Step perturbations

Fig. 1. A single step perturbation fails for the mutation of p-methylphenol to p-methoxyphenol because no configuration with a cavity large enough to accommodate the methoxy group is sampled in a simulation of p-methylphenol in water.

For constant energy simulations without temperature regulation, use heating steps of about 0.5 ps and a healing time of 20-30 ps. In gen eral, short h eating tim es and large temperature steps perturb th e initial system m ore than Ion gcr heating times and small tern -perature steps. 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

The basic FEP algorithm for ligand binding can be improved in several ways. One method is to use a nonphysical ligand that serves as the common reference state for a variety of ligands of interest [25]. This method, referred to as the one-step perturbation approach, appears to be quite successful even for complex and fairly diverse ligands [26],... 

Single-step perturbation methods have also been applied to electrostatic processes. One study probed the dielectric properties of several proteins at a microscopic level [41,42], Test charges were inserted at many different positions within or around each protein, and a dielectric relaxation free energy was computed, which is related to a microscopic dielectric susceptibility (see Sect. 12.3). 

Because the precipitation (or affinity binding step) perturbs the equilibrium of interest it has to be ensured that removal of one reactant is much faster than the dissociation or association of the binding partners. 

They were carried out over a 50 day period at different dilution rates and at different input substrate concentrations. In order to do these simulations more realistic, small fluctuations as well as drastic step perturbations were introduced alternately in the input concentrations (see Figures 15 to 18) and in the dilution rate (see Figure 14). The input variables X and were assumed negligible. The measurements of S, S2 and Pco- was calculated directly from the model (see Figures 19, 20 and 21). 

The real input concentrations shown in these figures were only used to validate hypothesis Hlg and to simulate the model (6) from which the measurements S , S2 and Pco2 were taken directly. These measurements are reported in Figures 43 to 45. The variables C j, X and were supposed to be negligible. As in the previous numerical example, small fluctuations as well as drastic step perturbations were alternatively introduced in the dilution rate and in the input concentrations to resemble actual industrial operating conditions. 

In spite of its implicit character, this expression is useful for kinetic purposes as it enables us to calculate kf(E) for any point on a measured E, t curve (the chronopotentiogram), being the response to a current step perturbation. 

Fig. 7. Examples of AE vs. t112 curves (response to small amplitude current step perturbation) for three different cases (a) slow charge transfer, (b) intermediate case, and (c) fast charge transfer. From ref. 32.

Another technique related to the principle of multi-potential step perturbation is square-wave polarography [52], However, because its theory is not essentially different from that of the much more popular a.c. method, it will not be treated here. 

Potential step perturbation chrono-amperometry, pulse polaro-graphy, and d.c. polarography... 

In more than one respect, the small-amplitude sinuosoidal a.c. method can be superior to the large-amplitude step methods for the study of coupled homogeneous reactions. First, the wide range of frequencies at which meaningful data can be obtained will correspond to an equally wide range of rate constants on which, in principle, information can be obtained. Second, the a.c. perturbation can be superimposed on a large-amplitude d.c. or step perturbation so that information in the time scale of the latter is incorporated as well. Moreover, this affords an internal check on the reliability of data interpretations. Finally, it is important... 

McBride, D.W., Jr., and Hamill, O.P., 1992, Pressure-clamp a method for rapid step perturbation of mechanosensitive channels, Pflugers Arch, 421, pp 606-612. 

Oostenbrink C, van Gunsteren WF. Single-step perturbations to calculate free energy differences from unphysical reference states limits on size, flexibility, and character. J Comput Chem 2003 24 1730-1739. 

An alternative approach widely used in polyatomic molecule studies is based on the Golden Rule and a perturbative treatment of the anharmonic coupling (57,62). This approach is not much used for diatomic molecules. In the liquid O2 example cited above, the Hamiltonian must be expanded to 30th order or so to calculate the multiphonon emission rate. But for vibrations of polyatomic molecules, which can always find relatively low-order VER pathways for each VER step, perturbation theory is very useful. In the perturbation approach, the molecule s entire ladder of vibrational excitations is the system and the phonons are the bath. Only lower-order processes are ordinarily needed (57) because polyatomic molecules have many vibrations ranging from higher to lower frequencies and only a small number of phonons, usually one or two, are excited in each VER step. The usual practice is to expand the interaction Hamiltonian (qn, Q) in Equation (2) in powers of normal coordinates (57,62) ... 

Chronoamperometry — Chronoamperometry belongs to the family of step techniques [i-iv]. In chronoamperometry the current is measured as a function of time after application of a potential step perturbation. If the potential is stepped from Iq, where no current flows, i.e., the oxidation or reduction of the electrochemically active species does not take place, to 2 where the current... 

The general aspect of this curve with an inflexion point is typical of the dynamic behaviour of a second-order overdamped system in response to a step perturbation, i.e. Heaviside function [4] A property of this curve is that the concentration reached at the plateau is equal to the concentration of the fluid entering R2. Data of these runs are summarized in table 2 and presented in Figures 3 and 4. 

Schelly, Z.A., and E.M. Eyring. 1971. Step perturbation relaxation techniques. J. Chem. Ed. 48 A639-654. 

In the case of an anay of band electrodes or interdigitated electrode structures, the width of each single electrode element and the gap between the electrode elements must be considered carefully in the sensor design. Interactions between electrode elements and their effects on the transient response to a potential step perturbation will directly affect the overall sensor output. When an interdigitated electrode structure is used, chemical cross-talk among the reactants and products in both electrode elements (cathode and anode) may occur, which will then influence the sensor output [3], Therefore, the relative location of the sensing elements is also an essential consideration. 

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