Molecular Poisson

A number of issues need to be addressed before this method will become a routine tool applicable to problems as the conformational equilibrium of protein kinase. E.g. the accuracy of the force field, especially the combination of Poisson-Boltzmann forces and molecular mechanics force field, remains to be assessed. The energy surface for the opening of the two kinase domains in Pig. 2 indicates that intramolecular noncovalent energies are overestimated compared to the interaction with solvent. 

Elamrani et al. 1996] Elamrani, S., Berry, M.B., Phillips Jr., G.N., McCammon, J.A. Study of Global Motions in Proteins by Weighted Masses Molecular Dynamics Adenylate Kinase as a Test Case. Proteins 25 (1996) 79-88 [Elcock et al. 1997] Elcock, A.H., Potter, M.J., McCammon, J.A. Application of Poisson-Boltzmann Solvation Forces to Macromolecular Simulations. In Computer Simulation of Biomoleeular Systems, Vol. 3, A.J. Wilkinson et al. eds., ESCOM Science Publishers B.V., Leiden... 

The Poisson equation has been used for both molecular mechanics and quantum mechanical descriptions of solvation. It can be solved directly using numerical differential equation methods, such as the finite element or finite difference methods, but these calculations can be CPU-intensive. A more efficient quantum mechanical formulation is referred to as a self-consistent reaction field calculation (SCRF) as described below. 

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

That the Poisson distribution results in a narrower distribution of molecular weights than is obtained with termination is shown by Fig. 6.11. Here N /N is plotted as a function of n for F= 50, for living polymers as given by Eq. (6.109). and for conventional free-radical polymerization as given by Eq. (6.77). This same point is made by considering the ratio M /M for the case of living polymers. This ratio may be shown to equal... 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

This relation was verified experimentally7 49 and it was shown that the degree of polymerization in a system containing "living polymers is independent of concentrations of initiator or monomer and of temperature. Furthermore, if all the growing centers were formed in a time much shorter than the time of polymerization, a Poisson molecular weight distribution would be obtained. Indeed, by using this technique samples of polystyrene were obtained for which MjMn = 1.04. 

The Debye temperature, can be calculated from the elastic properties of the solid. Required are the molecular weight, molar volume, compressibility, and Poisson s ratio.11 More commonly, do is obtained from a fit of experimental heat capacity results to the Debye equation as shown above. Representative values for 9o are as follows ... 

The behavior of the strain softened material resembles the behavior of rubberlike polymers. For instance, the Poisson s ratio of an ideally plastic material is also close to 0.5 [94, 95], Proper understanding of crack propagation involves the microscopic level. Apparently, the load is transmitted by the molecular strands [97] from one crosslink to the next crosslink, exactly, as it is in rubberlike materials. However, two things are different in strain softened polymers as compared to rubberlike materials ... 

The catalytic single-step Alfen process has a good space-time yield, and the process engineering is simple. The molecular weight distribution of the olefins of the single-step process is broader (Schulz-Flory type of distribution) than in the two-step Alfen process (Poisson-type distribution) (Fig. 2). As a byproduct 2-alkyl-branched a-olefins also are formed, as shown in Table 6. About... 

The molecular weight distribution of the obtained homologous fatty alcohols corresponds with a relatively narrow Poisson distribution and equals the molecular weight distribution of the olefins in the two-step Alfen process [38a]. 

Furthermore, the reaction scheme implies that the molecular weight distribution is Poisson-like — i.e. very narrow — as it had been shown earlier on theoretical basis by Flory 8), Gold 9), and Szwarc l0>. Even though two (or more) types of active species add monomer at very different rates, the polydispersity remains narrow, provided solvation/desolvation and ionic dissociation/association processes are fast U). 

The distinguishing feature of such a mechanism occurs in the fact that the growth of all polymer molecules proceeds simultaneously under conditions affording equal opportunities for all. (This will hold provided the addition of monomer to the initiator is not much slower than succeeding additions.) These circumstances are unique in providing conditions necessary for the formation of a remarkably narrow molecular weight distribution—much narrower than may be obtained by polymer fractionation, for example. Specifically, they are the conditions which lead to a Poisson distribution of the number and mole fraction, i.e. ... 

An attempt has been made by Spiering et al. [39,40] to relate the magnitude of the interaction parameter F(x) as derived from experiment to the elastic interaction between HS and LS ions via an image pressure [47]. To this end, the metal atoms, inclusive of their immediate environments, in the HS and LS state are considered as incompressible spheres of radius /"h and Tl, respectively. The spheres are embedded in an homogeneous isotropic elastic medium, representing the crystal, which is characterized by specific values of the bulk modulus K and Poisson ratio a where 0 < a < 0.5. The change of molecular volume A Fas determined by X-ray diffraction may be related to the volume difference Ar = Ph — of the hard spheres by ... 

Archontis, G. Simonson, T. Karplus, M., Binding free energies and free energy components from molecular dynamics and Poisson-Boltzmann calculations. Application to amino acid recognition by aspartyl-tRNA synthetase, J. Mol. Biol. 2001, 306, 307-327... 

A second type of anisotropic system is the biaxially oriented or planar random anisotropic system. This type of material is illustrated schematically in Figure 2A. Four of the five independent elastic moduli are illustrated in Figure 2B in addition there are two Poisson s ratios. Typical biaxially oriented materials are films that have been stretched in two directions by either blowing or tentering operations, rolled materials, and fiber-filled composites in which the fibers are randomly oriented in a plane. The mechanical properties of anisotropic materials arc discussed in detail in following chapters on composite materials and in sections on molecularly oriented polymers. 

Thermal initiation and ordinary bimolecular termination also occur during polymerization in addition to initiation by the dissociation of the adduct or the active polymer chain-end dissociation and reversible temination (formation of the dormant species). Therefore, the degree of the control of the molecular weight and the molecular weight distribution is determined by the ratio of the polymer chains produced under control and uncontrol. If the contribution of the thermal initiation and bimolecular termination is very small, the molecular weight distribution is close to the Poisson distribution, i.e., Mw/Mn=1 + 1/Pn, where Pn is the degree of polymerization. It was shown that when the number of... 

It is not uncommon for protons to be taken up or released upon formation of a biomolecular complex. Experimental data on such processes can be compared to computational results based on, for example, Poisson-Boltzmann calculations.25 There is a need for methods that automatically probe for the correct protonation state in free energy calculations. This problem is complicated by the fact that proteins adapt to and stabilize whatever protonation state is assigned to them during the course of a molecular dynamics simulation.19 When the change in protonation state is known, equations are available to account for the addition or removal of protons from the solvent in the overall calculation of the free energy change.11... 

A solvated MD simulation is performed to determine an ensemble of conformations for the molecule of interest. This ensemble is then used to calculate the terms in this equation. Vm is the standard molecular mechanics energy for each member of the ensemble (calculated after removing the solvent water). G PB is the solvation free energy calculated by numerical integration of the Poisson-Boltzmann equation plus a simple surface energy term to estimate the nonpolar free energy contribution. T is the absolute temperature. S mm is the entropy, which is estimated using... 

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