Elasticity and electrostatics of plectonemic DNA
We present a self-contained theory for the mechanical response of DNA in single molecule experiments. Our model is based on a 1D continuum description of the DNA molecule and accounts both for its elasticity and for DNA-DNA electrostatic interactions. We consider the classical loading geometry used in experiments where one end of the molecule is attached to a substrate and the other one is pulled by a tensile force and twisted by a given number of turns. We focus on configurations relevant to the limit of a large number of turns, which are made up of two phases, one with linear DNA and the other one with superhelical DNA. The model takes into account thermal fluctuations in the linear phase and electrostatic interactions in the superhelical phase. The values of the torsional stress, of the supercoiling radius and angle, and key features of the experimental extension-rotation curves, namely the slope of the linear region and thermal buckling threshold, are predicted. They are found in good agreement with experimental data.
Key words: elasticity; self-contact; twist-storing polymer; DNA electrostatics
Mechanics of the DNA molecule plays a key role in several biological processes at the cellular level. In several cases, the action of enzymes and proteins on DNA has been found to depend on the mechanical stress present in the molecule. For instance, the torsional moment in DNA controls the action of topoisomerases or RNA-polymerases (1, 2). In this context, experiments where forces and torques are applied to a single DNA molecule provide a remarkable opportunity to gain insights into the mechanics of DNA. We are here interested in extension-rotation experiments using either optical or magnetic tweezers (3, 4, 5, 6, 7), see Ref. (8) for a review. These experimental setups are equivalent from a mechanical perspective: a dsDNA molecule is fixed at one end on a glass pane while the other end is attached to a bead that pulls and twists it. In these experiments, traction and rotation are controlled differently: for the rotation mode, the twist angle is prescribed and the twist moment varies accordingly; for the stretching mode, the extension can vary although the pulling force is prescribed. DNA is under or over-wound and various molecule conformations are observed (6). In the present study, we focus on the over-winding of a dsDNA molecule under large imposed rotations: the molecule coils around itself in a helical way and forms plectonemes, as sketched in Fig. 1.
An important feature of the experimental loading curves is the linear decrease of the vertical extension of the molecule as a function of the imposed rotation. We have shown in previous studies (9, 10) that this behavior can be captured by a purely elastic rod model based on Kirchhoff-Love elastic rod theory. In this paper, we extend this model and investigate the combined effects of elasticity and electrostatics.
The response of plectonemic DNA under stress involves various physical phenomena such as elastic deformations, thermal fluctuations, electrostatic interactions, self-avoidance. Although some of these effects have been considered in the literature, a model addressing them all together is still lacking. Mechanical models of twisted rods in contact have been introduced, from an analytical (11) or numerical (12) perspective, but thermal fluctuations are not treated. A simplified analytical model, including some account for fluctuations but omitting contact forces in the plectonemic region, is proposed in Ref. (13). Statistical mechanics of plectonemic DNA has been approached, either analytically (14, 15) or numerically (16) using a Monte-Carlo method. The validity of some of these results was questioned in Ref. (17, 18); in addition, long-range potentials raise convergence issues that have not yet been overcome in Monte-Carlo simulations. A composite model, gathering results from torsionally constrained polymer (19) and Monte-Carlo simulations has recently been introduced (20). To date, this is the only model that confronts its predictions against experimental data. However it relies on an assumption on the supercoiling free energy that is not always valid (21, 22) and uses parameters extracted from Monte-Carlo simulations.
In this paper, we present a self-contained analytical model for the mechanical response of plectonemic DNA in extension-rotation experiments, which builds up on previous work (9). We focus on the plectonemic regime at large imposed rotations. This corresponds to the linear region in the experimental extension-rotation curves. Our elastic model accounts for DNA-DNA interactions in the plectonemic region and for thermal fluctuations in the tail regions, where they are dominant. It captures the main features of the experimental curves and allows quantitative comparison to experiments with no adjustable parameter.
The paper is organized as follows. In the next section we present our model and derive the equilibrium equations for a DNA molecule comprising plectonemes, for a generic interaction potential. In section III, we describe two representative DNA-DNA interactions potentials available in the literature, which we then plug into our model. The results are then compared with experimental data in section IV.
Our description of the DNA molecule is based on a coarse-grained representation (?). We introduce a continuum rod model whose mechanical behavior is similar to that of the molecule, and makes use of effective elastic and electrostatic properties obtained by smoothing out the details at a scale of several base pairs. We deal with an inextensible elastic rod with circular cross section, bending rigidity , and twisting rigidity . The loading geometry is that of Fig. 1, and applies to the experiments where the lower end of the molecule is clamped on a glass pane and the other end is subjected to a tensile force and rotated by turns (i.e. an angle ). The imposed rotation is achieved through a torsional moment . Note that the torsional moment has become accessible to experimental measurements (7) only recently.
The inextensible rod, of length , is parametrized by its arc-length , the origin being at the lower end. The rod centerline is described by a vector-valued function and its unit tangent . The geometric curvature of the rod is noted . The twist is noted : it describes the relative rotation of neighboring cross sections about the tangent . Note that the twist is a different quantity from the Frénet (geometric) torsion of space curves — the latter is irrelevant in the context of elastic rods.
We consider the geometry of the double stranded DNA sketched in Fig. 1, which is relevant to the plectonemic regime: two twisted, straight tails are separated by a plectonemic region composed of two identical and uniform helices. Note that each helix is itself a piece of the double stranded DNA molecule. For a large number of turns , the loop at the end of the plectonemes and the curved region connecting the tails to the plectonemes are much smaller than the tails and the helical parts, and hence are neglected. Even though we depict the plectonemic region as a single chunk for simplicity, our model applies equally well to the case where the plectonemes are distributed in several places along the molecule; then, the elastic rod is made up of two phases, one with linear DNA and the other one with plectonemes, and the plectonemic structure in Fig. 1 represents the plectonemic phase collectively. The molecule contour length spent in the tails phase and in the plectonemes are noted and respectively. They sum up to the total length . The plectonemic phase is characterized by its superhelical radius and its superhelical angle , which are assumed to be uniform: neither nor may depend on , although they depend on the loading. Curvature is zero in the straight tails, and takes a constant value in the plectonemes which can be evaluated using simple geometry. The integrated squared curvature, which enters into the bending energy, is then found to be:
see Ref. (9). Since the rod has a circular cross section, the twist is uniform, , as shown for instance in Ref. (23). The internal torsional moment in the rod is related to the twist by the constitutive law . Therefore, its value is constant along the rod, and equal to the torque applied by the bead. In what follows, we study the equilibrium of the rod and compute the parameters , , and as a function of the loading (pulling force and number of turns ) by minimizing the energy.
We derive the energy of the system as a function of the superhelical angle and radius , of the twist and of the plectonemic contour length . Equilibrium solutions and their stability will be derived later on by minimizing this energy. The experiments are performed under imposed end rotation: energy minimization is performed under the constraint that the number of turns imposed on the bead is equal to the link of the DNA molecule. Neglecting the writhe of the tails, the link can be written as (10, 9):
where stands for the chirality of the two helices of the plectonemic phase.
The total energy of the system is the sum of three terms, , where the first is the strain elastic energy, the second is the potential energy associated with the external load , and the last term accounts for DNA-DNA interactions between the two helices in the plectonemic phase. The strain elastic energy of the rod is defined as the sum of a bending term, proportional to the integrated squared curvature, and a stretching term, proportional to the integrated squared twist:
This energy captures the elastic behavior of the rod in response to applied forces and moments; it is zero in the natural (straight, twistless) configuration of the rod. The pulling force is described using a potential energy:
Here is the extension of the molecule along the direction of application of the pulling force. Since we assume the tails to be straight and neglect the curved region region connecting the tails and the plectonemes, the vertical extension of the filament reads and we can rewrite:
There is no need to consider any potential energy associated with the rotation of the end attached to the bead since the energy will be minimized for a given rotation of the bead using the constraint on the link.
In previous work (9) we solved this elastic rod model by assuming the superhelical radius to be prescribed and extracted its value from experimental data. Here, we take a more principled approach and complement the above elastic equations with a proper model for DNA-DNA interactions in the plectonemes; in particular, this makes it possible to predict the superhelical radius. These interactions are dominated by different physical effects depending on the separation distance between the two DNA superhelices. In the range of separations relevant to extension-rotation experiments, of order several nanometers, electrostatic effects dominate. In our model, interactions are limited to the plectonemic phase and are described by an energy contribution of the form:
This energy depends on the superhelical parameters and , and is proportional to the plectonemic contour length , and assumption valid when is much larger than .
The total energy of the system is defined as the sum of the elastic, potential and interaction contributions:
It will be minimized subjected to the end rotation constraint given by Eq. 2. This constraint provides an affine relation between and and so can be used to eliminate the quantity in favor of . Dropping the constant term in the energy, we obtain:
The total energy of the system, given by Eq. 8, takes into account the fixed end rotation since Eq. 2 has been used to eliminate . The equilibria of the rod are then directly given by minimization of with respect to its three arguments. Canceling the first variation of , that is writing , and , we obtain
In the first term of the last equation, we have eliminated in favor of using the constitutive relation , and thereby removed the twist rigidity from the equations (its value is not known with good accuracy).
The set of three nonlinear equations (9) must be solved for the three unknown values of the parameters , and at equilibrium, given the value of the external force . This requires an interaction potential to be specified, as is done in the forthcoming Section III. The set of equations (9) extends the model of Ref. (9), valid for non-penetrable tubes, to filaments in long-range interaction (such as electrostatic interactions).
Note that the equations (9) do not depend on the number of turns . As a result, their solution , , do not depend on either. The equations describes the equilibrium of two phases; increasing transfers some arc length from the tail phase to the plectonemic phase, without changing their properties. This invariance with respect to explains the presence of a linear region in the experimental curves, as shown in the next section.
The term in parenthesis in Eq. 9c is always positive. This shows that the sign of the chirality is opposite to that of : rotating the bead in the positive direction for instance, requires a positive torque , hence a negative by this equation (left-handed superhelices).
Vertical extension of the filament
In extension-rotation experiments the vertical extension of the filament is recorded while the number of turns is increased. The formula , valid for straight tails, does not holds in the presence of thermal fluctuations. We account for these fluctuations by introducing a rescaled quantity:
where the correcting factor is given by the worm-like chain theory (24) as the solution of:
Here is the Boltzmann constant and the absolute temperature. In order to write as a function of the number of turns , we use the equality in the right-hand side of Eq. 10, and use for the expression obtained by solving Eq. 2:
Recall that neither , nor depend on ; as a result, the extension depends linearly on the number of turns in the above equation. This linear dependence is a well-known feature of the experimental curves.
Iii DNA-DNA interactions
In the variational formulation exposed in the previous section we have introduced an energy describing DNA-DNA interactions. At moderate distances DNA-DNA interactions in solution mainly originate from electrostatic effects between the charged sites of the two molecules (phosphate groups) and between these charged sites and the counter and co-ions present in the solution. The theoretical analysis of the long and short range interactions between two poly-ions in solution has been the subject of numerous studies (25, 26) and there is currently a regain of interest in this question due to recent progress in single-molecule experiments — see (27) for a review. In the present model the interaction energy is specified independently of the mechanical behavior of the molecule. As a result, we can combine the elastic description of the previous sections with different theories for DNA-DNA interactions. In the following, we illustrate this approach using two representative interaction energies that can be found in the literature.
We favor interaction energies that can be expressed in closed analytical form and have no adjustable parameters; this enables us to make predictions and compare them to experiments, rather than to fit existing data. In the literature on DNA-DNA interactions (28, 27, 29), we picked two well-established models satisfying those requirements. The first one, , derives from the Poisson-Boltzmann equation and was obtained by Ubbink and Odijk (30); the second one, , is based on the counterion condensation theory (31) and was derived by Ray and Manning (32). These two models address the electrostatics of DNA in solution but their treatment of the interactions between DNA and the ions in solution differ substantially.
In their study of supercoiled DNA plasmids (30), Ubbink and Odijk derive an analytical expression for the electrostatic interaction energy between two interwound DNA molecules. Their work is based on the Poisson-Boltzmann framework (PB); in the computation of the electrostatic repulsion of the two charged molecules, the presence of the counter-ions and co-ions in solution is considered. It has been shown in Ref. (33) that the non-linear PB problem could be simplified to a linear one by considering screened (Debye-Huckel like) potentials and renormalized linear charge densities . The value of the effective charge is obtained by matching the solution of the non-linear PB equation with the solution of the linear PB equation in the far-field region. However there is no consensus on the exact value of this effective charge and the various numerical (34, 35) or analytical (36, 37) studies yield scattered results.
In Ref. (30), Ubbink and Odijk compute the electrostatic interaction energy per unit length as
|where the angular dependence reads|
Here, is the Boltzmann constant, the temperature in Kelvin, the effective linear charge density (in m), the Bjerrum length, and the Debye length. For a typical temperature we have , and for a monovalent salt concentration the Debye length is . The value of the effective charge depends on salt concentration, its value for a monovalent salt concentration is taken as according to Ref. (30).
The calculation of the interaction energy can be simplified by taking , hence , which amounts to consider two straight and parallel molecules; this approximation has been used for instance in Ref. (34). In the rest of the paper will refer to the potential obtained under this approximation, namely .
Ray and Manning model
The interaction energy derived by Ray and Manning in (32) is based on the counter-ions condensation theory (31). The authors examine the interaction of two straight and parallel DNA molecules with charged sites in solution (the dependence on the superhelical angle is neglected). The main point of the theory is to consider that part of the DNA bare charge is neutralized by the condensation of the counter-ions around the molecule. The energy is the sum of three terms: interactions between pairs of charged sites belonging to the same DNA segment, interactions between pairs of charged sides located on opposite segments, and adsorption energy of the counter-ions onto the molecule. Three cases are considered, namely short, intermediate, and long interaxial distances between the molecules. The short distance case, below the crystallographic radius of DNA, is not relevant to our analysis. The intermediate case introduces an adjustable parameter, which we try to avoid. Consequently we only use the long-range case, relevant for inter-distances larger than the Debye length; in our notations it writes:
where is the charge spacing parameter of the DNA molecule, and is the dimensionless charge density of DNA ( at ). The function is the modified Bessel function of the second kind and order 0.
We solve equations 9 for the superhelical radius , angle , and external torque , using one of the interaction energies , or . These equations are nonlinear and their roots are found numerically using a Newton-Raphson algorithm. We present the results for the superhelical variables and in Fig. 2, for the torsional moment in Fig. 3. We also plot derived quantities, to be defined later, such as the slope of the extension-rotation curves in Fig. 4, and the thermal buckling threshold in Fig. 6. We compare our results with the model of Ref. (20) and with experimental data. To ease comparison with our previous work (9), we use the same set of experimental data. These data were obtained on a 48 kbp lambda phage DNA molecule in a 10 mM phosphate buffer.
With the interaction energies used in the present paper, we find that the nonlinear equations have two roots below a threshold value of the force, and no root above. For a salt concentration , this threshold value of the force is using , using , and using ; all these values are above the maximum pulling force applied in typical experiments. We have studied the stability of the two solutions corresponding to the two roots of our equations when the force is below threshold, and found that that with lower and is unstable; the other one is stable. We study and plot the latter in the following.
When the force reaches its threshold value, an instability occurs and the stable solution disappears by merging with the unstable one. For larger forces, no stable solution exist and the two helical parts collapse. This may be related to the observation of tightly supercoiled configurations in Ref. (38). The collapse arises when the electrostatic interaction is no longer strong enough to sustain the applied force; the description of collapsed solutions would require an account of hard-core repulsion and other short-range forces.
The quantities and are plotted Fig. 2 as a function of the applied force .
The curves and obtained for the different interaction energies , or are close over the entire range of forces. As will be confirmed later, the predictions based on the different interaction models are all very similar.
As expected, the superhelical radius decreases with the pulling force; note that it becomes less than the Debye length for forces above . The superhelical angle is known to be a control parameter in the action of the topoisomerases (39). It is plotted in the inset of Fig. 2. In contrast with models of elastic tubes in contact (40), where increases monotonically and reaches the value asymptotically at large forces, we find here that it reaches a maximum well below and then decreases, due to long-range forces. This decrease has already been observed in Ref. (9), where the value of the superhelical angle was extracted from experimental data.
Recall that the torque applied by the bead in order to impose a rotation does not depend on the number of turns by equations. 9. This quantity is plotted in Fig. 3 as a function of the pulling force . We compare our results (i) with Marko’s Eq. 17 in Ref. (20); (ii) with a formula obtained by Strick et al. by approximating the plectonemes by a chain of circles (41); and (iii) with our previous study (9) based on hard-wall interactions.
Even though Refs. (20), (41) and (9) do not address long-range interactions, all the curves reveal a similar behavior: the moment increases monotonically with the applied force, with a decreasing slope. However, our results show that long-range interactions significantly increase the value of the moment required to achieve a given rotation.
Our model predicts that the derivative of the vertical extension with respect to is constant, i. e. that the extension-rotation curve is linear in the regime of large rotations that we consider. This linear regime is well-known experimentally, see Fig. 5. From Eq. 12, the slope is given by:
Its value is computed using the values of and obtained earlier by solving the equilibrium equations. We plot in Fig. 4 the slope as a function of the force, for the three interaction energies.
For comparison, we also plot the slope predicted by Marko’s model (20), and the slopes read off directly from experimental extension-rotation curves (these experimental data were kindly provided by V. Croquette and have appeared in Fig. 2 of our Ref. (9) and in Refs. (10) and (20)).
Our model shows good agreement with the experimental data which are reproduced in a more consistent manner, especially at low forces, than in Ref. (20). In this reference, hard-wall interactions are used with a supercoiling radius independent of the pulling force; this may be the cause of the poorer agreement with experimental data at low forces, when long-range interactions dominate.
A typical extension-rotation curve comprises two regions: a linear region for large , which we have been discussing so far, and a parabolic region at low studied in Ref. (42).
The central region is dominated by thermal effects and will not be addressed here. However, we can characterize the transition between the two regions. The number of turns at which the transition occurs is defined using the linear extrapolations shown in Fig. 5. This corresponds to the onset of the plectonemic regime. In our model, it is computed by setting in Eq. 2; this yields . Recall that the value of is computed from the equilibrium equations 9. We plot in Fig. 6 the value of as a function of the force and compare to the values extracted from the experimental curve as well as the value from the theory in Ref. (20).
V Discussion and conclusion
In Section IV, we found that the solutions
disappear above a threshold value of the pulling force. This can be
interpreted as the fact that the physical solution involves a very
short inter-distance, although we have retained the long-distance part
of Ray and Manning’s potential only. This can be cured in principle
by restoring a complete expression
We have presented an analytical model for DNA supercoiling in extension-rotation experiments. It is based on an elastic description of DNA deformations, carefully accounts for DNA-DNA interactions in the plectonemic region, and makes use of a valid formula for the link. DNA interactions are modelled using long-range forces computed from potentials available from the literature. Their description is compartmentalized from the rest of the theory, which makes it possible to test different interaction energies. We have used our model in combination with two interaction energies. These energies come from different physical contexts and are widely used in the literature. Using either one, we find good agreement with experimental data without adjusting any parameter. This suggests (i) that using yet another energy for electrostatic interactions would yield comparable results and (ii) that DNA tertiary structures are determined to a large extent by the elasticity of the molecule and do not depend heavily on the details of the interaction model. The extension-rotation experiments that are now routinely performed can then be viewed as a way to probe the elastic properties of the molecule. Given that the mechanics of DNA under combined twist and tension can be captured by a relatively simple analytical model, an interesting direction for future research is to extend this model to the mechanical action of proteins, such as RecA for example, on supercoiled DNA.
S. N. thanks Armand Ajdari (CNRS) for starting discussions on the subject of electrostatic repulsion in DNA supercoiling.
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