Using Multiple Substitution Matrices

The generalized affine gap model allows for both gaps and unaligned regions. Alternatively, gap scores can be dispensed with completely, in 

Our work is supported by a grant from NIH. Preparation of the manuscript was aided by stimulating discussions during a visit to NCBI. We thank Paul Talbert and Elizabeth A. Greene for comments. 

Bailey, T. L., and Gribskov, M. (1996). The megaprior heuristic for discovering protein sequence patterns. In Proceedings of the Fourth International Conference on Intelligent Systems for Molecular Biology, pp. 15-24. AAAI Press, Menlo Park. 

Dayhoff, M. O. (1979). Atlas of protein sequence and structure. Vol. 5, suppl. 3 National Biomedical Research Foundation, Washington, D.C. (pp. 345-358). 

3 National Biomedical Research Foundation, Silver Spring, Maryland (p. 33). 

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Substituting equation (5) into equation (2), multiplication by the internal basis functions and integration over (Vy, a, 3,0y, (l)y) gives a set of close-coupled equations in the translational f functions [23]. These can be solved by a variety of methods. We prefer to use the R-matrix method of Light and Walker [22]. 

After the first iteration rR = ctr, rRU = aRU,. On succeeding iterations we use ARX from Eq. (7) to obtain R = R j + ARX and AUX = — ARX to obtain U = Uo+AUx. Then substitution into Eq. (5) determines the updated rR, rRU,... in terms of the a matrices. The resulting formulas involve straightforward matrix multiplications and additions entirely within the local space. In Eq. (7) ARX is projected in terms of R,> and Uq. Later we will find it convenient to go back one step further to Ro and Uo. This is readily done by taking advantage of Eq. (4). 

In order to determine vibrational NLO properties efficiently it is necessary to carry out finite field geometry optimizations as we have seen. In principle, Eq. (35) can be used directly for this purpose. There are, however, practical considerations related to convergence of the self-consistent field (SCF) iterations. The most obvious iterative sequence is (i) determine the zero-field solution (ii) evaluate dC/dk, (iii) substitute dC/dk from the previous step into the TDHF equation (iv) solve for C(k) and return to step (ii) etc. until convergence is achieved. In order to carry out step (iv) the normalization condition C SC = 1 may be used to write dC/dk = [(5C/5A )OS]C. Then the multiplicative form of the field-free equation is preserved and the polarization matrix will remain Hermitian for all iterations. Investigations are underway to test the convergence properties of the above iterative sequence and to determine how the convergence properties depend upon the magnitude of the field as well as the number of -points that are sampled in the band structure treatment. 

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