Force matching procedure

The aim of the force matching procedure is to obtain the effective pair-force between CG sites using the force data obtained from a detailed atomistic molecular dynamics (MD) trajectory. The current implementation of the force-matching method closely follows the formulation from Refs. [23, 24], 

Assume that there are a total of N coarse-grained sites in the system for any one MD snapshot (p = l), with coordinates (r = jq, y, z ) and net forces, F), (where i = 1 - N) acting on them, and that these are known from the atomistic MD trajectory data. If /)r.) represents the force acting on the ith CG site due to 

Here the pair-force fj (r, r -) is unknown, so a model pair-force fij(r , rj, p, P2 pm) is chosen, which depends linearly upon m unknown parameters p, p2 - Pm- Consequently, the set of Eq. (8-2) is a system of linear equations with m unknowns p, P2 - - Pm- The system (8-2) can be solved using the singular value decomposition (SVD) method if n m (over-determined system), and the resulting solution will be unique in a least squares sense. If m n, more equations from later snapshots along the MD trajectory should be added to the current set so that the number of equations is greater than the number of unknowns. Mathematically, n = qN m where q is the number of MD snapshots used to generate the system of equations. 

It is important to note that model pair-forces for the interactions A-A, A-B, A-C, etc. are different from each other although they may have the same functional form. If required, the interaction between two non-bonded A CG sites (A-Anon-imncieci) can be treated differently from the interaction between two bonded A CG sites (A-A bonded)-In a system with A, B. .. E as chosen coarse-grained sites, 

Clearly, the total number of unknowns that need to be determined is m = a + +. .. + z and a solution set for parameters p, P2 pm is determined using the singular value decomposition or any other suitable method. The mean pair-force corresponding to the potential of mean force can be obtained in a systematic manner by averaging a number of sets of solutions for parameters p, P2 Pm obtained along the atomistic MD trajectory in which the phase space is sampled extensively. 

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The reduction in the number of degrees of freedom can lead to an incorrect pressure in the simulation of the coarse-grained systems in NVT ensembles or to an incorrect density in NPT ensembles [24], The pressure depends linearly on the pair-forces in the system, hence the effect of the reduced number of degrees of freedom can be accounted for during the force matching procedure [24], If T is the temperature, V the volume, N the number of degrees of freedom of the system, and kb the Boltzmann constant then the pressure P of a system is given by... 

Site-site W-W radial distribution function obtained from CGMD simulation and compared with that of the atomistic MD simulation using the force-matching procedure. 

So far, CG approaches offer the most viable route to the molecular modeling of self-organization phenomena in hydrated ionomer membranes. Admittedly, the coarse-grained treatment implies simplifications in structural representation and in interactions, which can be systematically improved with advanced force-matching procedures however, it allows simulating systems with sufficient size and sufficient statishcal sampling. Structural correlations, thermodynamic properties, and transport parameters can be studied. 

In the second step, parameters of renormalized interaction energies between beads are specified, defining the force field under which the system trajectory evolves. Interactions between beads could be determined by force matching procedures from atomistic interactions (Izvekov and Violi, 2006 Izvekov et al., 2005) or by fitting of experimental structural correlation functions (Marrink et al., 2007). 

An early version of a CG model with explicit solvent was developed by Smit et al., to study the dynamical interface between water and oil [26]. A similar strategy was also used by Goetz and Lipowsky [30] to simulate the self-assembly of a model surfactant into micelles and bilayers. Later, Klein and coworkers employed thermodynamic properties derived from atomistic simulations to develop a CG model for surfactants that includes the chemical structure [24, 31]. In this form, the procedure used to obtain the simplified potential functions of the CG model bears some level of similarity to the force-matching method used to fit simple potential functions for pairs of atoms against a fully electronic description [32]. Voth and coworkers later essentially followed this latter approach to also define an algorithm based on the force-matching procedure specific for CG-MD [33-35]. 

Briefly, the force matching procedure implies definition of a penalty funetion ... 

Figure 12.33 Curve matching procedure to obtain the network draw ratios for polyester fibres. (Reproduced with permission from Long, S.D. and Ward, I.M. (1991) Shrinkage force studies of oriented polyethylene terephthalate. ]. Appl. Polym. Sci., 42, 1921. Copyright (1991) John Wiley Sons, Inc.)...

The process of finding frame-to-frame correspondences can be formulated as the search of the nearest neighbour from one set of descriptors for every element of another set. It s called the matching procedure. There are two main algorithms for descriptor matching in OpenCV Brute-force matcher and FLANN-based matcher. In this case, the former has been due to its better permanence with ORB. 

You need to develop documented procedures that define your subcontractor evaluation and selection process and in certain cases this may result in several closely-related procedures for use when certain conditions apply. Do not try to force every purchase through the same selection process. Having purchasing policies that require three quotations for every purchase regardless of past performance of the current subcontractor is placing price before quality. Provide flexibility so that the policies and procedures complexity match the risks anticipated. Going out to tender for a few standard nuts and bolts would seem unwise. Likewise, placing an order for lm of equipment based solely on the results of a third party ISO 9000 certification would also seem unwise. 

Stereo view of the sixty propane minima (thick lines) obtained with the modified force field (see text) on the surface of the A peptide chain (medium lines) of the GCN4 leucine zipper (PDB code 2ZTA). Although the peptide chain was removed during the MCSS procedure, its backbone and hydrophobic side chains are also drawn (thin lines) to show how the propane minima match the aliphatic groups of chain B. Hydrophobic residues are labeled at their Ca atom.Five clusters of propane minima that do not match the hydrophobic side chain of the helix involved in the interhelical interactions are labeled from A (top right) to E (bottom center) and discussed in the text. 

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