A New Approach to Analyzing Gene Expression Time Series Data

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A New Approach to Analyzing Gene Expression Time
Series Data

• Ziv Bar-Joseph
• Georg Gerber
• David K. Gifford
• Tommi S. Jaakkola
• Itamar Simon

Learning Seminar Bioinformatics Other
Applications Prof. Nathan Intrator Presented
By Adam Segoli Schubert May 16, 2005
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Overview

• Gene Expression
• Time Series
• Statistical Analysis of Time-Series
• DNA Microarray
• Gene Expression Time-Series
• Analyzing Gene Expression Time-Series Data
• Estimating Unobserved Expression Values and Time
  Points
• What is a Spline?
• Using the Splines
• Parameters Analysis
• Aligning Time-Series Data
• Aligning Temporal Data Using Splines
• Results Unobserved Data Estimation
• Result - Aligning Temporal Data
• References

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Gene Expression
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Time-Series

• A series of values of variables taken in
  successive periods of time
• Time Points
• Sampling Intervals (constant / inconstant)
• A well established area in statistical analysis
  of data is dedicated to the study of time-series

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Statistical Analysis of Time-Series

• Two main goals
• Identifying the nature of the phenomenon
• Predicting unobserved values of the time-series
  variable

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DNA Microarray

• Allows the monitoring of expression levels of
  thaousands of genes under a variety of
  conditions.
• The data of microarray experiments is usually in
  the form of a large matrix.
• Very Expensive.

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Gene Expression Time-Series

• Determined by measuring mRNA levels or protein
  concentrations
• Commonly are very short (i.e. 4 to 20 samples)
• Usually unevenly sampled
• The measuring techniques are extremely
  noise-prone and/or subject to bias in the
  biological measurements.

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Analyzing Gene Expression Time-Series Data

• Estimating Unobserved Expression Values and Time
  Points
• Aligning Time-Series Data

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Estimating Unobserved Expression Values and Time
Points

• Row Average or Filling with Zeros
• Singular Value Decomposition (SVD)
• Weighted K-Nearest Neighbors
• Linear Interpolation

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A New Analysis Approach

• By using Cubic Splines.

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What is a Spline?

• A special curve defined piecewise by polynomials.
• Given k points ti called knots in an interval
  a,b with
• The parametric curve is
  called a Spline of degree n if
  and
• A Cubic Spline if n 3.

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Using the Splines

• We Obtain a continues time formulation by using
  cubic splines to represent gene expression
  curves.
• Spline control points are uniformly spaced.
• We constrain spline coefficients of co-expressed
  genes to have the same covariance matrix.

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Estimating Unobserved Data Using Splines

• Given c Genes Classes.
• - The gene i (of class j) value as
  observed at time t
• Can be written as

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Estimating Unobserved Data Using Splines

• Resampling gene I at any time t of an unobserved
  time point
• Estimating Missing Values
• Averaging of the observed values using the class
  covariance matrix , class average and
  the gene specific variation .
• Where are determined by a
  probabilistic model.

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Estimating Unobserved Data Using Splines
Parameters Analysis

• Yi Vector of observed expression values for
  gene i.
• Si Matrix mxq for m observations.

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Aligning Time-Series Data

• Dynamic Time Wraping
• Developed for voice recognition purposes at the
  70s.
• Dynamic Programming
• John Aach George M. Church
• operates on individual genes

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Aligning Temporal Data Using Splines

• Operates on a set of genes.
• Assume we have two spline curve for gene i
• We define a mapping function T(s) t
  
  
  

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Aligning Temporal Data Using Splines

• We Define the alignment error for each gene
• Alignment Limits
• Starting Point
• Ending Point

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Aligning Temporal Data Using Splines

• We define the error for a set of genes S of size
  n as
• - Weighted coefficients that sum to one
• (uniform / nonuniforn).

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Aligning Temporal Data Using Splines

• The Mapping function (T(s) t) can then be found
  by minimizing s value. Using standard
  non-linear optimization techniques.

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Results Unobserved Data Estimation

• Comparison of the new approach with
• Linear Interpolation
• Spline interpolation using individual genes
• K-Nearest neighbors (KNN)
• k 20

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Result - Aligning Temporal Data

• Aligned three yeast cell-cycle gene expression
  time series

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Thank You!

• Any Questions?

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References

• C. S. Moller-Levet. Clustering of Gene Expressiom
  Time-Series Data.
• Biology. Fifth Edition By Neil A. Campbell, Jane
  B. Reece, and Lawrence G. Mitchell.
• J. Aach and G. M. Church. Aligning gene
  expression time series with time warping
  algorithms. Bioinformatics, 17495-508, 2001.
• C. de Boor. A practical guide to splines.
  Springer, 1978.
• P. Dhaeseleer, X. Wen, S. Fuhrman, and R.
  Somogyi. Linear modeling of mrna expression
  levels during cns development and injury. In
  PSB99, 1999.
• G. James and T. Hastie/ Functional linear
  discriminant analysis for irregulary sampled
  curves. Jurnal of the Royal Statistical Society,
  to appear, 2001.
• Sharan R. and Shamir R. Algorithmic approaches to
  clustering gene expression data/ current topics
  in coputational Biology, To appear.
• O. Troyanskaya, M. Cantor, and et al/ Missing
  value estimation methods for dna microarrays.
  bioinformatics, 17520-525, 2001.