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The measurement of the 3-D "average propagator", P(r), from diffusion-weighted (DW) NMR or MRI data has been a "holy grail" in materials science and biomedicine, as P(r) provides detailed microstructural information, particularly about restriction, without assuming an underlying diffusion model. While Callaghan proposed a 3-D Fourier transform relationship between P(r) and the DW signal attenuation, E(q) [1], using it to measure P(r) from E(q) data is not currently feasible biologically or clinically, owing to the staggering amount of DW data required.To address this problem, we propose that computed tomography principles can be applied to reconstruct P(r) from DW signals. Moreover, this reconstruction can be performed efficiently using many fewer DW E(q) data as compared to conventional 3-D q-space MRI [1] or Diffusion Spectrum Imaging (DSI) [2] by employing a priori information about E(q) and P(r).

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Motivation: Obtaining accurate estimates of the numbers of protein-coding genes and protein domains in a proteome, and the number of protein domains in nature is a daunting challenge. Computational analysis of the protein domain sets in the proteomes of many species allows us to estimate these numbers and to find their evolution relationships.Results: We have analyzed the distributions of the number of occurrences of protein domains in sample proteomes of the 70 fully sequenced genome organisms of three major kingdoms of life: Archaea, Bacteria and Eukaryota. We found that a large fraction of the identified distinct protein domains (i.e., unique domains and homologous domain families) in these 70 proteomes (1051 (23%) out of 4493) are found in at least one organism in each of these kingdoms of life and that 43 (1%) of these domains are common to all the 70 organisms. All the observed domain occurrence frequency distributions for these 70 proteomes are well fitted by a family of Pareto-like functions, associated with the steady state distributions of a linear Markov random process. We present explicit * Corresponding author. 381 J. Biol. Syst. 2002.10:381-407. Downloaded from www.worldscientific.com by VIRGINIA COMMONWEALTH UNIV on 03/13/15. For personal use only. 382 Kuznetsov, Pickalov, Senko & Knott formulas that accurately predict the number of distinct protein domains and the number of protein-coding genes for a given organism as functions of the number of non-redundant domain-to-protein links in the proteomes. These functions allows us to predict that there are 42,740, 27,900, and 21,200 protein-coding genes/open reading frames in the human,A. thaliana, and mouse genomes, respectively. We also estimate that there are 5271, 2955, and 4915 distinct protein domains in the human, A. thaliana, and mouse proteomes, respectively, and about 5500 distinct protein domains in the entire "proteome world".