Analysis of previously published sets of DNA microarray gene expression data by singular value decomposition has uncovered underlying patterns or ''characteristic modes'' in their temporal profiles. These patterns contribute unequally to the structure of the expression profiles. Moreover, the essential features of a given set of expression profiles are captured using just a small number of characteristic modes. This leads to the striking conclusion that the transcriptional response of a genome is orchestrated in a few fundamental patterns of gene expression change. These patterns are both simple and robust, dominating the alterations in expression of genes throughout the genome. Moreover, the characteristic modes of gene expression change in response to environmental perturbations are similar in such distant organisms as yeast and human cells. This analysis reveals simple regularities in the seemingly complex transcriptional transitions of diverse cells to new states, and these provide insights into the operation of the underlying genetic networks.T he recent development of DNA microarray technology has enabled the genome-wide measurement of temporal changes in gene expression levels (1, 2). Analysis of the expression patterns obtained with large gene arrays has revealed the existence of groups or ''clusters'' of genes with similar expression patterns (3-6). Not surprisingly, gene clusters often contain genes that encode proteins required for a common function, and, hence, co-clustering has been helpful in identifying the functions of unknown gene products. However, such cluster analyses provide little insight into the relationships among groups of co-regulated genes or the behavior of biological networks as a whole.In this paper, we report the results of subjecting several large published gene expression data sets to singular value decomposition (SVD), a standard and straight-forward analytic procedure. We show that highly complex sets of gene expression profiles can be represented by a small number of ''characteristic modes'' that capture the temporal patterns of gene expression change. These modes are somewhat analogous to the characteristic vibration modes of a tuned violin string. The tone produced by the vibrating string can be entirely specified by the contributions of its characteristic vibration modes. We show here that a gene expression profile, similarly, can be precisely represented by specifying the magnitude and sign of the contribution of each of its characteristic modes. This type of ''spectral'' analysis yields a hierarchical interpretation of the expression data and provides insights into the nature and behavior of genetic networks. MethodsThe mathematical analysis is carried out straightforwardly by using SVD (7). The gene expression data of n genes, each measured at m discrete time points, may be written as an n ϫ m matrix, A. Following the procedures outlined in ref. 6, we have polished the data by requiring that the rows and columns have a zero mean by subtracting the mean values of the raw...

We describe the time evolution of gene expression levels by using a time translational matrix to predict future expression levels of genes based on their expression levels at some initial time. We deduce the time translational matrix for previously published DNA microarray gene expression data sets by modeling them within a linear framework by using the characteristic modes obtained by singular value decomposition. The resulting time translation matrix provides a measure of the relationships among the modes and governs their time evolution. We show that a truncated matrix linking just a few modes is a good approximation of the full time translation matrix. This finding suggests that the number of essential connections among the genes is small.

We present and implement a distance-based clustering of amino acids within the framework of a statistically derived interaction matrix and show that the resulting groups faithfully reproduce, for well-designed sequences, thermodynamic stability in and kinetic accessibility to the native state. A simple interpretation of the groups is obtained by eigenanalysis of the interaction matrix.

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In the case of Fig. I@), Le., when the edge is illuminated by a creeping mode excited by an earlier surface diffraction, we have n u d ( Q l ) = ui(L) TSc(q1, vl)eiUILMl/a T ( e ikR'1 re 9 1 9 92; V I , *Iud( Q 2 ) = ui(L) Tsc(ql, vl)eivlLM1/u T L (ql, 92; v l , vl)eblMIM2/" n eikI?2 . Ta(919 V I ) T a n ud(Q3)=ui(L)Tsc(ql, vl)eivlLw1/aTA2)(qI, q2; v l , pl)ei~~~w11w3'u n eikh?3 * Tm(12, P I ) -In the above expressions ui stands for the incident field, while the meanings of various geometrical parameters are shown in Fig. 1. p1 is the first zero of q2Ht1)(x) -iH;(l)(x) and the explicit expressions for the diffraction coefficients T, are found as listed below When a -t 03, the edge diffraction coefficient, is reduced to which is merely the coefficient related to the two-part impedance half-plane. REFERENCESClemmov, The Plane Wave Spectrum Representation of Electromagnetic Fields. Elmsford, N Y : Pergamon, 1966. R. Tiberio and G . Pelosi, "High-frequency scattering from the edges of impedance discontinuities on a flat plane," IEEE Tram. Antennas Propagat., vol. AP-31, no. 4, pp. 590-596, 1983. M. Idemen and L. B. Felsen, "Diffraction of a whispering gallery mode by the edge of a thin concave cylindrically curved surface," IEEE Trans. Antennas Propagat., vol. A€-29, pp. 571-579, 1981. T. B. A. Senior, "Half plane edge diffraction," Radio Sei., vol. 10, no. 6, pp. 645-650, 1975. A. Biiyiikaksoy and G . Uzgoren, "High frequency scattering by a cylindrically curved impedance surface," AEU, bd. 39, no. 6, pp. 351-358, 1983. M. Idemen, "Necessary and,sufficient conditions for a surface to be. an impedance boundary," AEU, bd. 35, no. 2, pp. 84-86, 1981. M. Idemen and E. Erdogan, "Diffraction of a creeping wave generated on a perfectly conducting spherical scatterer by a ring source," ZEEE Trans.Abstract-Using the Sierpinski gasket as an example, it has been shown here that the paraxial Fraunhoffer-zone diffracted field of a self-similar fractal screen also exhibits self-similarity. This also establishes that fractal structures can be used with great profit in problems involving array syntheses. Recently, it bas been shown that the Sierpinski gasket is itself a member of a much wider class of gaskets, and the potential for the use of fractal structures in electromagnetic (EM) problems may he vast indeed.

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Two distinct mechanisms underlying the existence of power-law distributions are presented: the distribution is stationary under the process of merging and splitting of classes and the distribution of the entities under study is invariant under changes of the classification scheme. We provide an explanation for the ubiquitous inverse n relationship in the species abundance relationship in ecology and the 1/n(2) distribution of company sizes based on the minimum impact principle.

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