A flexible and qualitatively stable model for cell cycle dynamics including DNA damage effects.

This paper includes a conceptual framework for cell cycle modeling into which the experimenter can map observed data and evaluate mechanisms of cell cycle control. The basic model exhibits qualitative stability, meaning that regardless of magnitudes of system parameters its instances are guaranteed to be stable in the sense that all feasible trajectories converge to a certain trajectory. Qualitative stability can also be described by the signs of real parts of eigenvalues of the system matrix. On the biological side, the resulting model can be tuned to approximate experimental data pertaining to human fibroblast cell lines treated with ionizing radiation, with or without disabled DNA damage checkpoints. Together these properties validate a fundamental, first order systems view of cell dynamics. Classification Codes: 15A68.

Quality control in combinatorial chemistry: determination of the quantity, purity, and quantitative purity of compounds in combinatorial libraries.

The quality of combinatorial libraries determines the success of biological screening in drug discovery programs. In this paper, we evaluate and compare various methods for measuring identity, purity, and quantity (yield) of combinatorial libraries. Determination of quantitative purity reveals the true library quality and often indicates potential quality problems before full-scale library production. The relative purity can be determined for every member in a large library in a high-throughput mode, but must be cautiously interpreted. In particular, many impurities are not observable by relative purity measurements using detectors such as UV(214), UV(254), and evaporative light-scattering detection. These "invisible" impurities may constitute a significant portion of the sample weight. We found that TFA, plastic extracts, inorganic compounds, and resin washout are among these impurities. With compelling evidence, we reach a conclusion that purification is the only way to remove "invisible" impurities and improve the quantitative purity of any compound even though some compounds may have a high relative purity before purification.

Partitioned and Hadamard Product Matrix Inequalities.

This note is partly expositor). Inequalities relating inversion with, respectively, extraction of principal submatriees and the Hadamard product in the two possible orders are developed in a simple and unified way for positive definite matrices. These inequalities are known, hut we also characterize the cases of equality and strict inequality. A by-product is, for example, a pleasant proof of an inequality due to Fiedler. In addition, it is shown that the Hadamard product preserves inequalities in a generalization of Schur's observation. In the process, many tools for dealing with the positive semi-definite partial ordering are exhibited.

Eigenset Generalizations of the Eigenvalue Concept.

For an n-by-n complex matrix A some generalizations of the eigenvalue-eigenvector equation A x = λ x , 0 ≠ x ∈ C n are investigated. These take the form A S = λ S  or  A S ⊆ λ S where S is a suhset of C n about which various assumptions are made. For example, it is shown that there exists a finite set S ⊂ C n , the sum of whose elements is not 0, such that AS = λS, if and only if λ is an eigenvalue of A in the usual sense. The requirement that the sum of the elements of S is not 0 should he viewed as a natural analog of the requirement x ≠ 0 in the classical eigenvalue-eigenvector equation.