Stopped-flow fluorescence kinetic study of protein sliding and intersegment transfer in the target DNA search process

Abstract

Kinetic characterizations of protein translocation on DNA are nontrivial because the simultaneous presence of multiple different mechanisms makes it difficult to extract the information specific to a particular translocation mechanism. In this study, we have developed new approaches for the kinetic investigations of proteins’ sliding and intersegment transfer (also known as ‘direct transfer’) in the target DNA search process. Based on the analytical expression of the mean search time for the discrete–state stochastic model, we derived analytical forms of the apparent rate constant k_app for protein-target association in systems involving competitor DNA and the intersegment transfer mechanism. Our analytical forms of k_app facilitate the experimental determination of the kinetic rate constants for intersegment transfer and sliding in the target association process. Using stopped-flow fluorescence data for the target association kinetics along with the analytical forms of k_app, we have studied the translocation of the Egr-1 zinc-finger protein in the target DNA association process. Sliding was analyzed using the DNA length-dependent k_app data. Using the dependence of k_app on the concentration of competitor DNA, we determined the second-order rate constant for intersegment transfer. Our results indicate that a major pathway in the target association process for the Egr-1 zinc-finger protein is the one involving intersegment transfer to a nonspecific site and the subsequent sliding to the target.

Introduction

Since Riggs et al. discovered amazingly rapid target location by the E. coli lac repressor in 1970,¹ the mechanisms that allow DNA-binding proteins to efficiently locate their target DNA sites have been studied both experimentally and theoretically (e.g. reviews in refs ^2–9). It was shown that nonspecific DNA binding plays an important role in increasing the efficiency with which proteins locate their specific target DNA sites. Berg et al. conceptually defined three major mechanisms for protein translocation on DNA:¹⁰ 1) sliding, 2) dissociation and re-association, and 3) intersegment transfer (also known as direct transfer). Sliding is the random walk of protein while being bound to DNA and can be regarded as one-dimensional diffusion. Translocations via dissociation and re-association are categorized into long-range translocation to an uncorrelated site and short-range translocation (known as ‘hopping’) to a nearby site. Intersegment transfer is direct transfer of protein from one DNA duplex to another (without going through the intermediary of free protein) via an intermediate where a protein molecule transiently bridges two DNA duplexes.

These distinct translocation mechanisms can coexist, and their relative contributions to the overall efficiency of a target DNA search should depend on proteins. For example, intersegment transfer is considered to be unlikely for proteins that cannot bridge two DNA duplexes transiently. The efficiency of each translocation mechanism should also depend on environmental factors, such as ionic strength, DNA density, DNA geometry, and the presence of other proteins. Characterizing a particular translocation mechanism is difficult due to the simultaneous presence of distinct mechanisms, particularly when molecular ensembles are measured in bulk solutions. Recently, remarkable advancements in single-molecule biophysics enabled the direct observation of protein sliding on DNA.^3,11–17 Elegant biochemical methods were also developed for kinetic investigations of one-dimensional search.^18–25 These methodological advances have substantially deepened the understanding of protein sliding on DNA. However, there is a lack of experiment-based knowledge about the kinetics of the other translocation mechanisms and the interplay between the distinct mechanisms during the target search process.

The least understood of the above-mentioned major translocation mechanisms is intersegment transfer. While many theoretical papers on the target DNA search have overlooked intersegment transfer, recent experimental studies clearly showed the significance of intersegment transfer, at least for several proteins.^26–34 As explained below, intersegment transfer appears to be a second-order process and was studied via biochemical or biophysical measurements of apparent exchange^29,33,34 or dissociation^27,30,32 rates as a function of DNA concentration. Although previous studies suggested the importance of intersegment transfer, its actual kinetic contribution to the target association process remains to be addressed.

In this paper, we present a new theoretical framework and experimental approaches to quantitatively investigate the kinetics of intersegment transfer and sliding in the target association process. Using this methodology, we study the target search kinetics of the inducible transcription factor Egr-1 (also known as Zif268), which plays important roles in the brain and the cardiovascular system. In the brain, Egr-1 is induced by synaptic signals and activates genes for long-term memory formation and consolidation.^35,36 In the cardiovascular system, Egr-1 serves as a stress-inducible transcription factor that activates the genes for defense responses against vascular stress and injury.^37,38 Egr-1 recognizes a 9-bp target DNA sequence, GCGTGGGCG, via three zinc-finger domains.³⁹ Within its short lifetime (~½ – 1 hour),³⁷ the induced Egr-1 protein regulates a particular set of genes, allowing the cells to rapidly respond to the stimuli. It is thus important to understand how Egr-1 efficiently scans DNA. Our current study provides insights into the roles of sliding and intersegment transfer in the target search by Egr-1.

Theory

Here we provide the theoretical framework for our stopped-flow kinetic method that is applicable to proteins that form a stable complex with their target DNA site. The studied systems involve three macromolecular components: the probe DNA, protein, and nonspecific competitor DNA (Figure 1A). The probe DNA contains a target site and a fluorescent group that is tethered to a position near the target. The fluorescence from the probe changes upon binding of the protein to the target site. The time course of the change is monitored immediately after a solution of the protein is rapidly mixed with a solution containing the probe DNA and the competitor DNA. We specifically deal with systems in which the total concentrations of the protein, the probe DNA, and the competitor DNA (P_tot, D_tot, and C_tot, respectively) satisfy the following:

$$D_{tot}«P_{tot}«C_{tot}.$$ [1]

Figure 1.

Experimental design and kinetic model used in this study. (A) Stopped-flow fluorescence assay of the target DNA search kinetics. (B) Discrete binding sites on the probe and competitor DNA duplexes. (C) Rate constants involved in our kinetic model for the target association process. The rate constants k_sl,N and k_sl,S for sliding and the rate constants k_off,N and k_off,S for dissociation are first-order rate constants (units, s⁻¹), whereas the rate constants k_IT,N and k_IT,S for intersegment transfer and the rate constants k_on,N and k_on,S for association are second-order rate constants (units, M⁻¹ s⁻¹). A rate constant for sliding is defined for sliding from one site to an adjacent site (i.e., shift by 1 bp). The rate constant k_sl,N is related to the one-dimensional diffusion coefficient D₁ for sliding, as indicated by Eq. 15. A complete set of rate equations for this kinetic model is given in the Supplemental Information.

These conditions can be used only for proteins that exhibit high specificity to their target. Because of these inequalities, the relevant second-order processes occur in a pseudo-first-order manner, which simplifies the kinetic analysis of experimental data.⁴⁰ The fluorescence time-course data are used to determine an apparent pseudo-first-order rate constant, k_app, for binding of the target to the protein. Due to the large excess of competitor DNA, only one protein can bind to the probe DNA, which allows for accurate kinetic measurement of sliding without interference due to the presence of multiple protein molecules on the same DNA. Moreover, the target DNA search in the presence of a large excess of competitor DNA represents a more realistic case and provides insights into the search processes in vivo.

Intersegment transfer as a phenomenological second-order process

Intersegment transfer of a protein between two DNA duplexes can be represented by:

$$D_b+P{D}_a\underset{k_{d1}}{\overset{k_{f1}}{\rightleftarrows }}{D}_{b}P{D}_a\underset{k_{f2}}{\overset{k_{d2}}{\rightleftarrows }}P{D}_b+D_a$$ [2]

where PD represents a protein-DNA complex; D_bPD_a, the DNA-bridging intermediate; k_f1 and k_f2, second-order rate constants for formation of the intermediate; and k_d1 and k_d2, first-order rate constants for dissociation of the intermediate into a protein-DNA complex and free DNA. If the intermediate is a transient and low-population state, the intersegment transfer appears to be a second-order process:

$$D_b+P{D}_a\underset{k_{IT,ba}}{\overset{k_{IT,ab}}{\rightleftarrows }}P{D}_b+D_a.$$ [3]

Using a steady-state approximation (Supplemental Information), the apparent second-order rate constants k_IT,ab and k_IT,ba are given by:

$$k_{IT,ab}=k_{f1}{k}_{d2}{\left(k_{d1}+k_{d2}\right)}^{-1}$$ [4]

$$k_{IT,ba}=k_{f2}{k}_{d1}{\left(k_{d1}+k_{d2}\right)}^{-1}.$$ [5]

If k_d1 = k_d2 and the duplexes D_a and D_b exhibit the same affinity, then k_IT,ab = k_IT,ba = ½k_fI = ½k_f2. Because of the second-order nature, the overall rate for intersegment transfer is proportional to the free DNA concentration. In fact, this nature has been essential in the experimental detection of intersegment transfer.^{27–30,32–34,41,42}

In contrast, the overall rate for translocation via dissociation and re-association is virtually independent of the free DNA concentration when the DNA concentration is higher than the dissociation constant K_d. This translocation is represented by:

$$D_b+P{D}_a\underset{k_{on,a}}{\overset{k_{off,a}}{\rightleftarrows }}{D}_b+P+D_a\underset{k_{off,b}}{\overset{k_{on,b}}{\rightleftarrows }}P{D}_b+D_a$$ [6]

When [D] » K_d, the rate limiting step for this scheme is the dissociation (a first-order process) because this inequality corresponds to k_off « k_on[D]. In this case, an increase in [D] does not affect the overall rate because the rate-limiting step is of first order. Thus, intersegment transfer can be distinguished experimentally from translocation via dissociation and re-association.

Discrete-state kinetic model for target DNA search

Our kinetic model for a target DNA search assumes discrete states in nonspecific protein-DNA association and is similar to the model used by Veksler and Kolomeisky.⁴³ The discrete nature of nonspecific protein-DNA association is supported by the recent solution NMR studies on nonspecific protein-DNA complexes.^{29,34,44–46} As shown in Figure 1B, the probe DNA in our model contains a total of L sites, of which only the m-th site from an edge is a target, and all the others are nonspecific sites. Competitor DNA contains a total of M nonspecific sites. Our kinetic model addresses dissociation, association, sliding, and intersegment transfer (Figure 1C). Based on the above-mentioned considerations, our kinetic model treats intersegment transfer as a second-order process between a protein-DNA complex and free DNA. Although intersegment transfer can occur between two distant sites on the same molecule if the DNA length is significantly longer than the persistence length (i.e., ~150 bp), such intra-molecular intersegment transfer is not considered here because only relatively short (< 150 bp) DNA duplexes are used in our experiments. The model involves the intrinsic (as opposed to apparent) association rate constants k_on,N and k_on,S, the dissociation rate constants k_off,N and k_off,S, the first-order rate constants k_sl,N and k_sl,S for sliding, and the second-order rate constants k_IT,N and k_IT,S for intersegment transfer (annotations N and S are for nonspecific and specific sites, respectively). These kinetic rate constants are schematically summarized in Figure 1C. A complete set of the rate equations for our kinetic model is given in the Supplemental Information. Numerical integration of the rate equations, which can readily be performed with a standard ordinary-differential-equation (ODE) solver, provides the populations of the individual species as a function of time (details given in the Supplemental Information). The ODE-based simulations allowed us to assess the analytical forms of k_app presented below.

Mean search time of the Veksler-Kolomeisky model

For systems involving neither competitor DNA nor intersegment transfer, Veksler and Kolomeisky derived a general analytical expression for the mean time T_P for a protein, which is initially in the free state, to reach the target:⁴³

$$T_P=\left\{L{k}_{out}+\left(L-S\right)k_{in}\right\}/\left(k_{out}{k}_{in}S\right).$$ [7]

In this expression, the relevant parameters are as follows:

$$k_{in}=L{k}_{on,N}{D}_{tot}$$ [8]

$$k_{out}=k_{off,N}$$ [9]

$$S=y\left(1+y\right)\left(y^{-L}-y^L\right)/\left\{\left(1-y\right)\left(y^{1-m}+y^m\right)\left(y^{1+L-m}+y^{m-L}\right)\right\}$$ [10]

$$y=1+\left(1/2\right)\left(k_{out}/k_{sl,N}\right)-{\left\{\left(k_{out}/k_{sl,N}\right)+\left(1/4\right){\left(k_{out}/k_{sl,N}\right)}^2\right\}}^{1/2},$$ [11]

where k_in is the rate constant for the free protein to bind to any site on the probe DNA; k_out, the rate constant for the bound protein to leave the probe DNA; L, the total number of sites; and m, the target position. Hereafter, this general analytical expression is referred to as the VK model.

k_app for systems involving competitor DNA in the absence of intersegment transfer

Now we address the systems involving competitor DNA under the conditions of D_tot « P_tot « C_tot. Owing to these inequalities, the association of proteins with the competitor DNA reaches quasi-equilibrium far more rapidly than the association with the target. Based on this assumption and the VK model, the apparent pseudo-first-order rate constant k_app for the target to bind to the protein in the absence of intersegment transfer is given by:

$$k_{app}=\frac{1}{T_P}\frac{f_P{P}_{tot}}{D_{tot}}\left\{1+\frac{K_{d,S}}{f_P{P}_{tot}}\right\},$$ [12]

where f_p is the fraction of the protein in the free state:

$$f_P=K_{d,N}{\left(K_{d,N}+\varphi M{C}_{tot}\right)}^{-1}.$$ [13]

The derivation of Eq. 12 is provided in the Supplemental Information. K_d,N (= k_off,N / k_on,N) is the dissociation constant for each nonspecific site, and K_d,S is the dissociation constant for the target. The parameter ϕ is the number of possible orientations for each nonspecific site. Due to structural pseudo-C2 symmetry for DNA, ϕ = 2 for proteins that bind as monomer; ϕ = 1 for symmetric dimers. When ϕ = 2 is used, microscopic parameters (e.g., K_d,N, k_off,N, k_IT,N) are defined for each orientation. Additional considerations on systems with ϕ = 2 are given in the Supplemental Information. The k_app constants from this analytical expression (Eq. 12) agree well with those from the ODE-based simulations (Figure 2A).

Figure 2.

Validation of the VK model-based analytical form of k_app (Eq. 12) for systems involving competitor DNA but with no intersegment transfer. (A) Rate constant k_app for target association as a function of the total number of sites L. The position of the target was set to m = 2. The dashed lines were obtained from the ODE-based numerical kinetics simulations. The solid lines were obtained using Eq. 12. k_sl,N = 10³, 10⁴, 10⁵, and 10⁶ s⁻¹ and k_off,N = 10 s⁻¹ were used. Under these conditions, effective sliding lengths λ are 10 (black), 32 (red), 100 (magenta), and 316 (blue) bp. (B) Rate constants k_app as a function of the target position m for probe DNA with L = 90. For this panel, k_sl,N = 10⁴ s⁻¹ and k_off,N = 10 s⁻¹ (an effective sliding length λ = 32 bp) were used. The dotted line is from the ODE-based numerical simulations; the solid line, from the VK model-based analytical form of k_app (Eq. 12); and the dashed line, from the modified Berg’s approximation (Eq. 16). For both panels A and B, D_tot = 2.5 nM; P_tot = 50 nM, K_d,N = 4 μM; K_d,S = 0.1 nM; C_tot = 2 μM; ϕ = 1; and M = 20.

VK model vs. Berg’s approximation

For systems without competitor DNA, Berg et al. showed that if the target site is located in the middle of rod-like DNA, the apparent rate constant k_app can be approximated by:¹⁰

$$k_{app}=2k_{on,N}{P}_{tot}\sqrt{\frac{D_1}{l^2{k}_{off,N}}}\tanh \left(\frac{L}{2}\sqrt{\frac{l^2{k}_{off,N}}{D_1}}\right)$$ [14]

where D₁ is a one-dimensional diffusion constant for sliding, and l is the distance between two adjacent sites (3.4 Å) along the DNA axis. This expression assumes an approximate continuum state for sliding, whereas the VK model assumes discrete states. It is straightforward to derive a simple relationship between D₁ and k_sl,N:

$$D_1=l^2{k}_{sl,N}.$$ [15]

Eq. 15 indicates that D₁ is equivalent to k_sl,N when D₁ is given in bp² s⁻¹. For systems involving competitor DNA, Eq. 14 can be modified to the following:

$$k_{app}=2k_{on,N}{f}_P{P}_{tot}\left\{1+\frac{K_{d,S}}{f_P{P}_{tot}}\right\}\sqrt{\frac{k_{sl,N}}{k_{off,N}}}\tanh \left(\frac{L}{2}\sqrt{\frac{k_{off,N}}{k_{sl,N}}}\right).$$ [16]

As demonstrated in Figure 2B, the modified Berg’s approximation (Eq. 16) provides an accurate k_app only if the target is located at m ≈ L / 2. This limitation occurs because Eqs. 14 and 16 assume that the antenna effect (i.e., enhancement of target association via association with nonspecific sites followed by sliding)^7,47 applies to the same degree for both sides of the target. When the target is located near the edge of the DNA, only one side provides a significant degree of the antenna effect, and therefore the search kinetics can slow down by up to ~2-fold.⁴³ The VK-model-based expression (Eq. 12) provides an accurate k_app for any m value, because this model accounts for different antenna effects for the two sides. This property of the VK-model-based expression is especially important when an extrinsic fluorescent group in close proximity to the target is used to detect the formation of the specific complex.

k_app for systems with competitor DNA in the presence of intersegment transfer

Under the condition of D_tot « P_tot « C_tot, the fraction of the protein bound to the competitor DNA (f_CP) is given by:

$$f_{CP}=\varphi M{C}_{tot}{\left(K_{d,N}+\varphi M{C}_{tot}\right)}^{-1}$$ [17]

If intersegment transfer is efficient, a major pathway in the target association kinetics can be the pathway from the competitor DNA to the target via intersegment transfer and subsequent sliding. Using the mean time T_CP for the search via this additional pathway, the k_app constant is given approximately by:

$$k_{app}=\left(\frac{f_P}{T_P}+\frac{f_{CP}}{T_{CP}}\right)\frac{P_{tot}}{D_{tot}}\left\{1+\frac{K_{d,S}}{f_p{P}_{tot}}\right\}$$ [18]

Because the VK model does not assume any particular mechanisms for protein’s arrival to and departure from the probe DNA, T_CP is given in the same form as T_P (i.e., Eqs 7–11), but with k_in and k_out substituted to:

$$k_{in,CP}=L{k}_{IT,N}{D}_{tot}$$ [19]

$$k_{out,CP}=k_{off,N}+\varphi M{k}_{IT,N}{C}_{tot}.$$ [20]

The rate constant k_out for T_P should also be substituted to k_out,CP for the systems involving the intersegment transfer mechanism because the protein on a nonspecific site of the probe DNA can leave the probe DNA via intersegment transfer as well (with a pseudo-first-order rate constant $\varphi M{k}_{IT,N}{C}_{tot}$). As shown in Figure 3, the k_app constants calculated using this analytical expression (Eq. 18) agree well with those from the ODE-based numerical simulations.

Figure 3.

Validation of the VK model-based analytical forms of k_app for systems involving competitor DNA and intersegment transfer (Eq. 18). (A) Rate constant k_app for target association as a function of the total number of sites L. A single orientation per site (i.e., ϕ = 1) was assumed. The sliding rate constant k_sl,N = 10⁵ s⁻¹ was used. The other conditions are exactly the same as those for Figure 2A except that intersegment transfer is taken into consideration. The rate constants for intersegment transfer are indicated. (B) Rate constant k_app for target association as a function of the total number of sites L under conditions identical to those for the panel A, except that 2 orientations per site are assumed (i.e., ϕ = 2). The kinetic and thermodynamic parameters are defined for each orientation. For both panels, the dashed lines were obtained from the ODE-based numerical kinetics simulations, whereas the solid lines were obtained with Eq. 18. Details of the ODE-based simulations are given in the Supplemental Information.

Sliding and dependence of k_app on length of the probe DNA

Although some previous studies used the DNA-length dependence of target association to investigate sliding,^{20,24,48–50} the previous analytical expressions (e.g. Eq. 14) were applicable only to systems involving neither competitor DNA nor intersegment transfer. With our analytical form of k_app, we can determine the rate constant k_sl,N for sliding from experimental data for systems involving competitor DNA and intersegment transfer. In Eq. 18, the dependence of k_app on k_sl,N arises from T_P and T_CP. Note that T_P and T_CP share exactly the same S (Eq. 10) and y (Eq. 11), which are the only parameters containing the rate constant k_sl,N for sliding. Because of this similarity, the length dependence for Eq. 12 with k_out substituted to k_out,CP is virtually indistinguishable from that for Eq. 18. The rate constants k_sl,N can be accurately determined from length-dependent k_app data via nonlinear least-squares fitting with Eq. 12, even for systems involving intersegment transfer. This is convenient for experimental research. An important parameter for sliding is the effective sliding length λ given by:⁴³

$$\lambda ={\left(k_{sl,N}/k_{out}\right)}^{1/2}.$$ [21]

Again, k_out should be substituted to k_out,CP if intersegment transfer is involved. As is evident from Eq. 11 (also from Eq. 16), λ is directly relevant to the length dependence of k_app. When L < λ, k_app is virtually proportional to L because of the extended antenna effect. As noted previously, an increase in L beyond 2λ does not enhance the antenna effect.⁷ Because this asymptotic length dependence is important for determining k_sl,N and λ, the length L should be varied with the maximum being larger than (or at least comparable to) λ.

Intersegment transfer and dependence of k_app on concentration of competitor DNA

In the absence of intersegment transfer, the k_app constant is virtually proportional to C_tot⁻¹ when ϕMC_tot » K_d,N as noted by Lin and Riggs.⁵¹ This proportionality is obvious from Eqs 12 and 13. At a higher concentration of competitor DNA, the protein can be trapped at nonspecific sites more easily, which slows down the target association process. However, intersegment transfer can counteract this trapping effect. In the presence of intersegment transfer, the escape of a protein from competitor DNA can become faster at a higher concentration of competitor DNA because intersegment transfer is a second-order process whose rate is proportional to the concentration of the DNA in the free state. When the f_CP / T_CP term in Eq. 18 is significantly larger than the f_P / T_P term, the dependence of k_app on C_tot becomes substantially weaker than the proportionality to C_tot⁻¹. By measuring k_app as a function of C_tot, the rate constant k_IT,N for intersegment transfer can be determined via nonlinear least-squares fitting with Eq. 18. The term involving f_CP / T_CP in Eq. 18 corresponds to the contribution of the pathway that involves intersegment transfer followed by sliding to the overall target association kinetics. Thus, the role of intersegment transfer in the target association process can be directly assessed.

Results

In the current study, we investigated the target search kinetics for the DNA-binding domain of human Egr-1. Hereafter, this protein construct comprising three zinc fingers is referred to as the ‘Egr-1 zinc-finger protein’ for simplicity’s sake. The association of the Egr-1 zinc-finger protein with DNA was monitored via the fluorescence from the fluorescent group that was covalently attached to a DNA terminus. We used the DNA duplexes shown in Figure 4.

Figure 4.

DNA duplexes used in the current study on the Egr-1 zinc-finger protein. (A) DNA duplexes used for kinetic measurements. The solid lines indicate chemically synthesized nucleotides, whereas the dotted lines indicate enzymatically extended nucleotides. (B) DNA duplexes used for affinity measurements. (C) PAGE of the fluorescent DNA duplexes, which were visualized via fluorescence from the covalently attached FAM or TAMRA.

Dissociation constants for nonspecific and specific complexes

Analyses of sliding and intersegment transfer based on the discrete kinetic model require the equilibrium constants for the nonspecific and specific complexes. We measured the affinities using 12-bp nonspecific and specific DNA duplexes with tetramethylrhodamine (TAMRA) attached to the 3’-terminus (Figure 4). The apparent K_d constants were determined from the fluorescence anisotropy as a function of protein concentration. TAMRA was used for the K_d measurements because it exhibits a large change in fluorescence anisotropy upon binding of the protein to DNA. We measured the apparent K_d for the 12-bp nonspecific DNA to be 0.58 ± 0.08 μM at 80 mM KCl (Figure 5). Because the previous structural studies for the specific and nonspecific DNA complexes of Egr-1 showed that Egr-1 binds to DNA as a monomer and covers 9 bp,^34,39 the 12-bp nonspecific DNA has 8 [= 2 × (12 – 9 + 1)] overlapping sites (this number corresponds to ϕM). Assuming that the apparent K_d can be approximated by k_off,N / (ϕMk_on,N) in this case where only one protein molecule can bind to the 12-bp DNA, the dissociation constant K_d,N for each nonspecific site was calculated to be 4.6 μM (= 0.58 × 8). This value was used in the calculations to determine k_sl,N and k_IT,N (see below). For the 12-bp specific DNA containing the target site, the affinity at 80 mM KCl was too strong to directly measure with the same fluorescence method. However, we were able to measure the dissociation constant K_d at higher ionic strengths (Figure 5). By extrapolating from these data along with the counterion condensation theory,^52,53 K_d,S at 80 mM KCl were estimated to be ~0.1 nM.

Figure 5.

Dissociation constant K_d for nonspecific and specific complexes between the Egr-1 protein and 12-bp DNA at various concentrations of KCl. These data were obtained by measuring the TAMRA fluorescence anisotropy as a function of protein concentration. For each complex, the solid line represents a linear extrapolation assuming a linear relationship (solid lines) between logK_d and log[KCl], which is based on the counterion condensation theory.^52,53

Change of FAM fluorescence intensity upon Egr-1 target association

To detect the binding of Egr-1 to the target site on the probe DNA duplexes, we monitored the change in fluorescence intensity of 5’-terminal fluorescein amidite (FAM), which was located near the Egr-1 target (at the second position from the terminus; see Figure 4). We chose this fluorescent probe for our stopped-flow kinetic experiments because the FAM-labeled DNA exhibited stronger fluorescence intensity (but weaker anisotropy) than the TAMRA-labeled DNA exhibited. The FAM emission spectra recorded for the 113-bp probe DNA in the presence and absence of the Egr-1 zinc-finger protein and competitor DNA are shown in Figure 6A. Although no spectral change was observed upon the addition of competitor DNA to the solution of probe DNA, a significant (up to ~18%) reduction in the fluorescence intensity was observed when the Egr-1 zinc-finger protein was added. This change in FAM fluorescence is presumably due to its close proximity to the DNA-bound protein, as observed for other proteins.^54,55 Varying the concentration of the Egr-1 zinc-finger protein, we measured the FAM fluorescence intensity from the probe DNA (2.5 nM) in the presence of 2 μM competitor DNA at equilibrium (Figure 6B). The titration data clearly show a high-affinity protein-DNA interaction with K_d « 1 nM at 80 mM KCl. This high affinity is consistent with the above-mentioned data for the 12-bp DNA containing a target site. Thus, these results indicate that the change in the FAM fluorescence intensity reflects the association of Egr-1 with the target (rather than other sites) on the probe DNA.

Figure 6.

FAM fluorescence data for the target association of the Egr-1 zinc-finger protein with the 113-bp probe DNA (D_tot = 2.5 nM) in the presence of 28-bp competitor DNA (C_tot = 2 μM). The buffer conditions were 10 mM Tris•HCl (pH 7.5), 80 mM KCl, and 200 nM ZnCl₂. (A) Change in the FAM emission spectra upon protein binding. The black solid line is the spectrum recorded for the probe DNA only; the green dotted line, for the probe DNA plus competitor DNA; and the magenta solid line, with the probe DNA, competitor DNA, and protein (P_tot = 100 nM). (B) FAM fluorescence intensity measured as a function of P_tot in the presence of the competitor DNA. (C) Stopped-flow time-course data of the FAM fluorescence intensity immediately after mixing the protein solution (P_tot = 50 nM) with the solution containing the probe and competitor DNA duplexes. The red curve represents the best fit to a mono-exponential function. (D) Protein-concentration dependence of the apparent pseudo-first-order kinetic rate constant k_app for target association. The apparent second-order rate constant for association was determined to be 4.5 × 10⁷ M⁻¹ s⁻¹. The error bars represent the standard deviations for 8 – 10 replicates.

Target association kinetics for the Egr-1 zinc-finger protein

Using a stopped-flow device, we recorded the time courses of FAM fluorescence intensity immediately after mixing the protein solution with a solution containing the probe and competitor DNA duplexes. In this experiment, the emission light that passed through a long-pass filter (515 nm cutoff) was detected without using monochromator. Although this configuration allowed higher sensitivity, the percentage change in the emission intensity upon the target association was smaller due to a higher non-fluorescent background. Figure 6C shows the fluorescence time-course data obtained under conditions of D_tot = 2.5 nM, P_tot = 50 nM, and C_tot = 2 μM. Under these conditions together with the above-mentioned equilibrium constants, 95% of the protein molecules are bound to the 28-bp competitor DNA at the quasi-equilibrium that occurs immediately after mixing, and 96% of the target on the 113-bp DNA is bound to the protein at the conclusion of the binding reaction. To analyze the kinetics of distinct translocation mechanisms, we measured the apparent rate constant k_app using various settings of L, P_tot, and C_tot (all satisfying D_tot « P_tot « C_tot). In all cases, the fluorescence intensity changed in a mono-exponential manner, and an apparent pseudo-first-order rate constant k_app was determined via mono-exponential fitting. Figure 6D shows the dependence of k_app on P_tot in the presence of competitor DNA (C_tot = 2 μM). As our analytical forms (i.e., Eqs. 12, 16, and 18) predict, the rate constant k_app was proportional to P_tot. The slope of the plot corresponds to an apparent second-order rate constant k_a for target association (k_a = 4.5 × 10⁷ M⁻¹ s⁻¹ for Figure 6D). Although k_a was used in some previous studies of protein translocation on DNA,^20,25,48 we used the pseudo-first-order rate constant k_app instead when investigating sliding and intersegment transfer, as demonstrated below. This makes the investigations quicker yet remains equivalent to the use of k_a because k_app was found to be proportional to P_tot for all other conditions tested (Supplemental Information). In all the following experiments, we used P_tot = 50 nM, because at this concentration, k_app was typically less than 5 s⁻¹ and could be measured precisely (with errors < 10%) with our current method.

Dependence on the length of the probe DNA

To investigate the sliding kinetics of Egr-1 using dependence of k_app on the length of the probe DNA, we prepared 33-, 48-, 63-, 83-, 113- and 143-bp DNA duplexes, each containing a single target site and a FAM probe, as shown in Figure 4. With the stopped-flow fluorescence method, the rate constant k_app for target association was measured with 50 nM protein, 2.5 nM probe DNA, and 2,000 nM nonspecific DNA duplex (28 bp) at 80 mM KCl (Figure 7). As predicted in Figures 2 and 3, our experimental k_app data showed that the length dependence was an increasing function with a steeper slope for shorter lengths and an obvious asymptote for longer lengths. For example, the k_app constant for the 63-bp probe DNA was 64% larger than that for the 33-bp probe DNA, whereas the k_app constants for the 143-bp and 113-bp probe DNA duplexes were identical within experimental error. Qualitatively, these results suggest that the effective sliding length λ is comparable to the DNA lengths used in this experiment.

Figure 7.

DNA-length dependence of the target association kinetics measured for the Egr-1 zinc-finger protein at 80 mM KCl (D_tot = 2.5 nM; P_tot = 50 nM; and C_tot = 2 μM). The pseudo-first-order rate constant k_app for target association is plotted as a function of the total number of sites on the probe DNA. The blue curve represents the best fit to Eq. 12. The effective sliding length λ is given by Eq. 21. Note that an increase in L beyond 2λ does not enhance the target association. The error bars represent the standard deviations for 8 – 10 replicates.

Kinetic rate constant for sliding of Egr-1 on DNA

The analytical forms of the k_app constants allow us to determine the sliding rate constant k_sl,N from the length-dependent k_app data for Egr-1 in the system involving a large excess of competitor DNA. As described above, the determination of k_sl,N does not require accurate information on intersegment transfer. For a DNA length > 30 bp, the effective diffusion coefficient (D_protein + D_DNA), which is relevant to the Smoluchowski limit, should be dominated by the three-dimensional diffusion coefficient D_protein for the Egr-1 zinc-finger protein (10 kDa; hydrodynamic radius, ~15 Å). Therefore, the length dependence of the diffusion coefficient D_DNA for DNA does not have to be taken into consideration in the analysis of the length-dependent k_app data for the investigation of sliding.¹⁰ Because Egr-1 binds to DNA as a monomer, Eq. 12 with ϕ = 2 was employed for nonlinear least-squares fitting to determine the rate constant k_sl,N. The parameter L was set to A – B + 1, where A is the total length of DNA in bp and B is the length of each nonspecific site. As mentioned above, B was set to 9 bp. Thus, the values of L were 25, 40, 55, 80, 105, and 135 for the 33-bp, 48-bp, 63-bp, 88-bp, 113-bp and 143-bp probe DNA duplexes, respectively. The target position was m = 2 for all the probe DNA duplexes. Calculation of k_app using Eq. 12 requires four parameters: k_sl,N, k_out, K_d,N, and K_d,S. In the current case with K_d,S / (f_CPP_tot) << 1, information about K_d,S is not essential in the calculation (see Eq. 12). With the experimentally obtained dissociation constant K_d,N, the rate constant k_on,N was treated as k_off,N / K_d,N, while the other two parameters, k_sl,N and k_out, were optimized via nonlinear least-squares fitting. The resultant best-fit curve is shown in Figure 7. The rate constant k_sl,N was determined to be (2.1 ± 0.2) ×10⁵ s⁻¹ for the sliding of the Egr-1 zinc-finger protein at 80 mM KCl. This rate constant corresponds to D₁ = (2.4 ± 0.1) ×10⁻² μm² s⁻¹. Using Eq. 21, the effective sliding length λ was determined to be 50 ± 2 bp.

Dependence on the concentration of the competitor DNA

Using the 113-bp probe DNA, we measured the apparent rate constant k_app for target association at six different concentrations of competitor DNA (C_tot = 0.5, 1.0, 2.0, 4.0, 6.0 and 8.0 μM). The results are shown in Figure 8A. The k_app constant was found to decrease upon increasing C_tot. As described above, k_app should be virtually proportional to C_tot⁻¹ if intersegment transfer is not involved in the target search process and ϕMC_tot » K_d,N. Although this inequality was satisfied, our data for the Egr-1 zinc-finger protein showed that k_app constant was not proportional to C_tot⁻¹. For example, our experiment showed that the ratio of the k_app constant at C_tot = 2.0 μM to that at C_tot = 8.0 μM was only 1.3, instead of 4. These results qualitatively suggest that intersegment transfer contributes significantly to the target association process.

Figure 8.

(A) Competitor DNA concentration (C_tot) dependence of the apparent rate constant k_app measured for Egr-1 (P_tot = 50 nM) at 80 mM KCl. The 113-bp probe DNA was used (D_tot = 2.5 nM). The circles represent the experimentally measured k_app constants. The error bars represent the standard deviations for 8 – 10 replicates. The solid red lines are the best-fit curve to Eq. 18, which accounts for intersegment transfer. The dotted green lines are the best-fit curve to Eq. 12, which does not account for intersegment transfer. (B) Contributions of major target association pathways under the current experimental conditions.

Kinetic rate constant for intersegment transfer of Egr-1

Using the C_tot-dependent k_app data, we determined the rate constant k_IT,N for intersegment transfer. The analytical form of k_app for systems involving intersegment transfer (Eq. 18) was employed for nonlinear least-squares fitting to determine k_IT,N. The calculation of k_app with Eq. 18 requires five parameters: k_IT,N, k_off,N, k_sl,N, K_d,N, and K_d,S. Again, in the current case with K_d,S / (f_CPP_tot) << 1, information about K_d,S is not essential in the calculation. As described above, k_on,N was treated as k_off,N / K_d,N. Because k_sl,N was determined from the length-dependent k_app data, only two parameters, k_IT,N and k_off,N, were optimized in the fitting calculation. This procedure provided an excellent fit to the experimental C_tot-dependence data (Figure 8A; solid curves). In contrast, fitting with Eq. 12 (no intersegment transfer) provided only a poor fit to the experimental C_tot-dependence data (Figure 8A; dotted curves). Because these two models for fitting differ in the degree of freedom, Akaike’s information criterion (AIC) was used for model assessment. The AIC values for the fittings with Eqs 12 and 18 were 187.9 and 7.4, respectively. These results clearly indicate that the model that accounted for intersegment transfer is far better to describe the C_tot-dependent k_app data. The rate constant k_IT,N for intersegment transfer was determined to be (1.0 ± 0.1) × 10⁶ M⁻¹ s⁻¹ for the Egr-1 zinc-finger protein at 80 mM KCl. The kinetic parameters determined in our current study are summarized in Table I.

Table I.

Kinetic parameters for translocation of the Egr-1 zinc-finger protein on nonspecific DNA sites determined from the stopped-flow fluorescence data.

Parameters	Values and uncertainties a)
- Sliding -
ksl,N (s−1) b)	(2.1 ± 0.1) × 105
D1 (μm2 s−1) c)	(2.4 ± 0.1) × 10−2
- Intersegment transfer -
kit,N (M−1 s−1) d)	(1.0 ± 0.1) × 106
- Dissociation -
koff,N (s−1) f)	10 ± 4
kout (s−1) g)	83 ± 3
- Association -
kon,N (M−1 s−1) h)	(2.3 ± 0.8) × 106
- Effective sliding length -
λ (bp) i)	50 ± 2

^a)Uncertainties in reported values were estimated with a 68% confidence interval in fitting.

^b)From the data shown in Figure 7.

^c)From k_sl,N along with Eq. 15.

^d)From the data shown in Figure 8.

^f)From the data shown in Figure 8.

^g)From the data shown in Figure 7. This actually corresponds to k_out,CP because of the strong presence of intersegment transfer.

^h)From k_off,N and K_d,N.

ⁱ⁾From k_sl,N and k_out along with Eq. 21.

Intersegment transfer → sliding as the major pathway for target association

Our methodology permits kinetic measurements of intersegment transfer between nonspecific DNA sites in the presence of the target site. Owing to this unique feature, we can directly examine the role of intersegment transfer in the target association process. The analytical form of k_app (Eq. 18) for the system involving competitor DNA and intersegment transfer includes the contributions of two major pathways for target association (Figure 8B). The $T_P^{-1}{f}_P{P}_{tot}{D}_{tot}^{-1}$ term in Eq. 18 represents the contribution from the pathway involving the association of the free protein with a nonspecific site on the probe DNA and the subsequent sliding to the target (“Pathway 1”). The $T_{CP}^{-1}{f}_{CP}{P}_{tot}{D}_{tot}^{-1}$ term in Eq. 18 represents the contribution from the pathway involving the intersegment transfer of competitor-bound protein to a nonspecific site on the probe DNA and the subsequent sliding to the target (“Pathway 2”). From the obtained kinetic rate constants, the relative contribution of Pathway 2 in the presence of 2 μM competitor DNA was calculated to be 92%. This result clearly indicates the importance of intersegment transfer in the presence of a large excess of competitor DNA.

Discussion

Extremely efficient intersegment transfer of Egr-1 between nonspecific DNA sites

Our current study shows that intersegment transfer of Egr-1 between nonspecific DNA sites is extremely efficient. The intersegment transfer rate constant k_IT,N is only two-fold smaller than the association rate constant k_on,N (Table I). In our previous NMR study, the second-order rate constant for the intersegment transfer of Egr-1 between two different 28-bp nonspecific DNA duplexes was measured at 20 mM KCl to be 3.6 × 10⁶ M⁻¹ s⁻¹.³⁴ This value corresponds to k_IT,N = 1.8 × 10⁵ M⁻¹ s⁻¹, assuming that the number of sites is M = 20 in 28-bp DNA (note that k_IT,N is defined for each pair of departure and arrival sites). Although the value of k_IT,N = 1.0 ×10⁶ M⁻¹ s⁻¹ from our current study is ~6-fold larger, the difference in ionic strength (20 mM vs. 80 mM KCl) can easily account for this discrepancy. In fact, in the case of the HoxD9 homeodomain, the kinetic rate constant for intersegment transfer at 60 mM NaCl was 10-fold larger than that at 20 mM NaCl.⁵⁶ Interestingly, the intersegment transfer of Egr-1 between the target DNA sites is > million-fold slower (k_IT,S = 0.8 M⁻¹ s⁻¹),⁴² presumably due to the absence of domain motions that facilitate formation of the transient DNA-bridging intermediate for intersegment transfer.³⁴ Our current fluorescence data and the previous NMR data are consistent in that both indicate the very efficient intersegment transfer of Egr-1 between nonspecific DNA sites.

Intersegment transfer as a mechanism to counteract trapping by high-concentration DNA

Nonspecific binding to DNA can substantially enhance the protein-target association via the antenna effect and reduced dimensionality.^2–9 However, trapping by nonspecific sites can substantially slow down the target association process at a high concentration of DNA.^4,7 For example, Eq. 12 for the systems in the absence of intersegment transfer indicate that k_app constant is proportional to C_tot⁻¹, which represents a stronger trapping effect at a higher DNA concentration. However, our data (Figure 8) clearly show that this dependence on C_tot becomes significantly weaker in the presence of intersegment transfer, allowing the protein to find its target site efficiently.

Although intersegment transfer was defined as direct transfer between two DNA chains via a transient DNA-bridging intermediate (Scheme 2), this intermediate has never been directly observed experimentally, as far as we know. Only coarse-grained molecular dynamics (CGMD) simulations directly inferred the DNA-bridging intermediates in intersegment transfer.^34,57–59 If the breakage of the bridging intermediate is the rate-limiting step, intersegment transfer should appear to be a first-order process that cannot be detected by DNA concentration-dependence experiments. In such a case, the DNA-bridging intermediate should be a stable species. Intersegment transfer should appear to be a second-order process (Scheme 3) when the formation of the bridging intermediate via collision of a nonspecific complex and free DNA is the rate-limiting step. In this case, the DNA-bridging intermediate is a transient, low-population species. Intersegment transfer observed in the current and previous^26–34 studies are of this kind. Our previous studies using NMR spectroscopy and CGMD simulations suggested that local dissociation of one of the three zinc fingers could allow Egr-1 to transiently bridge two DNA duplexes during intersegment transfer.^34,60 However, as Sidorova et al. recently suggested,³² the phenomenological second-order nature of intersegment transfer can be explained without assuming the presence of DNA-bridging intermediates. Regardless of whether the DNA-bridging intermediates are actually involved or not, the second-order nature of intersegment transfer is important because it allows the protein to counteract trapping by nonspecific sites at a high concentration of DNA.

Relevance to the target association process in vivo

Because the DNA density is extremely high in the nucleus (~100 mg/ml),⁶¹ the target search pathway involving intersegment transfer may also play a major role in the target association process in vivo, at least for Egr-1. The intersegment transfer between the two DNA ends of a nucleosome particle may occur efficiently because the two ends are separated by only ~60 Å in three-dimensional space.⁶² In fact, the CGMD simulations showed that Egr-1 could undergo rapid intersegment transfer between two DNA duplexes separated by this distance.³⁴ Thus, extremely efficient intersegment transfer, which was observed in vitro in our current and previous studies, may allow Egr-1 to effectively bypass nucleosome particles and to rapidly locate the target sites in vivo.

Egr-1’s sliding on DNA: Comparison with other proteins

The one-dimensional diffusion coefficient D₁ for sliding on DNA is known for some proteins. For example, D₁ = 4.8 × 10⁶ bp² s⁻¹ (0.55 μm² s⁻¹) for OGG1;¹² D₁ = 2.6 × 10⁶ bp² s⁻¹ (0.30 μm² s⁻¹) for p53;⁶³ D₁ = 4.9 × 10⁵ bp² s⁻¹ (5.7×10⁻² μm² s⁻¹) for MutSα;¹⁴ D₁ = 9.5 × 10⁴ bp² s⁻¹ (1.1×10⁻² μm² s⁻¹) for EcoRV;⁶⁴ and D₁ = 3.0 × 10⁴ bp² s⁻¹ (3.5×10⁻³ μm² s⁻¹) for EcoRI.²² In most of these studies, sliding on DNA was directly observed with single-molecule fluorescence techniques. As noted previously,^5,63 the single-molecule techniques could overestimate D₁ due to their limited spatial resolution, which makes it difficult to distinguish sliding and hopping. Thus, D₁ measurements with completely different principles are of practical importance. Our current bulk-solution study shows that D₁ = 2.1 × 10⁵ bp² s⁻¹ (2.4 × 10⁻² μm² s⁻¹) for the Egr-1 zinc-finger protein (Table I). This value is within the typical range for the D₁ coefficients. Unlike three-dimensional diffusion coefficients in solvent, the one-dimensional diffusion coefficient D₁ for proteins sliding on DNA appears to be virtually independent of the molecular size. The number of intermolecular hydrogen bonds and ion pairs between protein and DNA in nonspecific complexes could be more important determinants of D₁.

Conclusions

We have studied the kinetics of Egr-1’s sliding and intersegment transfer in the target DNA search process. We have derived the analytical forms of k_app that facilitate kinetic investigations of distinct translocation mechanisms in the presence of a large excess of nonspecific competitor DNA. The stopped-flow fluorescence data along with the analytical forms of k_app permit the determination of the rate constants for sliding and intersegment transfer for proteins that form a stable complex with their target. The most important finding in this work is that intersegment transfer plays a major role in the target association process. Although this role has been speculated in previous studies on intersegment transfer in the absence of the target, those studies could not provide information on the extent to which intersegment transfer can actually contribute to the target association kinetics. Owing to the analytical expressions for k_app, our current methodology provides the relative contributions of distinct translocation pathways to target association. For the Egr-1 zinc-finger protein, the major pathway involves the intersegment transfer to a nonspecific site and the subsequent sliding to the target on the probe DNA. The relative contribution of this pathway was 92% for the system with 2.5 nM 113-bp probe DNA and 2,000 nM 28-bp nonspecific competitor DNA at 80 mM KCl. The corresponding contribution of the pathway involving the free protein’s association with a nonspecific site and the subsequent sliding to the target was as small as 8%. Despite its significant contribution, translocation via intersegment transfer cannot be distinguished from translocation via dissociation and association unless the DNA concentration dependence is studied. This nature is most likely responsible for a delay in the understanding of intersegment transfer in the current field. We hope that our present work will stimulate further investigations on the target search pathways involving intersegment transfer for other transcription factors as well as for DNA repair/modifying enzymes.

Materials and Methods

Egr-1 zinc-finger protein:

The DNA-binding domain of human Egr-1 comprising three zinc fingers (residues 335–432) was expressed in E. coli BL21(DE3) and purified as described in our previous papers.^34,42 The Egr-1 solution was treated with 3 mM TCEP at 4 ˚C overnight to completely reduce the protein, and the buffer was extensively exchanged to 10 mM Tris•HCl (pH 7.5), 300 mM KCl, 1 mM 2-mercaptoethanol, and 200 nM ZnCl₂. The protein was quantified using a BCA assay kit (Pierce) or using UV absorbance at 280 nm and an extinction coefficient of 1490 mM⁻¹ cm⁻¹ (the results from these two methods were consistent). The protein solution was kept under argon gas until use.

Fluorescence-labeled probe DNA for kinetics measurements:

To prepare the DNA duplexes shown in Figure 4A, 33-mer single-stranded DNA with a FAM attached to the 5’-terminus was purchased from Integrated DNA Technology (IDT). Of the 33 bases, the first 12 bases contain an Egr-1 target sequence (9 bp) and the last 21 bases can hybridize with positions 1160–1180 of the pUC-19 plasmid. The DNA duplexes of 63, 88, 113, and 143 bp were prepared via PCR using Vent DNA polymerase (New England), the FAM-labeled 33-mer primer, a reverse primer, and pUC-19 as the template. The 48-bp duplex was prepared via hybridizing the FAM-labeled 33-mer and a singled-stranded DNA, followed by base filling with DNA polymerase. The 33-bp duplex was prepared via annealing of equimolar amounts of individual complementary strands. The FAM-labeled 33-bp duplex was purified via polyacrylamide gel electrophoresis (PAGE) using 4–20% gradient polyacrylamide / TBE gels (Invitrogen). The other duplexes (i.e., 48-, 63-, 88-, 113- and 143-bp) were purified through the following three procedures. First, the reaction mixture was loaded onto a Resource-Q anion-exchange column (GE Healthcare), and the DNA was eluted with a gradient of 0 – 1.5 M NaCl in a buffer of 50 mM Tris•HCl (pH 7.5) and 1 mM EDTA. The fractions containing the desired reaction product were concentrated and subjected to PAGE with 4–20% gradient polyacrylamide / TBE gels. The band of the desired PCR product was excised, crushed, and shaken at room temperature in a buffer of 10 mM Tris•HCl (pH 7.5) and 40 mM KCl for 24 – 36 hours to extract the DNA from the gel. The extracted DNA duplexes were further purified via a PCR purification kit (Qiagen).

Competitor DNA:

The 28-bp nonspecific competitor DNA used for the kinetic experiments was the same as that used for the NMR studies of the Egr-1 - nonspecific DNA complex. This DNA was synthesized and purified as described previously.³⁴

Stopped-flow fluorescence-based assay of target association kinetics:

The kinetics of target association was measured at 20 ˚C with an ISS PC-1 spectrofluorometer equipped with an Applied Photophysics Rx-2000 stopped-flow device. The following two solutions were mixed: Solution A: 50–200 nM protein in a buffer of 10 mM Tris•HCl (pH 7.5), 80 mM KCl, and 200 nM ZnCl₂; and Solution B: 5 nM FAM-labeled DNA and 1000–8000 nM 28-bp competitor DNA in the same buffer. The mixing ratio was 1:1; thus, the final concentrations of the protein and DNA became halved. The spectrofluorometer has two emission channels in a T-format: one with and the other without an emission-light monochromator. In the stopped-flow kinetics experiments, we used the emission channel with no monochromator for better sensitivity. The FAM fluorophore was excited at 460 nm, and the emission light that passed through a long-pass filter with a cutoff at 515 nm (Edmund Optics) was recorded. Immediately after stopping the flow for mixing, the time-course data of the fluorescence intensity were collected for 4–10 s with a time interval of 20–50 ms. Each experiment was repeated 8 to 10 times. As expected from the ODE-based numerical simulations, the time-course data of the fluorescence intensity were found to be mono-exponential. The apparent pseudo-first-order rate constant k_app for target association was determined from the experimental data via nonlinear least-squares fitting with f(t) = f_∞ + (f₀ – f_∞)exp(–k_appt), where f₀, f_∞, and k_app are optimized, and f(t) represents the fluorescence intensity.

Fluorescence anisotropy-based affinity measurements:

The two 12-bp DNA duplexes with a 3’-terminal TAMRA were prepared for affinity measurements (Figure 4B). HPLC-purified DNA strands from 250-nmol scale syntheses were purchased from Integrated DNA Technology. The complementary strands were mixed and annealed at 85 ˚C, and the duplex was purified by PAGE using 4–20% gradient polyacrylamide / TBE gels. The protein-titration experiments were performed at various concentrations of KCl in a buffer of 10 mM Tris•HCl (pH 7.5). The fluorescence anisotropy of the TAMRA-labeled DNA was measured at an excitation wavelength of 533 nm and emission wavelength of 580 nm using an ISS PC-1 spectrofluorometer. Because the K_d for specific complexes at low ionic strength was lower than the detection limit of the fluorescence method, we measured the K_d at higher ionic strengths (150, 200, 250, 300, 350, and 400 mM KCl) and extrapolated to the K_d at 80 mM KCl under the assumption of a linear relationship between log[KCl] and logK_d based on the counterion condensation theory.^52,53

Determination of the rate constants for protein translocation:

Using the Levenberg-Marquardt algorithm of MATLAB, the rate constant k_sl,N for sliding was determined from the length-dependent k_app data (Figure 7) via nonlinear least-squares fitting with Eqs. 7–13. The total numbers of binding sites, L for the probe DNA and M for the competitor DNA, were calculated as A – B + 1, where A is the total number of base pairs, and B is the number of base pairs covered by a protein molecule. Based on structural information,^34,39 B = 9 was used. Based on the sequences of the probe DNA duplexes (Figure 4), m = 2 was used. The values of k_off,N, and k_it,N were determined from the C_tot-dependent k_app data (Figure 8) via nonlinear least-squares fitting with Eqs 7, 10, 11, 13, and 17–20. These calculations used ϕ = 2 and k_on,N = k_off,N / K_d,N. The MATLAB scripts for the ODE-based simulations of the target search kinetics and those for determination of the kinetic parameters via fitting to experimental data are available as the ‘TDSK’ (target DNA search kinetics) package upon request to the corresponding author.