Multi-scale mechanical modeling of DNA

Snapshots of DNA centerlines reconstructed from cryo-EM data.

Related Articles

O. Gonzalez & J.H. Maddocks (2001) ``Extracting parameters for base-pair level models of DNA from molecular dynamics simulations,'' Theoretical Chemistry Accounts, vol. 106, pp. 76-82.


A. Stasiak, J. Dubochet, P. Furrer, O. Gonzalez & J. Maddocks (1999) ``DNA: Uncooked, al Dente, or Scotti?,'' Science, vol. 283, No. 5408 (March 12), p. 1641.


``On the statistical mechanics of rigid bodies and polymer chains,'' working manuscript.


``Statistical and dynamical models of DNA in solvent,'' working manuscript.

Overview

A fundamental problem in the mechanical modeling of DNA is to determine properties of structure (such as shape, stiffness,...) given sequence (TAAACG...) and environment (solvent, temperature,...). The notion of structure and environment depend on length scale, and my interest is when DNA structure may be described by a continuous curve and environment by a continuous fluid. In this case, an elastic rod subject to electrical, hydrodynamic and random loads provides a reasonable mechanical model of DNA. Electrical loads may be defined by a (screened) Coulombic interaction potential, hydrodynamic loads by slender-body theory, and statistical properties of random loads by a fluctuation-dissipation relation. My research is concerned with the general investigation of such a stochastic PDE model. I am particularly interested in how to determine rod model parameters (geometrical and constitutive) from short-scale, atomistic information. Moreover, I am interested in using the model to help explain experimental data on large-scale DNA configurations.

Problems of Interest

I. Determination of rod model parameters from molecular dynamics simulations

What is the unstressed shape and flexibility of a given DNA sequence in a given solvent? This information enters any reasonable model of DNA through the "material parameters" and is difficult to measure directly. For a continuum rod model, these parameters can be extracted from state-of-the-art molecular dynamics (MD) simulations of short DNA segments in solvent. The key idea to this stochastic multi-scale matching is to consider an intermediate system of interacting rigid bodies in a heat bath. This system has a dual interpretation: it is a coarse model of the all-atom system, and a fine (spatially-discrete) model of the continuum rod system. This dual interpretation can be exploited to connect rod parameters to MD data assumed to be in statistical equilibrium.


Our stochastic multi-scale matching is based on three observations. First, spatial discretization of the rod model leads to equations analogous to those governing a system of interacting rigid bodies in a heat bath. Second, the long-time statistical behavior of this rigid body system is governed by a known invariant measure. This measure provides explicit connections between the rigid body (or discretized rod) constitutive parameters and the averages of certain functions. Third, approximate numerical values for these averages can be deduced from statistical data produced by MD simulations. This leads to numerical values for the constitutive parameters. There are current plans to implement this program using MD data produced by the Lavery group (CNRS, Paris).

II. Interpretation of cryo-electron microscopy data of DNA molecules.

There are various experimental methods for studying DNA configurations. These methods range from indirect spectroscopy techniques in which DNA shape is extracted from spectroscopic signals, to direct techniques like classic electron microscopy (EM). In classic EM one can directly observe DNA shape, however, this observed shape is influenced by many "uncontrollable" factors. These factors are connected with the EM specimen preparation process in which DNA molecules are adsorbed onto supporting films (which may distort the DNA shape). The cryo-EM technique is a direct method for studying DNA configurations that in some sense overcomes, or bypasses, the drawbacks of classic EM. In the cryo-EM method, DNA molecules are suspended in a thin layer of fluid, which is then cryo-vitrified (flash-frozen). It is believed that the flash-freezing process does not greatly distort the DNA shape, and that the resulting data represents something like a snapshot of a thermally-fluctuating DNA molecule. A continuum rod model of DNA as described above will provide a basic tool with which to test these conjectures and better interpret cryo-EM data. For example, with the model we should be able to answer questions such as the following:

• Does a DNA molecule have sufficient time to change its shape during flash-freezing?

• Does the "ensemble of shapes" observed in cryo-EM data correspond to an equilibrium ensemble of the dynamics? If so, what is the "effective temperature"?

III. Probing unstressed shape and flexibility of DNA.

One indirect way to quantify unstressed shape and flexibility of DNA is through cyclization or looping probabilities of relatively short molecules (about 200bp). Roughly speaking, a cyclization probability is the probability (steady-state in time) that a linear piece of DNA will close to form a loop. Cyclization probabilities are believed to be very sensitive to unstressed shape and flexibility, and so would provide a means to validate, and perhaps even determine, shape and flexibility data. The basic idea here is to experimentally measure cyclization probabilities for a number of different DNA molecules, and then to match this data with a continuum rod model as described above. This program has already been implemented with a static equilibrium rod model (no solvent, zero temperature) with some success by Manning and coworkers. The computational data captures very well the overall trends in the experimental data, but there is a systematic error which is likely attributable to the fact that thermal fluctuations ("entropy effects") are being neglected. A continuum rod model of DNA that accounts for thermal fluctuations will provide a basic tool to better understand and exploit experimental data on cyclization probabilities.

Is it possible for torsional waves, or more generally, internal elastic energy, to become localized at specific sites along a thermally-fluctuating DNA molecule? Torsional waves may be excited along a DNA axis in the process of transcription. With a continuum rod model of DNA as described above it will be possible to study the motion of such torsional waves and determine if energy can be localized at specific sites depending on the presence of local areas of sharp natural curvature, rapid variation in material parameters, etc. It is believed that an understanding of a mechanism for energy localization, taken together with Benham's model for the denaturation of the DNA double helix, may lead to predictions as to the probability of localized DNA denaturation during transcription and other biological processes.