van Gunsteren, W.F. and Karplus, M., Effects of constraints on the dynamics of macromolecules, Macromolecules, 15, 1528--44, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....free energy will be derived in Section 4. We finally rewrite (1) as d dt q = M Gamma1 p d dt p = GammarU (q) Gamma G(q) T Kg(q) 4) with U(q) V (q) Gamma g(q) T Kg(q) 2 : Remark. Typical constraint methods for molecular dynamics use the (hard) constraints g(q) 0 [16] [17]. However, when applied to the bond angle bending, the resulting molecule becomes too rigid and transition rates are no longer reproduced correctly [17] For that reason we introduced in [14] the soft constraints (2) which maintain the flexibility of a molecule in terms of its bonds and ....

....U(q) V (q) Gamma g(q) T Kg(q) 2 : Remark. Typical constraint methods for molecular dynamics use the (hard) constraints g(q) 0 [16] 17] However, when applied to the bond angle bending, the resulting molecule becomes too rigid and transition rates are no longer reproduced correctly [17]. For that reason we introduced in [14] the soft constraints (2) which maintain the flexibility of a molecule in terms of its bonds and bond angles. 3 Mathematical Background First we rewrite (4) in local coordinates (q 1 ; q 2 ) defined by q 1 = g(q) q 2 = b(q) where b(q) is a vector valued ....

van Gunsteren, W.F. and Karplus, M., Effects of constraints on the dynamics of macromolecules, Macromolecules, 15, 1528--44, 1982.

....M Gamma1 P 2 V (Q) g(Q) T and constraint g(Q) 0. The flow of (27) can be shown to be symplectic (in a generalized sense [9] The solutions of (27) are also time reversible. The approximation (27) has been used, for example, in MD simulations to remove the bond stretching modes [17] [21]. Since g(Q) 0 constrains the local variable q 1 = g(Q) to its equilibrium value q 1 = 0, we call g(Q) 0 a hard constraint. Let us see now whether or not the approximation (27) yields indeed the smoothed dynamics of (1) up to terms of order O(ffl 2 ) To do so, we use local coordinates and ....

....constrained equations (29) yield satisfying results only for small enough values of ffl 2 . While, for example, ffl 2 0:01 for the force constants corresponding to bond stretching in molecular dynamics, one has to take finite size effects of ffl into account when looking at bond angle bending [21] where ffl 2 0:1 and ffi 0:1. In other words, the approximation hq 1 i p ffl = 0 has to be replaced by a more accurate one. This can be achieved by using dhp 1 i p ffl =dt = O(ffl 2 ) to derive hq 1 i p ffl = Gammaffl 2 K Gamma1 [r q 1 V (0; q 2 ) r q 1 V F (0; q 2 ) r q ....

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van Gunsteren, W.F. and Karplus, M., Effects of constraints on the dynamics of macromolecules, Macromolecules, 15, 1528--44, 1982. Smoothed Dynamics 22

....(13) which is Hamiltonian [13] on the constrained manifold M 0 = f(Q; P ) 2 IR 2n : g(Q) 0; G(Q)M Gamma1 P = 0 g (14) provided that the matrix (3) is invertible as assumed earlier. This approximation has been used, for example, in MD simulations to remove the bond stretching modes [22] [26]. The solutions on M 0 are now smooth. However, the approximation (8) introduces, in general, an error of order O(ffi) over bounded time intervals (see Section 6) While, for example, this error turns out to be not significant Smoothed Dynamics 7 for the covalent bond stretching the same ....

.... over bounded time intervals (see Section 6) While, for example, this error turns out to be not significant Smoothed Dynamics 7 for the covalent bond stretching the same formulation (13) yields qualitatively wrong results when applied to the bond angle bending or the harmonic dihedral bending [26]. The constrained formulation derived in this paper approximates the smoothed dynamics of (1) up to terms of order O(ffi ffl 2 ) over bounded intervals of time and provides therefore a qualitative improvement over (13) Furthermore, let hA(q)i denote the time average of an observable A(q) along ....

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van Gunsteren, W.F. and Karplus, M., Effects of constraints on the dynamics of macromolecules, Macromolecules, 15, 1528--44, 1982.

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