Spring, harmonic

In this section we describe the behavior of a ligand subjected to three types of external forces a constant force, forces exerted by a moving stiff harmonic spring, and forces exerted by a soft harmonic spring. We then present a method of reconstruction of the potential of mean force from SMD force measurements employing a stiff spring (Izrailev et al., 1997 Balsera ct al., 1997). 

To enable an atomic interpretation of the AFM experiments, we have developed a molecular dynamics technique to simulate these experiments [49], Prom such force simulations rupture models at atomic resolution were derived and checked by comparisons of the computed rupture forces with the experimental ones. In order to facilitate such checks, the simulations have been set up to resemble the AFM experiment in as many details as possible (Fig. 4, bottom) the protein-ligand complex was simulated in atomic detail starting from the crystal structure, water solvent was included within the simulation system to account for solvation effects, the protein was held in place by keeping its center of mass fixed (so that internal motions were not hindered), the cantilever was simulated by use of a harmonic spring potential and, finally, the simulated cantilever was connected to the particular atom of the ligand, to which in the AFM experiment the linker molecule was connected. 

As our first model problem, we take the motion of a diatomic molecule under an external force field. For simplicity, it is assumed that (i) the motion is pla nar, (ii) the two atoms have equal mass m = 1, and (iii) the chemical bond is modeled by a stiff harmonic spring with equilibrium length ro = 1. Denoting the positions of the two atoms hy e 71, i = 1,2, the corresponding Hamiltonian function is of type... 

The bead model for polymer simulations. The heads may he connected by stiff rods or by harmonic springs. 

The usual structure of off-lattice chain models is reminiscent of the Larson models the water and oil particles are represented by spheres (beads), and the amphiphiles by chains of spheres which are joined together by harmonic springs... 

The dependence on the nuclear positions is indicated by r and the dependence on the Drude positions is indicated by d. In Eq. (9-25) Ubond (r) is the intramolecular energy contribution from, typically, the bond lengths, valence angles, and dihedral angles, Ulj (r) is a Lennard-Jones 6-12 nonpolar contribution, Ueiect (r, d) represents all Coulombic interactions, atom-atom, atom-Drude, and Drude-Drude, and Useif (d) represents the atom-Drude harmonic bonds. The term Usey (d) arises from the harmonic spring separating the two charges and has the simple expression... 

For simplicity, let us consider a molecular system with a Hamiltonian J o(z) that is coupled to a harmonic spring with spring constant k and a time-dependent minimum r(t). The explicitly time-dependent Hamiltonian of the complete system is then... 

With neglect of the quantum effects that arise from the exchange of identical particles [147], (8.66) gives the exact quantum partition function in the limit P — oo. For finite P, Qp((3) is the canonical partition function of a classical system composed of ring polymers. Each quantum particle corresponds to a ring polymer of P beads in which neighboring beads are connected by harmonic springs with force... 

Figure 1. The N-particle ding-a-ling model. Odd particles can move freely in between two collisions, while even particles are bounded by a harmonic spring.

The diode model consists of two segments of nonlinear lattices coupled together by a harmonic spring with constant strength kint (see Fig. 6). Each segment is described by the (dimensionless) Hamiltonian ... 

The essential properties of incommensurate modulated structures can be studied within a simple one-dimensional model, the well-known Frenkel-Kontorova model . The competing interactions between the substrate potential and the lateral adatom interactions are modeled by a chain of adatoms, coupled with harmonic springs of force constant K, placed in a cosine substrate potential of amplitude V and periodicity b (see Fig. 27). The microscopic energy of this model is ... 

Exercise. Consider a string of point masses with harmonic springs between neighbors and fixed at the ends ... 

The potential for a particle bound to a massless one-dimensional harmonic spring is... 

The polarization of the harmonic spring is linear in the applied electric field. Anhar-monicities in Vx(t) are introduced by adding cubic or higher order terms in (x — x0) to V(x). In general,... 

The name of the method reflects the choice of the reference macrostate a crystalline solid comprising particles which do not interact with one another, but which are bound by harmonic springs to the sites of a crystalline lattice, / a, coinciding with that of the phase of interest. 

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