Boltzmann inversion

In structure matching methods, potentials between the CG sites are determined by fitting structural properties, typically radial distribution functions (RDF), obtained from MD employing the CG potential (CG-MD), to those of the original atomistic system. This is often achieved by either of two closely related methods, Inverse Monte Carlo [12-15] and Boltzmann Inversion [5, 16-22], Both of these methods refine the CG potentials iteratively such that the RDF obtained from the CG-MD approaches the corresponding RDF from an atomistic MD simulation. 

Both the inverse Monte Carlo and iterative Boltzmann inversion methods are semi-automatic since the radial distribution function needs to be re-evaluated at... 

RDFs calculated from a CG simulation using these initial interaction potentials differ from those calculated from the atomistic simulation. An inverse Monte Carlo algorithm is therefore used to iteratively refine these interaction potentials by correcting them by the difference between the CG and atomistic RDFs. This is essentially the same method that was used by Shelley et al to derive the parameters between the CG lipid head group particles, and is also similar to the Boltzmann inversion method, " which also uses an iterative procedure that uses RDFs measured from atomistic simulations to derive CG interaction potentials. Lyubartsev has used this method to fully parameterize his own CG model of DMPC. 

The NAST [16, 34] model represents each nucleotide by one pseudoatom at the C3 atom of the ribose group. NAST utilizes MD simulations and a force field parameterized from solved rRNA structures. NAST relies upon information from an accurate secondary structure and can also include experimental constraints. These constraints are modeled by a harmonic energy term. The bonded energy terms of distance, angle, and dihedral are further modeled by a harmonic potential, parameterized according to a Boltzmann inversion. Non-bonded interactions are modeled by a Lennard-Jones potential with a hard sphere radii of 5 A. Due to the low-resolution representation of one pseudoatom per nt, the conversion from the CG model to the all-atom model is complex and may produce steric overlaps. In order to overcome this difficulty, Jonikas et al. developed a program C2A [35] which is able to insert and minimize the all atom structure. 

In another approach, He et al. (He et al., 2013) proposed a 2-site per nucleotide (NARES-2P, nucleic acid united residue 2-point model) CG model where chain connectivity, excluded volume and base dipole interactions are sufficient to form helical DNA and RNA structures. This model was parametrized using a bottom-up strategy by employing a set of statistical potentials, derived from DNA and RNA structures from the Protein Data Bank, and the Boltzmann inversion method to reproduce the structural features. The base-base interactions were parametrized by fitting the potential of mean force to detailed all-atoms MD simulations using also the Boltzmann inversion approach. The respective potentials do not explicitly define the nucleic-acid structure, dynamics and thermod3mamics, but are derived as potentials of mean force. By detailed analysis of the different contribution to the Hamiltonian, the authors determined that the multipole-multipole interactions are the principal factor responsible for the formation of regular structures, such as the double helical structures. 

Development of a Coarse-Grained Water Forcefield via Multistate Iterative Boltzmann Inversion... 

Moore, T.C., lacovella, C.R., McCabe, C. Derivation of coarse-grained potentials via multistate iterative Boltzmann inversion. J. Chem. Phys. 140, 224104 (2014)... 

Bayramoglu, B., Faller, R. Coarse-grained modeling of polystyrene in various environments by iterative Boltzmann inversion. Macromolecules 45, 9205-9219 (2012)... 

Concerning the secrnid option to generate numerically a tabulated potential that closely reproduces a given melt structure, the iterative Boltzmann inversion (IBl) method [29,41, 51, 52] has been developed. 

This correction can be inserted into the Boltzmann-inversion iterations to adjust the pressure to the target value. 

Fig. 3 Chemical structure of P3 H I and Cgo with definition of CG beads l ). Right panels show selected CG nonbonded potentials betwetai CG sites dashed line), obtained through iterative Boltzmann inversion. The atomistic target radial distributirai functions solid line) and its CG analogous dotted line) are indistinguishable rai the figure scale. Reprinted with pcainission from [38]. Copyright 2010 American Chemical Society...

This is a sequential method, the simulations at atomistic and CG levels are treated separately. Potentials for collection of atoms i.e., CG beads) are derived from the all atomistic MD simulations. There are different ways to get the potentials for the coarse grained systems. Derivation of potentials from Boltzmann inversion of several bonded and non-bonded distributions (structure based) and force matching from all atomistic MD trajectories followed by their applications will be discussed in the following sections. 

Iterative Boltzmann Inversion Coarse Graining (IBICG). IBICG method was introduced and successfully applied to macromolecules by Kremer et and Muller-Plathe et jjjg method consists of steps like... 

Initially the coordinates of the CG beads are mapped from the atomistic details MD simulation trajectory, i.e., both the coordinate file and the trajectory file are been created for the CG beads. This CG trajectory derived from atomistic details MD trajectory is used to calculate the probability distribution of bonds, angles and dihedral, which further serve as reference distributions (or target distribution). The potentials for these bonded interactions are derived from above distributions by Boltzmann inversion. So the potential of mean force (PMF) for bonds, angles and dihedrals are represented in the following equation. 

The CG force field of the atactic PS should reproduce the same distributions of three bonds and six angles as mentioned above. It should also match to the intra and inter molecular RDFs extracted from the atomistic MD simulation trajectories. In this case, similar approach like PA-66 was adopted to calculate all the bonded and non-bonded potentials. For the bonded potential the bond and angle distributions were fitted to a combination of Gaussians (equation 6) and further Boltzmann inversion was performed. However, for non-bonded interactions the IBI technique was used. 

Figure 8 shows the resulting potentials at three temperatures. In principle, effective potentials obtained from the Boltzmann inversion [(12) and (13)] will contain entropic contributions and, hence, in general must be state-point dependent however. Fig. 8 suggests that in favorable cases this dependence is small. 

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