Cluster Analysis of Gene Expression Data

The expression levels of many thousands of genes can be measured simultaneously by DNA microarrays (chips). This novel experimental tool has revolutionized research in molecular biology and generated considerable excitement. A typical experiment uses a few tens of such chips, each dedicated to a single sample—such as tissue extracted from a particular tumor. The results of such an experiment contain several hundred thousand numbers, that come in the form of a table, of several thousand rows (one for each gene) and 50–100 columns (one for each sample). We developed a clustering methodology to mine such data. In this review I provide a very basic introduction to the subject, aimed at a physics audience with no prior knowledge of either gene expression or clustering methods. I explain what genes are, what is gene expression and how it is measured by DNA chips. Next I explain what is meant by clustering and how we analyze the massive amounts of data from such experiments, and present results obtained from analysis of data from colon cancer, brain tumors and breast cancer.     

Potts ferromagnets on coexpressed gene networks: identifying maximally stable partitions

Clustering gene expression data by exploiting phase transitions in granular ferromagnets requires transforming the data to a granular substrate. We present a method using the recently introduced homogeneity order parameter Lambda [H. Agrawal, Phys. Rev. Lett. 89, 268702 (2002)]] for optimizing the parameter controlling the "granularity" and thus the stability of partitions. The model substrates obtained for gene expression data have a highly granular structure. We explore properties of phase transition in high q ferromagnetic Potts models on these substrates and show that the maximum of the width of superparamagnetic domain, corresponding to maximally stable partitions, coincides with the minimum of Lambda.     

Resonance in a Quantum Chain

A resonance of a magnetic impurity with a quantum spin-chain is studied. A Fresnel diffraction pattern is found.     

Spectral Dependence of Rise Time in Gas Filled Phototubes: Implications concerning Possible Miniaturization of Gas Discharge Detectors of Electromagnetic Radiation

The rise time of a gas-filled photodiode at blue-green wavelengths is found to be less than that at red wavelengths. As photoionization of excited atoms in such devices increases response to light of decreasing wavelength over that provided by the photocathode, spectral differences in rise time are also attributed to the same mechanism. It is suggested that these phenomena decrease parasitic inductance of the tube. This implies that rise times in glow discharge detectors of EM radiation can be improved through miniaturization of inter-electrode spacing.     

Duality relations and equivalences for models with O(N) and cubic symmetry

Formal expression for high-temperature series are derived for models with O(N) and cubic symmetry, with a special form of nearest neighbor interactions on the honeycomb lattice. By deriving low-temperature series for a class of generalized solid-on-solid and cubic models, a duality relation is established. Equivalences between cubic and SOS type models are also found. In the large-N limit, the series reduce to those of the hard hexagon model.     

Neural networks: A biased overview

An overview of recent activity in the field of neural networks is presented. The long-range aim of this research is to understand how the brain works. First some of the problems are stated and terminology defined; then an attempt is made to explain why physicists are drawn to the field, and their main potential contribution. In particular, in recent years some interesting models have been introduced by physicists. A small subset of these models is described, with particular emphasis on those that are analytically soluble. Finally a brief review of the history and recent developments of single- and multilayer perceptrons is given, bringing the situation up to date regarding the central immediate problem of the field: search for a learning algorithm that has an associated convergence theorem.     

Ostwald Ripening in Two Dimensions: Treatment with Pairwise Interactions

Ostwald ripening is the last stage of the evolution of a system with two coexisting phases. It is a relatively simple nonequilibrium phenomenon with several interesting features. For example, as the system coarsens it goes through a scaling state, one which looks the same (up to an overall length scale, which grows) at all times. The dynamics of the problem can be mapped, in two dimensions, onto an evolving Coulomb system. In this work we present a brief summary of a novel theoretical approach to this problem, based on an analytic derivation (using a mean-field approach) of an effective two-body interaction between droplets of the minority phase. The resulting interacting many-body dynamics is solved by a very efficient numerical algorithm, allowing us to follow the evolution of more than 10⁶ droplets on a simple workstation. The results are in excellent agreement with recent experiments.     

The Entropy of a Binary Hidden Markov Process

The entropy of a binary symmetric Hidden Markov Process is calculated as an expansion in the noise parameter ε. We map the problem onto a one-dimensional Ising model in a large field of random signs and calculate the expansion coefficients up to second order in ε. Using a conjecture we extend the calculation to 11th order and discuss the convergence of the resulting series     

Lack of ultrametricity in the low-temperature phase of three-dimensional Ising spin glasses

We study the low-temperature spin-glass phases of the Sherrington-Kirkpatrick (SK) model and of the 3-dimensional short-range Ising spin-glass (3DISG). By using clustering to focus on the relevant parts of phase space and reduce finite size effects, we found that for the SK model ultrametricity becomes clearer as the system size increases, while for the short-range case our results indicate the opposite, i.e., lack of ultrametricity. Another method, which does not rely on clustering, indicates that the mean-field solution works for the SK model but does not apply in detail to the 3DISG, for which stochastic stability is also violated.