Design considerations and computer modeling related to the development of molecular scaffolds and peptide mimetics for combinatorial chemistry

Summary A critical issue in drug discovery utilizing combinatorial chemistry as part of the discovery process is the choice of scaffolds to be used for a proper presentation, in a three-dimensional space, of the critical elements of structure necessary for molecular recognition (binding) and information transfer (agonist/ antagonist). In the case of polypeptide ligands, considerations related to the properties of various backbone structures (-helix, -sheets, etc.; , space) and those related to three-dimensional presentation of side-chain moieties (topography; (chi) space) must be addressed, although they often present quite different elements in the molecular recognition puzzle. We have addressed aspects of this problem by examining the three-dimensional structures of chemically different scaffolds at various distances from the scaffold to evaluate their putative diversity. We find that chemically diverse scaffolds can readily become topographically similar. We suggest a topographical approach involving design in chi space to deal with these problems.     

Constructing nonresidues in finite fields and the extended Riemann hypothesis

We present a new deterministic algorithm for the problem of constructing th power nonresidues in finite fields , where is prime and is a prime divisor of . We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed and , our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomial-time bound holds even if is exponentially large. More generally, assuming the ERH, in time we can construct a set of elements that generates the multiplicative group . An extended abstract of this paper appeared in Proc. 23rd Ann. ACM Symp. on Theory of Computing, 1991.

     

On a problem of H. N. Gupta

It is shown that the axiom For any points x, y, z such that y is between x and z, there is a right triangle having x and z as endpoints of the hypotenuse and y as foot of the altitude to the hypotenuse, when added to three-dimensional Euclidean geometry over arbitrary ordered fields, is weaker than the axiom Every line which passes through the interior of a sphere intersects that sphere.     

Splitting the Pasch axiom

On the basis of the theory ^– of Pasch-free 2-dimensional geometry, Pasch's axiom is shown to be equivalent to the conjunction of the following two axioms: In any right triangle the hypotenuse is greater than the leg and If AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is greater than the segment BD. This represents an attempt to split the Pasch axiom with respect to ^–. Only the question whether the second of the above two axioms is really weaker than Pasch's axiom, remains open.     

Common supports as fixed points

A family S of sets in R ^d is sundered if for each way of choosing a point from rd+1 members of S, the chosen points form the vertex-set of an (r–1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R ^d , and for each partition (S , S ), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S from the members of S . This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.     

Critical behavior of self-avoiding walks on fractals

Using a new graph counting technique suitable for self-similar fractals, exact 18th-order series expansions for SAWs on some Sierpinski carpets are generated. From them, the critical fugacityx _c and critical exponents _SAW and _SAW are obtained. The results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied. We find for some carpets with low lacunarity that _SAW<0.75, thus violating the relation _SAW(fractal) > _SAW (d) for SAWs on the fractals which are embedded in ad-dimensional Euclidean space.     

Computational complexity of inner and outerj-radii of polytopes in finite-dimensional normed spaces

This paper studies the complexity of computing (or approximating, or bounding) the various inner and outer radii of ann-dimensional convex polytope in the space ⁿ equipped with an _p norm or a polytopal norm. The polytopeP is assumed to be presented as the convex hull of finitely many points with rational coordinates (V-presented) or as the intersection of finitely many closed halfspaces defined by linear inequalities with rational coefficients (-presented). The innerj-radius ofP is the radius of a largestj-ball contained inP; it isP's inradius whenj = n and half ofP's diameter whenj = 1. The outerj-radius measures how wellP can be approximated, in a minimax sense, by an (n — j)-flat; it isP's circumradius whenj = n and half ofP's width whenj = 1. The binary (Turing machine) model of computation is employed. The primary concern is not with finding optimal algorithms, but with establishing polynomial-time computability or NP-hardness. Special attention is paid to the case in whichP is centrally symmetric. When the dimensionn is permitted to vary, the situation is roughly as follows: (a) for general -presented polytopes in _p spaces with 1相似文献   

Inequalities for tails of adapted processes with an application to Wald's lemma

In this paper we introduce a new tail probability version of Wald's lemma for expectations of randomly stopped sums of independent random variables. We also make a connection between the works of Klass^{(18, 19)} and Gundy⁽¹¹⁾ on Wald's lemma. In making the connection, we develop new Lenglart and Good Lambda inequalities comparing the tails of various types of adapted processes. As a consequence of our Good Lambda inequalities we include the following result. Let {d _i }, {e _i } be two sequences of variables adapted to the same increasing sequence of σ-fields ? _n ↗?, (e.g., ? _n =σ({d _i } _i=1 ⁿ , {E _i } _i=1 ⁿ ), and letN?∞ be a stopping time adapted to {? _n }. Then for allp>0, there exists a constant 0<C _p <∞ depending onp only, such that $$\mathop {\overline {\lim } }\limits_\lambda \lambda ^p P\left( {\mathop {\sup }\limits_{1 \leqslant n \leqslant N} \left\| {\sum\limits_{i = 1}^n {d_i } } \right\| > \lambda } \right) \leqslant C_p \mathop {\overline {\lim } }\limits_\lambda \lambda ^p P\left( {\mathop {\sup }\limits_{1 \leqslant n \leqslant N} \left\| {\sum\limits_{i = 1}^n {e_i } } \right\| > \lambda } \right)$$ This result holds when the sequences are real, tangent, and either conditionally symmetric or nonnegative, or alternatively, if {d _i } is a sequence of independent random variables and {e _i } is an independent copy of {d _i }, withN a stopping time adapted to the filtration generated by {d _i } only. Other examples include Hilbert space valued differentially subordinate conditionally symmetric martingale differences. The result is true for more general operators applied to sequences as shown by an example comparing the square function of a conditionally symmetric sequence to the maximum of its absolute partial sums.     

Study of ab initio molecular data for inelastic and reactive collisions involving the H3+ quasimolecule

The lowest two ab initio potential energy surfaces (PES), and the corresponding nonadiabatic couplings between them, have been obtained for the H3+ system; the molecular data are compared to those calculated with the diatomic in molecules (DIM) method. The form of the couplings is discussed in terms of the topology of the molecular structure of the triatomic. The method of Baer is employed to generate "diabatic" states and the residual nonadiabatic couplings are calculated. The ab initio results for these are markedly different from the corresponding DIM data, and show the need to consider the third PES.