Chaire d'Analyse Appliquée

Continuum Rod model of DNA


Table of Contents

Extracting continuum parameters from experimental DNA data

(cf. Manning, Maddocks, and Kahn (1996) )

In order to apply a continuum rod model to DNA, one must extract appropriate continuum parameters (e.g. intrinsic shape, stiffnesses) from existing experimental or computational data:
X-ray crystal structures
molecular mechanics energy-minimization computations
molecular dynamics computations
minimum-energy shapes within the wedge-angle model

In all of these cases, the model underlying the data is not a continuum rod, but rather some discrete model, either an all-atom model (for the first three) or a base-pair-level model (for the last). The challenge is to design algorithms to translate information from these discrete models to the continuum model, where we can take advantage of rod theory and efficient computations. The next section gives an example of this parameter extraction, in the case where the underlying model is the wedge-angle model
 

The wedge-angle model

The wedge-angle model describes each DNA base-pair as a rigid body, or equivalently a 3D axis system or frame (d1,d2,d3). The origin of the frame is at the center of the base-pair, the d3 axis points to the next base-pair center, and the d1 axis tracks the rotation of the DNA sugar-phosphate double helix.

DNA intrinsic shape is incorporated into the wedge-angle model by allowing the minimum-energy stacking orientation of one frame on the previous frame to depend on the base-pairs involved. These minimum-energy stacking orientations have been computed by various researchers, using a mixture of real experiments and atomic simulations (although there is still considerable disagreement among these results).

Smoothing the wedge-angle intrinsic shape

Given a wedge-angle minimum-energy shape, we convert it to a continuous shape by a smoothing procedure involving a combination of data filtering and least-squares fitting. Many of the mathematical techniques in this smoothing are standard, but the particular combination required to produce a continuum shape with the same global curvatures as the wedge-angle shape but smoother local curvatures is specialized to this DNA problem.

The natural frame

In addition, there is an important transformation required to accommodate the rapid intrinsic twist of the DNA double-helix (which makes a full turn every 10.5 base-pairs); if not removed, this intrinsic twist causes undesirable rapid variations in the continuum rod parameters. Instead of using the real DNA d1 vector field of the intrinsic shape, we replace it by the natural, or parallel-transport, or zero-twist, frame D1. This natural frame D1(s) is uniquely defined (up to choice of D1(0)) by the centerline. For each value of the arclength parameter s, let w(s) denote the angle between d1(s) and D1(s) in the rod intrinsic shape.

The remarkable fact is that if the continuum rod is uniform and has circular cross-section (i.e. K1(s)=K1=K2(s) in the energy functional), then a static equilibrium of the rod with the real frame can be recovered exactly from a static equilibrium of the rod with the natural frame simply by rotating d1(s) (and d2(s)) by w(s). Said another way, the real-to-natural-frame transformation and the continuum equilibrium computation commute, so that we may adopt the following strategy:
(1) Transform from real frame to natural frame
(2) Compute equilibrium configuration
(3) Transform from natural frame to real frame

Real DNA, by virtue of its anisotropic base-pairs, probably does not have equal bending stiffnesses K1=K2 as assumed in the above argument. However, since DNA has large intrinsic twist, this anisotropy should be effectively averaged out over the length scale of several base-pairs, so that the above assumption is accurate (see Kehrbaum & Maddocks (1998) for proofs of results of this type).

Verification

The accuracy of the wedge-angle-to-continuum procedure has been verified as follows.

Given a DNA sequence, construct its wedge-angle intrinsic shape.
Apply the above procedure to produce a continuum intrinsic shape.
Compute the lowest-energy ring equilibrium with d1(0)=d1(1) within the continuum model for that intrinsic shape.
Take this continuum solution as an initial guess for the determination of the lowest-energy ring equilbrium with d1(0)=d1(1) within the wedge-angle model (this requires a high-dimensional constrained minimization not described here).

By this procedure, we directly measure the error introduced in our conversion to the continuum problem, which is quite small, both in energy (less than 0.5%) and in configurations (see the superpositions of centerlines and helices below):
 

Modelling DNA cyclization

Cyclization as an experimental probe of DNA mechanical properties

Many biophysicists are trying to understand the role of DNA intrinsic shape and flexibility in phenomena such as transcription. However, the direct measurement of these shapes and stiffnesses is often difficult, so instead we turn to indirect observations such as cyclization (the joining of two ends of a DNA molecule to form a looped molecule). Cyclization is a promising probe for DNA shape and flexibility because the energy of a cyclized molecule (and hence its probability of forming) depends quite sensitively on its intrinsic shape (e.g. C-shaped molecules form cycles much more readily than S-shaped ones) and stiffnesses (e.g. a very flexible sequence in just the right location can lead to a very low cyclization energy). We thus are faced with an inverse problem: given experimental cyclization energies, estimate parameters of shape and flexibility of the DNA by coupling the experiments to a model such as the continuum rod model.

Test problem: continuum predictions of cyclization probabilities (cf. Manning, Maddocks, and Kahn (1996) )

As a first test, we conducted a study with Jason Kahn (Dept. of Chemistry and Biochemistry, University of Maryland College Park) to solve the forward problem: given currently accepted intrinsic-shape and stiffness parameters for "normal" DNA (i.e. DNA containing no sequence thought to have properties in disagreement with the parameters), can the continuum computations recreate experimental cyclization probabilities? The test problem involved 11 molecules, of lengths between 150 and 160 base-pairs, each containing an A-tract (intrinsically bent by about 90o) as well as a CAP binding site (intrinsically bent by about 10o). Within the family of 11 molecules, the length of the region between the two bends is varied so that the spectrum between "C-shaped" and "S-shaped" molecules is covered.

Within the continuum model, a cyclized molecule is exactly a twisted ring equilibrium:

As described elsewhere, we compute the set of stable twisted ring equilibria as  is varied. The solution branches for three of the 11 molecules (branch color indicates the molecule) are superimposed in the figure below. The energy versus m3 of each ring equilibrium is plotted. In addition, the actual cyclized equilibria (those with  = 0) are indicated with circles. Stable equilibria are indicated with solid lines, and unstable equilibria with dashed lines.

For each molecule, the lowest-energy stable cyclized equilibrium should be the one seen experimentally. (If two stable cyclized equilibria are close enough in energy, they should both be seen experimentally. Further, in this case, the two equilibria are likely to have links differing by 1. Both of these predictions are confirmed by experimental results).

The energies of these lowest-energy stable cyclized equilibria are compared to the experimental results for the 11 DNA molecules in the figure below. The continuum energies are plotted as a function of the ratio of twisting stiffness to bending stiffness K3/K1, since the appropriate value of this parameter for DNA is under some debate. The table at the right shows the experimental results. The continuum model does not contain entropy effects, so its energies are a constant shift  down from the experimental free energies. The ordering and relative spacing of the energies match experiment to within experimental error, and if we include the entropic shift as a free parameter, the continuum results match experiment to within experimental error.

Computational cost

The entire process of starting from a base-pair sequence, constructing a continuum intrinsic shape, and computing from it cyclization equilibria, takes about 5 minutes for the above molecules. Further, once this process is completed, it takes only a few seconds to compute the cyclization equilibria at a different value of a parameter like K3/K1. Since the central computation in the interpretation of cyclization experiments is a search among DNA shape and flexibility parameters to best match experimental cyclization energies, the appeal of these continuum computations is clear.

Conclusions from test study

This test case thus demonstrates that apart from an entropic term, the continuum computations can solve the forward problem of predicting experimental cyclization energies given "correct" rod parameters. Even without the ability to compute this entropy shift, the continuum model offers a promising approach to solving the inverse problem, using the following strategy. Given experimental cyclization energies, use the continuum model to search within a space of DNA shape and stiffness parameters plus an entropic shift parameter to find the best fit to the experimental results. Then, as a verification, perform a single Monte Carlo simulation (or some other method that does account for entropy). Since the continuum computations are so much faster than Monte Carlo or similar computations (seconds per parameter step versus tens of hours), this strategy should be quite efficient.

Shape and Flexibility of the TATA box

(with John Maddocks and Jason Kahn)

Having established the value of the continuum approach for the forward problem of predicting cyclization energies, we come to the real goal: to use the continuum approach in inverse problems to extract mechanical properties of special DNA regions of biological interest.

We are currently pursuing this question, in collaboration with Jason Kahn, to study the TATA-box sequence (a key player in the initiation of transcription), both with and without the presence of the TATA-box-binding-protein. In each case, cyclization experiments are well different from what would be predicted from "standard" DNA shape and stiffness parameters, so we are seeking to use the continuum model to determine a shape and/or stiffness profile of the TATA-box region consistent with the experimental results. 

Multiplicity of DNA equilibria

(with Patrick Furrer and John Maddocks)

We are studying the relation between intrinsic DNA shape and multiplicities of stable nicked equilibria and stable cyclized equilibria.

Nicked DNA are DNA loops in which only one of the sugar-phosphate strands is closed. We are adopting the model for nicked DNA of Katritch and Vologodskii (1997) in which r(0)=r(1) and d3(0)=d3(1) for the rod, but no constraint is placed on the relative orientation of d1(0) and d1(1). Equivalently, we seek points in E vs. m3 cyclization bifurcation diagrams which are both stable and local minima in E. It is clear that real nicked DNA can not spin freely about its tangent vector as this model suggests, but we have adopted it for comparisons with the results of Katritch and Vologodskii.

When no intrinsic curvature is present, then because of the symmetries of the rod, there is a degenerate circle of (neutrally) stable nicked equilibria (corresponding to the bottom-most point in the perfect diagram), as well as degenerate circles of (neutrally) stable cyclized equilibria (which precise links are stable depends on the length of the DNA and the value of K3/K1).

Perturbation results show that for infinitesimal perturbations, these circles of degenerate stable equilibria are expected to perturb to a pair of equilibria, one stable and one unstable, unless two special integrals involving the rod intrinsic shape vanish. Thus, for the macroscopic intrinsic curvatures seen in real DNA, the likelihood of multiple stable equilibria should correlate well with the smallness of these two special integrals. This hypothesis has been verified via tests on large sets of DNA sequences. 

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last update : PF, july 20, 1998