DNA Modeling Reveals an Extended Lac Repressor Conformation in Classic In Vitro Binding Assays

Abstract

Protein-mediated DNA looping, such as that induced by the lactose repressor (LacI) of Escherichia coli, is a well-known gene regulation mechanism. Although researchers have given considerable attention to DNA looping by LacI, many unanswered questions about this mechanism, including the role of protein flexibility, remain. Recent single-molecule observations suggest that the two DNA-binding domains of LacI are capable of splaying open about the tetramerization domain into an extended conformation. We hypothesized that if recent experiments were able to reveal the extended conformation, it is possible that such structures occurred in previous studies as well. In this study, we tested our hypothesis by reevaluating two classic in vitro binding assays using a computational rod model of DNA. The experiments and computations evaluate the looping of both linear DNA and supercoiled DNA minicircles over a broad range of DNA interoperator lengths. The computed energetic minima align well with the experimentally observed interoperator length for optimal loop stability. Of equal importance, the model reveals that the most stable loops for linear DNA occur when LacI adopts the extended conformation. In contrast, for DNA minicircles, optimal stability may arise from either the closed or the extended protein conformation depending on the degree of supercoiling and the interoperator length.

Introduction

Protein-mediated DNA looping is a well-known gene regulation mechanism. A DNA loop essentially functions as a genetic switch by either inhibiting or promoting the activity of RNA polymerase over long lengths of DNA (1,2). A prime example of DNA looping is that induced by the lactose repressor protein (LacI) of Escherichia coli. LacI binds simultaneously to two distant operator sites on a single DNA molecule, causing the intervening DNA to form a loop that is subject to considerable bending and twisting. In this instance, the looped DNA represses RNA polymerase activity.

Extensive experimental studies have revealed fundamental mechanisms that control protein-mediated DNA looping. For instance, members of the Müller-Hill laboratory evaluated DNA loop formation in vitro over a large range of interoperator lengths (63–535 bp) using electron microscopy, nondenaturing polyacrylamide gel electrophoresis, and DNase I protection experiments (3). They found that changing the interoperator length basepair by basepair not only lengthened the intervening DNA, it also changed the relative torsional alignment (phasing) of the two Lac operators by ∼34° per basepair. As a consequence of operator phasing, loop stability was found to oscillate with a period of approximately a helical turn. Moreover, Krämer et al. (3) identified the optimal interoperator length/phasing that maximizes loop stability. The periodic dependence on the interoperator length/phasing has also been observed in vivo by determining the periodic repression efficiency of the lac operon (4–6). In addition, recent single-molecule studies (7,8) confirmed this periodic dependence on operator phasing.

The effects of supercoiling on DNA minicircles on loop stability and topology were investigated by Krämer et al. (9), who probed loop stability by adding Lac inducer (IPTG) or competitor DNA. The results show that the degree of supercoiling alters the interoperator length for optimal loop stability. They also probed the topology of the DNA loops using topoisomerase I relaxation assays, and concluded that LacI-induced looping can absorb twist up to one superhelical turn.

Mehta and Kahn (10) revealed the dependence of loop stability on intrinsically bent DNA sequences using sequences that incorporated A-tract domains. The location of the A-tract within the interoperator DNA strongly influenced loop stability, leading to hyper- and hypostable loops.

Key to understanding LacI-induced DNA looping and stability is the structure of Lac repressor protein, as known, for example, by the x-ray cocrystal structure of the repressor bound to oligonucleotide operators (11,12). This static image of LacI is a dimer of dimers, where the N-terminal headpiece forms a DNA-binding domain and the C-terminal tetramerization domain is composed of a (possibly flexible) four-helix bundle.

Despite these and other observations of LacI-induced DNA looping, many unanswered questions about this mechanism, including the role of protein flexibility, remain. Mounting evidence (both experimental and theoretical) suggests that LacI can adopt an extended conformation by pivoting the two dimers about the four-helix bundle. Both electron microscopy and x-ray scattering (13,14) reveal extended conformations in conjunction with the crystallographic closed or V-shaped structure mentioned above. The extended conformation was proposed to explain how short (∼150 bp) stable loops form with LacI even when the operators are out of phase (15). Compelling evidence has also been provided by single-molecule tethered particle motion (TPM) experiments that revealed a significant conformational change in LacI (15,16). These studies demonstrated that flexibility of the tetramerization domain (at the vertex of the V), as opposed to the two hinge domains (at the top of the V) that connect to the DNA-binding heads, is responsible for the observed results. Other studies that altered the mechanical stiffness of the protein by introducing hinge mutants (17) and cross-linking the protein to prevent opening of the tetramerization domain (18) revealed that looping was completely prevented. Evidence for LacI conformational changes from experiments using TPM (18) and atomic force microscopy (15) suggests that the two looped states interconvert (i.e., change binding topologies) without first unlooping. In bulk fluorescence resonance energy transfer experiments, investigators examined different DNA constructs in which Lac operators bracketed a sequence-directed bend, and the results suggest that the LacI-DNA complex exists in two states (possibly closed/extended or parallel/antiparallel) (19,20). Additional studies of other looping proteins showed that loop geometry and energy depend on the geometry of the protein (21–23).

Investigators have employed various coarse-grained DNA models (8,24–27) to probe the role of flexibility of the LacI tetramerization domain by introducing approximations for an extended LacI conformation. By contrast, an atomistic model of LacI coupled to a coarse-grained (elastic rod) model of DNA implicates flexibility of the hinge domains as the primary source of compliance in LacI (28). Given the mounting evidence for an extended LacI conformation, it is natural to hypothesize that this structure was active but overlooked in earlier experiments. A partial answer is found in the study of Swigon et al. (26), who employed a discrete model for DNA to compute the free energy of looping as a function of the interoperator DNA length, as in previous in vitro looping experiments (3,9). Computed results demonstrate that an extended conformation best agrees with DNase I cutting patterns and the interoperator lengths yielding optimal loop stability (3). A detailed analysis of the supercoiled DNA minicircle experiments of Krämer et al. (9) has not yet been conducted.

In this work, we reinterpret the classic in vitro studies (3,9) conducted by the Müller-Hill laboratory reviewed above. We do so by employing a computational rod model for the DNA-LacI complex that considers both the closed or V-shaped conformation of the LacI cocrystal structure and an open or extended conformation. Our work confirms the results of Swigon et al. (26) for linear DNA and extends them by adding a thorough analysis of loop stability and topology for the topoisomerase I relaxation assays of the looped and supercoiled DNA minicircles (9). The predicted loop stability aligns well with experimental observations and provides further evidence for the extended protein conformation. This new interpretation of these classic experiments emerges from a model that explicitly captures the three-dimensional bending and twisting of looped DNA. We open with a summary of this model.

Methods

The lengths of typical DNA loops are on the order of the DNA bending persistence length or longer, rendering analysis by molecular dynamics (MD) nearly impossible. To circumvent this limitation, investigators have developed coarser-grained DNA models by drawing from statistical mechanics and continuum (rod) mechanics. Continuum rod models are computationally efficient at the (long) length scales and timescales that govern looping (8,29), and can successfully predict the aforementioned oscillatory behavior of looping free energy (30). Initially, the rod model was limited by its inability to account for entropic contributions to looping free energy. However, extended models that account for both enthalpic and entropic free-energy contributions have recently been developed (31). A multiscale model that couples an elastic rod model of DNA with an all-atom description of the Lac repressor was demonstrated (28). Alternative modeling approaches include classical statistical mechanical descriptions (25) and discrete models in which each basepair is modeled as a rigid body elastically coupled to its nearest neighbors (26,32–35).

Herein, we employ a previously developed computational rod model to explore the effects of interoperator DNA length and alternative LacI conformations on the looping of both linear DNA and supercoiled DNA minicircles per the experiments of the Müller-Hill laboratory (3,9). A derivation of the computational rod model is provided in the Supporting Material and a brief summary is given below.

The continuum rod is fundamentally a coarse-grained model of DNA (36,37) that averages the elastic properties of DNA in describing the three-dimensional bending and twisting of the DNA helical axis on length scales of approximately a helical turn (3 nm) and longer. As described previously (29,38), we use averaged stiffness properties of DNA as determined by commonly accepted values of the bending and torsional persistence lengths (50 nm and 75 nm, respectively) (39,36). These stiffness properties are employed in the following elastic energy functional:

$$E\left(t\right)={\int }_0^L\frac{1}{2}\left[{\left\{\overrightarrow{\kappa }\left(s,t\right)\right\}}^T\mathbf{B}\left\{\overrightarrow{\kappa }\left(s,t\right)\right\}\right]ds,$$ (1)

where E(t) is the elastic strain energy (in units of kT), t is the independent time variable, s is the independent contour length variable (nm), B is a diagonal tensor that defines the (assumed constant) bending and torsional stiffness of DNA (nm-kT), and $\overrightarrow{\kappa }\left(s,t\right)$ is the curvature/twist vector of the helical axis (nm⁻¹).

The overall free-energy budget for looping includes contributions from 1), the elastic energy of deforming the DNA into the looped conformation (Eq. 1); 2), the entropic energy cost; 3), the electrostatic energy due to possible DNA self-interaction; 4), the free energy of binding to the operator sites; and 5), the free energy of deforming the protein into the looped state. In this study, we assume that the binding free energy and entropic cost (though nonzero) remain essentially constant from one loop to the next. Further, we assume that the free-energy change of LacI upon looping is constant. The electrostatic contribution, which is expected to remain modest and relatively constant (except possibly when very close contact is approached (40)), is neglected. Thus, among these contributions to the free energy, it is the elastic energy reported herein that most likely discriminates looped conformations.

The computational rod model is described by partial differential equations (PDEs) that govern the balance laws for linear and angular momentum for an infinitesimal rod (DNA) element (41,42). The PDEs are discretized via space-time finite differencing and augmented with boundary conditions that describe the binding of DNA to LacI. These boundary conditions are derived by aligning the DNA operator sites, representing the ends of the elastic rod model, with the operator-bound DNA in the LacI crystal structure (Protein Data Bank ID 1LBG (12)). Following Goyal et al. (42), we align rod-fixed reference frames (43) at the rod ends to corresponding basepair-fixed reference frames of the bound operator DNA (and at the site of the third basepair into the operators). Therefore, the protein crystal structure determines the position and orientation of the rod at its terminal ends. The computation begins from an initial condition with the DNA unbound and in a stress-free (straight) conformation and, after dynamic relaxation occurs, concludes with equilibrated DNA bound to LacI, satisfying the above boundary conditions. Analysis of the equilibrium conformation yields the enthalpic (elastic energy) cost of looping and the complete three-dimensional equilibrium geometry of the interoperator DNA.

The symmetric or ideal operators, found in the crystal structure, permit the protein to bind to the substrate DNA in multiple ways, yielding multiple possible binding topologies (42,43). To reduce the computational effort, we follow Lillian et al. (20) and assume that LacI is symmetric about its dyadic axis, which necessitates very small position and orientation corrections to the above boundary conditions from the original (slightly asymmetric) crystal structure. These approximations lead to three possible binding topologies for the closed LacI protein (denoted as P1, P2, and A1 in Fig. 1 a). Several topoisomers may also exist for each binding topology, often including two lowest-energy topoisomers (one overtwisted and one undertwisted) that do not induce self-contact of the intervening DNA. Ignoring the electrostatics (except in the neighborhood of near self-contact) remains a reasonable assumption (40), and therefore we analyze looped conformations without self-contact while recognizing their lower energetic cost (and hence greater likelihood) of formation. Therefore, our analysis of a single molecule (i.e., one interoperator length) requires computation of six looped conformations distinguished by three possible binding topologies each having under- and overtwisted topoisomers.

Figure 1.

(a) Three possible binding topologies for the closed protein for looping experiments of linear DNA. (Because of the assumed rotational symmetry of the protein, the A1 binding topology remains equivalent to the A2 binding topology.) (b) Three possible binding topologies for the closed protein for looping experiments of supercoiled DNA minicircles. The binding topologies follow from pairing a secondary loop (light gray/green) with the primary interoperator DNA loop (dark gray/purple) using the only available binding topology that remains after binding of the primary loop. (Due to protein symmetry, the primary + secondary combination is identical for the combinations A1+A2 (= A2 + A1)). The operator DNA is considered fixed by the protein and fills the gap between the two rod domains. (c) Approximation to the extended protein conformation (as introduced in the Discussion). Because of the assumed rotational symmetry of the protein, the P1^E primary loop is equivalent to the P2^E primary loop, and the primary + secondary combination is identical for the cases P1^E + P2^E (= P2^E + P1^E).

We simulate the LacI-DNA looping experiments of Krämer et al. (3) on free (unsupercoiled) DNA with interoperator DNA lengths ranging from 153 to 168 bp. Adopting the convention used by Krämer et al. (3), we define the interoperator DNA length as the length in basepairs between the centers of symmetry of the two ideal Lac operators. (In previous studies (20,42), we defined the interoperator length as the length in basepairs from the third basepair inside the DNA operators.) We assume that all sequences have a helical pitch of 10.5 bp and ignore any influence of the unlooped tail domains outside the interoperator DNA.

We also simulate the looping experiments of Krämer et al. (9) on supercoiled 452 bp DNA minicircles. In this instance, LacI binding generates two looped domains that are modeled independently. One of the DNA loops, termed the “primary” loop, ranges in length from 153 to 168 bp, whereas the “secondary” loop forms the remainder of the DNA minicircle. For instance, the 153 and 168 bp primary loops generate companion 299 and 284 bp secondary loops, respectively. Primary loops formed with the P1, P2, and A1 binding topologies necessarily form secondary loops with the P2, P1, and A2 binding topologies, respectively (see Fig. 1 b). Additionally, each loop that constitutes the minicircle possesses over- and undertwisted topoisomers. Therefore, a computation of 12 conformations (three binding topologies × two loops × two topoisomers) is required for each length of the primary interoperator loop. Furthermore, the minicircles of Krämer et al. (9) have an initial ΔLk of −2, −1, 0, and +1. Here, ΔLk is relative to the Lk of the topoisomer with the least superhelical stress, the ΔLk = 0 topoisomer. ΔLk is the sum of ΔTw (the relative amount of twist with respect to the relaxed state) and Wr (writhe as calculated according to Klenin and Langowski (44)). As with the computations for linear DNA, we ignore both (higher energy) inter- and intradomain DNA self-contact and protein contact. Consequently, we do not prescribe the ΔLk of the minicircle but calculate it from the computed equilibrium conformation.

To further interrogate the topology of LacI-DNA minicircle complexes, Krämer et al. (9) employed topoisomerase I enzymes. Specifically, they used topoisomerase I to relax some of the superhelical stress stored in the 452 bp DNA minicircle-loop complexes with initial ΔLk of −2, −1, 0, and +1. Gel migration assays were used to measure the ΔLk of the “relaxed” DNA minicircle-loop complexes. The degree of superhelical stress that remains after relaxation is an artifact of the topology of the DNA minicircle-loop complex before relaxation.

To simulate the topoisomerase I assays using the computational model, we employ several assumptions: 1), topoisomerase I enzymes relax the DNA with LacI remaining bound (i.e., the binding topology remains fixed); 2), the relaxed state is independent of the dynamic relaxation process (i.e., we do not model the actual dynamic pathway leading to the final equilibrium conformation); and 3), topoisomerase I allows a complex to relax to a lower-energy state, which may also increase ΔLk. Consequently, we consider all possible loops (P1+P2, P2+P1, and A1+A2) with fixed ΔLk as a population of reactants. To facilitate a direct comparison with experimental results, we present our data in the form of computationally predicted gel band distributions by employing the following relation:

$$P_i^x=\frac{e^{-E_i/kT}}{\sum _{j=1}^{N^x}{e}^{-E_j/kT}},$$ (2)

where N^x is the number of loop conformations that are possible in a given ensemble of reactants or products (which we denote by replacing x with either r or p, respectively), E_i is the computed elastic energy of the i^th loop conformation, and P^x_i is the proportion of molecules that form a loop with the i^th conformation.

To calculate the band grayscale intensity, we first calculate the proportion, P_i^r, of each reactant (loop conformation) using Eq. 2, where the ensemble is composed of N^r conformations possessing a given ΔLk. The ensemble of products for an individual reactant is composed of all N^p conformations with the same binding topology as the reactant. The product P_i^rP_j^p for each reactant-product pair constitutes the proportion of loops that relax to form the j^th conformation of the i^th reactant. Next, we sort the products by ΔLk, considering all reactants. Finally, to calculate the grayscale intensity for a prescribed ΔLk, we sum the proportions of all the reactant-product pairs that result in the given ΔLk.

Results

We open the Results section by focusing on calculations that assume the closed (V-shaped) protein conformation. A comparison of these theoretical results with experimental findings reveals significant shortcomings. In the Discussion, we then introduce results that assume the open protein conformation, which largely overcomes these shortcomings.

We first consider predictions for the looping of linear DNA, paralleling the experiments of Krämer et al. (3). We compute the elastic energy of LacI-DNA complexes as a function of the interoperator length over the experimental range of 153–168 bp. Fig. 2 presents the elastic energy required to form a loop for each of the three possible binding topologies, reproducing the results obtained by Swigon et al. (26) using a different modeling approach. We report the minimum energy of the over- and undertwisted topoisomers. The elastic energy for each binding topology is periodic, with an interoperator length of approximately the DNA helical repeat. Note that the A1 binding topology (blue circles) is energetically preferred for all interoperator lengths and has an energetic minima at 161 bp. The next-highest-energy binding topology is the P1 (green squares) with a minima at 159 bp, followed by the highest-energy P2 binding topology (black triangles) with minima at 156 and 167 bp. In the original gel data, sharp, highly resolved bands occur at 158 and 168 bp, and indicate that stable loops form at those locations.

Figure 2.

Elastic energy E (kT) as a function of interoperator DNA length (bp). The reported elastic energy is the minimum energy of the over- and undertwisted topoisomers. Shown are results for the A1 binding topology (blue circles), the P1 binding topology (green squares), and the P2 binding topology (black triangles). The curve with stars (red) corresponds to the P1^E binding topology with E_LacI = 0 kT as described in the Discussion. The reported energy is the elastic energy necessary to deform DNA into a loop with a given binding topology. For the extended conformation, the energy penalty associated with opening the protein is added to the elastic energy as a means of comparing binding topologies. As E_LacI is increased, the energy required to form a loop in the extended conformation increases and simply shifts the curve upward. The triangles below the axis indicate locations where the most stable loops form in the experiments. These results reproduce those obtained by Swigon et al. (26) using a different modeling approach.

Next, we consider computational results that parallel the experimental LacI-induced looping of DNA minicircles (9). Specifically, we consider supercoiled DNA minicircles that retain the previous interoperator lengths of 153–168 bp.

For each of the four minicircle topoisomers (ΔLk = {−2, −1, 0, +1}), we report in Fig. 3 the change in energy (ΔE) as a function of interoperator DNA length. The computed ΔE represents the difference in elastic energies of the minimum energy LacI-looped complex and the minimum energy of the unbound minicircle that possesses the given ΔLk. Although the entropic contribution may play a role, its contribution should be similar in the LacI-DNA complex and in the unbound minicircle because they are the same overall size. For each ΔLk topoisomer, we report the change in energy for the minimum energy complex, which may adopt different binding topologies as the interoperator length changes. Thus, the dashed lines indicate ΔE for all looped complexes among all three possible (closed protein) binding topologies in Fig. 1 b. The apparent discontinuities arise because the minimum-energy complexes possess different binding topologies. Minimum-energy looped complexes that possess the P1+P2 binding topology are denoted by dashed green lines, and those with the A1+A2 binding topologies are denoted by dashed blue lines. Looped complexes that possess the P2+P1 binding topology are never energetically preferred, due to the high energetic cost of the primary P2 loop in the P2+P1 complex.

Figure 3.

ΔE (kT) of the minimum energy loop as a function of interoperator DNA length (bp) for (a) ΔLk = −2, (b) ΔLk = −1, (c) ΔLk = 0, and (d) ΔLk = +1 topoisomers. Green, P1+P2; black, P2+P1; blue, A1+A2; red, P1^E+P2^E. The shaded regions in the ΔLk = −2 topoisomer indicate regions where self-contact occurs. The dashed lines represent minimum energy solutions considering only the closed-protein binding topologies. The solid lines represent minimum energy solutions considering both the closed- and extended-protein binding topologies with E_LacI = 0 kT. The extended LacI conformation is described in the Discussion. The P2+P1 binding topology is never energetically preferred, due to the high energetic cost of forming the primary P2 loop in the P2+P1 complex.

The energy of the unbound minicircle template serves as an energy datum for each topoisomer. Computation of the energy datum values follows a separate calculation in which a straight isotropic rod is bent into a circle and then twisted to achieve the requisite ΔLk. It therefore follows that the ΔLk = +1 and ΔLk = −1 topoisomers require equal energy (16.5 kT) to form, whereas the ΔLk = 0 topoisomer requires only the energy (6.6 kT) to bend the rod into a circle. Computation of the ΔLk = −2 topoisomer (29.4 kT) requires two turns of twist, which causes the circle to deform out of plane into a figure-8 shape. This conformation induces self-contact at the junction of the figure-8. For this special instance, we employ a previous form of our model that explicitly accounts for the electrostatic self-repulsion of DNA and prevents the cut-through in the figure-8 (45). However, the change in energy from this datum only dictates the offset on the y axes between each topoisomer in Fig. 3, and does not affect any other results in this work. Finally, for the ΔLk = −2 topoisomer, self-contact exists for several interoperator lengths, as illustrated by the shaded regions. Although self-contact is neglected, the equilibria in these regions require substantially higher formation energy due to the added electrostatic energy.

To further probe the loop topology, we investigate the topoisomerase I relaxation experiments of Krämer et al. (9). The computed relaxation products for the topoisomerase I relaxation experiments of Krämer et al. (9) are illustrated as a distribution of bands in the predicted gel of Fig. 4 (subscript 1), as described in Methods. Shown are the relaxation products as a function of interoperator length for each of the four topoisomers. The predicted band distributions do not estimate migration velocity and are constructed to mimic the appearance of experimental gel data.

Figure 4.

ΔLk after relaxation as a function of interoperator DNA length (bp) for (a) ΔLk = −2, (c) ΔLk = −1, (d) ΔLk = 0, and (e) ΔLk = +1 topoisomers. Subscript 1: Predicted band distributions assuming a closed protein. Subscript 2: Predicted band distributions now including the extended conformation P1^E with E_LacI = 0 kT as described in the Discussion. The band intensity is weighted on a grayscale based on the energy between the lowest-energy solution and possible higher-energy solutions (see Eq. 2). The shaded regions in the ΔLk = −2 topoisomer indicate regions where self-contact occurs. (b) A sketch of the original gel data for the ΔLk = −1 topoisomer from Krämer et al. (9). Highly resolved/sharp bands are shown in black, and faint bands are shown in gray.

Discussion

We begin by comparing the predicted results for looping of linear DNA with the experimental results of Krämer et al. (3). Although the binding topologies are not observable from the experiments, the rod model predicts the preferred binding topology/topologies. We assume that the minimum energy conformation considering all three possible binding topologies in Fig. 1 a is the complex that is most likely formed in the experiments. From Fig. 2, the A1 binding topology is energetically preferred regardless of the interoperator length. Moreover, the A1 binding topology has energetic minima, and therefore the most stable loops, at 161 bp. In the experiments, however, the most stable loops (indicated by sharp bands in the electrophoretic gel) occur at 158 and 168 bp. In our interpretation, we believe the theory should predict the optimal interoperator lengths to within approximately a basepair. A single-basepair difference may arise due to differences in our assumed helical repeat (10.5 bp) versus the actual helical repeat. By this criterion, the theory does not predict the location of the most stable loops.

Similarly, we compare the most stable loops for supercoiled minicircles of Krämer et al. (9) with the energetic minima from computations. The results of Fig. 3 clearly demonstrate that the degree of supercoiling strongly influences the location of the energetic minima (and therefore the interoperator length that yields the most stable looped complexes; see dashed lines).

A major conclusion from the experiments is that supercoiling changes the interoperator length requirement for stable loop formation. Our computations agree by predicting stable loops at 159 bp for the ΔLk = 0 topoisomer (notably, near the 158 bp interoperator length for linear DNA). If negative supercoiling is added, the location of the most stable loop increases by 4 bp. The model captures this trend and predicts stable loops at 162 bp for the ΔLk = −1 topoisomer, in accord with Swigon et al. (26). Experimentally, the location of the most stable loop for the ΔLk = −2 topoisomer lies at 163 bp. We report two minima near 155 and 166 (at the edge of the shaded region indicating contact), and note that although our computed minima are 3 bp away from the experimental minimum, they would likely remain even if self-contact (incurring a higher energetic cost) were incorporated into the model. Regardless, the shaded regions in Fig. 3 a indicate that self-contact likely plays a role in the ΔLk = −2 topoisomer. Conversely, adding positive supercoiling to the substrate DNA was experimentally shown to decrease the length of the interoperator DNA for the most stable loops by 3–4 bp. However, our computations for the ΔLk = +1 topoisomer predict an interoperator length of 159 bp, which is 5 bp away from the experimental value.

As with the linear DNA and supercoiled minicircles, the computations for the topoisomerase I assays yield mixed agreement with the experimental results. In agreement with a major conclusion of Krämer et al. (9), we predict that up to one superhelical turn can be constrained by LacI looping (i.e., we predict relaxation products having nonzero ΔLk in Fig. 4, subscript 1). Gel data at every interoperator length are available for the ΔLk = −1 topoisomer (as sketched in Fig. 4 b) and reveal single bands centered at about 158 and 168 bp (indicating the ΔLk = 0 topoisomer) and about 153 and 163 bp (indicating the ΔLk = −1 topoisomer). However, between these regions double bands exist (Fig. 4 b). The computations correctly predict the ΔLk = 0 topoisomer at 158 and 168 bp, the ΔLk = −1 topoisomer at 153 and 163 bp, and the stepwise transition with multiple products (gray bands) in between these limits (see, for example, the region near 160 bp). Although we can perform a comparison with gel data for the ΔLk = −1 topoisomer at every interoperator length, original gel data for each topoisomer are available only for interoperator lengths of 158 and 163 bp. The results obtained at these interoperator lengths are summarized in Table 1 and show disagreement with the ΔLk = +1 topoisomer. Complete gel data at all interoperator lengths for the ΔLk = −2, 0, and +1 topoisomers would, of course, enable a comprehensive comparison.

Table 1.

Summary of relaxation products at 158 and 163 bp, showing experimental and computed values

ΔLk	Relaxation products
	158 bp, experiment	158 bp, computed	163 bp, experiment	163 bp, computed∗
−2∗	0	0 (0)	−1
−1	0	0 (0)	−1	−1 (−1)
0	0	0 (0)	0/−1	−1 (+1)
+1	0	+1 (0)	0/−1	+1 (+1)

Forward slash separating values indicates multiple populations present in the experiments. Asterisks indicate that the model does not predict solutions at this location. Results including the extended conformation are shown in bold in parentheses.

Although a successful prediction for the ΔLk = −1 topoisomer relaxation was obtained, the predictions for the linear DNA and supercoiled minicircles do not capture the major experimental findings. We propose that including the LacI protein flexibility would resolve these disagreements for both linear DNA and the ΔLk = +1 minicircle. Allowing for an extended protein conformation would likely yield lower-energy loops, which would strongly affect our results.

In accordance with previous studies (8,26,27), we assume that the protein can rotate about the tetramerization domain and adopt an extended conformation in which the angle formed between the two dimers is ∼180°. Specifically, we assume that the protein is bistable and rigid in both the closed and extended conformations. As estimated by Swigon et al. (26), the change from the closed to extended conformation could involve a free-energy penalty of 1.8–9.4 kT (though likely closer to 1.8 kT). Let the extended conformation be denoted by P1^E with free-energy penalty E_LacI.

We now return to the linear DNA experiments and augment the boundary conditions previously associated with the closed protein conformation with those defined by the open/extended conformation (Fig. 1 c). In addition, we include an estimate of the cost of protein opening (E_LacI) in arriving at the energetic cost of looping for the P1^E binding topology. We treat E_LacI as an unknown parameter and vary it over an estimated range. In Fig. 2, the curve with stars (red) represents the new results for the limiting case E_LacI = 0 kT. Increasing E_LacI simply shifts this curve (the energetic cost of looping) upward relative to the previous results for the closed protein conformation. For low values of E_LacI (under ∼4 kT), the P1^E binding topology is energetically preferred over all previously considered binding topologies and nearly all interoperator lengths. Moreover, its energetic minima occur at 157 and 168 bp, which agrees well with the most stable loops (resolved/tight bands) at 158 and 168 bp observable in the gel data (3). By contrast, recall that the A1 binding topology is energetically favorable for the closed protein conformation but achieves a minimum at 161 bp, which is significantly different from the experimental observations. The facts that the P1^E binding topology is energetically preferred and possesses the correct minima for optimal loop stability provide compelling evidence that LacI adopts an extended conformation in these early experiments.

The looped complexes in the experiments exhibited noticeably different gel migration behavior (3), and this provides an opportunity for comparison with the model. We computed the radius of gyration of the LacI-(linear)DNA complex (Fig. S2) as a means of comparing size with migration velocity, and the results suggest that the P1^E (and not A1) binding topology is present.

We now return to the predicted looping of the DNA minicircles upon addition of the extended protein conformation; refer to the red lines in Fig. 3, which represent new results for the P1^E+P2^E binding topology for the limiting case E_LacI = 0 kT. The solid lines arise from considering solutions from all closed protein binding topologies and the extended conformation in arriving at the overall energetic minimum. Including the extended LacI conformation yields new energetically favorable loops at specific interoperator lengths for the ΔLk = +1, 0, and −1 topoisomers. Of more importance, for the ΔLk = +1 topoisomer, the energetic minima now occur at 155 and 165 bp when P1^E+P2^E is included, in full agreement with experimental observations. The minima for the cases ΔLk = −2 and −1 remain largely unchanged, whereas there is a modest change (from 159 bp to 157 bp) for the ΔLk = 0 case, which is also more closely aligned with experiment. The overall results, summarized in Table 2, clearly suggest that calculations that incorporate the extended protein conformation yield superior agreement with experimental observations. The energetic minima reported in Table 2 for E_LacI = 0 kT remain unchanged over the entire range E_LacI ≤ 6 kT. (As noted above, increasing E_LacI simply increases the energetic cost of looping, and only extreme values E_LacI > 13 kT render the P1^E+P2^E energetically unfavorable relative to the solutions for the closed LacI conformation for all interoperator lengths.

Table 2.

Interoperator DNA length for most stable loops, excluding and including the extended LacI conformation

ΔLk	−2	−1	0	+1
Experiment	163	161/162	157/158	154,165
Computed excluding P1E+P2E	155,166	162	159	159
Computed including P1E+P2E	155,166	162	157,168	155,165

Forward slash separating values indicates two neighboring stable complexes found in the experiments.

Of interest, the computed results for the ΔLk = −2 and −1 topoisomers exhibit little to no sensitivity to changes in E_LacI in contrast to the ΔLk = 0 and +1 topoisomers, which exhibit significant sensitivity. Thus, it appears that each topoisomer favors a specific binding topology. In particular, the ΔLk = −2 and −1 topoisomers largely prefer the A1+A2 binding topology of the closed protein. Conversely, the ΔLk = 0 and +1 topoisomers largely prefer the P1^E+P2^E topology (with some dependence on E_LacI).

A major contribution of the above analysis is the new (to our knowledge) light it sheds on the energetically preferred binding topology for the looping of both linear and supercoiled (minicircle) DNA. In the original work by Krämer et al. (9), it appears that changes in supercoiling were thought to be partitioned mainly into Tw (and not Wr), which, when absorbed by the minicircle, alters the helical repeat of the intervening DNA. Our data dispel the notion that the addition of basepairs to the intervening DNA sequence is simply a small perturbation of the same structure (Figs. 2 and 3). By contrast, the computations strongly suggest that the loops analyzed in the electrophoretic gels likely represent a variety of binding topologies that are dependent on the length of the intervening DNA. The insertion of a single basepair alters the phasing to such a degree that the loop may bind in a completely different orientation. Moreover, in the interactions between LacI and DNA minicircles, the length of the intervening DNA not only shifts the locations of the most stable loops, it also dictates which binding topology the complex prefers.

In light of the above discussion, we now also revisit the topoisomerase I relaxation experiments. Topoisomers in the extended conformation are now added in the thermal ensemble (see Eq. 2) with those of the closed protein in arriving at computationally predicted band distributions. Fig. 4 (subscript 2) presents the relaxation products as a function of interoperator length, and Table 1 (in bold) summarizes the relaxation products at 158 and 163 bp when the extended conformation is included.

The computed relaxation products for the ΔLk = −2 and −1 topoisomers show little to no sensitivity to the inclusion of the extended conformation in contrast to the significant sensitivity exhibited by the ΔLk = 0 and +1 topoisomers (compare windows in Fig. 4, subscripts 1 and 2). These trends are consistent with the above discussion of minicircle looping. In the original experiments, data obtained at every interoperator length are available only for the ΔLk = −1 topoisomer, which we show has little sensitivity to inclusion of the extended conformation. At 158 bp, the model now agrees with the experiments by predicting that all topoisomers relax to the ΔLk = 0 topoisomer. At 163 bp, however, the model predicts that the ΔLk = 0 and +1 topoisomers will mainly remain at ΔLk = +1, which is still inconsistent with experimental results (see Table 1). However, the sensitivity to the inclusion of the extended conformation in the ΔLk = 0 and +1 topoisomers provides a clear means of determining whether the extended conformation is present. If similar experiments can be carried out at every interoperator length for the ΔLk = 0 and +1 topoisomers, the model predictions can be directly tested to ascertain whether the extended conformation plays a role.

Conclusions

This study extends a computational rod model for DNA (20,42) to reinterpret classical experiments on LacI-induced DNA looping in linear DNA (3) and supercoiled DNA minicircles (9). The model can quantitatively predict the structural features of DNA-LacI interactions, as well as distinguish the possible binding topologies. Computed equilibrium conformations of the DNA-LacI complex replicate the experimental observations and provide detailed structural insights that extend well beyond what can be interpreted from the gel data alone. A comparison of theory with experiment provides substantial evidence that LacI adopts an extended conformation, and that this extended conformation was unknowingly present in the Müller-Hill experiments conducted more than 20 years ago.

The two experimental studies explored the effects of operator phasing on loop stability by varying the length of interoperator DNA from 153 to 168 bp. The computed optimal interoperator lengths for maximum loop stability align strikingly well with the experimental evidence. In addition, the calculations for linear DNA reveal that the most stable loops for linear DNA occur when LacI adopts the extended conformation. For supercoiled DNA minicircles, the calculations again predict the location of the most stable complexes for the ΔLk = −2, −1, 0, and +1 topoisomers. When we account for relaxation by topoisomerase I, the calculations confirm that one superhelical turn can be absorbed by LacI looping.