Rigid constraints

The foregoing approaches used an umbrella potential to restrain q. The pmf W(q) can also be obtained from simulations where q is constrained to a series of values spanning the region of interest [48,49]. However, the introduction of rigid constraints complicates the theory considerably. Space limitations allow only a brief discussion here for details, see Refs. 8 and 50-52. 

Bead-spring models without explicit solvent have also been used to simulate bilayers [40,145,146] and Langmuir monolayers [148-152]. The amphi-philes are then forced into sheets by tethering the head groups to two-dimensional surfaces, either via a harmonic potential or via a rigid constraint. 

As is apparent from Figure 10.1, an a-helical structure imposes fairly rigid constraints on the relative positions of successive residues in a peptide chain. Thus there is a loss of entropy that must be overcome energetically in order for an a-helix to form. To explain the underlying biophysics of this system, John Schellman introduced a theory of helix-coil transitions that is motivated by the Ising model for one-dimensional spin system in physics [180, 170],... 

For all its comparisons to 14C, the 41Ca method has potentially serious deficits that could easily put quite rigid constraints on the types of deposi-tional environments that could be expected to give straightforward results. For example, the fact that lithospheric rather than atmospheric production predominates raises the strong possibility that localized mixing and erosion... 

Electrons may ordinarily pass with readiness from one position in the outer shell to another. Nevertheless, they are held in position by more or less rigid constraints, and these positions and the magnitude of the constraints are determined by the nature of the atom and of such other atoms as are combined with it. 

It is tacitly assumed that all degrees of freedom are included in Eqs. (113) and (115). Thus, the formulas are valid only in the absence of constraints. The unconstrained contravariant metric tensor (denoted now for clarity by M )) can be used to obtain the A x, 4-dimensional contravariant metric tensor in the presence of P = 3N — A rigid constraints as [1,55]... 

In contrast, random displacements of individual sites of a chain (or a few neighboring sites), when feasible, can be a valuable tool (see Fig. 1). For this approach to be applicable, the chain backbone cannot have rigid constraints (e.g., rigid bonds and bond angles). It is particularly effective for coarse-grained models that allow wide fluctuations of individual sites around their bonded neighbors. Relevant examples are the bond-fluctuation model and certain bead-spring models such as that employed by Binder... 

Pro is an imino acid and the imino ring confers conformationed stability upon Pro, imposes rigid constraints on rotation about the N-C bond of the backbone, and disrupts the helical structure. The peptide bond preceding a Pro residue is more likely to adopt the cis configuration. Pro is an especially common residue at hairpin turns in globular proteins. 

If a component is at a different temperature than its surroimding attachments then stresses will develop. For example, a rod attached to a rigid constraint will be placed under a thermal stress if it is at a different temperature than the constraint. If the constraint is at temperature and the rod at temperature T, a strain a(TQ—r,) will develop in the rod, where a is the thermal expansion coefficient of the rod. If the rod is linearly elastic, the thermal stress ctj. will be given by aj=Ea Tf —T, where Eis Young s modulus. Clearly, the situation is more complex if the rod can creep, as these stresses may relax over time. For this chapter, it will be assumed that the ceramics are linearly elastic and isotropic, in order to set out the basic principles. 

To apply the theory to such general systems, we have to consider a system with rigid constraints. However, in this section we shall first consider the case in which there are no rigid constraints, i.e., the force SWdR is finite and well behaved. 

We now consider the case where the beads are subject to rigid constraints. This is necessary to deal with the problems of suspensions of a rigid body, or polymers with rigid constraints (such as the rodlike polymer, or the freely jointed model), but the reader who is interested only in flexible polymers can omit this section. 

Phenomenologically, the viscous stress is the stress which vanishes instantaneously when the flow is stopped. On the other hand the elastic stress does not vanish until the system is in equilibrium. The elastic stress is dominant in concentrated polymer solutions, while viscous stress often dominates in the suspensions of larger particles for which the Brownian motion is not effective. Whichever stress dominates, the rheological properties can be quite complex since both and are functions of the configuration of the beads and therefore depend on the previous values of the velocity gradient. Note that the viscous stress only appears in the system with rigid constraints.t... 

Last, in order to overcome some shortcomings in the KMC approach—such as the requirement for defining all transitions prior to the start of the simulation—linking KMC with molecular dynamics or with inputs from atomistic calculations would provide more quantitative support for the energy barrier calculations and allow for less rigid constraints on the simulated structure [44—46]. 

Thus, it is argued that tte rigid constraint model can provide exact results for important structural properties [49-51]. This assumption has been adopted by workers in the area of off-lattice polymer simulations [45,47,51], and will be retained in this work. However, we wish to point out that the validity of this assumption has been tested mostly on short chain molecules [55, 56, 58-60], and on a small protein [57]. As our computational facilities and methods improve, and it becomes possible to work with denser systons of longer chains, this assumption must be scrutinized closely and carefully, since the approximation might deteriorate with chain length. 

What remains to be done is to translate the configuration information of the pendulum, i.e., the rigid constraint between the pivot point and the point mass... 

Note that if the constraint would be elastic, the gyristors (GR) would stay in the model and represent the fictitious forces like the centrifugal force (in case of a rigid constraint, the corresponding velocity and thus the corresponding power is zero, such that the contribution becomes irrelevant for the behavior). 

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