Abstract
The free energy of solvation of a polypeptide or a protein can be expressed in terms of the accessible surface area of the molecule. Algorithms for energy minimization or for molecular dynamics, which involve the first derivatives of the energy, including the free energy of solvation, are commonly used in the conformational analysis of proteins. Discontinuities of the first derivatives, which occur in the accessible surface area and, hence, in the solvation energy, can cause serious numerical problems. In this paper, we describe all the situations in which the gradient of the molecular surface area becomes discontinuous.
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References
F.M. Richards, Ann. Rev. Biophys, Bioeng. 6 (1977)151.
D. Eisenberg and A.D. McLachlan, Nature 319 (1986) 199,
T. Ooi, M. Oobatake, G. Némethy and H.A. Scheraga, Proc. Natl. Acad. Sci. 84 (1987) 3086.
M.L. Connolly, J. Appl. Cryst. 16 (1983) 548.
T.J. Richmond, J. Mol. Biol. 178 (1984) 63.
G. Perrot and B. Maigret, J. Mol. Graph. 8 (1990)141.
G. Perrot, B. Cheng, K.D. Gibson, J. Vila, K.A. Palmer, A. Nayeem, B. Maigret and H.A. Scheraga, J. Comp. Chem. 13 (1992) 1.
L. Wessen and D. Eisenberg, Protein Sci. l (1992) 227.
B. von Freyberg and W. Braun, J. Comp. Chem. 14 (1993) 510.
K.D. Gibson and H.A. Scheraga, Mol. Phys. 62 (1987) 1247.
R. Sikorski,Advanced Calculus, Functions of Several Variables (Polish Scientific Publ., Warsaw, 1969).
A functionf of a variable çεR λ is a C∞ function in a neighborhood [13] of\(\zeta \), if all derivatives off of any order exist, and are continuous in this neighborhood; see also ref. [11], chap. IV, sect. 3, p.140, and sect. 5, p.150.
(i) A setΩ inR λ is open, if for any\(\zeta \) εΩ, there exists ε > 0, such that any point ζ εR λ distant from\(\zeta \) by less than ε, belongs toω; (ii) a setΓ inR λ is closed, ifR λ −Γ is open; (iii) if Λ is any set inR λ, then−Λ denotes the smallest closed set inR λ containing Λ, and is called a closure of Λ. (iv) if\(\zeta \) εR λ, then any open set inR λ containing this point is called a neighborhood (or an open neighborhood) of\(\zeta \). The above definitions can be extended to any spaceX (playing the role ofR λ), for which the distance between any two points inX is defined; precise definitions can also be found in ref. [11], chap. III, sect. 3.
Iff andF are functions defined on setsX andY, respectively, andX ⊂Y thenF is called an extension off, ifF(x) = f (x) for allx εX.
The λ-dimensional Lebesgue measureμ is defined for a class of the so-called measurable subsets ofR λ large enough to contain all the sets which may appear in practical problems; it is an extension of the intuitive notion of length, area, or volume, depending on λ, for virtually any sets which may be of practical use. For a precise definition see ref. [11], chap. VI.
The MSEED algorithm computes the quantity S0 (u).
Iff is a C∞ function of a variable ζεR λ, the functionF(ζ) =F(ζ1,...,ζλ) = ∫ζ1 0...∫ζλ 0 f (ξ1, ...,ξλ)dξ1...dξλ is called a primitive off.
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Wawak, R.J., Gibson, K.D. & Scheraga, H.A. Gradient discontinuities in calculations involving molecular surface area. J Math Chem 15, 207–232 (1994). https://doi.org/10.1007/BF01277561
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DOI: https://doi.org/10.1007/BF01277561