Applications of a theory of sequence dependent DNA ... - CiteSeerX

Applications of a theory of sequence dependent DNA elasticity that accounts for intramolecular electrostatic forces

Yoav Y. Biton & Bernard D. Coleman

Continuum Models and Discrete Systems 11, in press: (2008). Editors: D. Jeulin & S. Forest Publisher: Les Presses de l'Ecole des Mines de Paris

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Applications of a Theory of Sequence-dependent DNA Elasticity that Accounts for Intramolecular Electrostatic Forces Yoav Y. Biton* and Bernard D. Coleman** Department of Mechanics & Materials Science, Rutgers University, Piscataway, NJ 08854-8058, USA

Abstract A recently developed naturally discrete theory of DNA elasticity [Y.Y. Biton, B.D. Coleman, D. Swigon, J. Elasticity 87,187-210, 2007] takes into account the fact that the mechanical properties of a double helical DNA molecule are dependent on both the nucleotide sequence in the molecule and the concentration c of salt in the medium. The latter dependence arises from the fact that as each nucleotide base in a base pair is attached to the sugar-phosphate backbone chain of one of the two DNA strands that have come together to form the Watson-Crick structure, and each phosphate group in a backbone chain bears one negative electronic charge, two such charges are associated with each base pair, which implies that each base pair is subject to not only the elastic forces and moments exerted on it by its neighboring base pairs but also to long range electrostatic forces that are only partially screened out by positively charged counter ions. It is often the case that the theory implies that a DNA molecule can have several distinct locally stable equilibrium configurations at the same value of c. In the present paper examples are given of cases in which the theory predicts that the radius of gyration of the minimum energy configuration of a small (i.e., 549 base-pair) circularized DNA molecule (called a "DNA minicircle") has a remarkably strong dependence on c.

____________________ *[email protected] **[email protected]

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2 1. Introduction DNA molecules in the Watson-Crick double helical B form can often be treated as though they have rod-like structures obtained by stacking dominoes (called base pairs) one on top of another with each rotated approximately one-tenth of a full turn with respect to its immediate predecessor in the stack. A theory of DNA elasticity developed by Coleman, Olson, and Swigon [1] takes into account the observation that the step from one base pair to the next can be one of several distinct types, each having its own mechanical properties that depend on the nucleotide composition of the step (see, e.g., [2,3,4]). Applications of that theory are presented in reference [5]. A base pair is formed by joining together two nearly planar complementary nucleotide bases each of which is attached to one of two sugar-phosphate chains. At the present time it appears reasonable to assume that the elastic energy Ψ of a DNA configuration is the sum over n of the energy ψ n of interaction of the n-th and (n+1)-th base pairs and that ψ n is given by a function of six numbers, called the kinematical variables (for the n-th base pair step) that describe the orientation and displacement of the (n+1)-th base pair in the stack relative to the n-th. As each of the two bases in a base pair is covalently bound to the sugar-phosphate backbone chain of one of the two DNA strands that form the Watson-Crick structure, and each phosphate group bears one (negative) electronic charge, two such charges are associated with one base pair. The charges associated with two distinct base pairs exert on each other an electrostatic force of repulsion, the strength of which depends on the distance between them and the concentration c of salt in the aqueous solution of DNA. The dependence on c results from the fact that salt ions of positive charge form clouds around the negatively charged sites on the DNA and in so doing partially screen out the electrostatic interaction of each site with others. As a consequence, an increase in c decreases the repulsion of nonadjacent base pairs and weakens the tendency of electrostatic forces to straighten DNA molecules. Despite this partial screening, even under physiological conditions a DNA molecule is subject to intramolecular electrostatic forces that under appropriate circumstances (e.g., when the molecule has intrinsic curvature) cause equilibrium configurations to be sensitive to the concentration of salt in the medium. Thus DNA is not what, in the terminology of modern continuum mechanics, is called "a simple material", "a higher gradient material", or even a material with mechanical behavior that one can assume to be well approximated by the behavior of such materials. A theory [6], here referred to as the "BCS theory", of the equilibrium configurations of electrically charged and intrinsically curved rod-like structures was

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3 recently developed and proposed as a model of DNA molecules in aqueous solution. The theory goes beyond that of reference [1], which we may now call the "COS theory", in that the BCS theory accounts for and renders explicit the electrostatic interactions between nonadjacent base pairs. In both theories the variational equations of mechanical equilibrium for a DNA molecule with N+1 base pairs form a nonlinear system S of µN equations for the µN unknown kinematical variables characterizing a configuration, where µ , the number of such variables at each base-pair step, is in general 6, but reduces to 3 when two neighboring base pairs are (or are assumed to be) much stiffer for changes in their relative displacement than for changes in their relative orientation. When, as in [1] and [5], the electrostatic forces are not rendered explicit and the COS theory is employed, S is a system of weakly coupled equations. In marked contrast, in the new (i.e., BCS) theory, because the force on a base pair depends on the position in space of all the other base pairs in the DNA molecule, the µN × µN Jacobian matrix for S is full. In the new theory, the system S depends on the salt concentration c, and the influence of c on the geometry and stability of equilibrium configurations is a matter of fundamental importance. In the remainder of this paper, unless we state otherwise, the theory under discussion will be the BCS theory.

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4 2. Basic assumptions In the BCS theory, as in the COS theory, base pairs are represented by rectangular objects, and the energy of a duplex DNA molecule with N base–pair steps is determined when there is given, for each n, both the location x n of the center of the rectangle B n n n n that represents the n-th base pair and a right-handed orthonormal triad d1 , d 2 , d 3 that is embedded in the base pair as shown in Figure 1. The piecewise linear curve C composed of the line segments that connect the spatial points x 1 , x 2 ,..., x n , x n+1 ,... is called the axial curve of the molecule. The elastic energy Ψ of a configuration is taken to be the sum over n of the energy ψ n of interaction of the n-th and (n+1)-th base pairs, i.e., N

Ψ = ∑ψ , n

(2.1)

n =1

where ψ n , the elastic energy of the n-th base-pair step, is given by a function of the relative orientation and displacement of the (n+1)-th base pair with respect to the n-th, i.e., by a function of the components (with respect to the basis d1n , d 2n , d 3n ) of the vectors d n+1 and r n = x n+1 − x n . The components Dijn = d in ⋅ d n+1 of the vectors d n+1 form a j j j n n n 3 × 3 orthogonal matrix that is determined by three angles, θ1 , θ2 ,θ 3 , called the tilt, roll, and twist (see Figure 2). The displacement variables, ρ1n , ρ n2 , ρ n3 , called shift, slide, and rise, are related to the components ri n = r n ⋅ d in of r n by a coordinate transformation of 3'

d 2n +1

d 1n +1 d 3n+1

d 3n d 2n

5'

5'

rn

d 1n 3'

Figure 1. Schematic drawing of the n-th base-pair step showing the vectors r n , d nj , and d n+1 j . Each nucleotide base in the n-th base pair lies mainly on

one side of the plane containing d 1n and d 3n and at its darkened corner is covalently bonded to one of the two sugar phosphate chains. The direction of the orientation of each chain is indicated by a light-face arrow. The long edges shaded gray are in the minor groove of the DNA.

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5 the type ⌣

ρin = ρ i (θ1n ,θ 2n , θ3n , r1n , r2n , r3n ),

(2.2)

which can be written in the form 3

ρin = ∑ Rˆ ji (θ1n ,θ 2n , θ3n )r jn .

(2.3)

j =1

θ1 tilt

θ2 roll

θ3 twist

ρ1 shift

ρ2 slide

ρ3 rise

Figure 2. Schematic representations of the kinematical variables that describe the relative orientation and displacement of consecutive base pairs: θ1 and θ2 are angles of rotation about two perpendicular lines that lie in the midplane between the base pairs; θ3 is an angle of rotation about a line l perpendicular to the midplane; ρ1 and ρ2 are mutually perpendicular displacements in directions parallel to the midplane; and ρ3 is the displacement along l. Each drawing illustrates one of the kinematical variables for a case in which that variable has a positive value and the others (with the exception of ρ3) are set equal to zero.

The numbers Rnji = Rˆ ji (θ1n ,θ n2 ,θ 3n ) are the components of an orthogonal matrix, and the ⌣ functions ρi and Rˆ ji appearing are independent of n. The elastic energy ψ n of the n-th base-pair step is thus given by a functionψ˜ n of six kinematical variables:

ψ n = ψ˜ n (θ1n , θ2n ,θ 3n , ρ1n , ρ2n , ρ n3 ) .

(2.4)

It is a matter of importance in the molecular biology of DNA that the function ψ˜ n depends on the nucleotide composition of the n-th and (n+1)-th base pairs. It is generally assumed that ψ˜ n , as a function of (θ1n , θ2n ,θ 3n , ρ1n , ρ2n , ρ n3 ) alone, is independent of the 5

6 nucleotide composition of other base pairs, e.g., the (n-1)-th and (n+2)-th, from which it follows that a base-pair step can be one of ten different types. We shall make that assumption here. We note, however, that BCS theory remains valid in more general cases in which the function ψ˜ n is assumed to be influenced by the composition of base pairs other than the n-th and (n+1)-th. The kinematical variables θ in and ρin were defined with precision by El Hassan ⌣ and Calladine [3] in 1995. Properties of functions, such as ρi , that relate ρin and θ in to ri n and Dijn are discussed in detail in [1], [3], and [6]. Here, as in [6], we consider a DNA molecule in an aqueous solution with a concentration c of a monovalent salt (e.g., NaCl). We assume that no external forces or moments act on a base pair other than those that result from the long range electrostatic interaction of negatively charged phosphates in the same polymeric molecule. We further assume, as an approximation, that the two negative charges associated with a base pair are located at the barycenter of the base pair, which implies that the electrostatic energy Φ of the DNA molecule depends on the configuration through an equation of the form, Φ=

1 N +δ n ∑ϕ , 2 n =1

ϕ n = ϕ˜ n (x 1 ,..., x N +δ ) ,

(2.5)

in which ϕ n is the electrostatic energy associated with the n-th base pair and (i) δ = 0 if the DNA molecule is "closed", i.e., is "circularized" so that its ends are joined and aligned in such a way that they form a single base pair and the axial curve of the molecule has the topology of a circle or a knot, and (ii) δ = 1 if the molecule is "open", i.e., has two distinct ends (one at n = 1 and the other at n = N + 1) and an axial curve with the topology of a line. In the present theory we employ Manning's theory of charge condensation [7], which, as we shall discuss below, is in accord with equation (2.5). The total energy U of the DNA molecule is the sum of its elastic energy Ψ and its electrostatic energy Φ : U = Ψ + Φ.

(2.6)

In each configuration Ψ is given by the equations (2.1) and (2.4). We here follow the usual practice of assuming that the function ψ˜ n is a quadratic form in the excess tilt ∆ θ1n , the excess roll ∆ θ n2 , the excess twist ∆ θ3n , the excess shift ∆ ρ1n , the excess slide ∆ ρ n2 , and the excess rise ∆ ρ3n which quantities are defined by the relations

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θ in = oθ ni + ∆θ in ,

ρin = o ρ ni + ∆ρin ,

(2.7)

in which o θ in and o ρ in are intrinsic values, i.e., values appropriate to the stress-free state of the n-th base pair step. Thus 3

3

3

3

3

3

1 1 ψ = ∑ ∑ Fijn ∆θ in ∆ θ nj + ∑ ∑ Gijn ∆θ in ∆ρ nj + ∑ ∑ Hijn ∆ρin ∆ρ nj . 2 i=1 j=1 2 i=1 j=1 i=1 j=1 n

(2.8)

In accord with Manning's theory of charge condensation [7] we employ the following expression for the energy resulting from the electrostatic interaction between the charge associated with the n-th base pair and the charges associated with other base pairs: nm

n

ϕ =Q

∑ m≠ n m≠ n±1

e − κx , x nm

x nm = x n − x m .

(2.9)

(See also the discussion of Westcott et al. [8].) The Debye screening parameter κ (in -1 units of Å ) is given by the formula

κ = 0.329 c ,

(2.10)

in which c is the concentration of (monovalent) salt in moles per liter. For the constant Q we have Q = q 2 / 4πε 0 ε w ,

(2.11)

with ε 0 the permittivity of free space and ε w the dielectric constant of water. In accord with Manning's theory [7,9], q is set equal to 24% of the charge of the two phosphate groups associated with each base pair, i.e., q = 2 × 0.24e − , where e − is the charge of an electron. For each n the summation in (2.9) is taken over all m from 1 to N + δ that correspond to base pairs that are not adjacent to, or the same as, the n-th base pair. Nearest neighbors are omitted from that summation because here, as in [6], we take the position that the local elastic energy functions ψ˜ n −1 and ψ˜ n account for all interactions, including those of electrostatic origin, between the n-th base pair and the two that are adjacent to it.

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8 3. The equations of equilibrium

A configuration of a DNA molecule is said to be in equilibrium if for it the first variation of the total energy U vanishes for all variations that are compatible with the constraints imposed on the molecule. By setting this variation equal to zero we obtain, for n = 1,2,..., N , the equilibrium equations

π n := f n

n

−f

n

µ := m − m

n −1

n −1

+ gn = 0 , n

+r × f

n

=0,

(3.1a) (3.1b)

in which g n equals the total electrostatic force exerted on the n-th base pair by other base pairs in the same molecule and is given by the equation n

g =−

∂Φ . ∂x n

(3.2)

For n = 1,2,..., N , f n and m n are the (non-electrostatic) force and moment that the (n+1)-th base pair exerts on the n-th. In reference [1] it is shown that, for n = 1,2,..., N , the components min and fi n of m n and f n with respect to the local basis d1n , d n2 , d 3n are

⌣ ∂ψ˜ n ∂ρ j fi = ∑ n n , j=1 ∂ρ j ∂ri n

min

3

 ∂ψ˜ n Γijn  n  ∂θ j j =1 3

=∑

fi n = f n ⋅ d in ,

⌣ ∂ψ˜ n ∂ρ k  +∑ n n, ∂ ρ ∂ θ k j k =1  3

min = m n ⋅ d in ,

(3.3a)

(3.3b)

⌢ ⌢ where Γijn = Γij (θ1n ,θ 2n ,θ 3n ) with Γij a function independent of n whose form is known

(see the equation (A1) of reference [1] or the Appendix to reference [6]). For n = 1, the vectors f n −1 and m n −1 , i.e., the vectors f 0 and m 0 , can be preassigned as end conditions, or, for closed molecules and for open molecules that are subject to such end constraints as strong anchoring, f 0 and m 0 are determined together with the 6N kinematical variables θ1n , θ2n ,θ 3n , ρ1n , ρ2n , ρ n3 , n = 1,2,..., N , as a solution of the 6N equations (3.1) subject to 6 end conditions. For an open molecule free of external forces and moments

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9

m0 = 0 .

f 0 = 0,

(3.4)

Thus the equations (3.1), when taken together with (2.5), (2.9) - (2.11), and (3.2), form a system S of 6N equations for the 6N scalar variables θ1n , θ2n ,θ 3n , ρ1n , ρ2n , ρ n3 , n = 1,2,..., N . Remark: When we wrote the equation (2.5) which, because we employ Manning's charge condensation theory, yields the equations (2.9)-(2.11), we made an approximation that will be noticed by readers familiar with current ideas about DNA structure and base-pair level models of DNA elasticity. We assumed "that the two negative charges associated with each base pair are located at the barycenter of that base pair". In fact, those negative charges belong to the phosphate groups of two sugar phosphate chains that are at the surface of the DNA double helix and not on its axial curve. However, Manning's theory gives us reason for expecting that the condensed counter ions (which, in the case of DNA in an aqueous solution of monovalent salt, neutralize 76% of the charged phosphate groups) are not actually bound to the DNA macromolecule, i.e., are not localized at definite sites on that molecule, but are instead free to move in and out (and also along the axis) of a tube that has a diameter that depends on c but exceeds the 20Å steric diameter of the macromolecule. When one accepts the idea that the net effective charge of the DNA macromolecule is not to be thought of as arising from charges at preassigned points but rather as the result of rapid fluctuations in a region interior to a tube of specified radius, it appears reasonable to suppose that the assumption stated in quotes above does not introduce serious errors. For simplicity of discussion, from this point on we confine attention to the limiting case in which the DNA molecule is stiff with respect to the displacement parameters ρ1n , ρ n2 , ρ n3 that characterize shift, slide, and rise. In that limit ρ1n , ρ n2 , ρ n3 are considered preassigned constants, the equations (3.3) are replaced by

fn= f0−

n

∑ gm ,

(3.5a)

m=1

min

 ∂ψ˜ n Γijn  n  ∂θ j j =1 3

=∑

3

3

− ∑ ∑ ρ nk k =1 l=1

∂Rˆkl n  f , ∂θ nj l 

min = m n ⋅ d in ,

(3.5b)

and S becomes a system of 3N equations for the tilt, roll, and twist, i.e., for the 3N angular variables θ1n , θ2n ,θ 3n , n = 1,2,..., N , and the elastic energy ψ n associated with the n-th base-pair step is given by a function ψˆ n of θ1n , θ2n ,θ 3n :

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10

ψ n = ψˆ n (θ1n , θ2n ,θ 3n ) .

(3.6)

For the calculations presented below, we assume that ψˆ n is a quadratic form in ∆ θ1n , n n n n ∆ θ n2 , ∆ θ3 , which are defined as before. Thus, ρ1 , ρ 2 , ρ 3 , are here material constants, 3

3

1 ψ = ∑ ∑ Fijn ∆θ in ∆ θ nj , 2 i =1 j =1 n

(3.7)

and the moduli Fijn = Fjin , like the intrinsic parameters o θ in , are constants that depend on the nucleotide composition of the n-th and (n+1)-th base pairs. One may think of F11n and F22n as the local coefficients of rigidity for bending and of F33n as the corresponding coefficient for twisting. Quadratic energy functions of the form (3.7) (or 2.8) are approximations useful for small values of ∆ θin (or, in the case of (2.8), for small values of both ∆ θin and ∆ ρin ). A change in the choice of the direction of increasing n leaves θ 2 and θ 3 invariant but changes the sign of θ1 , which places no restriction on the coefficient F23n coupling roll to twist but does imply that, when the n-th base-pair step is such that it has on one strand either the sequence AT, TA, GC, or CG, the coefficients F12n and F13n coupling tilt to roll and tilt to twist are zero. Such coupling between twist and a mode of bending is not present in Kirchhoff's theory of rods, but crystallographic data indicate that for appropriate base-pair steps the coupling parameter F23 / F22 can be as large as 0.8. Calculations showing some striking implications such coupling are reported in references [1] and [5]. In the case we are considering, i.e., that in which ρ1n , ρ n2 , ρ n3 are constants, the equations of equilibrium are a system S of 3N nonlinear equations of the form

Ω (Θ ) = 0 ,

(3.8)

where 1

1

1

2

N

Θ =(θ1 ,θ 2 , θ3 ,θ 1 ,...,θ 3 )

(3.9)

and

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11

Ω = (µ11 , µ 12 , µ13 , µ12 ,..., µ3N ), Here µ n := m n − m n −1 + r n × f n , with f

n

µni = µn ⋅ d 1i .

(3.10)

as in (3.5a) and m n as in (3.5b).

In research presented in reference [6] a method was developed and employed to solve the system S that made use of the following fact: If one has a solution ∗ Θ of S for a value * c of the salt concentration c , one may seek to employ a Newton-Raphson procedure to find a solution for a nearby value of c . The linear system to be solved at each iteration then has the form

Ω(* Θ) + ∇Ω Ω(* Θ )[∆Θ Θ] = 0 ,

(3.11)

where ∆Θ Θ = Θ − * Θ . When, as here, long range electrostatic forces are taken into account the Jacobian matrix J = ∇Ω Ω(* Θ) is full, which for values of N greater than circa 100 renders unstable several conventional iteration schemes. However, a recipe presented in reference [10] for producing an analytical expression for each entry in J permits one to write a numerically stable code that efficiently yields solutions of the system S to within machine accuracy. An equilibrium configuration is said to be stable (also called locally stable or metastable) if for it the second variation of the energy U is positive. We write H for the (symmetric) Hessian matrix characterizing that second variation. Once one has an exact analytical expression for the (usually not symmetric) Jacobian matrix J , one may obtain, without difficulty, an exact expression for H, and by calculating the proper numbers of H determine whether the equilibrium configuration is stable.

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4. An open molecule We here show some of the computational results obtained for an open DNA molecule containing 450 base pairs (i.e., with N = 449 ) [6]. This particular molecule can be thought of as having been formed by the end-to-end joining of three 150 base pair DNA segments that are assumed to be homogeneous and to have uniform curvature such that in an intrinsic (stress-free) configuration each would have the shape of a circular ring. The example, like the one that will be examined in Section 5, illustrates the strong influence that changes in the salt concentration c can have on the equilibrium configurations of intrinsically curved DNA and yields a bifurcation diagram with regions in which two very different, but yet both locally stable, equilibrium configurations occur at the same value of c. In order to focus our attention on the role of electrostatic forces in determining the properties of equilibrium configurations, for these examples the DNA molecule was deliberately chosen to be non-shearable and homogeneous in its mechanical properties. As we take the parameters ρ1n , ρ 2n , ρ n3 to be preassigned constants and assume that the molecule is open and free of external forces and moments, the constitutive equation for ψ n here has the form (3.7) and the end conditions are as in (3.4). We assume (a) that the molecule is inextensible in the sense that for each base pair step the rise is a constant equal to 3.4 Å, and (b) that the molecule is homogeneous and transversely isotropic in the sense that the intrinsic shift and slide are everywhere zero, F11n and F22n are equal, and there is no coupling. Hence we have: n o ρ1

= 0,

n o ρ2

= 0,

n o ρ3

= 3.4 Å,

n = 1,..., N .

(4.1)

n = 1,..., N .

(4.2)

n F33 = 1.05F11n ,

(4.3)

and n F11n = F22 ,

n n F12n = F13 = F23 =0,

In the present case we put n

n

2

F11 = F22 = 0.0427 kBT / deg ,

with kB Boltzmann's constant and T the temperature which is taken to be 300 K. In addition we make assumptions about the parameters o θ in that are appropriate to a molecule with the property that each of its 150 base-pair subsegments has an intrinsic configuration that is circular in the sense that its axial curve is a perfect polygon of 150 sides. One way to obtain a model with these properties is to assume that the molecule is

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13 homogeneous with, for each n, no intrinsic tilt, an intrinsic roll equal to 2π / 150 (i.e., 2.4 degrees) and no intrinsic twist: n n o θ1 = o θ 3

=0,

n oθ 2

= 2π / 150 .

(4.4)

Of course in nature DNA obeys the Watson Crick model and the helical repeat length of the double helical structure is close to 10 base pairs, which is equivalent to saying that the values of the intrinsic twist o θ 3n are close to π / 5, i.e., to 36 degrees. A molecule with o θ 3n non-zero that has, in its stress free configuration, a shape close to that of a circle cannot have its intrinsic kinematical parameters constant (i.e., independent of n); for, if it did, the (constant) twist and bend would cause the axial curve to be helical and not planar and hence not a circle. A model for the equilibrium mechanics of intrinsically curved DNA was developed in reference [1]. The molecules treated there were composed of repetitions of unit segments of length 10 base pairs; in each such unit 5 base-pairs constituted a homogeneous intrinsically straight subsegment and the remaining 5 a homogeneous intrinsically curved subsegment, and thus each molecule had overall rod-like properties that were periodic rather than homogeneous. By appropriate adjustment of the value of o θ 2n (or o θ1n ) and the value of o θ 3n in the curved subsegments one was able to control the overall intrinsic curvature and hence the number of base pairs that would give a closed, circular, stress-free configuration. In a paper now in preparation we compare the results of calculations of equilibrium configurations for two cases: in Case I the equations (4.4) are assumed to hold; in Case II the recipe of reference [1] just described is followed with the unit segments (of 10 base pairs each) such that for the first 5 base-pair steps,

(o θ1n , o θ 2n , o θ3n ) = (0,0, 36

π 180

),

(4.5a)

and, for the remaining 5 steps, n

n

n

(o θ1 , o θ 2 , o θ3 ) = (0, 7.413

π 180

, 35.568

π 180

).

(4.5b)

In Case II the DNA molecule has the property that each of its 150 base-pair subsegments (i) has an average helical repeat length of 10 base pairs, (ii) becomes an o-ring when it is closed (by joining its ends), and (iii) is nearly (but not perfectly) planar in its stress-free configuration. (When a recipe of this general type is followed a perfectly planar nonchiral structure cannot be attained with o θ1n = 0 and with the equations (4.1) holding.) In that forthcoming paper it will be observed that the equilibrium configurations we calculate for Case I (using (4.4)) have axial curves close to axial curves calculated for Case II (using (4.5a&b)).

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14

H c = 1.97×10-3

H c=1

H c =10-1

P c = 2×10-4

H c = 2×10-2

U = 15.14

U = 699.69 U =111.61 U = 235.57 U = 464.65

Figure 3. Four helical configurations and one planar configuration that are global minimizers of the total energy U (in kBT) at the indicated (molar) salt concentration c. The lines of view in the upper and the lower drawings are perpendicular. The scale of length is the same for each case shown here and in figures 4 and 5. The configuration on the branch of planar configurations is labeled P; those on the branch of approximately helical configurations are labeled H. The configurations shown here and in Figures 3 and 4 are all drawn to the same scale.

For the 450 base pair molecules considered here, the equations of equilibrium (3.1), with f n and m n as in (3.4) and (3.5), yield a system S of 1347 nonlinear equations for the 1347 unknown kinematical variables comprising Θ (as in equation (3.9)). A detailed discussion of the bifurcations of equilibria of the 450 base pair open DNA molecule is given in [6], where bifurcation diagrams are presented with the (monovalent) salt concentration c the bifurcation parameter. We here emphasize results that show the remarkable dependence of the molecule's equilibrium configurations on c. To a good approximation a DNA molecule is a tube with a cross-sectional diameter of 20 Å and is so represented in Figures 3-7. In its intrinsic configuration, the 450 base pair DNA molecule under consideration would have an axial curve with the form of a triply wound (planar) circle, but because the molecule cannot penetrate itself,

14

15 that configuration is not physically attainable. For values of c less than 1.2 M (1.2 molar) our calculations showed the existence of stable equilibrium configurations for which the axial curve of the molecule is close to a helix with a pitch greater than 20 Å, and such configurations, as they are free from self-penetration, are physically attainable. The sensitivity of globally stable equilibrium configurations to changes in salt concentration is illustrated in Figure 3, in which we show the minimum energy configurations at five values of c. Four of these configurations are for values of c of interest to experimenters and lie on a primary branch labeled H called the "helical branch" because the configurations in it have an approximately helical shape. Each configuration in H is a global minimizer of U for its value of c. Also shown in Figure 3 is a planar configuration which lies in the stem branch P at c = 2 × 10−4 M (on the right hand side of the figure). The configurations in P are stable only for values of c less than a critical value c * = 3.26 × 10−4 M and were found to be the global minimizers of the total energy U for those values of c [6]. The "helical configurations" in the primary branch H exist only for c greater than (its value at the bifurcation point) c * ; for those values of c, the planar configurations are unstable. In Figure 4 we show, for two values of c that exceed c * , the (globally stable) configuration in H and the (unstable) configuration in P. (When a configuration in a branch is unstable we place a superscript pound sign after the label # for that branch; thus the (planar) stem branch is labeled P where c < c* and P where c < c* .) As was discussed in [6] a primary branch, labeled Es, made up of "symmetric everted helical configurations" was found. The configurations in Es are stable only in the range 1.95 × 10 −3 < c < 4.84 × 10−3 M. Two secondary branches labeled E α and E β originate off the branch Es. The branch E α contains only unstable configurations, while the branch E β has at least one stable configuration at each value of c in the range 4.84 × 10 −3 < c < 1.01× 10 −1 M. Shown in Figure 5 are three distinct equilibrium configurations at c = 3 × 10 −3 M; although they have values of the total energy U that differ by less than 1.5 kBT , they differ in their stability properties: one, labeled H, is a global minimizer of the energy (for c = 3 × 10 −3 M), another, labeled Es, is (locally) # stable, and the third, labeled E α is unstable. We hope we have presented enough information about the dependence on c of the equilibrium states of this particular DNA molecule to encourage readers to look at reference [6] where they will find a more detailed, but yet introductory, description of (i) the computational scheme employed here to calculate equilibrium configurations of DNA molecules and (ii) the bifurcation diagram it yields in the rather elementary case of an open molecule with uniform intrinsic curvature and homogeneous mechanical properties.

15

16

H c = 5×10-4

P# c = 5×10-4

U = 608.54

H c = 5×10-3

U = 608.38

P# c = 5×10-3

U = 373.47

U = 367.94

-4

-3

Figure 4. Two configurations at c = 5×10 M and two at c = 5×10 M. The # configurations in P are labeled P as they are unstable. When the assumptions made here hold, configurations on the branch H exist for c > c* and are globally stable. In -4 the present case c*=3.26×10 M. Planar configurations on the branch P are stable only if c < c*.

16

17

H c = 3×10-3

Es c = 3×10-3

Eα# c = 3×10-3

U = 421.86

U = 421.93

U = 420.57 -3

Figure 5. Three distinct configurations that can occur at c = 3×10 M. The configuration on H is globally stable. At this particular value of c the configuration on Es is locally stable, and the one on Eα is unstable and hence is # labeled Eα .

17

18

5. Closed molecules A closed DNA molecule with N base pairs is here treated as though it is one with N + 1 base pairs that has its two terminal base pairs in coincidence and hence obeys the end conditions :

x 1 = x N +1 ,

d 1i = d iN +1 ,

i = 1,2,3.

(5.1a)

These end conditions give rise to 6 independent constraints that when combined with the equations (3.8), or equivalently (3.1b) and (3.5a), i.e. µ n : = m n − m n −1 + r n × f n = 0 with f

n

= f

0

− ∑m =1 g m , yield a system of 3 N + 6 nonlinear equations for the 3N n

variables θ1n , θ2n ,θ 3n , n = 1,2,..., N , and the 6 components of f 0 and m 0 . We take the components, ℓ i = (x N +1 − x 1 ) ⋅ d 1i , of the end-to-end vector x N +1 − x 1 to be zero and +1 set the components, Qij = d i1 ⋅ d N , of the orthogonal transformation Q relating d 1i j

+1 to be determined by a triplet of angles (ζ1 ,ζ 2 , ζ3 ) defined so that, if ζ1 =ζ 2 = 0 to d N j and ℓ i = 0 , then the base pairs 1 and N + 1 lie in the same plane, and the end-to-end twist

angle, ζ 3 , becomes the angle that d1N +1 makes with d11 . When this is done, the end conditions (5.1a) become the relations ℓi = 0 ,

ζj = 0,

i, j = 1,2,3 .

(5.1b)

The conditions (5.1a&b) do not completely determine the topology of the molecule. For, one can, in theory, nick a single strand of the molecule, twist by a complete turn the base-pair on one side of the nick while holding fixed the base pair on the other side, and then join together the two ends of the nicked strand to form a closed molecule for which (5.1a&b) still hold. Such a procedure changes by one the linking number, Lk , of the molecule. The linking number is a topological invariant equal to the number of times the (either) one of the two (closed) strands is linked to the axial curve C. (For details see [11] and [12].) As the molecule can be closed in such a way that its axial curve has the topology of a knot, not only its linking number but also its knot type must be specified. We here consider a closed, transversely isotropic, DNA molecule with N = 549 base pairs that obeys the assumptions (4.1), (4.2), (4.3) and has the intrinsic curvature of a molecule of No base-pairs (with No < N ) that, if one could turn off the electrostatic forces by, say, raising the salt concentration to a high enough level, would become an oring if its ends were brought together, properly aligned, and sealed. In other words, the

18

19 closed 549 base pair molecule under consideration is assumed to be such that each of its subsegments of No base pairs has an intrinsic configuration that is circular in the sense that for it the axial curve is a perfect polygon of No sides. For the calculations reported here we have put No = 220 .

(5.2)

This value of No corresponds to an intrinsic curvature close to one observed [13,14,15,16] in segments of kinetoplast DNA in species of Trypanosomatidae. For simplicity we assume that, for the (hypothetical) molecules we consider, o θ1n , o θ 2n , o θ 3n , obey the relations n o θ1

=

2π 2π sin( (n − 1 2)), No ho

n oθ 2

=

2π 2π cos( (n − 1 2)) , No ho

n oθ 3

=

2π ho

(5.3)

in which n = 1,..., N and ho is a material parameter equal to the number of base-pair steps per helical turn in the intrinsic configuration. We further assume that the ratio N / ho is an integer. Attention will be focused on three of the many possible topologies that a molecule with these properties may have: Topology A: In this case the molecule has the topology of a knot-free circular ring, and when it is constrained so that its axial curve lies in a plan with no self crossing the excess twist is everywhere zero: ∆ θ3n = 0 for n = 1,..., N . The assumptions we have made imply that the integer o Lk = N / ho is the linking number of the molecule. If this molecule is cut open and resealed to obtain a molecule with a different topology, say one with linking number Lk , then the number,

∆Lk = Lk − o Lk ,

(5.4)

is called the excess link of the new molecule. Of course, for molecules with topology A, ∆Lk = 0. Topology B: This is the topology obtained when the molecule is closed in such a way that the result is a knot-free circular ring that has, when its axial curve lies in a plan with no self crossing, ∆ θ3n = −2π / N for n = 1,..., N . For this molecule, ∆Lk = −1. Topology C: This is the topology of molecule that is closed in such a way that its axial curve has the topology of a trefoil knot with ∆Lk = 3 .

19

20

In the present theory the transverse isotropy of a DNA molecule obeying (4.1) and (4.2) is such that, when the simplifying assumptions (4.1), (4.2), and (5.3) are made, the equilibrium configurations have axial curves and mechanical properties that are independent of the choice of intrinsic twist angle, o θ 3n , i.e., of the choice made for o Lk = N / ho . Hence, without further loss of generality, the value of o Lk for the closed molecules under consideration can be set equal to zero by passing to the limit where ho → ∞ , and the equation (5.3) reduces to n o θ1

=0,

n oθ 2

=

2π , No

n oθ 3

=0,

n = 1,..., N

,

(5.5)

Thus we have again, as in Section 4, the case of a homogeneous molecule with zero intrinsic twist. From this point on the homogeneous closed molecule under consideration obeying (4.1) , (4.2), (4.3), and (5.5) will be referred to as H549A, H549B, or H549C in accord with whether it has the topology A, B, or C. For H549A there is a trivial solution of the equations (3.8) for each value of c. In that solution, here called the equilibrium ring configuration, θ1n , θ2n ,θ 3n , are independent of n and are given by,

θ1n = 0 ,

θ 2n =

2π , N

θ 3n = 0 ,

n = 1,..., N .

(5.6)

It can be shown that the values of the kinematical variables in (5.6) satisfy the equations of equilibrium (3.1b) and (3.5a) only when the vectors f 0 and m 0 associated with the end conditions obey the relations,

f0=

g 2

(cot(π / N )d 13 − d 11),

m 0 = F22n (2π / No − 2π / N )d 12 , (5.7)

in which g , the magnitude of the electrostatic force g n acting on the n-th base pair, is independent of n as is F22n , in view of (4.3). A calculated bifurcation diagram for H549A is shown in Figure 6 which gives the normalized radius of gyration, R / Rring , of equilibrium configurations as a function of the molar salt concentration c. Here, by definition, the radius of gyration, R , is 1 N  n n R =  ∑ ( x − x ) ⋅( x − x )  N n=1 

1/2

,

x=

1 N n ∑x . N n =1

(5.8)

20

21 The stem branch, labeled Ring in Figure 6, contains only the trivial ring configuration for which the axial curve is a perfect polygon of N equal sides, and hence its radius of gyration, Rring , is equal to the radius of the circle circumscribing that polygon. Although others were calculated, we show only those branches of equilibrium configurations for which there is a significant range of values of c for which the equilibrium configurations are stable. Branch intervals made up of stable and unstable equilibria are drawn as solid lines and dashed lines, respectively. Analysis of the stability of closed equilibrium configurations (a description of the method employed is given in the thesis of Biton [10]) shows that the ring configuration is globally stable for values of c less than the critical value c # = 2.32 × 10 −3 M and is unstable for values of c greater than c # . The primary branch St originates at a pitchfork

Figure 6. Bifurcation diagram showing the normalized radius of gyration R/Rring as a function of c for the closed 549 base-pair molecule H549A for

which ∆Lk = 0 and the intrinsic curvature is such that each 220 base-pair subsegment has a stress-free circular configuration when closed to form a ring. Some representative equilibrium configurations are drawn next to their corresponding branches. Here and in Figure 7 configurations are drawn to a single scale which, however is not the same for the two figures. 21

22 bifurcation point on the stem branch where c = c # . The configurations in Stb have the form of symmetrically buckled rings. The axial curves of those configurations have two perpendicular mirror symmetry planes and a shape similar to that of the curve that divides the surface of a tennis ball into two identical halves (hence the designation Stb). For c in the interval c # < c < 1.34 × 10 −1 M, the configurations in Stb are stable; for c > 1.34 × 10 −1 M they are unstable. For c # < c < 3.48 × 10−2 the configurations in Stb were found to be globally stable. The secondary branch CII originates from a sub-critical pitchfork bifurcation at the point in Stb where c = 1.34 × 10−1 M. The configurations in CII have a "hidden symmetry". (A term employed by Domokos and Healey [17] in their discussion of the symmetry of closed homogeneous Kirchhoff rods). The axial curves of those configurations have a two-fold (proper) rotational symmetry. The configurations in the sub-branch of CII between the bifurcation point at c = 1.34 × 10−1 M (for which R / Rring < 0.501 and the turning point at c = 2.78 ×10−2 M (for which R / Rring = 0.483) are unstable. The configurations in CII in the lower region of the fold (where R / Rring < 0.483) are stable. In this lower region of CII , as c increases the equilibrium configurations collapse (and, in particular, R / Rring decreases) toward a shape, which in the limit of high salt concentration, has the form of a triply wound configuration with R / Rring close to 1/3. (Because the steric diameter of the cross sections of the rod-like DNA molecule are not zero, as c increases without limit the value of R / Rring approaches a number strictly greater than 1/3). Easily performed experiments with a closed elastic ring show that by applying a (twist) torque on one pole of the ring while anchoring the opposite pole one can get an equilibrium configuration that has the shape of a chair. There is a literature on the theory of the stability of symmetric chair configurations of an o-ring free of intramolecular electrostatic forces. Charitat and Fourcade [18], using the Kirchhoff theory of inextensible, transversely isotropic rods, and Coleman, Olson, and Swigon [1], using their theory of sequence dependent DNA elasticity for the case in which the present equations (3.5) - (3.7), (4.1), (4.2), and (4.5) hold, studied the stability of such configurations as a function of the ratio ω of the twist modulus to the bending modulus. (In the notation used here and in [1]: ω = F33 / F11 .) These equilibrium configurations are called symmetric because they have a mirror symmetry plan and an axis of two-fold rotation symmetry. In [1] it was shown that, in the absence of electrostatic forces, the symmetric chair configurations are stable if and only if ω < 1. (Their study was for the case in which the equations (3.5) - (3.7), (4.1), (4.2), (4.5) all hold; which implies that the closed rod-like structure is, as they state, an o-ring.) These observations led us to expect the existence of equilibrium chair configurations of H549A. After a thorough search for such configurations, we did find one, and were then able to calculate the branch labeled Schair

22

23 and study the nature of the equilibria in it. In bifurcation diagrams in which c is the bifurcation parameter, the branch Schair is not connected to the stem branch Ring or to any branches connected to that branch. (We found however that in a bifurcation diagram in which the end-to-end twist angle, ζ 3 , is the bifurcation parameter, the branch Schair is directly connected to the stem branch Ring, and may be considered a primary branch.) The symmetric chair configurations of H549A in Schair are stable only when c is less than the critical value c # # = 4.54 × 10−3 . The secondary branch CI originates at the pitchfork bifurcation point in Schair at c = c # # . The configurations in CI have a two fold improper rotation symmetry and they are stable in the range c > c ## in which they exist. As c increases, the configurations in CI collapse toward a shape of a triply wound configuration for which R / Rring is close to 1/3. For c > 3.48 × 10−2 the configurations in CI were found to be globally stable with values of the total energy U = Ψ + Φ that are lower than the corresponding configurations in CII (calculated at the same value of c) by less than 1 k BT . An analysis of bifurcations of equilibria performed for H549B and H549C similar to that just described for H549A revealed the dependence of their equilibrium configurations on c. The influence of the value of c on the configurations that are global minimizers of the energy is depicted in Figure 7, where we show the calculated minimum energy configurations of the three molecules for four values of c. Since, as we have just remarked for H549A, when for that molecule the salt concentration c exceeds 3.48 × 10 −2 M, the differences between the values of the total energy of the equilibrium configurations in CI and CII are very small, the stable configurations in both of those -1 branches are shown for c=10 M (which is not far from the physiological value of the ionic strength) and for c=5 ×10-1M . The reader will note that for the two knot free molecules with uniform curvature that we have considered here, the length and intrinsic curvature are such that as c increases without bound, the global minimizer of the energy U approaches a triply wound structure if ∆ Lk is an even number and a doubly wound structure if ∆ Lk is an odd number. For H549B and also the trefoil knot H549C, for each value of c the global minimizer is in the stem branch, and as c increases R / Rring decreases and the configuration approaches one that is doubly wound with R / Rring close to 1/2. We note that for the problems we have discussed here the range of variables was such that the equilibrium configurations did not show self-contact, but, of course, for these particular molecules self-contact does occur when the salt concentration is high enough. When an intrinsically straight elastic rod obeying the Kirchhoff theory of homogeneous rods, without the electrostatic forces that we have emphasized here, is closed and given the topology of a knot it exhibits self-contact in each of its equilibrium configurations. For such rods a theory of the equilibrium states of knots, with the

23

24 emphasis on trefoil knots, is now available [19]. -3

10 M

-2

10

M

-1

10 M

-1

5 × 10 M

H549A ∆Lk = 0

H549B ∆Lk = -1

H549C Trefoil knot ∆Lk = 3

Figure 7. Minimum energy configurations of the closed 549 base-pair DNA molecules H549A with ∆Lk = 0, H549B with ∆Lk = -1, and H549C with the topology of a trefoil knot. For each of these molecules, the intrinsic curvature is such that each 220 base-pair subsegment has a stress-free circular configuration when closed to form a ring. The ring configuration of H549A shown on the upper -3 left corner is the global minimizer at c=10 M; the buckled ring configuration in -2

the branch Stb is the global minimizer at c=10 M. The two triply-wound

-1

collapsed configurations in the branch CI are the global minimizers at c = 10 M -1

and c=5 × 10 M and are shown above the two configurations in the branch CII which have values of U that are greater than those of the corresponding global -1 -1 minimizers by 0.35 kBT at c=10 M and by 0.2 kBT at c=5 × 10 M.

24

25 Acknowledgements We thank Professor David Swigon for valuable discussions. This research was supported by the National Science Foundation under Grant DMS-0514470.

References

1. B.D. Coleman, W.K. Olson, & D. Swigon, Theory of sequence-dependent DNA elasticity, J. Chem. Phys., 118, 7127-7140 (2003). 2. V.B. Zhurkin, Y.P. Lysov, & V. Ivanov, Anisotropic flexibility of DNA and the nucleosomal structure. Nucleic Acids Res. 6, 1081-1096 (1979). 3. M.A. El Hassan & C.R. Calladine, The assessment of the geometry of dinucleotide steps in double-helical DNA: a new local calculation scheme. J. Mol. Biol. 251, 648-664 (1995). 4. W.K. Olson, M. Bansal, S.K. Burley, R. E. Dickerson, M. Gerstein, S. C. Harvey, U. Heinemann, X. Lu, S. Neidle, Z. Shakked, C. Wolberger, & H. M. Berman, A standard reference frame for the description of nucleic acid basepair geometry, J. Mol. Biol. 313, 229-237 (2001). 5. W.K. Olson, D. Swigon, & B.D. Coleman, Implications of the dependence of the elastic properties of DNA on nucleotide sequence, Phil. Trans. Roy. Soc. Lond. A, 362, 1403-1422 (2004). 6. Y.Y. Biton, B.D. Coleman, & D. Swigon, On bifurcations of equilibria of intrinsically curved, electrically charged, rod-like structures that model DNA molecules in solution, J. Elasticity, 87, 187-210 (2007). 7. G.S. Manning, Limiting laws and counterion condensation in polyelectrolyte solutions: I. Colligative properties, J. Chem. Phys., 51, 924-933 (1969). 8. T.P. Westcott, I. Tobias, & W.K. Olson, Modeling self-contact forces in the elastic theory of DNA supercoiling, J. Chem. Phys., 107, 3967-3980 (1997). 9. M.O. Fenley, G.S. Manning, & W.K. Olson, Approach to the limit of counterion condensation, Biopolymers, 30, 1191-1203 (1990). 10. Y. Y. Biton, Bifurcations of equilibria in DNA elasticity, Doctoral Dissertation, Graduate Program in Mechanics, Rutgers University, New Brunswick, New Jersey (October 2007). 11. J.H. White, Self-linking and the Gauss integral in higher dimensions, Am. J. Math., 91, 693-728 (1969).

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26 12. J.H. White, An introduction to the geometry and topology of DNA structure. In: Mathematical methods for DNA Sequences, CRC, Boca Raton, Florida 225-253 (1989). 13. J. C. Marini, S.D. Levene, D.M. Crothers, & P.T. Englund, Bent Helical Structure in Kinetoplast DNA, Proc. Natl. Acad. Sci., USA, 79, 7664-7668 (1982). 14. S. Diekmann & J.C. Wang, On the sequence determinants and flexibility of the kinetoplast DNA fragment with abnormal gel electrophoretic mobilities. J Mol Biol. 186, 1–11, (1985). 15. T.E. Haran, J.D. Kahn, & D.M. Crothers, Sequence elements responsible for DNA curvature, J. Mol. Biol., 244, 135-144, (1994). 16. J. Griffith, M. Bleyman, C.A. Rauch, P.A. Kitchin, & P.T. Englund, Visualization of the bent helix in kinetoplast DNA by electron microscopy. Cell, 46, 717–724 (1986). 17. G. Domokos & T.J. Healey, Hidden symmetry of global solutions in twisted elastic rings, J. Nonlinear Sci. 11, 47-67 (2001). 18. T. Charitat & B. Fourcade, Metastability of a circular o-ring due to intrinsic curvature. European Phys. J. B, 1, 333-336 (1998). 19. B.D. Coleman & D. Swigon, Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids, Phil. Trans. Roy. Soc. Lond. A, 362, 1281-1299 (2004).

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