^{1,a)}and Ralf Everaers

^{1}

### Abstract

It is a standard exercise in mechanical engineering to infer the external forces and torques on a body from a given static shape and known elastic properties. Here we apply this kind of analysis to distorted double-helical DNA in complexes with proteins: We extract the local mean forces and torques acting on each base pair of bound DNA from high-resolution complex structures. Our analysis relies on known elastic potentials and a careful choice of coordinates for the well-established rigid base-pair model of DNA. The results are robust with respect to parameter and conformation uncertainty. They reveal the complex nanomechanical patterns of interaction between proteins and DNA. Being nontrivially and nonlocally related to observed DNA conformations, base-pair forces and torques provide a new view on DNA-protein binding that complements structuralanalysis.

This work was supported by the chair of excellence program of the Agence Nationale de la Recherche (ANR). The authors wish to thank R. Lavery for helpful discussion.

I. INTRODUCTION

II. BACKGROUND

III. DNA NANOMECHANICS

IV. METHODS

A. Rigid base-pair parameter sets

B. Restrained relaxation

C. Implementation and visualization

V. RESULTS

VI. DISCUSSION

A. Robustness of nanomechanics analysis

B. Force scale and validity range

C. Protein forces may exceed critical forces for DNAstructural transitions

D. Sharp elastic energy peaks result from balanced pairs of force and torque

E. Directions of deformation and force are nontrivially related

F. Forces require DNA-protein contacts, but contacts do not always transmit force

G. DNA is deformed by a combination of local forces and propagated tension

VII. CONCLUSIONS

## Figures

Constraint force and externally applied force in a stereotyped double-well free energy landscape and thermal distribution (solid lines). Under an external force , the landscape is tilted (dashed lines).

Constraint force and externally applied force in a stereotyped double-well free energy landscape and thermal distribution (solid lines). Under an external force , the landscape is tilted (dashed lines).

Rigid base-pair model. Atomic coordinates are reduced to the base-pair center position and orientation degrees of freedom, represented as bricks. Conformations of the I*ppo*-I complex before (left) and after (right) prerelaxation within a range of , see Sec. IV.

Rigid base-pair model. Atomic coordinates are reduced to the base-pair center position and orientation degrees of freedom, represented as bricks. Conformations of the I*ppo*-I complex before (left) and after (right) prerelaxation within a range of , see Sec. IV.

Force and torque pairs acting on DNA to produce an excess of one base-pair step parameter in each row. Torque vectors shown in blue and force vectors in red. The same deformation of the *central* base-pair step can be produced by external force and torque pairs attacking directly (left column), at the nearest-neighbor base pairs (middle column), or seven base pairs away (right column). Sequence-averaged MP parameters. For plots of the base-pair step parameters associated with these equilibrium shapes, see Fig. supp-1.

Force and torque pairs acting on DNA to produce an excess of one base-pair step parameter in each row. Torque vectors shown in blue and force vectors in red. The same deformation of the *central* base-pair step can be produced by external force and torque pairs attacking directly (left column), at the nearest-neighbor base pairs (middle column), or seven base pairs away (right column). Sequence-averaged MP parameters. For plots of the base-pair step parameters associated with these equilibrium shapes, see Fig. supp-1.

TFIIA(not shown)-TBP-DNA complex with force and torque vectors from three different TBP-DNA crystal structures (left). Corresponding energy, force magnitude, and torque magnitude profiles (right). Here and in the following figures, linear forces are shown as red arrows and torques as blue arrows; base pairs are represented as numbered small boxes with the following sequence coloring: “A,” red; “T,” blue; “G,” green; “C,” yellow; the two viewpoints are rotated by 90° around the vertical axis. Sequence (, base pair 1)-GGGGGGGCTATAAAAGG-(, base pair 17). Allowed relaxation range in all complexes. MP parameter set. The three-dimensional representations of base pairs, force, and torque vectors used for this figure are available, as detailed in Sec. IV (supplementary material data S1).

TFIIA(not shown)-TBP-DNA complex with force and torque vectors from three different TBP-DNA crystal structures (left). Corresponding energy, force magnitude, and torque magnitude profiles (right). Here and in the following figures, linear forces are shown as red arrows and torques as blue arrows; base pairs are represented as numbered small boxes with the following sequence coloring: “A,” red; “T,” blue; “G,” green; “C,” yellow; the two viewpoints are rotated by 90° around the vertical axis. Sequence (, base pair 1)-GGGGGGGCTATAAAAGG-(, base pair 17). Allowed relaxation range in all complexes. MP parameter set. The three-dimensional representations of base pairs, force, and torque vectors used for this figure are available, as detailed in Sec. IV (supplementary material data S1).

Complex of lac repressor and DNA. NMR structure of 1 out of 20 conformers with ensemble mean force and torque vectors (left). Two magnified views of the encircled region, middle column, with force and torque vectors of all conformers. Ensemble energy, torque, and force magnitude profiles (right); the ensemble standard deviation profiles and of force and torque vectors are shown in black. MP parameter set. Sequence (, base pair 1)-GAATTGTGAGCGGATAACAATTT-(, base pair 23). The three-dimensional representations in data S2.

Complex of lac repressor and DNA. NMR structure of 1 out of 20 conformers with ensemble mean force and torque vectors (left). Two magnified views of the encircled region, middle column, with force and torque vectors of all conformers. Ensemble energy, torque, and force magnitude profiles (right); the ensemble standard deviation profiles and of force and torque vectors are shown in black. MP parameter set. Sequence (, base pair 1)-GAATTGTGAGCGGATAACAATTT-(, base pair 23). The three-dimensional representations in data S2.

I*ppo*-I DNA complex. The points of single-strand cuts in the functional complex are indicated. Relaxation range . The MP parameter set was used for vectors and MP (solid line) and P (dashed line) parameter sets for profiles (right). Sequence (, base-pair 1)-TGACTCTCTT AAGAGAGTCA-(, base pair 20). Three-dimensional representations in data S3.

I*ppo*-I DNA complex. The points of single-strand cuts in the functional complex are indicated. Relaxation range . The MP parameter set was used for vectors and MP (solid line) and P (dashed line) parameter sets for profiles (right). Sequence (, base-pair 1)-TGACTCTCTT AAGAGAGTCA-(, base pair 20). Three-dimensional representations in data S3.

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