Wrapped-Around Models for the Lac Operon Complex

Abstract

The protein-DNA complex, involved in the lac operon of enteric bacteria, is paradigmatic in understanding the extent of DNA bending and plasticity due to interactions with protein assemblies acting as DNA regulators. For the lac operon, two classes of structures have been proposed: 1), with the protein tetramer lying away from the DNA loop (wrapped-away model); and 2), with the protein tetramer lying inside the DNA loop (wrapped-around model). A recently developed electrostatic analytical model shows that the size and net charge of the Lac protein tetramer allow the bending of DNA, which is consistent with another wrapped-around model from the literature. Coarse-grained models, designed based on this observation, are extensively investigated and show three kinds of wrapped-around arrangements of DNA and a lower propensity for wrapped-away configurations. Molecular dynamics simulations of an all-atom model, built on the basis of the most tightly collapsed coarse-grained model, show that most of the DNA double-helical architecture is maintained in the region between O3 and O1 DNA operators, that the DNA distortion is concentrated in the chain beyond the O1 operator, and that the protein tetramer can adapt the N-terminal domains to the DNA tension.

Introduction

Gene regulation is often achieved by interaction of DNA with proteins in both eukaryotes and prokaryotes: a paradigm of it is the lac repressor-lac operator interaction (1,2). The lac operator displays three operator sites: the principal operator O1, located upstream of the lacZ gene; the auxiliary second operator O2, located 401 basepairs (bps) downstream from O1 in the lacZ gene; and the third auxiliary operator O3, 93 bps upstream from O1 and at the end of lacI gene (3–5). The lac repressor is a protein of 360 amino acids that associates into a homotetramer, organized in a dimer of dimers linked in by a four-helix bundle at the C-terminus of each monomer subunit. This assembly has a V-shape, with two different contacts with DNA operators at the tips of the V (3,4). The binding of the lac repressor tetramer (LacI) to the lac operators inhibits transcription of the genes necessary to the cell metabolism's ability to use lactose as an energy source. In the presence of the binding to the repressor of an inducer, as natural as allolactose or as gratuitous as isopropyl β-D-1-thiogalactopyranoside, the conformation of the repressor is specifically changed to reduce the affinity for the operator and the expression of the genes is activated again (2,6,7).

In many cases (4,8,9) genetic regulation is better mediated by proteins having two separate binding sites recognizing two separate DNA sequences: this implies a looping of the DNA partner between the two binding sequences (10). Less is known about the actual configuration of the complex LacI-DNA, in particular in the case when one of the LacI DNA binding domains is related to the primary operator O1 and the other to the weaker auxiliary operator O3: one of the main questions is related to the organization of the 93-bp DNA O3-O1 segment in a wrapping-around model or in a wrapping-away model (10–12). Qualitative arguments have been reported that support one or the other model.

In Table S3, the sequence of O3-O1 DNA (113 bps) is reported (5, 12, 13) with the O3 and O1 operators in boldface and the central bases in the O3 and O1 operators indicated in italic (please note that in Table 12 of Swigon et al. (12), C18 is missing). The strand D1 is indicated and the strand D2 is complementary.

In Fig. 1, some of the known data about the LacI-DNA complex are summarized. The protein assembly, represented in Fig. 1 with monomers in different shaded intensities, has been built via computer simulations (14) based on low-resolution x-ray data (4). The structure of the protein assembly displayed in the figure is that published in the Protein DataBank (PDB ID code 1Z04). Two pairs of assembled N-terminal domains (top left and right, respectively) seize the two DNA operators, O3 and O1, respectively. On the bottom, the bundle of C-terminal α-helices constitutes the junction between the four proteins. The array of light- and dark-shaded spheres represent 95 bps of DNA wrapped around the protein tetramer and away from the protein tetramer, respectively, both in a left-handed superhelical turn. The DNA bead positions displayed in the figure were assigned by manipulation of coarse-grained models that are reported in this work. The topology of the wrapping (the left-handed superhelix) is arbitrary, even if the propensity for a left-handed superhelical wrapping of DNA is shown by the nucleosome core particle (15). The location of the O3 and O1 DNA operators are approximately those of the short double-helical segments contained in the nuclear magnetic resonance (13) structures of LacI-DNA complexes reported in the literature and in the PDB.

Figure 1.

Schematic representation of the two proposed configurations for the DNA in the lac operon complex: the left-handed wrapped-away (dark shaded) and wrapped-around (light shaded) configurations. Each DNA bp is represented as a sphere with radii of 0.5 nm, and the non-hydrogen atoms in the protein are represented as sticks with different shading intensities for each of the four monomers in the tetramer (P1 is dark shaded, P2 is medium shaded on the left, P3 is light shaded and P4 is medium shaded on the right). The VMD program (28) was used for this and the following molecular drawings.

Besides the many theoretical and numerical studies based on possible wrapped-away configurations (12,16,17), no direct experimental evidence on the structure of the entire O3-O1 DNA segment has been obtained so far, to our knowledge. Indirect structural information is provided, for instance, from recent single molecule fluorescence resonance energy transfer (SM-FRET) experiments (18). The distance between fluorescent probes (the inter-dye distance) introduced in DNA constructs forming particularly stable complexes with the LacI protein assembly was estimated to be ∼3.5 nm in the most compact configuration (ET efficiency of ∼90%). This result, combined with results of gel-electrophoresis, suggests that the complex can assume a closed-type, wrapped-away conformation. Based on numerical modeling (16), this conformation was modeled as a compact wrapped-away configuration with antiparallel topology of the DNA operator-protein headpiece interaction, and with negligible interactions between the inter-operator DNA region and the protein cores.

Quantitative information about the behavior of the entire DNA chain has also been obtained indirectly. After extended thermodynamic experiments, Tsodikov et al. (11) proposed a DNA wrapping path around LacI and a possible suggestive wrapping-around model motivated by large aspecific electrostatic interactions between DNA and LacI. The proposed model (Fig. 8 in (11)) is almost identical to the schematic dark-shaded structure, built with a coarse-grained model, that is described in this work and displayed in Fig. 1.

To exploit the stability of LacI-DNA complex structures where DNA is wrapped around the compact form of the LacI protein assembly, we first start from the model proposed by Tsodikov et al. (11) to calculate, quantitatively, the electrostatic interactions LacI-DNA and their consequences in supporting the wrapping-around model, using our recent general analytical approach to protein-DNA electrostatic interactions (19,20). Secondly, we built configurations of a coarse-grained model of DNA wrapped around the atomic structure of the protein assembly. This simplified model allows the molecular dynamics simulation of the whole pathway from configurations with 93 bps wrapped around the protein assembly toward wrapped-away configurations of 133 bps. Finally, we propose a possible atomic structure for the whole O3-O1 bp DNA segment. This segment is wrapped around the protein assembly, thermally relaxing, via empirical molecular dynamics (MD) simulations, an all-atom structure that includes water solvent and sodium counterions. This is built based on the tightly wrapped-around, coarse-grained model.

From the different models here applied, we conclude that the wrapped-away configurations, so often reported in the literature, are not likely, and that the process of DNA unwrapping from the protein assembly could be better represented as a transition between differently wrapped-around configurations.

Methods

In this section, we first briefly summarize the equations that describe the electrostatic free energy for the formation of a complex between an elastic model of DNA and a spherical macroion competing with small monovalent counterions dissolved in a continuum solvent model. Based on the simple electrostatic approach, a significant wrapping of a DNA segment around the protein tetramer is predicted. We then built an all-atom model of a 113-bp DNA segment in a wrapped-around configuration to monitor the effects of the large bending operated on DNA by the wrapping-around constraint. To build this configuration, an ad hoc bead model for a DNA chain of 133 bps is adopted, with each DNA bp modeled as a unique spherical Lennard-Jones site with a point charge in its center. This model is used to build a bead necklace approximating the path of different kinds of wrapped-around DNA configurations, approximately starting from the O3 binding site of the LacI protein tetramer and ending with the O1 binding site of the same protein assembly. The rigid structure of the protein assembly reported in the PDB (14) is used for this construction.

Once each bead of selected configurations obtained with the coarse-grained model is resolved into an all-atom DNA bp, using the sequence of the O3-O1 chain in the Escherichia coli bacteria (5,12), the distortions introduced in the DNA structure are relaxed by standard MD simulations.

The electrostatic model

We follow the generalized electrostatic model developed recently by us (20) to approach DNA-protein interactions, and applied to: the linker DNA N-terminal histone H3 tail (19); the DNA-histone octamer interactions in the chromatin nucleosome core particle; the CSB active remodeling of chromatin by wrapping and unwrapping of DNA that depend on ATP binding; and, finally, the DNA bending in archaea by binding to the histonelike protein MC1 (20). The approximation of this analytical approach is described in detail in the Supporting Material. Here is a brief summary.

DNA is modeled as a negatively charged elastic cylinder of radius a (for DNA a = 1 nm) and the LacI tetramer as a positively charged sphere of radius R₀ and charge Z₀. The estimated values of R₀ and Z₀ are, respectively, 4.5 nm and +20 |e| (see Supporting Material). The Bjerrum length and Debye screening constant are those for physiological conditions, corresponding to a water solution at room conditions of a monovalent salt of 0.1 M concentration. The persistence length, l_p, of DNA is 50 nm. The curvature radius of DNA wrapped around the protein assembly, R, is 5.5 nm. The fundamental variable for the free energy is l, the length of DNA wrapped around the sphere model of the protein-DNA complex. The charge of the complex, Z, is a function of l that enters into the free energy.

The binding free energy of this DNA-sphere model is calculated as the difference of the free energy of the model with a length l of DNA wrapped on the sphere, to that of isolated straight DNA of length L and sphere:

$$\Delta F\left(l\right)=F_s\left(L-l\right)+F_c\left(l\right)+F_{c-s}\left(l\right)+F_b\left(l\right)-F_s\left(l\right)-F_c\left(0\right)\text{.}$$ (1)

The meanings of the different contributions are explained in the Supporting Material and are here summarized.

The first term, F_s(L – l), is the electrostatic free energy in the condensation regime of the piece of straight DNA of length L – l (20–23).

The second term, F_c(l), is the sum of the usual electrostatic charging free energy of a sphere of global charge Z(l) in the absence of counterion condensation on the sphere and (when the charge Z(l) is strongly positive or negative) the entropic contribution of condensation. The approximation of uniform distribution of the charge Z(l) on the spherical surface enables the simple evaluation of this important entropic contribution.

The third term, F_c–s(l), describes the electrostatic interaction of straight DNA and the complex sphere-wrapped DNA.

The last term, F_b(l), is the elastic energy necessary to bend a piece of DNA of length l with a radius of curvature R = R₀ + a, corrected by the additional electrostatic repulsion due to DNA bending.

The optimal length l^∗ of the DNA wrapped around the sphere is now obtained by searching the minimum of ΔF(l). Alternatively to l^∗, the wrapping at thermodynamic equilibrium may be described as numbers of turns (or fraction) of DNA around the protein:

$$n_t^{\ast }=l^{\ast }/\left(2\pi R\right)\text{.}$$ (2)

It is noteworthy that the approximations involved in this model hold for a salt concentration near physiological conditions (20,21,24).

The bead model of DNA

In this second model, the LacI atomic structure (PDB ID 1Z04 (14)) was used and kept rigid. The protonation state of protein residues is the same assumed in the electrostatic model (see discussion in the previous subsection). The AMBER PARM94 force field (25) was adopted for the protein atoms. Each DNA basepair was assumed as an isotropic charged Lennard-Jones site. A chain of 133 beads was assembled by connecting beads next in the sequence with stretching and bending forces. The force-field parameters and the simulation protocol adopted to acquire trajectories for a 133-bp DNA bead model moving around the rigid protein assembly is described in detail in the Supporting Material.

One of the energy-minimized bead configurations is displayed in Fig. 2. This structure contains the essential features of the wrapped-around configuration in Fig. 1 (see also Fig. 8 in (11)) and it is, therefore, a reasonable starting point for the wrapped-around model simulation. The displayed structure was used as the template structure for simulating the process of unwrapping via restrained MD simulations at room temperature. The positions of the first and last 20 beads were used as the equilibrium positions for external harmonic forces. A first biased simulation was performed with the first and last beads of the ds-DNA chain restrained toward the positions, respectively, of first and last beads in the DNA model displayed in Fig. 2. Then, a second simulation was performed with the restraints applied to the same DNA beads, but with the equilibrium position of the harmonic forces moved symmetrically on beads 2 and 132 of the target structure of Fig. 2. The equilibrium positions for the first and last beads of DNA were then slowly symmetrically moved, in another 18 steps, toward the two approximate centers of the O3 and O1 operators (the beads number 20 and 113, joining the dark- and light-shaded beads displayed in Fig. 2), through the 20 different positions displayed in dark shading in the figure. By using this procedure, the DNA chain slowly moves from the statistics where the chain is tied together with the protein assembly, toward the statistics where the chain is less tied and is free to move far from the protein assembly.

Figure 2.

The minimal energy configuration derived for the coarse-grained 133-bp ds-DNA model wrapped around the rigid LacI protein assembly. For clarity, each DNA bp is represented as a sphere with one-half of the 1-nm radius used in the model. The beads that are used as equilibrium positions for harmonically restrained MD simulations are displayed in dark shading. The beads 21 and 113 (11 and 103 in the 113-bp notation), initially closest to the location of O3 and O1 DNA operators, respectively, are displayed in light shading. O3 is at the bottom-left, O1 is on top-left. The nonhydrogen atoms in the protein are represented as sticks with different shading intensity for each of the four monomers in the tetramer (as in Fig. 1).

The details of several sets of transitions (each occurring through the 20 MD simulations described above) are described in the Supporting Material, provided together with the equation that allows an estimate of the free energy change during the transition from the most-tied to the less-tied configurations.

Resolving the coarse-grained model into atomic details

The DNA bead model displayed in Fig. 2 is transformed into an all-atom model by replacing each of the 113 central beads, among the displayed 133, with a pair of DNA nucleotides. The procedure here summarized is described in detail in the Supporting Material, together with a summary of definitions of standard geometrical parameters adopted in the discussion of DNA structural relaxation.

During the writing of the article, a flexible web-oriented tool to perform the construction of DNA chains according to a sequence of geometric parameters was made available and documented (26,27). The slightly different procedure we adopted here is more suited to follow a pathway for the chain derived by coarse-grained models.

Results

The electrostatic model

In Table 1, the number of wrapped DNA turns are reported for the minima of the free energy of Eq. 1 and for several values of Z₀ (15, 20, 25, and 30) with fixed values of R₀ = 4.5 nm and c_s = 0.1 M. For convenience, the corresponding values obtained by charge neutralization (l = l_iso) are reported as well. The total length of DNA (wrapped l plus straight L – l), L, is kept constant during minimization and it is not relevant, provided the condition L > l is satisfied. The free energy difference always displays a minimum at values of l = l^∗ >> l_iso, thus indicating an important contribution of counterion decondensation to the wrapping of DNA around the protein. This contribution increases with Z₀ to a larger extent than the contribution derived from the Z₀ neutralization operated by the DNA wrapping. The difference l^∗ – l_iso simply corresponds to the length of DNA negatively overcharging the sphere. The value of n_{t, iso} accounts for the small adsorption of DNA operators around the protein N-terminal domains (the protein headpieces), while a further factor between 6 and 9 in the amount of wrapped DNA turns is mainly due to counterions decondensation from DNA operated by aspecific electrostatic attraction between the positively charged protein assembly and the negatively charged DNA chain.

Table 1.

Number of wrapped DNA turns with different values of protein charge

Z0	nt∗	nt, iso
15	0.67	0.07
18	0.68	0.09
20	0.69	0.10
25	0.71	0.12

The value n_t (from Eq. 2) for the wrapped-around model obtained with the electrostatic model, R₀ = 4.5 nm, a constant total DNA length L, and several values of Z₀ = ∼+20. Values n_{t; iso} are derived by charge neutralization of the complex.

The value of n_t^∗ = 0.7 obtained with the estimated values of R₀ = 4.5 nm and Z₀ = 20, at the physiological conditions of c_s = 0.1 M, must be compared with the value of 0.9 that can be roughly estimated by the contour length going from O3 to O1 in the x-ray (4), model (14), and nuclear magnetic resonance (13) structures. Most of the wrapping of DNA around the protein assembly hypothesized on the basis of thermodynamic data (11) is due to aspecific electrostatic interactions, while other specific interactions can provide further stabilization and the dissipation of the energy accumulated in internal distortions in the DNA chain and in the protein assembly.

The charge of the protein assembly can be modulated by the environment. A slight increase of pH above 7 decreases the probability of finding His residues in the protonated state, thus decreasing the positive charge of the protein assembly. In these conditions, the wrapped-around configuration may be destabilized, eventually affecting the regulation of genes transcription.

The coarse-grained model

The equilibrium positions of the first and last beads of the 133-bp coarse-grained model of ds-DNA were moved symmetrically through the two lines formed by, respectively, the first and last 20 beads of the minimized structure represented in dark shading in Fig. 2. Therefore, 20 differently restrained MD simulations were performed, one for each of the two symmetric equilibrium locations of first and last beads. Moreover, as explained in the Methods and in more details in the Supporting Material, each set of simulations was repeated, changing several parameters: the DNA intrinsic stiffness, the charge on each bp of the DNA model, and the dielectric permittivity.

Among the 80 MD simulations performed for the coarse-grained model of ds-DNA of 133 bps moving around the frozen protein assembly and with the dielectric permittivity of 80, only two simulations explore configurations corresponding to a wrapped-away arrangement. In Figs. 3–6, representative configurations obtained with all the MD simulations are displayed. The inter-dye distance measured by SM-FRET experiments (18) was here assigned to the distance between bp 30 and 104 of the DNA model (see the Supporting Material). After the inspection of the distribution of the assigned inter-dye distance (see Fig. S1, and related text in the Supporting Material), the configurations obtained are divided into four classes:

• Type A, with the DNA chain wrapped around the protein assembly and both the operators approximately interacting with the respective protein headpieces;

• Type B, with the DNA chain wrapped-around, but with one of the two operators moved away from the initial position;

• Type C, with the DNA wrapped-around and following a different pathway around the protein assembly compared to type A; and

• Types D and E, with the DNA chain differently wrapped-away from the protein assembly.

The initial DNA structure (also used for the running equilibrium positions of the first and last beads; see Methods for details) is displayed in small spheres. The protein assembly, whose structure is kept fixed, is displayed in lines. The collected DNA structures are represented as larger spheres. In Table 2 the mapping between simulations and the representative structures is summarized.

Figure 3.

Wrapped-around configurations, close to the initial model (configurations of type A in Table 2), obtained by MD simulations at T = 300 K and with ɛ_r = 80. The protein assembly is fixed in space and it is represented in lines; the initial DNA model, used as a template for the first and last beads of the model, is represented in small beads; the configurations obtained after 1 ns of simulation is represented in large spheres. In panel legends, the two integer numbers are, respectively, column and row of Table 2. The inter-dye distance is reported in nanometers within brackets.

Figure 4.

Wrapped-around configurations, with DNA pathway rotated with respect to the initial model, obtained after 1 ns of MD simulation at T = 300 K and ɛ_r = 80 (configurations of type C in Table 2). The drawings follow the same scheme detailed in Fig. 3.

Figure 5.

Wrapped-around configurations, with one of the two DNA operators moved far from the protein assembly (configurations of type B in Table 2), after 1 ns of MD simulation at T = 300 K and ɛ_r = 80. The drawings follow the same scheme detailed in Fig. 3.

Figure 6.

Wrapped-away configurations, with the DNA wrapped away from the protein assembly (configurations of types D and E in Table 2), after 1 ns of MD simulation at T = 300 K and ɛ_r = 80. The drawings follow the same scheme detailed in Fig. 3.

Table 2.

DNA configurations obtained via quasistatic unwrapping

MD simulation	Models
	1	2	3	4
1	A (Fig. 3a)	B	A (Fig. 3c)	B
2	A	B	A	B
3	A	D (Fig. 6a)	A	B
4	A	A (Fig. 3b)	A	C
5	A	A	A	B (Fig. 5b)
6	A	C	A	B
7	A	C	C	B
8	A	C	A	B
9	A	B	A (Fig. 3d)	A
10	A	B	C	B
11	A	B	C	B
12	A	B	C	B
13	A	B	C	B
14	A	B (Fig. 5a)	C	C
15	C	B	C	C
16	C	A	C	C
17	C	B	C	A
18	C	C (Fig. 4b)	C	E
19	C	C	C	E (Fig. 6b)
20	C (Fig. 4a)	C (Fig. 4c)	C (Fig. 4d)	B (Fig. 5c)

Final configurations obtained with each of the 80 MD simulations with dielectric permittivity of 80 discussed in the text. Models: 1 = low intrinsic stiffness, low DNA charge (−1/2); 2 = low intrinsic stiffness, high DNA charge (−2); 3 = high intrinsic stiffness, low DNA charge; 4 = high intrinsic stiffness, high DNA charge. Configurations: A = wrapped-around configurations, with both DNA operators close to the protein headpieces; B = wrapped-around configurations, with one operator close to the protein headpieces and one farther; C = wrapped-around configurations, with both DNA termini close to the protein headpieces, but DNA superhelical pathway rotated compared to the initial pathway; D = wrapped-away configuration, but with the protein assembly still within the superhelical coil of DNA; and E = wrapped-away configuration, with the protein assembly outside the DNA coil.

The most evident effect of the large charge density on DNA is to increase the DNA stiffness, but with a negligible effect on the protein-DNA interactions. This can be observed, for example, by comparing the type A structures displayed in Fig. 3 b with that in Fig. 3 d: the two structures are very similar, displaying similar long-range chain stiffness, but with different short-range stiffness. As the net charge on the protein assembly is relatively small (+20), the van der Waals short-range interactions between proteins and DNA dominate over the protein-DNA electrostatic interactions, and they are strongly modulated by the chain stiffness and the constraints acting on the chain termini (here included via harmonic forces).

Table 2 shows that there is a larger occurrence of configurations with the DNA pathway drastically changed compared to the initial one (structures of type C, Fig. 4) and with one of the DNA operators moved far from the initially close protein headpiece (structures of type B, Fig. 5), when the chain stiffness is increased (models 2–4 compared to model 1) and when the tightening of the chain around the protein assembly is released (going from constraint 1 to constraint 20). Only in model 4, when the DNA chain stiffness is maximal, does the chain find a pathway toward properly wrapped-away configurations (Fig. 6 b). This occurs only when the constraint acting on the DNA chain termini is almost completely released (simulations 18–20 of model 4). When the DNA pathway is of type C (Fig. 4), the DNA chain following the region interacting with the protein headpiece formed by one protein dimer, soon interacts with the core of the other dimer.

The free energy for changing the statistics of constraint 1 to that of constraint 20 is ∼−600 RT for models 1–3 and −800 RT for model 4. The change in average potential energy, ΔU, upon the same change in the position of harmonic restraints (i.e., the unwrapping of DNA from the protein assembly) can be decomposed into different contributions to make comparisons with elastic models reported in the literature (see (12,14,16) and the discussions therein). ΔU is in the range −200 ÷ −150 RT for models 1 and 3, whereas it is in the range +100 ÷ +150 RT for models 2 and 4. For these two latter models, the potential energy change is positive because favorable van der Waals interactions with the protein in the wrapped configurations cannot be fully exerted, due to the electrostatic contribution to the DNA chain stiffness. The electrostatic interactions between DNA beads in this coarse-grained model, especially when net charges on DNA beads are large (models 2 and 4), alter the elastic properties of the DNA chain. Nevertheless, the contribution of electrostatic forces in modulating the chain compaction when the DNA chain is wrapped around an oppositely charged protein assembly must be included and the Coulomb approach, with a dielectric permittivity mimicking bulk water, is the most straightforward approach. The model in Swigon et al. (12), which includes the change in electrostatic repulsion within DNA charges upon DNA binding to LacI in the closed form (the model P1, with 130 bps of DNA), displays a change of potential energy upon binding within 25–30 RT units, compared with the values for ΔU of our model 1, once the electrostatic and van der Waals contributions of protein-DNA interactions are removed. This quantity, for model 1 (only wrapped-around configurations), goes from 260 to 320 RT units, while it reaches values of 3 RT when wrapped-away configurations are included (model 4). This shows that the configurations collected in this work span energies that are similar to those of other elastic models when wrapped-away configurations are concerned, whereas energies 10-times larger are obtained when wrapped-around configurations are analyzed.

The change in the average DNA bending energy upon unwrapping is in the range +50 ÷ +100 RT for models 1 and 3. For model 2, the bending energy of DNA is almost constant, whereas for model 4, when the wrapped-away configurations are obtained, the change from statistics 1 to statistics 20 is −150 RT, with an average bending energy of 250 RT when the statistics span the wrapped-away configurations described above (see Table 2). As in model 4, the intrinsic chain stiffness was 10-times larger than the usual value adopted in elastic models (the value adopted, for instance, in models 1 and 2), a realistic average bending energy of the wrapped-away configurations sampled by the statistics of our model can be estimated as 25 RT. This value is twice the minimal value obtained as bending energy for the elastic model of Goyal et al. (16) (11 RT units for the P1F model of 142 bps, the model closer to the model reported in our work) and it is consistent with the largest value obtained for the same model by forcing extreme twisting conditions (30 RT; see Figs. 7 and 8 in (16)). A similar range (10 ÷ 40 RT) for the elastic energy was obtained in Balaeff et al. (14) for different kinds of wrapped-away configurations. In summary, for different binding topologies and sequences, the elastic energy of elaborated models (like those reported in the literature for the same system) is in the range of 10 ÷ 100 RT units. With the simple model reported in this work, we obtain, for the intrinsic elastic contribution, values in the range of 25 ÷ 320 RT.

The changes in free energy and in average potential energy reported above, however, show that, upon releasing the tightening of DNA to protein, the increase of conformational entropy of the DNA model greatly dominates over the energy contribution to DNA-protein binding (this latter within −150 and +200 RT). Interestingly, the free energy change for accessing entirely wrapped-away configurations, like those accessed by model 4, is only slightly larger than that where wrapped-away configurations are not accessed at all (800 RT compared to 600 RT). The contribution of 200 RT to the free energy when wrapped-away configurations are accessed is due almost entirely to the release of DNA bending energy (150 RT, see paragraph above) upon wrapping away the DNA chain from the protein assembly.

It can be observed that if one-half of the counterions eventually condensed on the DNA surface of the 133-bp chain is ejected in the solvent and assuming that each of these counterions can be condensed in a volume of 1:100 that of the molar volume in the bulk solution, the corresponding free energy contribution would be ∼133RT log(1/100) ∼ −600 RT. This shows that the free energy change estimated for the most reasonable chain models (models 1–3, 600 RT) can be entirely compensated by one-half of the counterion condensation.

The barrier against the complete unwrapping of the DNA from the protein assembly (from configurations of types A–D to configurations of type E) is mainly due to the difficulty in moving both the DNA chain termini away from the corresponding protein headpieces. This difficulty is shown by the large weight of configurations with one DNA operator moved away from the headpiece (type B), compared to the weight of configurations where both the operators are moved away at the same time (type C or D). This effect is due to the residual strong binding of DNA with the protein surface, persisting even for straight DNA segments that occur when the tightening constraint is released (Fig. 4, c and d, and Fig. 5). When the constraint is released, the chain starts wobbling between two equivalent configurations with a single operator in the headpiece region. When the DNA stiffness is lower, the rotated wrapped-around pathways of type C, displayed in Fig. 4, are almost equivalent to the configurations of type B, displayed in Fig. 5 (see also Table 2, where the exchange between these two kind of configurations, B and C, is frequently represented).

As for structural information, the coarse-grained model allows a comparison with results obtained by SM-FRET experiments (18) and related modeling (16). Even if the SM-FRET experiments estimate an inter-dye distance of 3.5 nm, i.e., smaller than the lower limit found in our statistics (∼6 nm, see Fig. 4 a), most of the configurations of type A, B, and C obtained with models 1–3 (Fig. S1) display a significant population around an inter-dye distance of 8 nm. This latter distance, also obtained with the wrapped-away antiparallel elastic models (16), is consistent with a 10% ET efficiency in FRET experiments. Interestingly, all the wrapped-away configurations obtained with coarse-grained model 4 display inter-dye distance >10 nm, i.e., with a negligible ET efficiency in FRET experiments.

Analysis of the relaxed all-atom configurations

From the analysis of time evolution described in detail in the Supporting Material, the last 0.6 ns of the simulation with the protein atoms not fixed in space can be considered as a trajectory describing a wobbling protein-DNA complex. The time evolution before this time range is necessary to safely relax the stress induced in the DNA, in the protein tetramer, and in the counterions cloud, by the initially built model for the DNA chain that is wrapped around the protein assembly. The forthcoming assessment of the protein tetramer near the DNA chain is a very slow process, for which an even larger system should be simulated, or an ad hoc coarse-grained model should be developed.

In this subsection, the analysis of the structure of the protein-DNA interface (mostly concentrated in the regions of O3 and O1 operators, bp 1–22 and 93–113, respectively) is also summarized.

The DNA chain maintains its wrapped configuration around the protein tetramer during the entire simulation. The final structure of the protein-DNA complex obtained with the simulation is displayed in Fig. S4 from two different points of view. A summary of notations and deviations from canonical B-DNA measured for each bp in DNA is reported in the Supporting Material. The structural details more relevant in monitoring the global stability of the double-helical structure of DNA are discussed below.

By analyzing the hydrogen bonds involving the DNA bases, identified according to the method reported in the Supporting Material, we found that 80% of hydrogen bonds characteristic of the B-DNA double helix are kept in the wrapped-around configuration. Completely broken inter-basepair hydrogen-bond networks are in bps 1, 7–10, 23, and 105–113, while for bps 52, 72, and 80–83 only a partially demolition of the hydrogen-bond network occurs. The short 90–93 segment forms noncanonical hydrogen bonds that still connect the two strands. As for the double-helical DNA structure, the most demolished regions are the terminal regions 1–11 and 105–113. Only a few defects—23, 90, 91, 92, 93—are present in the 93-bp-long segment 12–104.

For the B-DNA, the solvent-accessible surface area (SASA) of each nucleotide (nt), averaged over the nonterminal nt, is 1.67 ± 0.08 nm², and the total SASA, for the whole chain, is 382 nm². For the DNA in the protein-DNA complex, the average SASA is 1.7 ± 0.4 and the total SASA is 378 ± 10. The larger dispersion of SASA per nt in the complex is due to the averaging over nts exposed to the solvent and buried in the protein complex. However, despite the large DNA bending, the distortions of the DNA double-helical structure, and the extrusion of some bases into the solvent, the solvent accessibility of DNA in the complex is the same as that of a linear B-DNA with the same nucleotide sequence. This effect is due to the large distance of the DNA chain from the protein assembly in those regions not involved in short-range interactions (for example, the DNA operators). The DNA-protein separation is kept large by the significant DNA bending stiffness in the all-atom model, resembling that of the coarse-grained models 3 and 4.

The analysis of the distortion in DNA double-helical structure due to specific interactions with the protein, is discussed in detail in the Supporting Material. With the single exception of bp 23, the 12–104 region has a distorted double-helical structure, with the stress that is induced in the DNA by the large bending transferred into backbone distortions (mainly of γ dihedral angle and furanose ring). On the other hand, the double-helical structure in the segments 1–10 and 105–113 (O3-LS and O1-RS, respectively) is demolished, and hydrophobic clusters are formed, which involve Tyr side chains together with salt bridges between phosphate groups and Lys side chains. The occurrence of these protein-DNA interactions strongly depends on the initial location of the DNA grooves in the all-atom model construction. A wider statistics is necessary for a reliable simulation of these interactions, but, interestingly, the model shows that regions outside the O3 and O1 operators are more affected than the region in between.

Conclusions

We have developed an analytical model of the electrostatic interactions between the lac repressor tetramer, assumed as a sphere, and a double-stranded DNA, assumed as an elastic cylinder. This model gives a quantitative estimate of the nonspecific electrostatic free energy (which was originally suggested by Tsodikov et al. (11)) that is responsible for Tsodikov's wrapping-around model of the complex LacI-DNA, based on extended thermodynamics experiments. At physiological conditions, a generic wrapping of 0.7 turns around the sphere is obtained based on the interactions between the negative phosphate charges of DNA and the positive charges of LacI mediated by salt. This contribution to the wrapping is what can be expected from taking into account that the remaining contributions (necessary to reach 0.9 turns corresponding to the contour length from the two operons O3 and O1) should come from specific protein-DNA interactions (like those with the DNA-binding domains at the tips of the V-shaped LacI tetramer).

The simple coarse-grained model of the 133-bp ds-DNA, moving around the compact and rigid form of the LacI protein assembly, shows that wrapped-away configurations of DNA are rarely represented and that their contribution provides approximately one-third of the stabilization free energy associated to the more frequent interconversion between differently wrapped-around configurations. When the constraint of the interaction between O3 and O1 DNA operators and the protein headpieces is released, the DNA chain tends to adopt conformations with the chain crossing from the headpiece formed by one dimer toward the core of the other dimer. The transition between conformations with ∼93 DNA bps wrapped around the protein assembly toward conformations with 133 bps approximately wrapped-around, is more likely than the complete unwrapping of DNA. Moreover, the free energy change for transitions between differently wrapped configurations can be compensated for by the mechanism of counterion condensation, not included in the coarse-grained model. The intrinsic bending elastic energy accumulated in the DNA chain in wrapped-around configurations is larger than in more elaborated elastic models (12,14,16). Nevertheless, both for wrapped-away and wrapped-around configurations the bending energy is within a factor 10 from the maximal elastic energies obtained for the differently wrapped-away configurations estimated by the other models reported in the literature. This difference in elastic energy can be compensated for by entropic contributions, due to DNA chain disorder and to counterion condensation.

The resolution of the simple electrostatic model into an all-atom model shows that the DNA can be accommodated in a wrapping-around fashion, with slight distortions both in DNA and in the protein assembly. The most dramatic structural changes of the wrapped DNA chain around the protein tetramer are localized in regions 7–11 and 105–113 of the simulated model. These two regions are approximately one-half of the O3 and O1 operators, respectively. Our results show that, within a simulation time that allows the local relaxation of the entire assembly from the largely distorted initial construction, the bp range 12–104 (93 bps) is substantially in a double-helical structure, with most of the distortion concentrated in the DNA backbone. Moreover, the DNA relaxation is accompanied by a significant rearrangement of protein headpieces, as it was observed for the wrapped-away model including protein flexibility reported in the literature (17). These DNA and protein distortions are found sufficient to adapt the DNA double helix to a largely bent pathway around the protein assembly.

Even though the wrapped-away configuration cannot be excluded by these calculations, we observe that now there are at least four independent motivations for the wrapped-around model:

• 1.The extended thermodynamic experiments of Tsodikov et al. (11), who first motivated the wrapped-around model of the complex LacI-DNA;
• 2.The simple electrostatic argument, giving a substantial wrapping around LacI;
• 3.The MD simulations of simple coarse-grained models, that show frequent interconversions between wrapped-around configurations and rare unwrapping of the DNA chain;
• 4.The molecular dynamics relaxation of the all-atom model, which shows a maintained double-helical structure of the DNA chain in the 12–104 region.

Supporting Material

Details of the three models and related methods are available at http://www.biophysj.org/biophysj/supplemental/S0006-3495(10)00356-5.