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17
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-
22244483490
-
-
note
-
Throughout the paper, subscripted Greek letters identify macrostates and are unrelated to regular Greek letters such as the inverse temperature β. Summations over these macrostate indices are over all macrostates.
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-
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20
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0023323551
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' S. Lee, M. Karplus, D. Bashford, and D. Weaver, Biopolymers 26, 481 (1987).
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A. Heidenreich, I. Schek, D. Scharf, and J. Jortner, Z. Phys. D 20, 227 (1991).
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Heidenreich, A.1
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Jortner, J.4
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24
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22244478354
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in Modeling of Biomolecular Structures and Mechanisms, edited by A. Pullman, J. Jortner, and B. Pullman of Kluwer Academic, Boston
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P. Ahlstrom, J. Lausmaa, P. Löfgren, and H. Berendsen, in Modeling of Biomolecular Structures and Mechanisms, edited by A. Pullman, J. Jortner, and B. Pullman, Vol. 27 of The Jerusalem Symposia on Quantum Chemistry and Biochemistry (Kluwer Academic, Boston, 1994).
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J. A. Board, L. V. Kale, K. Schulten, R. D. Skeel, and T. Schlick, IEEE Comput. Sci. Eng. 1, 19 (1995).
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Board, J.A.1
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28
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0029633152
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J. D. Madura, J. M. Briggs, R. C. Wade, M. E. Davis, B. A. Luty, A. Ilin, J. Antosiewicz, M. K. Gilson, B. Bagheri, L. R. Scott, and J. A. McCamman, Comput. Phys. Commun. 91, 57 (1995).
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Madura, J.D.1
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35
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4944261446
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in edited by G. R. Fleming and P. Hänggi World Scientific, Singapore
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E. Pollak, in Activated Barrier Crossing: Applications in Physics, Chemistry and Biology, edited by G. R. Fleming and P. Hänggi (World Scientific, Singapore, 1993), p. 5-41.
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Pollak, E.1
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47
-
-
0004086603
-
-
The problem is similar to that of determining the relationship between the molecular wave functions and the localized basis sets in quantum chemistry. However, the quantum chemistry problem is simply solved (e.g., by MO-LCAO approximation) because the localized basis functions are roughly factorizable in terms of atomic orbitals [Hartree-Fock-Ruthaan approximation; W. A. Benjamin, New York This is not true for complicated multidimensional systems where the localized conformational macrostates are not factorizable in terms of the individual atomic coordinates
-
The problem is similar to that of determining the relationship between the molecular wave functions and the localized basis sets in quantum chemistry. However, the quantum chemistry problem is simply solved (e.g., by MO-LCAO approximation) because the localized basis functions are roughly factorizable in terms of atomic orbitals [Hartree-Fock-Ruthaan approximation; R. G. Parr, The Quantum Theory of Molecular Electronic Structure, (W. A. Benjamin, New York, 1964)]. This is not true for complicated multidimensional systems where the localized conformational macrostates are not factorizable in terms of the individual atomic coordinates.
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(1964)
The Quantum Theory of Molecular Electronic Structure
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-
Parr, R.G.1
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48
-
-
22244436321
-
-
note
-
α̇ J [P]dR, where P is the projection operator defined in Eq. (24). The current generated by the eigenfunction components that are outside the subspace corresponds to fast relaxation of the probability distribution within, not between, macrostates.
-
-
-
-
50
-
-
22244485761
-
-
note
-
β, resulting in an unbounded spectrum of relaxation rates which would lead to a singularity in the rate at t=0.
-
-
-
-
51
-
-
0042754937
-
-
in Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding: DIMACS Workshop, edited by P. Pardalos, D. Shalloway, and G. Xue of American Mathematical Society, Providence, RI
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B. W. Church, M. Orešič, and D. Shalloway, in Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding: DIMACS Workshop, edited by P. Pardalos, D. Shalloway, and G. Xue, Vol. 23 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science (American Mathematical Society, Providence, RI, 1996), pp. 41-64.
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DIMACS Series in Discrete Mathematics and Theoretical Computer Science
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-
-
Church, B.W.1
Orešič, M.2
Shalloway, D.3
-
52
-
-
22244454459
-
-
note
-
obs, so the minimum decay condition can equivalently be imposed in either the occupation or observation representation.
-
-
-
-
54
-
-
22244449365
-
-
note
-
See Ref. 16 for further discussion. If H is invariant under a continuous symmetry group (e.g., rigid body translation and rotation), we will only be interested in group-invariant macrostates; therefore the sums in Eq. (24) can be restricted to group- invariant eigenfunctions, and degeneracies associated with irreducible representations of the group of rank >1 will not enter. If H is invariant under a discrete group of transformations, the natural requirement that all group-transformation-related macrostates be included in the set of macrostates (i.e., that the macrostates form a basis for a representation of the group) implies that the macrostate subspace will include either all or none of the degenerate eigenfunctions belonging to an irreducible representation of the group. Thus, there will be no symmetry-induced ambiguity in the subspace specification.
-
-
-
-
55
-
-
22244477145
-
-
note
-
This variational equation for σ is equivalent to the Rayleigh-Ritz variational equation for the first excited eigenfunction; see Ref. 16.
-
-
-
-
56
-
-
22244485100
-
-
note
-
The detW term ensures that the node is located between the two macrostates; the denominator will become small if σ is varied so much that the nodal surface does not separate the macrostates. However, this does not occur during normal numerical variation.
-
-
-
-
57
-
-
22244493026
-
-
note
-
0 as R→∞ [A. Ulitsky and D. Shalloway (in preparation)]. Thus, by Eq. (15), the window and transition functions will go to constants.
-
-
-
-
58
-
-
22244455470
-
-
note
-
Note that erf(0.83x)-tanh(x)<0.02∀x. Thus the functions provide essentially equivalent parameterizations for the variational condition.
-
-
-
-
59
-
-
22244464281
-
-
note
-
For a system with N degrees-of-freedom, both the nodal surface and the transition "plane" are (N-1)-dimensional manifolds.
-
-
-
-
60
-
-
22244457109
-
-
note
-
Since η is a vector, Eqs. (40) represents N+1 conditions where N is the number of degrees-of-freedom.
-
-
-
-
62
-
-
22244440579
-
-
note
-
X and Z instead of the characteristic harmonic frequency.
-
-
-
-
64
-
-
0011779827
-
-
in edited by G. R. Fleming and P. Hänggi World Scientific, Singapore
-
A. Nitzan and Z. Schuss, in Activated Barrier Crossing: Applications in Physics, Chemistry and Biology, edited by G. R. Fleming and P. Hänggi (World Scientific, Singapore, 1993), pp. 42-81.
-
(1993)
Activated Barrier Crossing: Applications in Physics, Chemistry and Biology
, pp. 42-81
-
-
Nitzan, A.1
Schuss, Z.2
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66
-
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0004161838
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-
Cambridge University Press, Cambridge
-
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1992).
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(1992)
Numerical Recipes in Fortran
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Press, W.H.1
Teukolsky, S.A.2
Vetterling, W.T.3
Flannery, B.P.4
-
68
-
-
22244449033
-
-
note
-
This method (see Ref. 66) was chosen because it uses only the objective function and does not require derivatives which are relatively expensive to accurately compute by Monte Carlo integration.
-
-
-
-
69
-
-
22244471421
-
-
note
-
The integrals in the right-hand-sides of Eqs. (47) were computed to 3% accuracy as determined by the variance (Ref. 65).
-
-
-
-
70
-
-
22244438599
-
-
note
-
XY<0.03.
-
-
-
-
71
-
-
22244462774
-
-
note
-
ξ=0.03.
-
-
-
-
73
-
-
22244436978
-
-
note
-
c.
-
-
-
-
75
-
-
22244473388
-
-
note
-
1 to 1% accuracy.
-
-
-
-
81
-
-
22244477462
-
-
in preparation
-
D. Shalloway (in preparation).
-
-
-
Shalloway, D.1
-
83
-
-
22244454808
-
-
note
-
t) from the transition region conformations resulting from the replacement of atoms 3 and 4 with other pairs of adjacent atoms.
-
-
-
-
84
-
-
35248857016
-
-
This procedure described in, was used. While multiple reaction paths are anticipated in multidimensional systems only one path connecting the two isoforms of the six-particle cluster was detected in 100 trials starting with different initial conditions.
-
This procedure described in A. Ulitsky and R. Elber, J. Chem. Phys. 92, 1510 (1990) was used. While multiple reaction paths are anticipated in multidimensional systems only one path connecting the two isoforms of the six-particle cluster was detected in 100 trials starting with different initial conditions.
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(1990)
J. Chem. Phys.
, vol.92
, pp. 1510
-
-
Ulitsky, A.1
Elber, R.2
-
85
-
-
22244482839
-
-
note
-
As an additional check, transition rates were also calculated using the mean first-passage time evaluated from Langevin dynamics trajectories. Similar results were obtained.
-
-
-
-
86
-
-
22244460892
-
-
note
-
2 using Eq. (49b) since only the coordinates of one atom enter Eq. (61).
-
-
-
-
90
-
-
22244442500
-
-
note
-
0exp[-βw(q)]dq/∫exp[-βw(q)]dq, where w(q) is the potential of mean force along an ad hoc preselected direction and q=0 is the maximizer of w(q) in the barrier region.
-
-
-
-
94
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-
22244457731
-
Variable-scale coarse-graining in macromolecular global optimization
-
in edited by L. Biegler, T. Coleman, A. R. Conn, and F. Santosa, Springer, New York
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D. Shalloway, Variable-scale coarse-graining in macromolecular global optimization, in Proceedings of IMA Conference on Large Scale Optimization, edited by L. Biegler, T. Coleman, A. R. Conn, and F. Santosa, (Springer, New York, 1997).
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(1997)
Proceedings of IMA Conference on Large Scale Optimization
-
-
Shalloway, D.1
-
96
-
-
22244460560
-
-
note
-
β≈1, but this has not been proven.
-
-
-
-
97
-
-
22244485447
-
-
note
-
α to approach zero or one, but this has not been proven.
-
-
-
-
98
-
-
0003498504
-
-
Academic, New York
-
I. S. Gradshteyn and I. M. Ryzik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994).
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(1994)
Table of Integrals, Series, and Products, 5th Ed.
-
-
Gradshteyn, I.S.1
Ryzik, I.M.2
-
101
-
-
22244452391
-
-
note
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0).
-
-
-
|