Linear Modeling of Genetic Networks from Experimental Data

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Linear Modeling of Genetic Networksfrom
Experimental Data

• E.P. van Someren, L.F.A. Wessels and M.J.T.
  Reinders
• ISMB 00.
• Talk by Kyu-Baek Hwang

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Abstract

• Topic
• Modeling regulatory interactions between genes
• Linear genetic networks
• Gene expression data
• The dimensionality problem (contribution of this
  paper)
• The number of genes gtgt the number of measured
  time points ? many solutions that fit the
  training data
• Prototypical genes (by clustering) ? biological
  genetic networks are sparse and redundant.
• Experiments
• An artificial dataset
• S. cerevisiae yeast cell-cycle dataset

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Exploitation ofDNA Microarray Datasets

• DNA microarray ? simultaneous measurements on the
  expression levels of thousands of genes
• Infer functionality of genes based on this new
  massive datasets.
• Clustering and pattern recognition techniques
  (NNs and SVMs)
• The regulatory interactions between genes
• Boolean networks, Bayesian networks, linear
  networks, neural networks, and differential
  equations
• Data sparseness problem inherent in the analysis
  of microarray data ? as few parameters as possible

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Linear Networks

• The basic linear model
• ?? 1
• where xj(t) represents the activity level of gene
  j at time point t, ri,j represents how strongly
  gene i controls gene j and N is the total number
  of genes under consideration.
• Prototypical genes ? hierarchical clustering
• Tackling the dimensionality problem
• Input and output sharing among genes involved
  within a gene family or pathway
• Genes are estimated to interact with four to
  eight other genes.

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The Modeling Approach
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Preprocessing Step I Thresholding

• Eliminate insignificant signals (genes).
• Due to experimental noises
• Gene expression levels in different cultures
  under similar conditions
• Vary up to ratio of two
• Gene expression levels in different cultures
  under different conditions
• Vary up to ratio of two to five
• Genes with profiles that remain below an absolute
  value of two ? do not participate in regulation
• Reduce the dimensionality problem.
• Avoid learning erroneous relationships.

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Preprocessing Step II Normalization

• If two signals share the actually (?) same
  characteristics, these two signals should be very
  similar after normalization. (Euclidean distance
  vs. Pearson correlation)

Used in the experiments
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The Linear Model Calls Clustering

• A set of measurements of gene expression levels
  at consecutive time points.
• The linear model is learned by Gaussian
  elimination.
• P a particular solution
• H a basis of homogeneous solutions
• F a set of free variables

More details on the whiteboard.
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Clustering ? Prototypes

• Find groups (clusters) of signals based on the
  similarity.
• Conceptualize the data by representing each
  cluster with a proper prototype.
• Selection of distance measuring metric is very
  important.
• Clustering method complete linkage hierarchical
  clustering based on the Euclidean distance
  measure.
• Prototype
• Reduction of noises in the gene expression levels
• The mean value of all the signals in one cluster
  (RMS)

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Prototypes

• Transforming signals to prototypes
• The inverse

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Experiments
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An Artificial Linear System

• An artificial linear system with five genes.
• R5 matrix in graphical representation.

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Expansion of the System

• Replication of R5 to R25 matrix.
• The (i, j)-th 5 ? 5 sub-matrix in R25 is
  constructed by placing r5i, j on the diagonal
  with all other positions in the sub-matrix
  occupied by zeros.

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Time Response for the 25 ? 25 System

• Initial values of genes in the same cluster were
  set to the more similar values than the values of
  genes in the other clusters. (20 time points)

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Estimation of the Model (1/2)

• Experimental steps (k 1 25)
• 1. The set of prototypes, Yk associated with the
  clusters in Ck was determined.
• 2. The weight matrix, , corresponding to
  each clustering was determined from Yk.
• 3. Given the complete model and the initial
  state, approximations to the original signals can
  be computed as follows
• One-step approximation
• Free-run approximation

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Estimation of the Model (2/2)

• Experimental steps (continued)
• 4. The mean squared errors (MSE) were computed.
  (Eos,k, Efr,k)
• 5. The weighted prototype MSE, Ewp,k is computed.

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Error Curves

• Error curves as a function of the number of
  clusters

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The Resulting Model

• The resulting model (analysis of multiple causes
  is not easy.)

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Real Experiments Yeast Data Set

• Gene expression profiles extracted from the 2467
  genes in the budding yeast S. cerevisiae by
  Eisen.
• Considered conditions
• Mitotic cell division cycle, sporulation and
  temperature and reducing shocks
• Thresholding
• For the ALPHA subset 18 time points with 45
  genes.
• For the CDC15 subset 15 time points with 113
  genes.

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The Effect of Normalization

• Normalization
• If the cluster size is equal to or greater than
  one less than the time step size, the prototype
  free run error is zero. (the limit of
  over-constrained condition)

The one-step MSE of all datasets and the four
kinds of normalization
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Error Curves on the CDC15 Dataset

• Fitting the linear model on the CDC15 subset.
• The error curve

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The Resulting Model on CDC15

• The resulting model

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Error Curves on the ALPHA Dataset

• Fitting the linear model on the ALPHA subset.
• The error curve

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The Resulting Model on ALPHA

• The resulting model

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Resulting Model without Normalization

• The resulting model without normalization (more
  reasonable ?)

Mating
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Summary

• The dimensionality problem was tackled by
  clustering.
• Biologically sound.
• Linear models were used to represent the
  relationships between the resulting
  gene-prototypes.
• Good balance between the model complexity and the
  accuracy.
• No intrinsic semantics was found.