative arguments (Slutsky and Mirny, 2004) that the mean search time is minimum when the protein spends equal times bound to the DNA and freely diffusing in the bulk.
Here, we more precisely address this question of minimizing the mean search time with respect to the one-dimensional frequency . This is the only specially "adjustable" (depending strongly on the structure of the protein) parameter: ' depends on the properties of the environment and will not vary significantly from one protein to another. The one-dimensional diffusion coefficient D is a specific quantity, and optimizing the search time with respect to this parameter is trivial: D should be as large as possible (note that D and are assumed to be independent).
The sign of the derivative at = 0 of the mean search time gives the criterion for having a minimum as
 | (13) |
= L + M,
 | (14) |
limit, or more precisely for 
For intermediate values of , boundary effects become important and the minimum can be significantly different.
The value at the minimum is particularly interesting. We compare it to the case of pure sliding where 
 | (15) |
- and D-values obtained in Results and for a DNA substrate of length 106 bp, the mean target site localization time is given when pure sliding is 1000-fold greater than that predicted by our model.
Further quantitative features of reactive pathways
In this paragraph, we compute two quantities which characterize more precisely the nature of the reactive paths. These quantities are of special interest as they could be experimentally measured using single-molecule techniques.
The first quantity is the distribution p(N) of the number of visits on DNA required before reaching the target site. We recall that in the initial state the protein is bounded to the DNA, therefore N 
1. The distribution can be obtained by slightly modifying the expression of the first passage density Eq. 5:
 | (16) |
 | (17) |
 | (18) |
 | (19) |
is a succession of approximately N one-dimensional excursions of average duration 1/ and N three-dimensional excursions of average duration 1/'x).
The second interesting quantity is the average number of distinct basepairs visited before the protein reaches its target site. In our continuous description, this corresponds to the average span 
of the one-dimensional motion. For sake of simplicity, the target is here assumed to be centered on the DNA strand of half-length L. The average span can be expressed as the integral over the position x on the DNA of the probability that x has been visited before reaction. One then obtains
 | (20) |
is the first passage density at x with adsorbing conditions at x = 0, whose Laplace transform will be explicitly computed in the next section in the context of competitive targets. Anticipating formula Eq. 27, the span finally reads
 | (21) |
-dependence, is easily cleared up. The span appears to grow monotonously from 
at = 0 to L for 
. This monotonicity, as opposed to the existence of a minimum for the mean search time, is a striking feature of this quantity, plotted in Fig. 5.
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 | (22) |
The average search time then reads (we only give the case L = M for sake of simplicity),
 | (23) |
large enough, the minimum is obtained for
 | (24) |
min 
' holds true only for 
For larger frequencies ', we have min 4'2r2/D. The value of the search time at the minimum 
is modified. For r small we get
 | (25) |
 | (26) |
Case of two competitive target sites
The biological system (Stanford, et al., 2000) consists in integrating two target sites for the restriction enzyme EcoRV on a 690 bp linear DNA substrate. The position along a DNA strand of the first target site, which will be called target 1, is fixed and equals 120 bp. The second target site, which will be called target 2, has been placed at 54 bp, 200 bp, and 387 bp from the first target site. Thus, three substrates (Fig. 6) were used to analyze the kinetics of DNA cleavage. Each assay was carried out at a very low concentration of enzyme with regard to the concentration of DNA. For higher concentration of enzyme, the probability for two—or more—molecules acting on a same DNA strand would be non-negligible. The cleavage of DNA produces different lengths of DNA. An enzyme can cut target 1, target 2, or both, resulting in five lengths of fragments. The authors observed the initial formation of four of these: A, BC, C, and AB types.
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Conditional search time density
To get a better understanding of this process we first study analytically the distribution of the search time t of one target, for instance 2, knowing that no reaction occurred at target 1. We denote by 
this conditional search time density averaged over the initial condition. We make use of the general method developed in the first section to derive this quantity. Indeed, this problem involves a combination of three-dimensional excursions and one-dimensional motions, its peculiarity being that the one-dimensional motion is a constrained diffusion, as reaction with target 1 is excluded. It suffices then to rewrite formula Eq. 6 as
 | (27) |
is the Laplace transform of the first passage density at 2 avoiding 1 for a standard one-dimensional diffusion, and corresponds to the last excursion before finding the target 2. In turn, the term proportional to 
is the Laplace transform of the survival probability density, and comes from the succession of nonreactive excursions on DNA. Theses quantities are obtained by standard methods, considering successively the initial condition on fragment A (with mixed boundary conditions), B (with absorbing boundary conditions), and C (with mixed boundary conditions). This finally yields to
 | (28) |
 | (29) |
Preference and processivity
To get quantitative measurements of the pathway of the enzyme, the authors of Stanford et al. (2000) introduced two concepts: preference and processivity. The value of the preference P quantifies the preferential use of the target 2 by EcoRV. The P-value is experimentally obtained by taking the ratio of the initial formation rate 
AB of AB substrates (resulting from cleavage at the target site 2), over the initial formation rate BC of BC substrates (resulting from cleavage at the target site 1):
 | (30) |
C – AB)/(C + AB). One can define a symmetric quantity in the same manner, which is the processivity factor of the reaction with the target 1 and then target 2, fp12 = (A – BC)/(A + BC), and then the total processivity factor which represent the fraction of both processive actions,
 | (31) |
1, 2, 21, and 12, which are defined by the following elementary reactions, instead of substrate rate production:
 | (32) |
dx/(L + M). The enzyme concentration is chosen sufficiently small so that multiple encounter events are negligible. Consequently, a fragment BC (or AB) can be cut into B and C (or A and B) only if the enzyme which cleaves the DNA molecule to give BC (or AB) remains on this fragment (the probability of this event, depending in detail on the chemical mechanism, will be denoted pinit) and then finds the site 2 (or 1). The reaction rates are then
 | (33) |
 | (34) |
2 and 21 are straightforwardly obtained by permutation of symbols 1 and 2. One is now able to derive the processivity and preference factors. |
RESULTS |
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from the comparison of the theoretical preference to experimental data. Then, using the value of ' which satisfies the optimal searching time (this assumption is justified below), we deduce several quantities related to the enzyme pathway which links the first target site to the second one. Last, by comparing the analytical expression of the processivity factor to experimental data, we introduce a dynamic-associated parameter: the probability that after an excursion the enzyme will associate to the same DNA substrate it has left, r.
Preference
The preference for the target site 1 over site 2 is given by
 | (35) |
x = dx/dt is the rate for forming the species x, which can be measured experimentally. Explicitly,
 | (36) |
The best fit is obtained for 
bp–1. For a representative fast one-dimensional diffusion coefficient D = 5 x 105 bp2/s (Erskine et al., 1997), the one-dimensional frequency is = 37.5 s–1. Then the average time spent by the restriction enzyme on DNA per visit equals 0.027 s and the average distance scanned per visit (
) is 260 bp. Using Eq. 21, we obtain a representative average number of distinct sites visited on the DNA during the searching process, 
bp.
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', which depends strongly on experimental conditions, such as DNA concentration. It could be obtained experimentally as the protein/DNA association rate, and we here choose a typical value corresponding to the optimal search strategy, i.e., = '. This assumption is supported by the fact that the target site localization is several orders-of-magnitude faster than the diffusion limit. Using the same calculation as from Eqs. 5–12 without averaging on the initial position of the enzyme, we obtain the mean time needed by the restriction enzyme to go from the target 1 to the target 2,
 | (37) |
The average number of DNA visits before the processive cleaving is, using Eq. 19's formula, 
The same quantities for the other intertarget site distances, namely 200 bp and 387 bp, are, respectively, 
; and 
Processivity
Using the previous results, the processivity factor takes the form
 | (38) |
i.e., the probability to ever reach 1 starting from 2. The crucial point is about the dilution approximation, hence we treat the case of one single enzyme. We take into account the fact that during each three-dimensional excursion the protein can escape, therefore being definitely lost. We introduce by r the probability of return after a three-dimensional excursion. Rigorously this quantity depends on physical parameters such as the DNA length and the typical size of its attractive domain. As the lengths of DNA substrates are constant in the experiments of Stanford et al. (2000) for which b + c = 570 bp, we consider a constant r. We finally obtain
 | (39) |
is given by the Eq. 11 with L = c and M = b, and 
is the Laplace transform of the first passage density at 2, starting from 1 which is given by Eq. 10 with x = M = b,
 | (40) |
obtained previously, there are two unknown parameters: pinit and r. They can be determined from the experimental data (Fig. 8); the best fit is obtained for pinit = 0.5 and r = 0.85. However, these values cannot be very accurate, as used to be the case when estimating two parameters by fitting experimental data with theoretical results.
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CONCLUSION |
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did for a single target site. The model presented here permits us to obtain numerous quantities determining the pathway followed by a restriction enzyme in finding one target site or two competitive target sites on DNA, by a series of one-dimensional diffusion periods (sliding) followed by three-dimensional excursions (jumping). The corresponding mean search time shows that such a two-step process is faster than pure sliding or pure three-dimensional diffusion. The existence and the optimization of such a search time is discussed. The length dependence of the optimum was obtained.
Using the preference data from assays on EcoRV (Stanford et al., 2000), we quantify the parameter characterizing the pathway of EcoRV, namely the one-dimensional residence frequency . Other quantities were extracted from this parameter: the mean distance scanned by the restriction enzyme during one binding event (260 bp), the distribution of the number of visits on DNA before cleaving the target site, and the average number of distinct DNA sites visited. It should be noticed that the small value of the mean distance scanned might be due to the assumption of a perfect reactive target site which leads to an overestimated . In fact, an imperfect reactive target site would decrease the preference. Using the data on processivity for EcoRV, we introduce two secondary parameters characterizing the detailed pathways of the restriction enzyme after DNA cleavage. These parameters come into play when more than one target site is present on the DNA. The first parameter is the probability for the enzyme to stay (after cleavage with a target site) on the DNA strand which harbors the second target site. It was assumed that this probability equals one-half as the DNA sequences which border the target site are almost symmetric. Our best fit suggest that the probability is fairly 0.5, justifying the common assumption. The second parameter r is the probability for the enzyme to rebind on the cleaved DNA strand it had left during an excursion. Because of the short length of DNA substrates, it is assumed that the enzyme is "lost" after the dissociation from the DNA. This means that the enzyme rebinds unvisited DNA substrates after each three-dimensional excursion. Therefore, this probability had been previously assumed to be negligible. Our model reveals that this probability is high (0.85) which shows that the enzyme frequently rebinds to the same DNA substrate. The high value of r may be explained by the fact that the fragment length (which is here b + c = 570 bp) is significantly larger than the persistence length (150 bp). The configuration of the DNA is therefore close to a globule, in which the protein can be trapped and hence escape with a rather low probability. However, r may be overestimated because of our assumption of neglecting the correlations between the starting and finishing points of the three-dimensional excursions. Indeed, these correlations would result (for small values of the intertarget distance b) in increasing the processivity factor, and therefore lowering r. Note that an imperfect reaction would lower the processivity, as in this case the enzyme can pass through the target site without a reaction, therefore increasing the probability of a definitive departure from the DNA strand.
The present model classifies the stochastic pathway followed by a restriction enzyme searching for its target site, by quantifying the dynamical parameters. Our work is in the framework of stochastic dynamics which dictates the biological processes occurring in the highly structured and crowded medium of in vivo systems. Moreover, this model can be helpful for generic situations where a protein has to find a target site on a DNA substrate, e.g., the numerous transcription factors needed to trigger the gene activation.
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ACKNOWLEDGEMENTS |
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Submitted on May 11, 2004; accepted for publication June 14, 2004.
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J. S. van Zon, M. J. Morelli, S. Tanase-Nicola, and P. R. ten Wolde Diffusion of Transcription Factors Can Drastically Enhance the Noise in Gene Expression Biophys. J., December 15, 2006; 91(12): 4350 - 4367. [Abstract] [Full Text] [PDF]
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T. Hu, A. Yu. Grosberg, and B. I. Shklovskii How Proteins Search for Their Specific Sites on DNA: The Role of DNA Conformation Biophys. J., April 15, 2006; 90(8): 2731 - 2744. [Abstract] [Full Text] [PDF]
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I. M. Sokolov, R. Metzler, K. Pant, and M. C. Williams Target Search of N Sliding Proteins on a DNA Biophys. J., August 1, 2005; 89(2): 895 - 902. [Abstract] [Full Text] [PDF]
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The two dashed lines correspond to the limit cases when there is no sliding (straight line, 
and pinit = 0.5: one for