Identifying the binding mode of a molecular scaffold

We describe a method for docking of a scaffold-based series and present its advantages over docking of individual ligands, for determining the binding mode of a molecular scaffold in a binding site. The method has been applied to eight different scaffolds of protein kinase inhibitors (PKI). A single analog of each of these eight scaffolds was previously crystallized with different protein kinases. We have used FlexX to dock a set of molecules that share the same scaffold, rather than docking a single molecule. The main mode of binding is determined by the mode of binding of the largest cluster among the docked molecules that share a scaffold. Clustering is based on our 'nearest single neighbor' method [J. Chem. Inf. Comput. Sci., 43 (2003) 208-217]. Additional criteria are applied in those cases in which more than one significant binding mode is found. Using the proposed method, most of the crystallographic binding modes of these scaffolds were reconstructed. Alternative modes, that have not been detected yet by experiments, could also be identified. The method was applied to predict the binding mode of an additional molecular scaffold that was not yet reported and the predicted binding mode has been found to be very similar to experimental results for a closely related scaffold. We suggest that this approach be used as a virtual screening tool for scaffold-based design processes.     

Reinforcement problems for elliptic equations and variational inequalities

Summary Consider the Dirichlet problem for an elliptic equation in a domain , with coefficients having discontinuity on a surface . Suppose divides into _{1 2}(₂ the inner core), the thickness of ₁ is of order of magnitude , and the modulus of ellipticity in ₁ is of order magnitude ₁. The asymptotic behavior of the solution is studied as 0, ₁ 0, provided lim (/₁) exists. Other questions of this type are studied both for elliptic equations and for elliptic variational inequalities.The second author is partially supported by National Science Foundation Grant 7406375 A01. The third author is partially supported by National Science Foundation Grant MC575-21416 A01.     

Dimensionality Reductions in ℓ<Subscript>2</Subscript> that Preserve Volumes and Distance to Affine Spaces

Let X be a subset of n points of the Euclidean space, and let 0 < ε < 1. A classical result of Johnson and Lindenstrauss [JL] states that there is a projection of X onto a subspace of dimension O(ε^-2 log n) with distortion ≤ 1+ ε. We show a natural extension of the above result to a stronger preservation of the geometry of finite spaces. By a k-fold increase of the number of dimensions used compared with [JL], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show how to embed a subset of size n of the Euclidean space into a O(ε^-2 log n)-dimensional Euclidean space, so that no set of size s ≤ k changes its volume by more than (1 + ε^s-1. Moreover, distances of points from affine hulls of sets of at most k - 1 points in the space do not change by more than a factor of 1 + ε. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL]. Our method can be applied to many problems with high-dimensional nature such as Projective Clustering and Approximated Nearest Affine Neighbour Search. In particular, it shows a first polylogarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [F], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.     

The emergence of kinship behavior in structured populations of unrelated individuals

The paper provides an explanation for altruistic behavior based on the matching and learning technology in the population. In a infinite structured population, in which individuals meet and interact with their neighbors, individuals learn by imitating their more successful neighbors. We ask which strategies are robust against invasion of mutants: A strategy is unbeatable if when all play it and a finite group of identical mutants enters then the learning process eliminates the mutants with probability 1. We find that such an unbeatable strategy is necessarily one in which each individual behaves as if he is related to his neighbors and takes into account their welfare as well as his. The degree to which he cares depends on the radii of his neighborhoods. Received June 1996/Revised version October 1998     

Probabilistic methods in partial differential equations

Results on the asymptotic behavior of solutions of the Cauchy problem∂u/∂t=Lu ast → ∞ are stated, both for nondegenerate and degenerate elliptic second order operatorL. The Dirichlet problem for degenerateL is also studied. The methods used depend on a detailed study of the behavior of solutions of stochastic differential equations. This work was supported partially by National Science Foundation Grant NSF GP-28484.     

A hyperbolic inverse problem arising in the evolution of combustion aerosol

Communicated by J. B. McLeod