Explicit counterion simulations

The predicted necklace conformations have been confirmed by simulations using only the Coulomb repulsion of the backbone charges [75], a Debye-Hiickel potential [75] and by simulations using the full Coulomb potential and explicit counterions [76]. In the following we will only review the simulations using explicit counterions. 

Klos JS, Sommer JU (2011) Monte Carlo simulations of charged dendrimer-linear polyelectrolyte complexes and explicit counterions. J Chtan Phys 134 204902... 

Molecular simulations of polyelectrolyte solutions at finite concentrations with explicit counterions and salt ions require special handling of the Coulombic interaction between the charges. These simulations are commonly performed under periodic boundary conditions to ensure that small sample surface effects are suppressed and that the results obtained for systems with various numbers of particles can be considered as approximations of properties of the hypothetical infinite system. The earliest computer... 

To investigate the effect of the Debye-Huckel approximation on the solution properties, Stevens and Kremer [152] performed molecular dynamics simulations of salt-free solutions of bead-spring polyelectrolyte chains in which the presence of counterions was treated via a screened Coulomb potential, and compared the results with their simulations with explicit counterions [146,148]. To elucidate the effect of the Debye-Hiickel approximation, the dependence of the mean square end-to-end distance, R ), osmotic pressure, and chain structure factor on polymer concentration was examined. Stevens and Kremer found that (i ) tends to be larger at low densities for DH simulations and is smaller at higher densities. However, the difference in (i ) between DH simulations and simulations with explicit counterions is within 10%. This trend seems to be a generic feature for all N in their simulations. The functional form and density dependence of the chain structure factor are very close in both simulations. The most severe Debye-Huckel approximation affects the dependence of the osmotic pressure on polymer concentration. It appears that in the DH simulations not only is the magnitude of the osmotic pressure incorrect, but also the concentration dependence is wrong. 

MD studies on B and Z forms of DNA with explicit waters and explicit counterions were reported by Swamy and Clementi.i G-C and A-T decamer sequences in their B forms were surrounded by a rectangular box with 1500 water molecules and 20 K+ ions. In addition, a G-C dodecamer in its Z form with 1851 water molecules and 24 K+ ions was studied. Water molecules in these studies were four-centered MCY waters. These simulations were carried out for a total of 7 ps, with the first 3 ps serving as an equilibration period. The DNA in all cases was rigid, and only the ions and the waters were allowed to execute motions. The dynamical behavior of those ions showed them to be strongly bound to the DNA with restricted mobilities, a conclusion different from what counterion condensation theory and other simulations tend to suggest. The exploration of space by the counterions around DNA in the short time scale of this study was insufficient to permit the derivation of general conclusions about the ion mobilities, however. 

Yin, D.-W., Yan, Q., de Pablo, J.J. Molecular dynamics simulation of discontinuous volume phase transitions in highly-charged crosslinked polyelectiolyte networks with explicit counterions in good solvent. J. Chem. Phys. 123, 174909 (2005). doi 10.1063/1. 2102827... 

Essential for MD simulations of nucleic acids is a proper representation of the solvent environment. This typically requires the use of an explicit solvent representation that includes counterions. Examples exist of DNA simulations performed in the absence of counterions [24], but these are rare. In most cases neutralizing salt concentrations, in which only the number of counterions required to create an electrically neutral system are included, are used. In other cases excess salt is used, and both counterions and co-ions are included [30]. Though this approach should allow for systematic smdies of the influence of salt concentration on the properties of oligonucleotides, calculations have indicated that the time required for ion distributions around DNA to properly converge are on the order of 5 ns or more [31]. This requires that preparation of nucleic acid MD simulation systems include careful consideration of both solvent placement and the addition of ions. 

Nevertheless, there is still much work to do in this field. The inclusion of solvent and/or counterions is just at the beginning, and solvent effects have been included with continuum models only. In the next years we will probably arrive to dynamically simulate the whole polymerization process in the presence of the counterion and of explicit solvent molecules. As for the experimental issues which have been not rationalized yet computationally, we remark that still it is not easy to model the relative activity of different catalysts, and even to predict if a certain catalyst will show any activity at all. Moreover, copolymerizations still represent an untackled problem. However, considering the pace at which the understanding of once obscure facts progressed it is not difficult to predict that also these challenges will be positively solved. 

Figure 20.6 Comparative cumulative distribution functions to assess the amount of counterion accumulation around the Tar—Tar complex, A-form RNA, and B-form DNA, respectively. Data were obtained from molecular dynamics simulations with explicit representations of water molecules and ions. The ordinate quantifies the number of counterions accumulated around the macroions within contours that have AG less than or equal to the value on the abscissa.

For the Tar—Tar kissing loops, the P—B calculations are unable to discern their propensity to accumulate counterions accumulation at the loop—loop interface (data not shown). This is because the fully hydrated ions as defined by the Stem layer cannot penetrate into the central cation binding pocket (data not shown). Similarly, the axial spine of counterion density observed in the A-RNA helix (Fig. 20.5) is not captured by the P—B calculation (Fig. 20.7). No noticeable sequence specificity is observed in the counterion accumulation patterns in the P—B calculations, even though the sequence effects are explicitly represented in the P—B calculation through the appropriate geometry and assignment of point-charges. This is because the sequence specificity observed in the molecular dynamics simulations usually involves first shell interactions of base moieties with partially dehydrated ions, which cannot be accurately represented in the P—B framework. 

N = 1000, a = lA,l = l0A,o = lNtm = 100 A, rj = 0.01, cE = 0.01 M, s2 = 1000 A2, = 80, r = -1.0, w = 0.0, and v = 1.0 A8. For small reservoirs, most counterions are not able to leave the brush hence the overall charging is small. This situation is not unlike some molecular dynamics simulations,20 where the counterions were explicitly taken into account, but the size of the system was limited (because of computational requirements). However, when the brush is embedded in an infinitely large reservoir and chemical equilibrium is reached, there are no counterions left in the brush, and a stronger stretching is achieved. 

Car-Parrinello techniques have been used to describe classical variables whose behavior, like quantum electrons in the Born-Oppenheimer approximation, is nearly adiabatic with respect to other variables. In simulations of a colloidal system consisting of macroions of charge Ze, each associated with Z counterions of charge —e, Lowen et al. [192] eliminated explicit treatment of the many counterions using classical density functional theory. Assuming that the counterions relax instantaneously on the time-scale of macroion motion, simulations of the macroion were performed by optimizing the counterion density at each time step by simulated annealing. 

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