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* Departments of Molecular Biophysics and Biochemistry and Chemistry, Yale University, New Haven, Connecticut; and Institute of Biomedical Sciences and Technology and Department of Molecular and Cell Biology, University of Texas at Dallas, Richardson, Texas
Correspondence: Address reprint requests to Stephen Levene, Dept. of Molecular and Cell Biology, University of Texas at Dallas, PO Box 830688, Richardson, TX 75083-0688. Tel.: 972-883-2503; Fax: 972-883-2409; E-mail: sdlevene{at}utdallas.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORETICAL METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORETICAL METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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) and Matthews (2)). In Escherichia coli, DNA looping represses gene expression at the ara, gal, lac, and deo operons (3–6) and activates transcription from the glnALG operon (7). The size of DNA loops formed in these systems varies between 
100 and 600 basepairs. In eukaryotes, a variety of transcription factors bind to enhancers that are hundreds to several thousands of basepairs away from their promoters and interact with RNA polymerases directly or through mediators to achieve combinatorial gene regulation (8). DNA looping is required to juxtapose two recombination sites in intramolecular site-specific recombination (9–11) and is also employed by a number of restriction endonucleases such as SfiI and NgoMIV, which recognize and cut two copies of well separated cognate sites simultaneously (12–14). The biological importance of DNA loop formation is underscored by the abundance of architectural proteins in the cell, such as HU, IHF, and HMG, which facilitate looping by bending the intervening DNA between protein-recognition sites (15). Moreover, DNA looping has been shown to be subject to regulation through the binding of effector molecules that alter protein conformation or protein-DNA interactions (16).
Two characteristics of DNA looping have been demonstrated by in vitro and in vivo experiments. One is cooperative binding of a protein to its two cognate sites, which can be demonstrated by footprinting methods (17). DNA looping can increase the occupancies of both binding sites; in particular, it can significantly enhance protein association to the lower-affinity site because of the tethering effect of DNA looping. This is a general mechanism by which many transcription factors recruit RNA polymerases in gene regulation. Another hallmark is the helical dependence of loop formation (1,3), which arises because of DNA's limited torsional flexibility and the requirement for correct torsional alignment of the two protein-binding sites. Although many methods have been developed to directly observe DNA looping in vitro, such as scanning-probe (7) and electron microscopy (18), and single-molecule techniques (19), assays based on helical dependence have been the only way to identify DNA looping in vivo. In these experiments, the DNA length between two protein binding sites is varied and the yield of DNA loop formation is monitored, for example, by the repression or activation of a reporter gene (20). Using this helical-twist assay, DNA looping in the ara operon was first discovered (3).
Our knowledge about the roles of DNA bending, twist, and their respective energetics in DNA looping has come largely from analyses of DNA cyclization (1,21,22). Shore et al. first showed that circularization efficiencies of DNA fragments, which are quantitatively described by J factors, oscillate with DNA length and therefore torsional phase (23,24). The J factor is defined as the free DNA end concentration whose bimolecular ligation efficiency equals that of the two ends of a cyclizing DNA molecule (25). For short DNA fragments J factors are limited by the significant bending and twisting energies required to form closed circles, whereas for long DNA, the chain entropy loss during circularization exceeds the elastic-energy decrease and reduces the J factor. Because of this competition between bending and twisting energetics and entropy, there is an optimal DNA length for cyclization (26). Analogous behavior has been expected for DNA looping, especially with respect to the helical dependence discussed above.
Quantitative analyses of DNA looping and cyclization are challenging problems in statistical mechanics and have been largely limited to Monte Carlo or Brownian dynamics simulations (27–31). Analytical solutions are available only for some ideal and special cases. An important contribution in this area is the theory of Shimada and Yamakawa (32), which is based on a homogeneous and continuous elastic rod model of DNA. This theory has been applied extensively to DNA cyclization (23,33) and also to DNA looping (21,22,34). The Shimada-Yamakawa theory makes use of a perturbation approach, in which small configurational fluctuations of a DNA chain around the most probable configuration are accounted for in the evaluation of the partition function.
The elastic-equilibrium conformation is obvious for the homogeneous DNA circle studied by Shimada and Yamakawa. However, the search for the elastic-energy minimum of homogeneous DNA molecules with complex geometry, such as in DNA looping, supercoiling, and the case of inhomogeneous DNA sequences containing curvature and nonuniform DNA flexibility, is not trivial (4,35,36). We have developed a statistical-mechanical theory for sequence-dependent DNA circles and applied it to the problem of DNA cyclization, combining computation of the equilibrium conformation with subsequent evaluation of thermodynamic quantities using a harmonic approximation (26). In this model DNA configuration is described by parameters defined at dinucleotide steps, i.e., tilt, roll, and twist (18), which allows straightforward incorporation of intrinsic or protein-induced DNA curvature at the basepair level. As in Shimada and Yamakawa's method, the theory takes advantage of small fluctuations around one stable mechanical configuration in small DNA circles (e.g., <1000 bp). Once the mechanical equilibrium configuration under certain constraints is found with an iterative algorithm, fluctuations around the equilibrium conformation can be taken into account with the harmonic approximation. The new method is much more computationally efficient than Monte Carlo simulation, has comparable accuracy, and has been applied successfully to analyze experimental results from DNA cyclization (26).
Here we extend this theory to DNA looping. The basis of the extension is to treat the protein subunits as connected rigid bodies and to allow for a limited number of degrees of freedom between the subunits. Motions of the subunits are assumed to be governed by harmonic potentials and an associated set of force constants, neglecting the anharmonic terms often required for proteins undergoing large conformational fluctuations among their modular domains. Indeed, the use of a harmonic approximation is supported by the success of continuum elastic models that are based only on shape- and mass-distribution information in descriptions of protein motion (37). Similar to the description used for individual DNA basepairs in the model, protein geometry and dynamics are described by three rigid-body rotation angles (tilt, roll, and twist). Therefore, DNA looping can be viewed as a generalization of DNA cyclization in which the protein component is characterized by a particular set of local geometric constraints and elastic constants. This treatment not only unifies the theoretical descriptions of DNA cyclization and looping, but also allows consideration of flexibilities at protein-DNA and protein-protein interfaces and application of the concepts of linking number and writhe. In previous work, proteins were considered rigid, and their effects on DNA configuration were represented by a set of constraints applied to DNA ends (1,38,39). With the approach described here, programs developed for analyzing DNA cyclization can be used to analyze DNA looping with only minor modifications.
Our method is most applicable to the problem of short DNA loops, in which the free energy of a wormlike chain is dominated by bending and torsional elasticity. As in the previous theory of cyclization, possible modes of DNA self-contact and contacts between protein and DNA at positions other than the binding sites are not considered (26). For large loops, contributions to the free energy from chain entropy and DNA-DNA contacts can become highly significant. Several alternative treatments of DNA looping have appeared recently. One of these addresses the excluded-volume contribution to DNA looping within large open-circular molecules (40), whereas two others consider the effect on looping of traction at the ends of a DNA chain (41,42). None of these treatments includes helical phasing effects on DNA looping. In contrast, a method based on the Kirchhoff elastic-rod model, which includes the helical-phase dependence, has been presented (39,43). However, this approach does not include thermal fluctuations per se and therefore is not directly applicable to calculations of the J factor. The comprehensive treatment of small DNA loops described here is thus far unique to the extent that it accounts for sequence- and protein-dependent conformational and flexibility parameters, thermal fluctuations, and helical phasing effects.
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THEORETICAL METHODS |
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TOPABSTRACTINTRODUCTIONTHEORETICAL METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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). Here we focus only on modifications necessary to treat DNA looping. In this work the protein subunits that mediate loop formation are modeled exclusively as two identical and connected rigid bodies, as shown in Fig. 1. There are three additional sets of rigid-body rotation angles that are defined in addition to those for dinucleotide steps: two sets for the interfaces between protein and the last (DP) and first (PD) basepairs of the DNA and one set for the interface between the two protein domains (PP), where the symbols in parentheses are used to indicate the corresponding angles through subscripts. The local Cartesian-coordinate frames for protein subunits are defined such that their origins coincide with vertices of a circular chain and their z axes point toward the next vertex in succession. Thus protein dimensions can be modeled in terms of a noncanonical value for the helix rise corresponding to particular segments within a circular polymer chain.
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bp = 3.4 Å, throughout. Most of our analysis focuses on the basic phenomenon of DNA looping apart from any effects of DNA-sequence-dependent structure or flexibility. Therefore, all calculations used canonical mechanical parameters for duplex DNA: helical twist 
0 = 34.45°, a sequence-independent twist-angle standard deviation, or twisting flexibility, 
= 4.388°, and standard deviations, or bending flexibilites, for all tilt and roll angles, 
and 
, respectively, of 4.678° (equivalent to a persistence length of 150 bp). Except for specific cases where intrinsic DNA bending is considered, the average values of tilt and roll are taken to be zero. Computations were carried out on a Dell laptop with 1 GHz Pentium III CPU and 256 Mb memory. The program source code is written in Fortran 90 and is available upon request.
Simplified protein geometries and flexibility parameters
For DNA loops with either zero or nonzero end-to-end distances, constraints are directly applied to the DNA ends, as in the case of DNA cyclization. We modeled DNA loops formed during site synapsis using protein-dependent parameters 
and 
The angle 
was considered an adjustable parameter that we denote the axial angle and, unless specified, all other protein-related angular parameters were set equal to 0°. In these cases, the DNA ends (the centers of two protein-binding sites on DNA) are separated by twice the protein-arm length 
and displaced from one another along the +x direction, or toward the major groove of DNA. Projected along the x axis, the axial angle is the included angle between the tangents to the DNA at the two protein-binding sites and is altered by varying the twist between protein subunits (Fig. 1, b and c). An axial angle equal to 0 corresponds to antiparallel axes at the ends as shown in Fig. 1 a. The case of a rigid protein assembly is modeled by setting the standard deviations of the DP, PP, and PD sets of rigid-body rotation angles to 1 x 10–8°.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHEORETICAL METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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 | (1) |
is the intrinsic DNA twist and 
an arbitrary angle related to the unconstrained torsional degree of freedom of DNA. The coefficients Ai are given by
 | (2) |
 | (3) |
 | (4) |
The coefficients in Eq. 4 were obtained by fitting the space curve corresponding to the DNA helical axis that gives the minimum elastic energy conformation of DNA loops of different sizes as follows: a0 = –335.0142, a1 = 2318.881, a2 = –1299.164, a3 = –4483.366, a4 = 38169.74, and a5 = –54753.5. The error for end-to-end distances computed using Eq. 1 is <2% of the DNA length from 50 bp to 100 bp, and <0.5% of that from 100 bp to 500 bp. The torsional phase angle between two ends is 
The entire loop lies in a plane, and the angle between the normal vector of the plane and the x axis of the external coordinate frame can be shown to be 
The expressions for 
and 
suggest that 
is related to DNA bending isotropy. Loop configurations with different -values are related to each other by globally twisting DNA molecules. Since the orientation of the first basepair is fixed, this global twist is equivalent to rotation of the loop plane, which corresponds to the rotational symmetry met in DNA cyclization of homogeneous DNA with bending isotropy (26) . Therefore, J factors for configurations with different -values are identical.
If DNA looping needs to be torsionally in-phase, only two degenerate loop configurations are available, breaking the rotational symmetry. These loop geometries can be expressed by Eq. 1 with two different -values, i.e., 
and 
which satisfy the torsional phase requirement 
In contrast to DNA cyclization, no twist change is involved in forming these ideal DNA loops for any DNA length and thus the helical dependence vanishes in this case. From the expression given above for it is clear that the helical axes of the two loops are coincident and their directions are reversed. Fig. 2 shows the bending profile of the loop configuration corresponding to 
for a 150-bp DNA. Surprisingly, the maximal J factor occurs at approximately the same DNA length, or 460 bp (data not shown), as in DNA cyclization (26). This can be partly explained by the fact that the total bending magnitude of the loop is 290°, close to a full circle, instead of 180°.
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and 
) with zero end-to-end distance and in-phase torsional alignment as initial configurations, two mechanical equilibrium configurations are obtained by using the iterative algorithm described previously (26). The J factor in Fig. 3 a is the sum of separate J factors calculated for the two configurations. Note that in all cases involving configurations that differ in linking number, equilibration between the two forms requires breakage of at least one of the protein-DNA interfaces. The contributions from each of these configurations are shown in detail for the case where the ends are separated by 10 bp. Interestingly, the length dependence of J computed from the individual configurations are out of phase and have a periodicity of two helical turns, which results from the half-twist dependence of the phase angles and 
However, their sum displays a periodicity of one helical turn. Fig. 3, b and c, shows two such configurations for DNA molecules that are torsionally in-phase (N = 210 bp) or out-of-phase (N = 215 bp).
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DNA looping in synapsis
Intramolecular reactions of most site-specific recombination systems (9–11) and a number of DNA restriction endonucleases, such as SfiI and NgoMIV (12), proceed through protein-mediated intermediate structures in which a pair of DNA sites are brought together in space and the intervening DNA is looped out. The intermediate nucleoprotein complex involved in site pairing and strand cleavage (and also exchange, in the case of recombinases) is termed the synaptic complex. In these systems, two characteristic geometric parameters are of interest: the average through-space distance between the sites and the average crossing angle between the two ends of the loop, which we denote the axial angle. The latter quantity can be described in terms of the twist angle between the protein domains, PP (Fig. 1 b), and we shall use these terms interchangeably. Here we focus on the effect of protein geometry on DNA looping, leaving consideration of protein flexibility to the following section.
Fig. 4 shows the helical dependence of looping (Fig. 4 a) and the elastic-minimum configuration of DNA loops (Fig. 4 b) for different values of the axial angle. The most prominent feature of these results is that the phase of the helical dependence is shifted as a function of the axial angle, characterized by a relative global shift of the curve along the x axis. This implies that DNA looping does not always occur most efficiently when two sites are separated by an integral number of helical turns, as has been suggested for some simple DNA looping systems studied previously. The axial angle also globally modulates J factors, which is apparent from the vertical shift in the J versus length curve and effects on the amplitude of the helical dependence. The torsion-angle-independent value of J, averaged over a full helical turn, decreases with increasing axial angle, whereas the amplitude of the helical dependence increases. The above observations can be qualitatively explained by analogous results from DNA cyclization. As in cyclization, DNA forms loops most efficiently when the number of helical turns in the loop is close to an integer value. It is therefore appropriate to consider this issue in terms of the linking number for the looped conformation, Lk, which involves contributions from the geometries of both the protein and DNA.
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the axial angle, will shift the phase of the helical dependence relative to that of the DNA alone. For a loop with N = 179 bp and PP = 0, the total twist is simply equal to that for the DNA loop. Because this loop has 17.0 helical turns, only one loop topoisomer contributes to the J factor. The value of J is a local maximum at PP = 0 and, as shown in Fig. 5 a, decreases monotonically for both PP > 0 and PP < 0. Contributions to J from other topoisomers of the 179-bp loop are <5% over the range –135° > PP > +120°. The twist for the planar equilibrium conformation of a 173-bp loop is 16.5 helical turns; thus, there are two alternative loops that can be efficiently formed (Fig. 5 a): either a loop with Ht,loop = 17.0 and 
or a loop with Ht,loop = 16.0 and 
The J value at PP = 0 is a local minimum and there is a bimodal dependence on axial angle for loops in which the DNA twist is half-integral. We investigated the phase shift of the J factor and found that this quantity is a nonlinear function of the axial angle. From Fig. 4 a, the calculated phase shifts for 60° and 120° axial angles relative to 0° are 52° and 103°, respectively. Moreover, the local maxima for the total J curve for N = 173, shown in Fig. 5 a, are located at –58.5° and 63°, positions that are not in agreement with predicted angle values based solely on Ht,loop –166° and 194°, respectively).
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,30,44,45). In the case of protein-mediated looping, nonzero values of the axial angle impose an intrinsically nonplanar conformation on the DNA. The relative contributions of loop writhe and twist for the Lk = 16 topoisomer of a 173-bp loop are shown as a function of axial angle in Fig. 5 b.
In Fig. 5 c, we plot the axial-angle-dependent values of the bending and twisting free energies for the Lk = 16 topoisomer and their sum, which is the total elastic free energy of the loop. The minimum value of the total elastic energy occurs at PP = –58.5°, coincident with the position of the J factor maximum for this topoisomer (Fig. 5 a). This mechanical state can be achieved with very little twist deformation of the loop, but at the expense of significant bending energy. Further reduction of the axial angle requires even less twisting energy; however, the bending energy increases monotonically. In contrast, for PP > –58.5°, somewhat less bending energy is required, but the twisting energy begins to increase significantly with increasing axial angle. Since the sense of the bending deformation for PP > 0 opposes the needed reduction in loop linking number, the elastic energy cannot be decreased by increasing the axial angle. The only way that the loop geometry can compensate for this is through twist deformation. This asymmetry arises because we are considering the contribution of only one loop topoisomer to the elastic free energy.
Effects of binding-site symmetry
Twofold symmetry in the DNA-binding domain of a protein or the sequence of its cognate site may allow formation of two alternative looping geometries that conserve protein-DNA contacts. This is shown in Fig. 6, which depicts looping mediated by the type-II restriction endonuclease SfiI as an example. SfiI binds two copies of its recognition sequence and, in the presence of Mg2+, catalyzes the concerted cleavage of all four DNA strands. These symmetric recognition sequences can be juxtaposed via two alternative loop geometries whose axial angles differ by 180°. If the reaction steps subsequent to synapsis are independent of DNA orientation, in particular that of the spacer sequence shown in Fig. 6, then the overall efficiency of the reaction should be related to the total J factor of the two geometries. Fig. 7 shows the configuration corresponding to these geometries and the helical dependence of the individual and total J factors for 179-bp DNA loops. From Fig. 7, a and c, it can be seen that the J dependences for the two geometries are out of phase, and may shift relative to each other along the vertical axis.
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Two alternate looping geometries and their out-of-phase contributions to loop formation were observed for SfiI by native polyacrylamide gel electrophoresis of the synaptic complexes (12). Our calculations not only explain the observation, but also predict that the SfiI tetramer binds its two cognate sites to form a crossed DNA structure, because the two DNA looping geometries have similar contributions in the helical-twist assay. Interestingly, a cocrystal structure of a similar endonuclease, NgoMIV, with two copies of its recognition site (14) exhibits an axial angle of –60° (or equivalently 120°), instead of 90° (or –90°). The difference may be due to either slightly different architectures for protein-DNA and protein-protein association of the two endonucleases, or flexibility at the protein-protein interfaces that allows preferences for axial angles to be perturbed by crystal-packing forces. It would be informative to investigate DNA loop geometries for the NgoMIV system in solution by a helical-twist assay similar to that done with SfiI.
Effects of protein flexibility
DNA loops as small as 40 to 70 bp have been widely observed in experiments, for example, in LacR- and AraC-mediated DNA looping (20,46). For the case of AraC, no lower limit on site spacing was found. Because it is energetically unlikely for DNA to form such small loops with rigid end constraints, plasticity in the protein assembly has been proposed to explain these observations. Protein plasticity can manifest itself through two mechanisms: multiple stable protein states with different configurations, as demonstrated by LacR (31), and protein conformational flexibility around specific states. To investigate the potential effects of protein elasticity, we considered DNA loop configurations mediated by the extended conformation of the LacR tetramer (31). The minimum-elastic-energy conformation of this protein structure corresponds to DP = PD = 90.0° with all other DP, PD, and PP parameters set equal to 0°.
Fig. 8, a and b, shows loop configurations for 137-bp and 53-bp DNAs in which roll and twist fluctuations were permitted at the protein-protein and both protein-DNA interface steps. Both DNA and protein configurations change with protein flexibility. By increasing protein flexibility, the system alleviates the need for strong DNA bending required to close the loop. Remarkably, DNA loops as small as 30 bp can easily be formed, as evidenced in the J dependence shown in Fig. 8 b. Higher protein flexibilities enable looping more dramatically for short DNAs because the bending energy required by looping is sharply decreased, but less dramatically for longer DNA segments because of the greater loss of chain entropy. Consequently, the peak corresponding to optimal DNA loop length is shifted to smaller values when protein flexibility is increased. This comparison not only explains the experimental observations of looping for short DNAs, but also provides a convenient method to assess the effective degree of protein flexibility from the optimum DNA length for loop formation.
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,47). To examine the general effects of DNA curvature on the helical dependence, we carried out calculations for a single kink of 36° or 72° introduced at different positions in loops mediated by the extended LacR conformation. The calculated dependence of J factor on DNA length is shown in Fig. 9. Depending on kink position, a kink can shift the phase and change the amplitude of the helical dependence. If the kink position is held constant, an increase in kink magnitude increases the amplitude of the helical dependence. The strong dependence of DNA looping on kink position and magnitude creates an opportunity for flexible genetic control by architectural proteins through DNA bending. The helical dependence of DNA looping modulated by DNA bending differs significantly from that in DNA cyclization, in which the relative position of a single kink does not alter the J factor and amplitude of the helical dependence. This comparison suggests that it is intrinsically inaccurate to determine DNA torsional rigidity by DNA looping if intrinsic bending is present. It also implies that a rigorous examination of effects of DNA bending on looping requires an additional phasing assay, in which the position of induced DNA curvature is systematically varied to determine the optimal location of the bending locus (48).
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONTHEORETICAL METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONTHEORETICAL METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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FOOTNOTES |
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Submitted on July 14, 2005; accepted for publication November 28, 2005.
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TOPABSTRACTINTRODUCTIONTHEORETICAL METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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where 
and 
are the tilt and roll, respectively, of the ith dinucleotide step.
(0°, dashed line) and 
(180°, solid line), as described in the text. (b and c) Stereo models of the two equilibrium configurations for 210-bp (b) and 215-bp (c) antiparallel DNA loops with end-to-end separation equal to 10 bp. The 210- and 215-bp DNA correspond to an adjacent peak and valley, respectively, of the curve in a. Conformations shown in blue correspond to 
Ht, and writhe of the loop formed by the Lk = 16 topoisomer for N = 173 bp as a function of axial angle. Excess twist is computed from the expression Ht, loop – 16, where Ht, loop is the loop helical turn value described in the text, and depends linearly on the axial angle. The writhing number of the loop was calculated using the method of Vologodskii (28
nGGCC, where n is any base and the arrow indicates the location of the cleavage site. Each subunit of the tetrameric SfiI binds to one GGCC sequence and two dimers bound at two distant copies of the recognition site associate to form a synaptic complex by looping the intervening DNA. Four DNA backbones are then cut in concert to release the looped DNA. The two geometries are related by reversing the intrinsic DNA direction at the binding site if the protein is rigid, forming a negative (left) or positive (right) crossing according to the right-hand rule. Due to the twofold symmetry of the protein dimer, this reversal does not affect protein-DNA interactions. Given the angle shown in the figure, the protein twist 
and 
for the left and right configurations, respectively.
and –135°. (d) Stereo models corresponding to the equilibrium loop geometries in c. DNA length was equal to 179 bp in all cases and the end-to-end separation was equal to 40 bp.
