Asymmetric periodic potential

We assume that, after the protein has entered the proteasome, the protein-proteasome interaction is characterized by a spatially periodic asymmetric potential U(x) with the period P equal to the distance between amino adds in the protein. In reality there is a basic periodicity, namely the periodidty of the protein (or peptide) backbone, that is superposed by a nonperiodic (in our sense irregular) part that is attributed to the amino add-spedfic residues. Below we consider also the influence of the nonperiodic constituent. The spatial asymmetry results from breaking the symmetry by entering the proteasome from one end, as well as from the C — N asymmetry of the protein (or peptide) backbone. Figure 14.3 (left) plots several examples of such asymmetric periodic potential. The detailed form of the asymmetric periodic interaction potential is of less importance for this qualitative study. 

Fig. 14.4 Left different examples of asymmetric periodic potential U(x) with the period P = 1. Only three periods are shown. From top to bottom U (x) is a saw-tooth function with smoothed angles (for details see [25] U(x) = —(0.25sin(2jrx/P) +...

Fig. 8 A power stroke model for a bacterial flagellar motor that has been described in terms of two asymmetric sawtooth potentials whose spatial periods are out of phase with one another. In the original depiction [24], only the long green arrows pointing downward in the shaded reaction windows were drawn for the chemical (proton binding and release) transitions, but this approximation represents a logically impossible limit...

Remoissenet, M. and Peyrard, M., Soliton dynamics in new models with parameterized periodic double-well and asymmetric substrate potentials, Phys. Rev. B, 29 (6), 3153-3166, 1984. 

Furthermore, the situation becomes even worse for an asymmetric potential like that in (3.18), because at low temperature nearly the entire period p is spent on dwelling in the potential well (see appendix A), so that lim -oo < >ins = 0- In other words, unless the potential is strictly symmetric, the transition state position x tends to the minimum of the initial state It is natural to expect that the centroid approximation will work well when x does not deviate too far from x. To summarize, the centroid method is an instructive way to describe in a unique TST-like manner both the high [T > T ) and fairly low [T < T ) temperature regions, but it does not give a reliable estimate for k. ... 

In most systems the substrate electrodes are larger than the powered electrodes. This asymmetric configuration results in a negative dc self-bias voltage Vdc on the powered electrode. Without that, the difference in electrode areas would result in a net electron current per RF period [134, 169]. It has been shown that the ratio of the time-averaged potential drops for the sheaths at the grounded (V g) and the powered electrode (Vsp) are inversely proportional to a power of the ratio of the areas of the two electrodes (Ag, Ap) [134, 170-172] ... 

In the presence of reflection symmetry with respect to the diagonal of the potential-energy surface, as in symmetric molecules or in the four-disk scatterer, Burghardt and Gaspard have shown that a further symmetry reduction can be performed in which the symbolic dynamics still contains three symbols A = 0, +, - [10]. The orbit 0 is the symmetric-stretch periodic orbit as before. The orbit + is one of the off-diagonal orbits 1 or 2 while - represents a half-period of the asymmetric-stretch orbit 12. Note that the latter has also been denoted the hyperspherical periodic orbit in the literature. 

Approximately 25% of all patients with hypertrophic cardiomyopathy (HCM) have latent left ventricular outflow obstruction with an intraventricular gradient (I). Pathophysiologic features are asymmetric hypertrophy of the septum and a systolic anterior movement of the anterior leaflet. Medical treatment includes betablockers, and calcium antagonists of the verapamil type. Approximately 5— 10% of the patients with outflow obstruction are refractory to such negative inotropic therapy (2). Positive inotropic drugs such as digitalis or sympathomimetics are strictly contraindicated. In the presence of atrial fibrillation, anticoagulation therapy should be started. Since endocarditis is more common in patients with HCM because of turbulence in the left ventricle, prophylactic antibiotics should be administered for periods of potential bacteraemia. 

Considering the highly processive mechanism of the protein degradation by the proteasome, a question naturally arises what is a mechanism behind such translocation rates Let us discuss one of the possible translocation mechanisms. In [52] we assume that the proteasome has a fluctuationally driven transport mechanism and we show that such a mechanism generally results in a nonmonotonous translocation rate. Since the proteasome has a symmetric structure, three ingredients are required for fluctuationally driven translocation the anisotropy of the proteasome-protein interaction potential, thermal noise in the interaction centers, and the energy input. Under the assumption that the protein potential is asymmetric and periodic, and that the energy input is modeled with a periodic force or colored noise, one can even obtain nonmonotonous translocation rates analytically [52]. Here we... 

The exact mechanism arises in the process of inverse pre-dissociation, as discussed in detail by Herzberg (1966). During an atom-molecule collision, the reactants interact with one another subject to the relevant potential energy surface. The lifetime of this excited intermediate is on the order of molecular vibrational periods, or 10 s. The lifetime is a complex function of the chemical reaction dynamics, which in turn depends on the number of available states. In this specific instance, there is a state dependence for the isotopically substimted species. Ozone of pure has a Cav symmetry and has half the rotational complement of the asymmetric isotopomers. As a result, it was suggested that the extended lifetime for the asymmetric species leads to a greater probability of stabilization. While these assumptions are valid for a gas phase molecular reaction, they do not sufficiently account for the totality of the experimental ozone isotopic observations. Reviews by Weston (1999) and Thiemens (1999) have detailed the physical-chemical reasons. 

The observation that a macromolecular brush gets stretched as the side chains get adsorbed on a flat surface provides a means to stimulate molecular motility by desorption of the brush molecule or a segment of it. If the molecule is in a subsequent period allowed to relax to the adsorbed stretched state it will eventually do a step forward. This is depicted schematically in Figure 28 as a sort of a creep motion. Here, the desorbed state might be characterized as an excited state whose formation requires input of energy. In the case that the structure of the surface and of the molecule favor relaxation into a distinct direction, i.e., in the case of an asymmetric potential, the motion of the molecule can be become directed. 

For such an asymmetric structure as dimethoxymethane with one torsion angle fixed at 60 °, additional sine terms are necessary, in order to account for the lack of symmetry about 0 = 180° (see Ref. 94). The individual components V,(0), V2(0), and V3(0) of the total potential function V(0) can be identified with specific physical effects of similar periodicity. For example, the onefold term, V,(0) = 0.5 V ( 1 — cos 0), moves from a maximum v ue to a minimum value as 0 changes by 180 °. The same variation with torsion shows dipolar or steric interactions. The twofold term, V2(0), changes from a maximum to a minimum as 0 changes by 90°. This periodicity frequently corresponds to the change of delocalization interactions. Finally, the threefold term, V3(0), moves from a maximum value to a minimum value as it changes by 60°. This is generally attributed to the intrinsic torsion potential. 

Figure 11 Differential dihedral potential in ethane. The ab initio second derivatives of the energy with respect to the dihedral angle t h are plotted vs. Thh> showing the periodic asymmetric behavior of this coordinate. Although a computational observable, this curve is not experimentally accessible. Recalculated from ref. 118.

In the original work of Bergsma et al., the 5 2 reaction system was placed in a simulation unit cell with 64 fully flexible water molecules. The intermolecular water potential was taken to be that developed by Watts, 0 and periodic truncated octahedron boundary conditions were used. To generate initial conditions, the asymmetric stretch coordinate of the reaction system was constrained to its transition state value and all other degrees of freedom were equilibrated. In general, 500 fs of dynamics was run both forward and backward in time from each initial condition after releasing the constraint on the... 

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