Chaire d'Analyse Appliquée                    The DNA modelling COURSE

Session 2

Introduction

The next several weeks of class will be concerned with developing the static continuum elastic rod model of the DNA mini-circle cyclization problem that was outlined in the first days lectures.

The mathematics that will be introduced involves the calculus of variations, Hamiltonian formulations of the Euler-Lagrange equations, bifurcation theory, numerical methods for ODE two-point boundary value problems (including numerical symmetry breaking), and the theory of the second variation in isoperimetric calculus of variations problems.

We start with a description of the mechanics model, known as the Cosserat theory of rods, and the quaternion parameterization of proper rotation matrices. We then describe the mathematical methods, and finally we discuss how to fit the parameters of our model to describe DNA itself, and make some computations.

The sum total of all this material will occupy several weeks.

A configuration of a Cosserat rod is a parameterized space curve ${\bf r}(s)$ along with a parameterized family of right handed, orthonormal triads $\d_i(s)$, i.e. three unit vectors which for each $s$ satisfy the constraints $$\d_i\cdot \d_j = \delta_{ij},\ \ \ \d_3=\d_1\times \d_2$$ with $\delta_{ij}$ being the Kronecker delta function.

Throughout the course Roman indices will be assumed to run through the values 1, 2 and 3 with repeated indices summed, unless the context or text indicates otherwise.

These orthonormality relations mean exactly that the $3\times 3$cosine matrix ${\cal R}$ of components of the three vectors $\d_i$ with respect to any fixed (right handed) system of orthonormal basis vectors, say $\i_i$, is a proper rotation matrix (i.e. a rotation matrix with determinant $+1$) or (a representative of) an element of the group $SO(3)$.

Notice that the frame $\{\d_i\}$ is defined externally to the curve ${\bf r}$, i.e. the frame contains additional information to the centerline. Thus the $\{\d_i\}$ frame has a different status from say the Frenet-Serret frame which is made up from the tangent, normal and binormal to the curve, and which is defined by the curve ${\bf r}(s)$ and its derivatives.

The curve ${\bf r}$ should be thought of as the centerline of the rod (say the average of the two backbones in the case of DNA) while the frame $\{\d_i\}$ should be regarded as specifying the orientation of each cross-section of the rod (or say some smooth interpolation of the orientation of each set of base pairs in the case of DNA).

The parameter $s$ will be given a precise physical interpretation in a little while, but is, roughly speaking, distance along the centerline of the rod.

Kinematics

With no loss of generality we may define the vector $\v(s)$ via $$\v(s)\equiv \r'(s)$$ where $\r'$ denotes the derivative of $\r$ with respect to the parameter $s$. And we will denote the components $v_i$ of $\v$ with respect to the triad $\{\d_i\}$ by $$v_i\equiv\v\cdot\d_i$$

As the $\d_i$ are an orthonormal basis they satisfy kinematic equations of the form

$$\d_i'=\u\times\d_i.$$ That is the vector $\u$ is an `angular velocity' but representing `evolution' with respect to the parameter $s$ instead of with respect to time.
 

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It is a simple exercise (do it!) to derive the existence of the vector $\u$, which is sometimes called the Darboux vector.

(Hint for one way: because the triad $\{\d_i\}$ is a basis at each $s$there exist functions $c_{ij}(s)$ such that $\d_i'=\sum c_{ij}\d_j$. Differentiating the orthonormality conditions on the $\d_i$then give conditions on the $c_{ij}$ which imply that there are only three independent non-zero $c_{ij}$, and they have the property that they are equivalent to taking a vector cross-product, i.e. the Darboux vector exists.)
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We also introduce the components of the Darboux vector $\u$ with respect to the basis $\{\d_i\}$ through

$$u_i=\u\cdot\d_i$$ In fact one can see that $$u_i=\frac{1}{2}\epsilon_{ijk}\mathbf{d}'_j\cdot\mathbf{d}_k$$ where $\epsilon_{ijk}$ is the alternating tensor.

The components $u_i(s)$ determine the frame $\{\d_i(s)\}$ (up to a single arbitrary rotation specifying initial conditions for the frame) through integration of the (nine, scalar) differential equations

$$\d_i'=\u\times\d_i=\sum_j u_j (\d_j\times\d_i).$$ or $$\d_i'=\epsilon_{jik}u_j\d_k$$

Once the frame $\{\d_i(s)\}$ is known, the three component $v_i(s)$determine the centerline ${\bf r}(s)$ (up to a single arbitrary translation specifying initial conditions on the centerline) through integration of the equations

$${\bf r}'=\v=\sum_iv_i\d_i$$

Thus the six scalar functions $u_i(s)$ and $v_i(s)$, determine the configuration of the rod up to a single arbitrary rigid body motion (i.e. a translation plus a rotation).

For this reason the six functions $u_i(s)$ and $v_i(s)$ can, and will, be called a set of strains for the rod.

There is certainly quite some arbitrariness in the definitions of the strains, which arises from an arbitrariness in the relation of the frame $\{\d_i(s)\}$, and the parameter $s$, to the material making up the rod.
 

The reference state and the adapted framing

This arbitrariness is usually removed by specifying particular reference strains $\hat u_i(s)$ and $\hat v_i(s)$ describing a reference state.

We will usually assume the reference state to be a minimum energy or unstressed configuration (the precise definitions of which will arise later).

Usually (but not necessarily) the parameter $s$ is chosen to be arc-length along the centerline of the reference curve $\hat{\bf r}(s)$, in which case the reference strains satisfy

$$\vert\hat\r'\vert=\vert\hat\v(s)\vert=1,$$ and in any other configuration $\vert\v(s)\vert-1$ is a measure of the local extension (or stretch) when positive, or of the local compression when negative.

Usually (but not necessarily) the reference unit vector field $\hat\d_3(s)$ is chosen to be parallel to the tangent vector to the reference centerline $\hat{\bf r}(s)$ so that provided $s$ has been chosen to be arc-length in the reference state, we have for all $s$ that

$$\hat{\bf r}'=\hat\d_3$$ The triad $\{\hat\d_i(s)\}$ is then called an adapted framing of the curve $\hat {\bf r}(s)$.

In this situation the $3$-components of the strains are distinguished from the $1$ and $2$ components, and it makes sense to describe $v_1$and $v_2$ as shear strains, and (somewhat imprecisely) $v_3$ as a stretch. Similarly $u_1$ and $u_2$ are described as bending strains, while $u_3$ will be described as the twist (strain).

Here are some examples of typical strains:

As a matter of physical modelling, when interpreting mathematical rod models as an approximation of long slender deformable bodies, it is implicitly assumed that the whole three-dimensional shape of the body can be reconstructed to a good approximation from knowledge of the centerline and the frame, using a known, locally-defined reconstruction rule at each value of the parameter $s$.

For example, in the context of DNA it is implicitly assumed that the location of each atom making up the $N$th base pair can be well-approximated once the frame, and centerline at the arc-length corresponding to the $N$th base are known.

The arbitrariness in the choice of reference strains $\hat u_i(s)$ and $\hat v_i(s)$ can be regarded as an arbitrariness in the definition of the local reconstruction rule, but once a reference configuration, or equivalently a local reconstruction rule at each value of $s$, has been set, then the same local reconstruction rule is used in any configuration of the rod.

For this to be a reasonable approximation, it is implicit assumed that the differences between the strains in any physically relevant configuration and in the reference configuration, are quite small, and for this reason rod theory is sometimes described as a linearized or small strain theory.

However this does not mean that the differences in configurations are small, because small differences in strains over a relatively large distance $s$ can give big differences in both centerline and frame. In particular rod theory keeps full geometrical nonlinearities.
 

The inextensible, unshearable rod

Much of the classic work on rod theory, and in particular rod theories applied to modelling DNA, assume an inextensible, unshearable rod.

In an inextensible, unshearable rod, the strains $v_i$ in any configuration equal the strains $\hat v_i$ in the reference configuration, i.e. they satisfy the constraints

$$v_i(s)\equiv\hat v_i(s).$$

For an inextensible, unshearable rod it almost always makes sense to choose the parameter $s$ to be arc-length in the reference configuration, because it is then also arc-length in any configuration.

Similarly it almost always makes sense to choose the reference framing to be adapted so that $\hat{\bf r}'(s)=\hat\d_3(s)$, and

$$\hat v_1(s)=\hat v_2(s)=0, \qquad \qquad \hat v_3(s) =1$$ because the framing is then adapted in any configuration i.e. ${\bf r}'(s)=\d_3(s)$ in all allowed configurations.

In fact for an inextensible, unshearable rod the configuration space can be viewed as a centerline ${\bf r}(s)$ parametrized by arc-length $s$,along with a single unit vector field, $\d_1(s)$ say, which is everywhere orthogonal to the centerline ${\bf r}(s)$.

In other words a ribbon.

An orthonormal (adapted) right handed triad can then be constructed from any configuration of a ribbon from the definitions

$$\d_3 \equiv {\bf r}',\qquad \qquad \d_2 \equiv \d_3\times \d_1,$$ and the $u_i$, as defined before, now form a full set of strains for the problem.

Even for an inextensible, unshearable rod, with an adapted framing, there remains some freedom as to which adapted frame is best chosen as the reference configuration. This is a topic that will be of some importance in our modelling of DNA, and we will return to it later.
 

last update 17.11.1998
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