Eigensolvers for analysis of microarray gene expression data

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Eigensolvers for analysis of microarray gene
expression data

• Andrew Knyazev (speaker) and Donald McCuan
• Image from http//www.biosci.utexas.edu/mgm/people
  /faculty/profiles/VIresearch.jpg
• Supported by NSF DMS 0728941. In collaboration
  with CU MCD Biology.

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Eigensolvers for DNA microarrays

• Crash course on gene expression
• Microarrays---a massively parallel experiment
• Clustering why?
• Clustering how?
• Spectral clustering
• Connection to image segmentation
• Eigensolvers for spectral clustering

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Crash course on gene expression 1/3

• Genes in DNA code for proteins
• Protein formation in a cell involves
• Transcription of DNA to mRNA (messenger RNA)?
• Translation of mRNA to a protein
• When proteins are being formed for a gene this is
  called gene expression

DNA sense strand .... ATA CGT
... antisense strand .... TAT GCA
... mRNA .... AUA CGU ... Protein .
... Ile Arg ...
transcription
translation
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Crash course on gene expression 2/3Image
Courtesy cnx.org
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Crash course on gene expression 3/3

• Gene expression in a cell depends on many
  factors, e.g., developmental stage, nutrition,
  environment, and diseases, so the level of gene
  expression may vary
• Knowing how genes are expressed helps to
  understand cellular processes and diagnose
  diseases
• Measurement of the concentration of proteins in a
  cell is complicated, so the concentration of mRNA
  is used instead, assuming that most mRNA created
  is actually translated to a protein
• DNA Microarrays (e.g., Affymetrix GeneChip
  arrays) measure the level of mRNA in a sample

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Microarrays-massively parallel experiment 1/5
Affymetrix GeneChip DNA
Microarrays Image Courtesy
Affymetrix
Affymetrix GeneChip DNA Microarrays
Image Courtesy Affymetrix
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Microarrays-massively parallel experiment 2/5

• GeneChip oligonucleotide sequences are
  photo-lithographed on a quartz wafer in a pattern
  of 10 micrometers dots.
• Oligonucleotide sequences (oligos) probes 25
  nucleotide chains for selected parts of a gene
  complementary to mRNA.
• GeneChips are manufactured to include all
  currently known and predicted genes of a
  particular organism, e.g., H. sapience. The
  information about physical locations of oligo
  probes for each gene on the chip is contained in
  the .cdf file.
• A sample of mRNA extracted from cells of an
  organism after pre-processing is hybridized with
  GeneChip giving PM and MM values which
  characterize genes expressions in the cells.

For every gene there are 11-20(depending on chip
design) of different oligo probes called perfect
matches (PM). In addition, there are mismatch
oligos (MM) corresponding to each of the PMs that
differ in the middle base pair.
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Microarrays-massively parallel experiment 3/5

• Labelled cRNA targets derived from the mRNA of an
  experimental sample are hybridized to oligo
  probes.
• During hybridization, complementary nucleotides
  line up and bind together via hydrogen bonds in
  the same way as two strands of DNA bound
  together.
• The chip is then scanned with a laser giving the
  amount of each mRNA species represented.
• Image Courtesy cnx.org

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Microarrays-massively parallel experiment 4/5

• A pool of mRNA is extracted from the cells of an
  organism and converted to a Biotin labelled
  strand (cRNA) that binds to the oligo probes on
  the GeneChip during hybridization.
• The higher the concentration of a particular mRNA
  in the testing pool---the greater the
  hybridization level of the PM probes and thus the
  amount of the hybridized material on the
  processed GeneChip.
• Then a fluorescent stain is applied that binds to
  the Biotin and the GeneChip is processed through
  a scanner that illuminates each dot of the
  GeneChip with a laser, causing dots to fluoresce.
• The image data of the scanned probe array is
  stored in a .dat file. The Affymetrix GCOS
  software processes the .dat file and generates a
  .cel file, containing all numerical data of the
  GeneChip experiment, e.g., probe locations and PM
  and MM intensities. The processing involves
  computing a square grid locating the dots for
  probes, intensity normalization, using internal
  controls, and detecting the outliers.
• More sophisticated .dat--gt.cel algorithms,
  e.g., taking into account the cRNA saturation,
  are being developed elsewhere.

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Microarrays-massively parallel experiment 5/5

• The PM and MM values are not normally used
  directly for high-level statistical analysis,
  instead they are first converted into the gene
  expression values, which involves
• Detecting unreliable data by comparing PM and MM
• Adjustment for background and noise
• Calculating the single array gene expression
  intensities, basically by averaging adjusted PM
  values for each probe set
• Alternatively, the Comparison Analysis
  (Experiment versus Baseline arrays) detects and
  quantifies changes in gene expressions between
  two arrays, applying normalization of data and
  using the Signal Log Ratio algorithms.
• Either way, the absolute or comparison gene
  expression values are stored in a .chp file,
  which serves as the input for high-level
  statistical analysis. Typically, multiple
  GeneChip tests are performed giving multiple
  .chp files with gene expression values.

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Clustering why?

• When conducting microarray experiments there are
  multiple microarrays involved typically
• Studying a process over time, e.g., to measure
  the response to a drug or food.
• Looking for differences between states, e.g.,
  normal cells versus cancer cells.
• A typical goal is Finding Gene Networks, i.e.,
  groups of genes that change expression
  inter-dependently across samples. Having a
  significantly large number of microarrays, we
  want to reverse engineer the regulatory network
  that controls gene expressions. We need computer
  clustering on the microarray data to select a
  small (ideally) number of co-expressed genes of a
  gene network. Separate experiments using gene
  knockout on the selected genes can then be
  performed to confirm the discovered regulatory
  network biologically.

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Clustering how?

• There is no good widely accepted definition of
  clustering. The traditional graph-theoretical
  definition is combinatorial in nature and
  computationally infeasible. Heuristics rule!
• Many clustering techniques and methods are known,
  e.g.,
• Hierarchical clustering/partitioning
• K-means (centroids)
• Self-organizing maps (partitioning vectors)
• Force-directed placement
• Principal Components Analysis (PCA)?
• Spectral clustering/partitioning using Fiedler
  vectors
• Some good and popular free open source software,
  e.g., METIS and CLUTO (Karypis Lab).
• We focus on PCA and spectral clustering.

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Spectral clustering
Images Courtesy Russell, Ketteriung U.
A 4-degree-of-freedom system has 4 modes of
vibration and 4 natural frequencies partition
into 2 clusters using the second eigenvector

• A adjacency matrix
• D degree matrix
• Laplacian matrix L D A
• Fiedler eigenvectors Lx?x
• N-cut eigenvectors Lx?Dx (smallest) are the
  largest for
• PCA Markov walks AxµDx with µ1-?. D-1A is
  raw-stochastic and describes the walk
  probabilities.

Example Courtesy Blelloch CMU
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5
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www.cs.cas.cz/fiedler80/
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Rows sum to zero
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Connection to image segmentation

• Image pixels serve as graph vertices. Weighted
  graph edges are computed by comparing pixel
  colours.
• Here is an example displaying 4 Fiedler vectors
  of an image

We generate a sparse Laplacian, by comparing
neighbouring pixels here when computing the
weights for the edges. Genes correspond to
vertices in microarrays, but we have to compare
all genes, possibly getting a Laplacian with a
large fill-in.
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Eigensolvers for spectral clustering

• Our BLOPEX-LOBPCG software has proved to be
  efficient for large-scale eigenproblems for
  Laplacians from PDE's and for image segmentation
  using multiscale preconditioning of hypre
• The LOBPCG for massively parallel computers is
  available in our Block Locally Optimal
  Preconditioned Eigenvalue Xolvers (BLOPEX)
  package
• BLOPEX is built-in in http//www.llnl.gov/CASC/hyp
  re/ and is included as an external package in
  PETSc, see http//www-unix.mcs.anl.gov/petsc/
• On BlueGene/L 1024 CPU we can compute the Fiedler
  vector of a 24 megapixel image in seconds
  (including the hypre algebraic multigrid setup).

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Work in progress/future work

• Our current test cases are to analyze
• Affymetrix GeneChip data from Marina Kniazeva et
  al. PLoS Biology, 2004.
• Microarray data from Liang Zhang et al.,
  Molecular Cell, 2007.
• Our future work will involve developing prototype
  spectral clustering software for Microarrays in
  the Bioinformatics toolbox in MATLAB, writing a
  Microarray analysis driver for our BLOPEX
  library, and testing on large-scale publicly
  available Microarray data.