Free-Energy Calculations for Semi-Flexible Macromolecules: Applications to DNA Knotting and Looping

Abstract

We present a method to obtain numerically accurate values of configurational free energies of semiflexible macromolecular systems, based on the technique of thermodynamic integration combined with normal-mode analysis of a reference system subject to harmonic constraints. Compared with previous free-energy calculations that depend on a reference state, our approach introduces two innovations, namely the use of internal coordinates to constrain the reference states and the ability to freely select these reference states. As a consequence, it is possible to explore systems that undergo substantially larger fluctuations than those considered in previous calculations, including semiflexible biopolymers having arbitrary ratios of contour length L to persistence length P . To validate the method, high accuracy is demonstrated for free energies of prime DNA knots with 20 L P = and 40 L P = , corresponding to DNA lengths of 3000 and 6000 base pairs, respectively. We then apply the method to study the free-energy landscape for a model of a synaptic nucleoprotein complex containing a pair of looped domains, revealing a bifurcation in the location of optimal synapse (crossover) sites. This transition is relevant to target-site selection by DNA-binding proteins that occupy multiple DNA sites separated by large linear distances along the genome, a problem that arises naturally in gene regulation, DNA recombination, and the action of type-II topoisomerases.

Loop-closure Kinetics Reveal a Stable, Right-handed DNA Intermediate in Cre Recombination

“…44 Here we employ an extension of this approach based on normal-mode analysis (NMA) to fit a structural model of the Cre-loxP synaptosome to the experimentally measured J-factor values, In recent work we developed a general method, termed TI-NMA, for computing the conformational free energy of DNA tertiary structures and nucleoprotein assemblies based on a combination of thermodynamic integration (TI) and NMA. 45 is the temperature); in terms of J this error is approximately within a factor of two. 46 We account here for this theoretical deviation as well as potential uncertainties in the exact value of Kb by introducing an adjustable multiplicative constant α relating computed J-factor curves to those obtained by applying NMA to our structural model, atomic-force and electron microscopy.…”

“…44 Here we employ an extension of this approach based on normal-mode analysis (NMA) to fit a structural model of the Cre-loxP synaptosome to the experimentally measured J-factor values, In recent work we developed a general method, termed TI-NMA, for computing the conformational free energy of DNA tertiary structures and nucleoprotein assemblies based on a combination of thermodynamic integration (TI) and NMA. 45 By augmenting NMA with TI it is possible to determine the conformational free energies of a wide range of nucleoprotein structures (and those of other macromolecules) across length scales.…”

“…where n is the number of base pairs in the Cre-loxP mediated DNA loop. NMA allows one to calculate the free energy and to identify the principal modes of vibration of macromolecular structures fluctuating about a mechanical-energy minimum and is mathematically equivalent to the harmonic approximation 45,46. The present structural model of the Cre-loxP synaptosome regards the looped nucleoprotein complex as a Cre half-tetramer anchoring the base of a DNA loop n base pairs in size.…”

Kinetic Pathways of Topology Simplification by Type-II Topoisomerases in Knotted Supercoiled DNA

“…Following previous studies (18,19,21,37) circular duplex DNA is modeled as a discrete semi-flexible chain with N extensible segments of mean length 0 10 b = nm, corresponding to a total chain length of 0 L Nb = ; in this work we use 200 N = corresponding to 6-kbp DNA (each segment has approximately 30 bps). The potential energy of a chain conformation is given by The twisting energy constant is given by…”

“…where C is the torsional rigidity constant of DNA; using Equilibrium ensembles of chains with fixed knot type K and linking number difference Lk Δ were generated by Monte Carlo (MC) simulation. In our procedure, chain conformations evolved by crankshaft rotations and stretching moves of sub-chains (21), and sub-chain translations, or reptations, along the local chain axis; the purpose of reptation moves was to increase the probability of extrusion and resorption of superhelix branches (37).…”