Analysis of time-course gene expression data

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Analysis of time-course gene expression data

• Shyamal D. PeddadaBiostatistics Branch
• National Inst. Environmental
• Health Sciences (NIH)Research Triangle Park, NC

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Outline of the talk

• Some objectives for performing long series
  time-course experiments
• Single cell-cycle experiment
• A nonlinear regression model
• Phase angle of a cell cycle gene
• Inference
• Open research problems
• Multiple cell-cycle experiments
• Coherence between multiple cell-cycle
  experiments
• Illustration
• Open research problems

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Objectives

• Some genes play an important role during the
  cell division cycle process. They are known as
  cell-cycle genes.
• Objectives Investigate various characteristics
  of cell-cycle and/or circadian genes such as
• Amplitude of initial expression
• Period
• Phase angle of expression (angle of maximum
  expression for a cell cycle gene)

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Phases in cell division cycle
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A brief description

• G1 phase
• "GAP 1". For many cells, this phase is the
  major period of cell growth during its lifespan.
• S ("Synthesis) phase
• DNA replication occurs.

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A brief description

• G2 phase
• "GAP 2 Cells prepare for M phase. The G2
  checkpoint prevents cells from entering mitosis
  when DNA was damaged since the last division,
  providing an opportunity for DNA repair and
  stopping the proliferation of damaged cells.
• M (Mitosis) phase
• Nuclear (chromosomes separate) and cytoplasmic
  (cytokinesis) division occur. Mitosis is further
  divided into 4 phases.

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Single, long series experiment
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Whitfield et al. (Molecular Biology of the Cell,
2002)

• Basic design is as follows
• Experimental units Human cancer cells (HeLa)
• Microarray platform cDNA chips used with approx
  43000 probes (i.e. roughly 29000 genes)
• 3 different patterns of time points (i.e. 3
  different experiments)
• One of the goals of these experiments was to
  identify periodically expressed genes.

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Whitfield et al. (Molecular Biology of the Cell,
2002)

• Experiment 1 (26 time points)
• Hela cancer cells arrested in the S-phase using
  double thymidine block.
• Sampling times after arrest (hrs)
• 0 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 18 20 22
  24 26 28 32 36 40 44.

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Whitfield et al. (2002)

• Experiment 2 (47 time points)
• Hela cancer cells arrested in the S-phase using
  double thymidine block.
• Sampling times after arrest (hrs)
• every hour between 0 and 46.

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Whitfield et al. (2002)

• Experiment 3 (19 time points)
• Hela cancer cells arrested arrested in the
  M-phase using thymidine and then by nocodazole.
• Sampling times after arrest (hrs)
• 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
  36.

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Whitfield et al. (2002)Phase marker genes

• Cell Cycle Phase Genes
• ------------------ -------
• G1/S CCNE1, CDC6, PCNA,E2F1
• S RFC4, RRM2
• G2 CDC2, TOP2A, CCNA2, CCNF
• G2/M STK15, CCNB1, PLK, BUB1
• M/G1 VEGFC, PTTG1, CDKN3, RAD21

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Questions

• Can we describe the gene expression of a
  cell-cycle gene as a function of time?
• Can we determine the phase angle for a given
  cell-cycle gene? i.e. can we quantify the
  previous table in terms of angles on a circle?
• What is the period of expression for a given
  gene?
• Can we test the hypothesis that all cell-cycle
  genes share the same time period?
• Etc.

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Profile of PCNA based on experiment 2 data
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Some important observations

1. Gene expression has a sinusoidal shape
2. Gene expression for a given gene is an average
   value of mRNA levels across a large number of
   cells
3. Duration of cell cycle varies stochastically
   across cells
4. Initially cells are synchronized but over time
   they fall out of synchrony
5. Gene expression of a cell-cycle gene is expected
   to decrease/decay over time. This is because
   of items 2 and 4 listed above!

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Random Periods Model (PNAS, 2004)

• a and b background drift parameters
• K the initial amplitude
• T the average period
• the attenuation parameter
• the phase angle

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Fitted curves for some phase marker genes
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Whitfield et al. (2002)Phase marker genes

• Phase Genes Phase angles (radians)
• -------- ------- ------------------------
• G1/S CCNE1, CDC6, PCNA,E2F1 0.56,
  5.96, 5.87, 5.83
• S RFC4, RRM2 5.47, 5.36
• G2 CDC2, TOP2A, CCNA2, CCNF 4.24, 3.74, 3.55,
  3.25
• G2/M STK15, CCNB1, PLK, BUB1 3.06,
  2.67, 2.61, 2.51
• M/G1 VEGFC, PTTG1, CDKN3, RAD21 2.66, 2.40,
  2.25, 1.81

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A hypothesis of biological interest

• Do all cell cycle genes have same T and same
  but the other 4 parameters are gene specific?
• i.e.

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An Important Feature

• Correlated data
• Temporal correlation within gene
• Gene-to-gene correlations

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Test Statistic

• Wald statistic for heteroscedastic linear and
  non-linear models
• Zhang, Peddada and Rogol (2000)
• Shao (1992)
• Wu (1986)

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The Null Distribution

• Due to the underlying correlation structure
• Asymptotic approximation is not
  appropriate.
• Use moving-blocks bootstrap technique on the
  residuals of the nonlinear model.
• Kunsch (1989)

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Moving-blocks Bootstrap

• Step 1 Fit the null model to the data and
  compute the residuals.
• Step 2 Draw a simple random sample (with
  replacement) from all possible blocks , of a
  specific size, of consecutive residuals.

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Moving-blocks Bootstrap

• Step 3 Add these residuals to the fitted curve
  under the null hypothesis to obtain the bootstrap
  data set
• Step 4 Using the bootstrap data fit the model
  under the alternate hypothesis and compute the
  Wald statistic.

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Moving-blocks Bootstrap

• Step 5 Repeat the above steps a large number of
  times.
• Step 6 The bootstrap p-value is the proportion
  of the above Wald statistics that exceed the Wald
  statistic determined from the actual data.

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Analysis of experiment 2

• The bootstrap p-value for testing
• using Experiment 2 data of Whitfield et al.
  (2002) is 0.12.
• Thus our model is biologically plausible.

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Statistical inferences on the phase angle
Multiple experiments
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Some questions of interest

• How to evaluate or combine results from multiple
  cell division cycle experiments?
• Are the results consistent across experiments?
• How to evaluate this?
• What could be a possible criterion?

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Data

• RPM estimate of phase angle of a
  cell-cycle gene g
• from the experiment.

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Representation using a circle

• Consider 4 cell cycle genes A, B, C, D. The
  vertical line in the circle denotes the reference
  line. The angles are measured in a
  counter-clockwise.
• Thus the sequential order
• of expression in this
• example is A, B, D, C.

A
B
C
D
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Coherence in multiple cell-cycle experiments

• A group of cell cycle genes are said to be
  coherent across experiments if their sequential
  order of the phase angles is preserved across
  experiments.

B
A
D
B
Exp 2
D
A
C
D
C
C
Exp 3
B
A
Exp 1
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Geometric Representation

• We shall represent phase angles from multiple
  cell cycle experiments using concentric circles.
• Each circle represents an experiment.
• Same gene from a pair of experiments is connected
  by a line segment.
• A figure with non-intersecting lines indicates
  perfect coherence.
• If there is no coherence at all then there will
  be many intersecting lines.

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Example Perfectly Coherent
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Example Perfectly Coherent
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Example No coherence
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Estimated Phase Angles

• Due to statistical errors in estimation, the
  estimated phase angles from multiple cell cycle
  experiments need not preserve the sequential
  order even though the true phase angles are in a
  sequential order.

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How to evaluate coherence?
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Some background on regression for circular data
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Experiment B
Experiment A
Question Can we determine a rotation matrix A
such that we can rotate the circle representing
Experiment A to obtain the circle representing
Experiment B?
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Angle of rotation for a rigid body

• Yes! By solve the following minimization problem

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Determination of Coherence Across k Experiments
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The Basic Idea

• Consider a rigid body rotating in a plane.
  Suppose the body is perfectly rigid with no
  deformations.
• Let denote the 2x2 rotation
  matrices from
• experiment i to i1 (k1 1). Then
• Alternatively

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The Basic Idea

• Equivalently, if
• Then under perfect rigid body motion we should
  have

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Problem!

• In the present context we do NOT necessarily have
  a rigid body!
• Not all experiments are performed with same
  precision.
• The time axis may not be constant across
  experiments.
• Number of time points may not be same across
  experiments.
• Etc.

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Example Not a rigid motion but perfectly
coherent
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Consequence

• Rotation matrix A alone may not be enough to
  bring two circles to congruence!
• An additional association/scaling parameter may
  be needed as see in the previous figure!

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Circular-Circular regression model for a pair of
experiments (Downs and Mardia, 2002)

• For , let
  denote a pair of
• angular variables.
• Suppose is von-Mises distributed
  with
• mean direction and concentration
  parameter

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Circular-Circular Regression Model (Downs and
Mardia, 2002)
The regression model is given by the link function
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Back to the toy examples
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Determination Of Coherence

• Suppose we have K experiments, labeled as
• 1, 2, 3, , K. Let denote the angle of
  rotation
• for the regression of i on j for a group of g
  genes.
• Compute
• Note .

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Determination Of Coherence

• We expect under no coherence
• to be stochastically larger than
• under coherence.

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Comparison of Cumulative Distribution Functions
Blue line Coherence Pink line No Coherence
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Determination Of Coherence

• For a given data compute
• Generate the bootstrap distribution of
• under the null hypothesis of no coherence.

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Bootstrap P-value For Coherence

• Let denote the angle of rotation
  using
• the bootstrap sample. Then the P-value is

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Illustration Whitfield et al. data

• There are 3 experiments. The phase angles of each
  gene was estimated using Liu et al., (2004)
  model.
• A total of 47 common cell-cycling genes were
  selected from the three experiments.

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Estimates

• The estimated values of interest are
• Note that

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Conclusion

• Since the bootstrap P-value lt 0.05, we conclude
  that the three experiments are coherent.

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Statistical inferences on the phase angle- Some
open problems
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Estimation subject to inequality constraints

• It is reasonable to hypothesize that for a normal
  cell division cycle, the p phase marker genes
  must express in an order around the unit circle.
• Thus they must satisfy

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Open problems- data from single experiment

• How to estimate the phase angles subject to the
  simple order restriction?
• More generally - wow to estimate the phase angles
  subject isotropic simple order restriction?
• How to test the above hypothesis? What are the
  null and alternative hypotheses?

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Open problems data from multiple experiments

• How do we estimate the phase angles from multiple
  experiments under the order restriction on the
  phase angles of cell cycle genes?
• What are the statistical errors associated with
  such an estimator?
• How to construct confidence intervals and test
  hypotheses?

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Acknowledgments

• Delong Liu (former Post-doc at NIEHS)
• David Umbach (NIEHS)
• Leping Li (NIEHS)
• Clare Weinberg (NIEHS)
• Pat Crocket (Constella Group)
• Cristina Rueda (Univ. of Valladolid, Spain)
• Miguel Fernandez (Univ. of Valladolid, Spain)