An analytical method to connect open curves for modeling protein-bound DNA minicircles

Abstract

We introduce an analytical method to generate the pathway of a closed protein-bound DNA minicircle. We develop an analytical equation to connect two open curves smoothly and use the derived expressions to join the ends of two helical pathways and form models of nucleosome-decorated DNA minicircles. We find that the simplest smooth connector which satisfies the boundary conditions at the end points and the length requirement for such connections to be a quartic function on the xy-plane and linear along the z-direction. This is a general method which can be used to connect any two open curves with well defined mathematical definitions as well as pairs of discrete systems found experimentally. We used this method to describe the configurations of torsionally relaxed, 360-base pair DNA rings with two evenly-spaced, ideal nucleosomes. We considered superhelical nucleosomal pathways with different levels of DNA wrapping and allowed for different inter-nucleosome orientations. We completed the DNA circles with the smooth connectors and studied the associated bending and electrostatic energies for different configurations in the absence and presence of salt. The predicted stable states bear close resemblance to reconstituted minicircles observed under low and high salt conditions.

1. Introduction

The 2 m of DNA found in almost every human cell must be folded to fit in the nucleus of the cell (~6 μm in diameter [1]). The first step in this compaction involves the coiling of ~150 base pairs (bp) of negatively charged DNA around a core of eight positively charged histone proteins—two each of histones H2A, H2B, H3, and H4—to form a nucleosome, the basic building block of chromatin [2]. Nucleosome positioning determines the structure of chromatin and the expression which regulates the functional behavior of a cell. Knowing how the superhelical stretches of DNA in nucleosomes are connected to one another is very important in understanding the folded structure of chromatin. Assuming the path of DNA in each nucleosome as a three-dimensional (3D) open curve, the problem becomes how to connect the end of one curve smoothly to the end of the next. In this paper we introduce an analytical method to connect any two nearby well defined 3D open curves smoothly. This is a general method which can be applied to connect curves of interest with well defined mathematical definitions as well as discrete systems found experimentally, such as the sets of atomic coordinates determined by x-ray crystallography or cryo-electron microscopy. Without loss of generality we can assume that the smooth connector has a constant slope along an arbitrary direction, taken here to be the cylindrical axis (z-axis) of one of the nucleosomes. We formulate a 3D smooth pathway by decomposing the trajectory of the connector along the z-axis and onto a two-dimensional (2D) xy-plane. We find that the simplest possible smooth connector that can satisfy the boundary conditions of the end points of two open curves has a quartic polynomial trajectory on the 2D plane. Here we consider smooth pathways of B DNA in which the centers of base pairs align with the DNA symmetry axis and fall along the trajectory of the curve (see figure 1). The combined set of nucleosomal DNA pathways and connector curves thus describe a relaxed, twist-free DNA system.

Figure 1.

Schematic of two open curves representing the left-handed circular helical pathways of two nucleosomes on a DNA minicircle. The thin curved line depicts the DNA pathway that smoothly connects the exit of the right helix to the entry of the left helix. The upper image highlights the distance d along the global x-axis separating the centers of the two nucleosomes and the rotation α of the right nucleosome about the x-axis. The lower image shows the pathway of the smooth connector described by a quartic equation on the xy-plane.

We applied this approach to small nucleosome-decorated DNA minicircles similar to chromatin fragments previously characterized by electron microscopy under low and hight salt conditions [3]. DNA rings of this length, known as extrachromosomal microDNAs [4], occur outside the normal chromosome architecture of various life forms [5, 6], with the sequences from some mammalian cells bearing nucleosomal signatures, e.g., AA, AT, and TT dinucleotides repeated at ~10-bp intervals along the length of many circles [4, 7]. Moreover, DNA rings long enough to carry a single gene appear to play important roles in the rapid adaptation of cancer cells to changes in tumor environment [8, 9]. We compared the shapes of the predicted low-energy structures of the nucleosome-decorated DNA to the shapes determined experimentally. We considered ideal, cylindrically shaped nucleosomes with different levels of DNA wrapping and allowed for different orientations of the nucleosomes with respect to one another. We ranked the configurations in terms of their elastic and electrostatic energies. We considered the electrostatic contribution in the presence and absence of salt, and given that the generated DNA configurations are torsionally relaxed and the chain is assumed to be inextensible, we only determined the bending contribution to the elastic energy. The predicted low-energy configurations of DNA resemble the experimental observations. Our findings suggest that the uptake of salt may increase the wrapping of DNA on the nucleosome.

2. Theory

Suppose we have two nearby open curves ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$, and we want to connect one end of ${\mathcal{C}}_1$ located at ${\mathcal{r}}_1$ with slope ${\mathcal{m}}_1$ smoothly to an end of ${\mathcal{C}}_2$ located at ${\mathcal{r}}_2$ with slope ${\mathcal{m}}_2$. For a connector ℓ to be smooth it should start from curve ${\mathcal{C}}_1$ at ${\mathcal{r}}_1$ with ${\mathcal{m}}_1$ and end at curve ${\mathcal{C}}_2$ at ${\mathcal{r}}_2$ with ${\mathcal{m}}_2$. Without loss of generality we can assume that ℓ has a constant slope along an arbitrary direction, taken here to be the z-axis. This assumption reduces the problem from finding ℓ in 3D to finding the projection of ℓ onto a 2D plane, i.e., ℓ_xy on the xy-plane, and also makes it possible to connect the arc length of ℓ_xy to the arc length of ℓ in 3D by using the Pythagorean theorem.

We are going to develop an expression for the total arc length, or contour length, of the connector and for the coordinates of points at specified fractional contour lengths. Given that the length of a curve is invariant to rotation and translation, the result will be valid in any coordinate system. So the problem is to find an equation of a curve y(x) in 2D which starts from an initial position, r_i = (x_i, y_i) with initial slope $r_{\text{i}}^{\prime }=r^{\prime }\left(x_{\text{i}},y_{\text{i}}\right)=y^{\prime }\left(x_{\text{i}}\right)$, and ends at a final position, r_f = (x_f, y_f) with final slope $r_{\text{f}}^{\prime }=r^{\prime }\left(x_{\text{f}},y_{\text{f}}\right)=y^{\prime }\left(x_{\text{f}}\right)$. There is also a length constraint, which is the minimum value that must be satisfied by the equation of the connector. Depending on the problem, the contour can be either continuous or discrete. For instance, if the connector is used to describe a DNA pathway, we need to incorporate its discrete length S in our treatment. The value of S is determined by the number of base pairs N and the distance s = 3.4 Å between successive base pairs, i.e., S = Ns. Therefore the function y(x) should satisfy five conditions: two boundary conditions for the initial and final positions, two for the initial and final slopes, and a constraint on the length. The equation should also be transferable from one coordinate system to another. Therefore the function y(x) must be well defined and differentiable within (−∞, +∞). Given these constraints, the simplest representation of a smooth connector is a polynomial of order four with five coefficients:

$$y(x)=a_4{x}^4+a_3{x}^3+a_2{x}^2+a_{1}x+a_0{x}_{\text{i}}⩽x⩽x_{\text{f}}.$$ (1)

2.1. Connector contour length

The constraints on the ends and length of the connector uniquely define the five coefficients a₄, a₃, a₂, a₁, and a₀ in equation (1), which in turn uniquely define the smooth connecting function y(x). The four boundary conditions—i.e., y_i = y(x_i), y_f = y(x_f), $y_{\text{i}}^{\prime }=y^{\prime }\left(x_{\text{i}}\right)$, and $y_{\text{f}}^{\prime }=y^{\prime }\left(x_{\text{f}}\right)$—lead to a set of four equations with five unknowns:

$$\{\begin{array}{l}{a}_4{x}_{\text{i}}^4+a_3{x}_{\text{i}}^3+a_2{x}_{\text{i}}^2+a_1{x}_{\text{i}}+a_0=y_{\text{i}}\hfill \\ a_4{x}_{\text{f}}^4+a_3{x}_{\text{f}}^3+a_2{x}_{\text{f}}^2+a_1{x}_{\text{f}}+a_0=y_{\text{f}}\hfill \\ 4a_4{x}_{\text{i}}^3+3a_3{x}_{\text{i}}^2+2a_2{x}_{\text{i}}+a_1=y_{\text{i}}^{\prime }\hfill \\ 4a_4{x}_{\text{f}}^3+3a_3{x}_{\text{f}}^2+2a_2{x}_{\text{f}}+a_1=y_{\text{f}}^{\prime }.\hfill \end{array}$$ (2)

By treating a₄, a₃, a₂, a₁, and a₀ as five unknowns and the different values of x_i, x_f, y_i, y_f, $y_{\text{i}}^{\prime }$, and $y_f^{\prime }$ as known coefficients, we can use Gauss–Jordan elimination to find four of the unknown coefficients in terms of the fifth one, e.g., a₃, a₂, a₁, and a₀ as linear functions of a₄,

$$\left[\begin{array}{cccccc}{x}_{\text{i}}^4& x_{\text{i}}^3& x_{\text{i}}^2& x_{\text{i}}& 1& y_{\text{i}}\\ x_{\text{f}}^4& x_{\text{f}}^3& x_{\text{f}}^2& x_{\text{f}}& 1& y_{\text{f}}\\ 4x_{\text{i}}^3& 3x_{\text{i}}^2& 2x_{\text{i}}& 1& 0& y_{\text{i}}^{\prime }\\ 4x_{\text{f}}^3& 3x_{\text{f}}^2& 2x_{\text{f}}& 1& 0& y_{\text{f}}^{\prime }\end{array}\right]\Rightarrow \{\begin{array}{l}{a}_3=c_{3a}{a}_4+c_{30}\hfill \\ a_2=c_{2a}{a}_4+c_{20}\hfill \\ a_1=c_{1a}{a}_4+c_{10}\hfill \\ a_0=c_{0a}{a}_4+c_{00},\hfill \end{array}$$ (3)

where c_3a, c₃₀, c_2a, c₂₀, c_1a, c₁₀, c_0a, and c₀₀ are expressed as follows:

$$\begin{array}{l}{c}_{3a}=-2\left(x_{\text{f}}+x_{\text{i}}\right),\\ c_{30}=-\frac{2\left(y_{\text{f}}-y_{\text{i}}\right)+\left(x_{\text{i}}-x_{\text{f}}\right)\left(y_{\text{f}}^{\prime }+y_{\text{i}}^{\prime }\right)}{{\left(x_{\text{f}}-x_{\text{i}}\right)}^3},\\ c_{2a}=-\left(x_{\text{i}}^2+4x_{\text{i}}{x}_{\text{f}}+x_{\text{f}}^2\right),\\ c_{20}=-\frac{1}{{\left(x_{\text{f}}-x_{\text{i}}\right)}^3}(-3x_{\text{f}}{y}_{\text{f}}-3x_{\text{i}}{y}_{\text{f}}+3x_{\text{f}}{y}_{\text{i}}+3x_{\text{i}}{y}_{\text{i}}+x_{\text{f}}^2{y}_{\text{f}}^{\prime }+x_{\text{f}}{x}_{\text{i}}{y}_{\text{f}}^{\prime }\\ -2x_{\text{i}}^2{y}_{\text{f}}^{\prime }+2x_{\text{f}}^2{y}_{\text{i}}^{\prime }-x_{\text{f}}{x}_{\text{i}}{y}_{\text{i}}^{\prime }-x_{\text{i}}^2{y}_{\text{i}}^{\prime }),\\ c_{1a}=-2\left(x_{\text{i}}^2{x}_{\text{f}}+x_{\text{i}}{x}_{\text{f}}^2\right),\\ c_{10}=-\frac{1}{{\left(x_{\text{f}}-x_{\text{i}}\right)}^3}(6x_{\text{f}}{x}_{\text{i}}{y}_{\text{f}}-6x_{\text{f}}{x}_{\text{i}}{y}_{\text{i}}-2x_{\text{f}}^2{x}_{\text{i}}{y}_{\text{f}}^{\prime }+x_{\text{f}}{x}_{\text{i}}^2{y}_{\text{f}}^{\prime }+x_{\text{i}}^3{y}_{\text{f}}^{\prime }\\ -x_{\text{f}}^3{y}_{\text{i}}^{\prime }-x_{\text{f}}^2{x}_{\text{i}}{y}_{\text{i}}^{\prime }+2x_{\text{f}}{x}_{\text{i}}^2{y}_{\text{i}}^{\prime }),\\ c_{0a}=x_{\text{i}}^2{x}_{\text{f}}^2,\\ c_{00}=-\frac{1}{{\left(x_{\text{i}}-x_{\text{f}}\right)}^3}(3x_{\text{f}}{x}_{\text{i}}^2{y}_{\text{f}}-x_{\text{i}}^3{y}_{\text{f}}+x_{\text{f}}^3{y}_{\text{i}}-3x_{\text{f}}^2{x}_{\text{i}}{y}_{\text{i}}-x_{\text{f}}^2{x}_{\text{i}}^2{y}_{\text{f}}^{\prime }\\ +x_{\text{f}}{x}_{\text{i}}^3{y}_{\text{f}}^{\prime }-x_{\text{f}}^3{x}_{\text{i}}{y}_{\text{i}}^{\prime }+x_{\text{f}}^2{x}_{\text{i}}^2{y}_{\text{i}}^{\prime }).\end{array}$$ (4)

Upon substitution of the coefficients a₃, a₂, a₁, and a₀ in the expression for the connector in 2D, y becomes a function of both x and the parameter a₄, i.e., y = y(x, a₄),

$$y\left(x,a_4\right)=a_4{x}^4+\left(c_{3a}{a}_4+c_{30}\right)x^3+\left(c_{2a}{a}_4+c_{20}\right)x^2+\left(a_{1a}{a}_4+a_{10}\right)x+a_{0a}{a}_4+a_{00}.$$ (5)

The length of the curve that satisfies the four boundary conditions in 2D is also a function of the parameter a₄:

$$\begin{gathered} S^{2D}\left(a_4\right)={\int }_{x_{\text{i}}}^{x_{\text{f}}}\sqrt{1+{\left[y^{\prime }\left(x,a_4\right)\right]}^2}\text{d}x \\ ={\int }_{x_{\text{i}}}^{x_{\text{f}}}\sqrt{1+{\left[4a_4{x}^3+3\left(c_{3a}{a}_4+c_{30}\right)x^2+2\left(c_{2a}{a}_4+c_{20}\right)x+\left(c_{1a}{a}_4+c_{10}\right)\right]}^2}\text{d}x. \end{gathered}$$ (6)

The minimum length of the trajectory in the xy-plane, is found by first determining the value of a₄ for which the derivative of S^2D vs a₄ vanishes,

$$\frac{\partial S^{2D}}{\partial a_4}={\int }_{x_{\text{i}}}^{x_{\text{f}}}\frac{y^{\prime }\left(\frac{\partial y^{\prime }}{\partial a_4}\right)}{\sqrt{1+y^{\prime 2}}}\text{d}x=0.$$ (7)

In general, equation (7) may have two numerical solutions, positive and negative: $a_4^{\text{opt}+}$ and $a_4^{\text{opt}-}$, corresponding to two possible smooth connectors. Depending on the boundary conditions the two solutions may converge to one. A system with N connections will in general have 2^N possible combinations of connections.

The value(s) of a₄ found upon solution of equation (7), $a_4^{\text{opt}}$, can then be used to obtain the minimum theoretical value(s) of the length along the connector(s) in 2D, S^2D, as:

$$S_{min}^{2D\text{-theory}}=S^{2D}\left(a_4^{\text{opt}}\right)={\int }_{x_{\text{i}}}^{x_{\text{f}}}\sqrt{1+{\left[y^{\prime }\left(x,a_4^{\text{opt}}\right)\right]}^2}\text{d}x\text{.}$$ (8)

Since we assumed that the slope of the connector in the z-direction is constant, the value of $S_{\min }^{2D\text{-theory}}$ and the minimum theoretical length of the connector in 3D, $S_{\min }^{3D\text{-theory}}$, are related by the Pythagorean theorem:

$$S_{\min }^{3D\text{-theory}}=\sqrt{{(S_{\min }^{2D\text{-theory}})}^2+{\left(z_{\text{f}}-z_{\text{i}}\right)}^2.}$$ (9)

2.2. Continuous connector

The value of $S_{\min }^{3D\text{-theory}}$ is the minimum length required for a continuous connector to satisfy the initial and final boundary conditions. If the problem requires a connector with a length greater than the minimum length, i.e., $S^{3D\text{-continuous}}>S_{\min }^{3D\text{-theory}}$, one should first find the projection of the larger curve in 2D, S^{2D-continuous}, by the Pythagorean theorem:

$$S^{2D\text{-continuous}}=\sqrt{{\left(S^{3D\text{-continuous}}\right)}^2-{\left(z_{\text{f}}-z_{\text{i}}\right)}^2.}$$ (10)

The parameter $a_4^{*}$ associated with this length can be obtained by solving equation (6):

$$S^{2D}\left(a_4^{*}\right)={\int }_{x_{\text{i}}}^{x_{\text{f}}}\sqrt{1+{\left[y^{\prime }\left(x,a_4^{*}\right)\right]}^2}\text{d}x.$$ (11)

After finding the value of $a_4^{*}$ associated with the projected curve of greater length, we can use equation (3) to find the values of a₃, a₂, a₁, and a₀. These parameters fully specify the equation of the longer connector in 2D, i.e., equation (1). The same procedure can be used to determine the equation of the connector of minimum length.

As noted above, the length along the connector ℓ in 3D is related to that of its projection ℓ_xy on the xy-plane. Thus for every x ∈ (x_i, x_f) there is a corresponding contour length along ℓ in 3D:

$$\ell (x)=\sec (\gamma ){\int }_{x_{\text{i}}}^x\sqrt{1+{\left[y^{\prime }(x)\right]}^2}\text{d}x,$$ (12)

where γ is a constant angle determined by the slope of ℓ in the z-direction:

$$\gamma ={\cos }^{-1}\left(\frac{S^{2D\text{-continuous}}}{S^{3D\text{-continuous}}}\right).$$ (13)

The equation of the connector ℓ in 3D can then be expressed as:

$$\{\begin{array}{l}x=x{x}_{\text{i}}⩽x⩽x_{\text{f}}\hfill \\ y=a_4{x}^4+a_3{x}^3+a_2{x}^2+a_{1}x+a_0\hfill \\ z=z_{\text{i}}+\frac{\ell (x)}{S^{3D\text{-continuous}}}\left(z_{\text{f}}-z_{\text{i}}\right).\hfill \end{array}$$ (14)

2.3. Discrete connector

The connector for a discrete system such as DNA is not necessarily equal to $S_{\min }^{3D\text{-theory}}$. The value of $S_{\min }^{3D\text{-theory}}$ can be any real number, not necessarily the length of the discrete system. The length of a discrete connector is in general described as:

$$S=n\delta ,$$ (15)

where n is the number of repeating units which comprise the connector and δ is the length of each repeating unit. In order to treat the discrete system, we need to find the minimum number n_min of repeating units in the system that is compatible with the minimum theoretical contour length. In order to find n_min, we divide $S_{\min }^{3D-\text{theory}}$ by the value of δ. Since the quotient is not necessarily an integer, we round the quotient up to the nearest integral value. That is to say we find the ceiling of the quotient i.e.,

$$n_{\min }=\lceil \frac{S_{\min }^{3D\text{-theory}}}{\delta }\rceil .$$ (16)

We then use n_min to find the actual minimum discrete length which satisfies the boundary conditions:

$$S_{\min }^{3D\text{-discrete}}=n_{\min }\delta .$$ (17)

We then use the Pythagorean theorem to find $S_{\min }^{3D\text{-discrete}}$:

$$S_{\min }^{2D\text{-discrete}}=\sqrt{{\left(n_{\min }\delta \right)}^2-{\left(z_{\text{f}}-z_{\text{i}}\right)}^2.}$$ (18)

The value of the parameter a₄ associated with the minimum discrete length can be obtained by solving equation (6):

$$S_{\min }^{2D\text{-discrete}}\left(a_4\right)={\int }_{x_{\text{i}}}^{x_{\text{f}}}\sqrt{1+{\left[y^{\prime }\left(x,a_4\right)\right]}^2}\text{d}x.$$ (19)

If the problem requires a discrete connector with a length greater than the minimum length, i.e., n > n_min, one should first find S^2D-discrete by the Pythagorean theorem:

$$S^{2D\text{-discrete}}=\sqrt{{(n\delta )}^2-{\left(z_{\text{f}}-z_{\text{i}}\right)}^2.}$$ (20)

The parameter $a_4^{*}$ can then be found by solving equation (6):

$$S^{2D\text{-discrete}}\left(a_4^{*}\right)={\int }_{x_{\text{i}}}^{x_{\text{f}}}\sqrt{1+{\left[y^{\prime }\left(x,a_4^{*}\right)\right]}^2}\text{d}x\text{.}$$ (21)

After finding the value of $a_4^{*}$ associated with the projected curve of greater length, we can use equation (3) to find the values of a₃, a₂, a₁, and a₀. These parameters fully specify the equation of the longer connector in 2D, i.e., equation (1). The same procedure can be used to determine the equation of the connector of minimum length.

As noted above, the length along the connector ℓ in 3D is related to that of its projection ℓ_xy on the xy-plane. Thus, for every discrete contour length along ℓ in 3D, jδ for j ∈ [0, n], there is a corresponding x _j ∈ [x₀, x_n] which can be found by solving the following integral for x _j:

$$\ell \left(x_j\right)=j\delta =\sec (\gamma ){\int }_{x_0}^{x_j}\sqrt{1+{\left[y^{\prime }(x)\right]}^2}\text{d}x,$$ (22)

where γ is a constant angle determined by the slope of ℓ in the z-direction:

$$\gamma ={\cos }^{-1}\left(\frac{S^{2D\text{-discrete}}}{S^{3D\text{-discrete}}}\right).$$ (23)

The equation of the connector ℓ in 3D can then be expressed as:

$$\{\begin{array}{l}{x}_j=x_{j}0⩽j⩽n\hfill \\ y_j=a_4{x}_j^4+a_3{x}_j^3+a_2{x}_j^2+a_1{x}_j+a_0\hfill \\ z_j=z_i+\frac{j}{n}\left(z_{\text{f}}-z_{\text{i}}\right).\hfill \end{array}$$ (24)

2.4. DNA model

Nucleosome-bound DNAs.

We model DNA rings containing two cylindrically shaped ideal nucleosomes. As illustrated in figure 1, the nucleosome-bound portions of DNA are represented by left-handed circular superhelices and the intervening, protein-free linker DNA by smooth connectors (equation (24)). The center of one nucleosome is taken to lie at the origin with its cylindrical axis along the global z-axis. The nucleosome is oriented such that its dyad axis runs parallel to the global x-axis. The coordinates of the DNA on the reference nucleosome are thus:

$$\{\begin{array}{l}x=r\cos \theta \hfill \\ y=r\sin \theta \hfill \\ z=-p\theta ,\hfill \end{array}$$ (25)

where r is the radius of the superhelix, p is the pitch, and θ is the cylindrical rotation of DNA about the superhelical axis. The value of θ ranges from an initial angle of θ_i to a final angle of θ_f, which respectively correspond to initial and final coordinates of the superhelix, i.e., the locations of the centers of the first and last nucleosome-bound base pairs.

The second nucleosome is separated from the center of the first one by a distance d along the global x-axis and is rotated by an angle α about the same axis (see figure 1). The coordinates of the second nucleosome-bound DNA are:

$$\{\begin{array}{l}x=r\cos \theta +d\hfill \\ y=r\sin \theta \cos \alpha -p\theta \sin \alpha \hfill \\ z=-r\sin \theta \sin \alpha -p\theta \cos \alpha ,\hfill \end{array}$$ (26)

where θ is the cylindrical rotation of the second nucleosome about its superhelical axis. The value of θ ranges from initial and final angles corresponding to initial and final coordinates of the second superhelix. The choice of initial and final values determines the orientation of the second nucleosome with respect to the first.

The equation of a smooth connector, i.e., equation (24), provides a representation of a flexible, protein-free pathway of the linker DNA that connects the terminus of one nucleosome to the start of another nucleosome. Two such connectors are required to form the desired nucleosome-DNA assembly. Here we assume connectors of the same length leading to evenly spaced nucleosomes. Under certain conditions the two connectors may self-intersect. Since it is not physically possible that the two connectors pass into one another, we set the coordinates and the tangents of both connectors at the crossing point as the boundary conditions of new midway points along each connector. We then use the initial boundary conditions and the midway conditions to split each initial connector into two parts. Knowing the equations of the nucleosomal and protein-free DNA we can find the coordinates of the points representing the centers of the base pairs along the DNA assuming that the double helix adopts the B form.

Bending Energy.

We consider the deformations of DNA resulting from its interaction with proteins, i.e. the wrapping of the double-helical structure along a superhelical pathway, and the molecular distortion required to connect successive nucleosomes. We measure the deformation of DNA compared to a naturally straight, inextensible, linearly elastic, isotropic rod with circular cross section. The energy associated with deformation of such a rod is expressed as a sum of bending and twisting contributions. The minicircle is assumed here to have at least one single-stranded scission and is thus torsionally relaxed. The energy associated with deformation is then simply the bending energy.

From the equations of the smooth connector and the circular superhelix, we can determine the components of the tangent to the curve at any point. Using the components of the tangents we can calculate the angle of bending Δη_i at each base-pair step, i.e., $\mathrm{\Delta }{\eta }_i={\cos }^{-1}\left(\frac{t^i\cdot t^{i+1}}{|t^i||t^{i+1}|}\right)$ where tⁱ is the tangent vector, |tⁱ| is its magnitude, i = 1 − N, and N is the total number of base pairs. The total bending energy Ψ associated with the configuration of the DNA is then given by:

$$\mathrm{\Psi }=\frac{1}{2}{k}_{\text{B}}T\sum _i\left(\frac{\mathrm{\Delta }{\eta }_i^2}{\langle \mathrm{\Delta }{\eta }^2\rangle }\right),$$ (27)

where the bending stiffness of individual base-pair steps is consistent with the known ~50 nm persistence length of DNA [10]. The energy is raised by $\frac{1}{2}{k}_{\text{B}}T$ when the direction of a step deviates from its equilibrium rest state by its root-mean-square fluctuation ⟨Δη²⟩^1/2, here taken to be 4.82°.

Electrostatic Energy.

We also estimate the electrostatic energy by considering the interactions of the negatively charged phosphate groups on the DNA backbone. The two charges associated with each base pair are merged into a single charge of twice the magnitude of the phosphate and placed, for simplicity, at the center of the base pair. The electrostatic energy of DNA is taken to be the sum of all pairwise interactions between the charges on different base pairs:

$$\mathrm{\Phi }=\sum _{j=2}^N\sum _{i=1}^{j-1}\frac{\delta q_i\delta q_j{\text{e}}^{-\kappa r^{ij}}}{4\pi {ϵ}_r{ϵ}_0{r}^{ij}},$$ (28)

where δqi and δqj are the respective charges associated with the ith and jth base pairs, r^ij is the distance between the centers of those base pairs, ϵ_r is the relative permittivity of water at 300 K (~80), and $\kappa =0.329\sqrt{C_{\text{m}}}$ is the Debye screening parameter for monovalent salt, such as NaCl, of molar concentration C_m. For protein-free DNA, we assume 76% charge neutralization associated with the screening of DNA charges by counterions [11], corresponding to 2 × 0.24e⁻ = 7.70 × 10⁻²⁰ C per base pair. For protein-bound DNA, we assume 86% charge neutralization associated with the combination of counterions and the net charge of the nucleosome, corresponding to 2 × 0.14e⁻ = 4.49 × 10⁻²⁰ C per base pair as deduced from molecular dynamic (MD) simulation of nucleosomes [12].

Writhing Number.

The writhing number Wr(C) is a topological property of a curve C which is invariant under translation and rotation and measures the chiral distortion of the curve. A 3D curve has different projections when viewed from different angles and its Wr is the average of the number of positive and negative self-crossings over all projections [13]. The writhing number of the nucleosome-DNA assembly is computed here using the formulation of Swigon et al [14] for a closed discrete curve:

$$Wr=\frac{1}{2\pi }\sum _{(i,j),j>i}{w}^{ij},$$ (29)

where:

$$w^{ij}={\xi }^{i,j}+{\xi }^{i+1,j+1}-{\xi }^{i,j+1}-{\xi }^{i+1,j},$$ (30)

and ξ^{i, j} is the solution of the following pair of equations:

$$\{\begin{array}{l}\cos {\xi }^{i,j}=\frac{r^{ij}\times t^j}{\left|r^{ij}\times t^j\right|}\cdot \frac{t^i\times r^{ij}}{\left|t^i\times r^{ij}\right|}\hfill \\ \sin {\xi }^{i,j}=-\frac{r^{ij}}{\left|r^{ij}\right|}\cdot \left(\frac{r^{ij}\times t^j}{\left|r^{ij}\times t^j\right|}\times \frac{t^i\times r^{ij}}{\left|t^i\times r^{ij}\right|}\right).\hfill \end{array}$$ (31)

2.5. Results

We generated analytical and computational representations of a torsionally relaxed, 360-bp DNA ring with two evenly-spaced ideal nucleosomes. We considered nucleosomes with different levels of DNA wrapping, 1.5 and 1.75 superhelical turns, and allowed for different inter-nucleosome orientation angles α over the range 0° to 90°. We completed the DNA circle with two (or four) linkers defined by the smooth connectors in equation (24). We then calculated the bending energy, the electrostatic energy, and the writhing number of the DNA for each configuration. In our initial calculations the molar concentration C_m of monovalent counterions was taken to be 0.1 M.

Table 1 reports the values of the coefficients (a_μk, μ = 0–4, k = 1–2) describing the two smooth connectors for minicircles containing two nucleosomes, each with 1.5 turns of DNA, and oriented at different values of α. As α changes the boundary conditions of the two protein-bound portions of the DNA minicircle change. Therefore for every value of the angle α we get unique values for (a_μk, μ = 0–4, k = 1–2). The reported values of d in the table are the shortest distances between nucleosome centers that satisfy the boundary conditions.

Table 1.

Minimum separation distances and coefficients^a of smooth connectors between two evenly spaced nucleosomes, each with 1.5 turns of DNA, on a torsionally relaxed, 360-bp minicircle as a function of the orientation angle α.

α(°)	d (Å)	a41 (Å −3)	a31 (Å −2)	a21 (Å −1)	a 11	a01 (Å)	a42 (Å −3)	a32 (Å −2)	a22 (Å −1)	a 12	a02 (Å)
0	200.94	1.61 × 10−7	6.47 × 10−5	6.47 × 10−3	−4.42 × 10−3	43.29	−1.61 × 10−7	−6.47 × 10−5	−6.47 × 10−3	4.42 × 10−3	−43.29
15	202.64	1.54 × 10−7	6.15 × 10−5	6.05 × 10−3	−5.85 × 10−3	46.24	−1.54 × 10−7	−6.15 × 10−5	−6.05 × 10−3	5.85 × 10−3	−46.24
30	203.66	1.50 × 10−7	6.01 × 10−5	5.93 × 10−3	−7.37 × 10−3	46.04	−1.50 × 10−7	−6.01 × 10−5	−5.93 × 10−3	7.37 × 10−3	−46.04
45	203.66	1.56 × 10−7	6.33 × 10−5	6.41 × 10−3	−9.08 × 10−3	42.70	−1.56 × 10−7	−6.33 × 10−5	−6.41 × 10−3	9.08 × 10−3	−42.70
60	203.66	1.44 × 10−7	6.02 × 10−5	6.38 × 10−3	−1.01 × 10−2	36.46	−1.44 × 10−7	−6.02 × 10−5	−6.38 × 10−3	1.01 × 10−2	−36.46
75	202.64	1.42 × 10−7	6.10 × 10−5	6.85 × 10−3	−1.11 × 10−2	27.72	−1.42 × 10−7	−6.10 × 10−5	−6.85 × 10−3	1.11 × 10−2	−27.72
90	200.94	1.40 × 10−7	6.23 × 10−5	7.46 × 10−3	−1.17 × 10−2	17.10	−1.40 × 10−7	−6.23 × 10−5	−7.46 × 10−3	1.17 × 10−2	−17.10

^aThe first subscript of each coefficient is its order (equation (1)) and the second one refers to the order of connectors. In the second subscript, values 1 and 2 refer respectively to the connectors joining the terminus of the second nucleosome to the start of the first nucleosome and the terminus of the first nucleosome to the start of the second.

Figure 2 presents the bending energy Ψ and electrostatic energy Φ of a series of nucleosome-decorated DNA minicircles as a function of the angle α. The lowest bending energy occurs when α = 60° and the lowest electrostatic energy when α = 75°. Variation in the bending energy is greater than that in the electrostatic energy for the assumed choice of parameters (ϵ_r = 80, κ = 0.104, 74% charge neutralization on DNA linkers, and 86% neutralization on nucleosomal DNA) and thus determines the location of the lowest value of the total energy. It is possible that for different values of ϵ_r, κ, and charge neutralization the electrostatic energy can overwhelm the bending energy and shift the angle of lowest energy to a higher value, e.g., less neutralization, smaller ϵ_r, and/or smaller κ. In this figure we present the two energies in different scales to show that they attain their lowest values at different values of α. On the top of every column is an image showing a top-down view of the upper (first, i.e., equation (25)) nucleosome, with its cylindrical axis coincident with the global z-axis of the system. As evident from the images, in the state of lowest bending energy the two connectors, on average, follow a straighter pathway and thus a smaller bending energy is stored in the configuration. Since the superhelical structure of the nucleosomal DNA does not change when the angle α changes, the bending energy associated with both nucleosomal DNAs is constant and therefore makes no contribution to the change in total bending energy. The pathways of both connectors, however, change with α. As expected, the average bending energy per base pair does not change for the nucleosomal DNA with change in α but the values of the bending energy for the connectors do change (see table S-I) (https://stacks.iop.org/JPA/53/435601/mmedia). The energies of the two connectors are equivalent due to the symmetry of the system.

Figure 2.

Molecular images illustrating the changes in overall folding and graphs of the associated bending Ψ and electrostatic Φ energies of a torsionally relaxed, 360-bp DNA minicircle with two evenly spaced nucleosomes, each wrapping 1.5 turns of DNA, as a function of the angle α between nucleosome axes. The two dark gray cylinders in each molecular image represent the histones proteins wrapped by DNA. See table S-II for numerical values.

Figure 3 depicts the writhing number Wr of the same nucleosome-decorated DNA along with two sets of molecular images. The two sets correspond to different views of the same molecular images shown in figure 2. The lower set in figure 3 depicts side views obtained by rotating the images in figure 2 by −90° about the global x-axis, i.e., the long axis of the system parallel to the downward direction of the page. The resulting view is that along the global +y-axis with the first nucleosome remaining on the top. The upper set of images in figure 3 are front views obtained by rotating the images in figure 2 by −90° about the global y-axis. The viewing direction then lies along the global −x-axis with the first nucleosome on top and in the back.

Figure 3.

The writhing number and alternate views of the nucleosome-DNA assemblies presented in figure 2. See table S-II for numerical values.

As evident from the plotted values, the magnitude of the writhing number increases monotonically with increase in α. The increase in magnitude corresponds to a greater number of self crossings in the chain. The added crossings are not evident in the (lower) side views in the figure nor in those in figure 2. The front views, however, clearly show the increase in self crossings when α increases.

Figure 4 shows the variation in energy and DNA configuration of a torsionally relaxed, 360-bp DNA minicircle with two nucleosomes, each wrapping 1.75 turns of DNA. We again let α change from 0°–90° at 15° increments. The molecular images are shown from the same view points—i.e., top, side, and front—used in figures 2 and 3. As expected, the magnitudes of the writhing numbers are greater than those found in figure 3 for nucleosomes wrapping less DNA. The self crossings that give rise to the increased magnitude of Wr are evident from all three molecular views in figure 4. The changes in the writhing number with increase of α, however, are comparable in magnitude to those found for nucleosomes wrapping less DNA and show a similar monotonic increase in magnitude with α. The bending energy also monotonically increases when α increases and all three sets of molecular images clearly show that the smooth connectors become more bent with increase in α.

Figure 4.

Molecular images illustrating the changes in overall folding and graphs of the associated bending Ψ and electrostatic Φ energies, and the writhing number Wr of torsionally relaxed, 360-bp DNA minicircles with two evenly spaced nucleosomes, each wrapping 1.75 turns of DNA, as a function of the angle α. See table S-III for coefficients of the smooth connectors and table S-IV for numerical values of plotted data.

As above, the bending energy per base pair stored in the nucleosomal DNA is constant and the change in total bending energy arises from the two connectors (see table S-I). The changes in average bending energies of the two connectors are equivalent for values α = 0°–30° due to the symmetry of the system. For α = 45°–90° the two connectors self intersect and their contact forces break the symmetry and change the average bending energies.

The electrostatic energy exhibits a local minimum when α = 75°. Here again since the variation of the bending energy is larger than that of the electrostatic energy for the chosen parameters, the total energy follows the bending energy and monotonically increases with α. It is again possible that for different values of ϵ_r, κ, and/or charge neutralization the electrostatic energy will dominate the bending energy and the configuration will adopt a local minimum at α = 60°.

3. Discussion

The low energy configurations of torsionally relaxed DNA minicircles determined in this work show a close resemblance to the shapes of chromatin constructs reconstituted under low and high salt conditions [3]. Figure 5(a) presents a molecular image of the predicted DNA pathway of a 360-bp minicircle, or so-called dimer, with two evenly spaced nucleosomes, each wrapping 1.5 turns of DNA. The configuration of the DNA in the absence of salt, found when the nucleosomes are oriented at an angle α = 60°, is very similar to the electron microscopic image of the dimer observed under low salt conditions (figure 5(b)). Figure 5(c) shows a molecular image of the predicted DNA pathway of the same dimer with each nucleosome wrapping 1.75 turns of DNA. The configuration, found at 100 mM monovalent salt concentration when the nucleosomes are oriented at an angle α = 0°, is very similar to the electron microscopic image of the dimer observed under similar conditions (figure 5(d)). The correspondence of the models with the observed images suggests that the addition of salt increases the wrapping of DNA around the nucleosomes from ~1.5 to ~1.75 turns. The model, however, does not take account of the torsional stress in the reconstituted minicircles, which may contribute to the observed pathways. The experimental constructs are negatively supercoiled, with a linking number difference of −2, which could possibly give rise to the observed increased crossings of DNA.

Figure 5.

Predicted configurations of torsionally relaxed 360-bp DNA minicircles bearing two evenly spaced nucleosomes compared with images of chromatin rings (dimers) of comparable size (359-bp) reconstituted under low and high salt conditions. (a) Predicted DNA pathway, in the absence of salt, with nucleosomes wrapping 1.5 turns of DNA and oriented at an angle α = 60° with respect to one another; (b) electron microscopic image of the corresponding dimer at low salt; (c) predicted DNA pathway, with each nucleosome wrapping 1.75 turns of DNA and oriented at an angle α = 0° under 100 mM monovalent salt concentration; (d) electron microscopic image of the dimer in the presence of 100 mM NaCl. Images in (a) and (c) viewed along direction angles {0.4, −1.5, 3.0} and {0.2, −0.8, 3.3}, respectively. Figures (b) and (d) reprinted from [3], Copyright 1988, with permission from Elsevier.

The next step in our studies will be to take the twist of DNA into account, and to treat the connector DNA as an elastic rod as opposed to a simple curve. Base pairs can be placed on the curves described herein by making use of the local Frenet–Serret trihedra [15] and determining the angle though which the base pairs rotate at each dinucleotide step [16]. The focus of this paper, however, is on the methodology needed to connect two open curves representing arbitrary stretches of protein-bound DNA and the energy of the relaxed DNA linkers. The incorporation of base pairs will allow us to determine the twists of successive residues and the linking number of the minicircle as a whole. The treatment of individual base pairs will also allow us to consider discrete nucleosome structures, such as those with sharp kinks and dislocations deposited in the Protein Data Bank [17]. The boundary value treatment described here can be immediately used to determine the smooth curve that connects the terminus of one set of coordinates to the start of the next and readily adapted to study the configurational properties of large, multinucleosome assemblies such as the simian virus 40 (SV40) minichromosome [18]. The method depends only on the values of the coordinates and slopes of the initial and final points. The arc length which connects the two objects to a desired length, a scalar, and the slopes, which are vectors, are invariant under rotation and translation of the system.

It is important to note that the scope of our results is limited to smooth DNA connections between two symmetrically spaced nucleosomes and the agreement of our results with experiment may reflect the short length and stiffness of the linking DNA. Here the thermal energy of the DNA is negligible since the lengths of the linkers are short. Our method can be used to describe the effect of thermal energy associated with each base pair. In doing so, one would consider new boundary conditions for each base pair as it moves due to its thermal energy. The smooth connectors can, in general, be split into as many connecting segments as there are neighboring base pairs. We can also use our approach to introduce non-equilibrium states with sharp kinks at selected locations, and use these trial states as the starting points of further optimization or simulation. The case of a DNA minicircle with one nucleosome is beyond the scope of this paper and requires a different treatment which will be described elsewhere.