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Algorithms forSmoothing Array CGH data

Kees Jong (VU, CS and Mathematics)Elena Marchiori (VU, Computer Science)Aad van der Vaart (VU, Mathematics)Gerrit Meijer (VUMC)Bauke Ylstra (VUMC)Marjan Weiss (VUMC)

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Tumor Cell

Chromosomes of tumor cell:

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CGH Data

Clones/Chromosomes

Copy#

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Naïve Smoothing

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“Discrete” Smoothing

Copy numbers are integers

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Why Smoothing ?• Noise reduction

• Detection of Loss, Normal, Gain, Amplification

• Breakpoint analysis

Recurrent (over tumors) aberrations may indicate:–an oncogene or –a tumor suppressor gene

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Is Smoothing Easy?

Measurements are relative to a reference sample

Printing, labeling and hybridization may be uneven

Tumor sample is inhomogeneous

•vertical scale is relative

•do expect only few levels

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Smoothing: example

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Problem Formalization

A smoothing can be described by• a number of breakpoints • corresponding levels

A fitness function scores each smoothing according to fitness to the data

An algorithm finds the smoothing with the highest fitness score.

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Smoothing

breakpoints

levelsvariance

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Fitness Function

We assume that data are a realization of a Gaussian noise process and use the maximum likelihood criterion adjusted with a penalization term for taking into account model complexity

We could use better models given insight in tumor pathogenesis

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Fitness Function (2)CGH values: x1 , ... , xn

breakpoints: 0 < y1< … < yN < xN

levels:

error variances:

likelihood:

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Fitness Function (3)

Maximum likelihood estimators of μ and 2 can be found explicitly

Need to add a penalty to log likelihood tocontrol number N of breakpoints

penalty

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Algorithms

Maximizing Fitness is computationally hard

Use genetic algorithm + local search to find approximation to the optimum

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Algorithms: Local Search

choose N breakpoints at random

while (improvement)

- randomly select a breakpoint

- move the breakpoint one position to left

or to the right

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Genetic Algorithm

Given a “population” of candidate smoothings create a new smoothing by

- select two “parents” at random from population- generate “offspring” by combining parents

(e.g. “uniform crossover” or “union”)- apply mutation to each offspring- apply local search to each offspring- replace the two worst individuals with the offspring

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Experiments• Comparison of

– GLS

– GLSo

– Multi Start Local Search (mLS)

– Multi Start Simulated Annealing (mSA)

• GLS is significantly better than the other algorithms.

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Comparison to Expert

expert

algorithm

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Relating to Gene Expression

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Relating to Gene Expression

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Algorithms forSmoothing Array CGH data

Kees Jong (VU, CS and Mathematics)Elena Marchiori (VU, CS)Aad van der Vaart (VU, Mathematics)Gerrit Meijer (VUMC)Bauke Ylstra (VUMC)Marjan Weiss (VUMC)

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Conclusion

• Breakpoint identification as model fitting to search for most-likely-fit model given the data

• Genetic algorithms + local search perform well• Results comparable to those produced by hand

by the local expert• Future work:

– Analyse the relationship between Chromosomal aberrations and Gene Expression

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Example of a-CGH Tumor

Clones/Chromosomes

Value

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a-CGH vs. Expression

a-CGH• DNA

– In Nucleus

– Same for every cell

• DNA on slide• Measure Copy

Number Variation

Expression• RNA

– In Cytoplasm

– Different per cell

• cDNA on slide• Measure Gene

Expression

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Breakpoint Detection

• Identify possibly damaged genes:– These genes will not be expressed anymore

• Identify recurrent breakpoint locations:– Indicates fragile pieces of the chromosome

• Accuracy is important:– Important genes may be located in a region

with (recurrent) breakpoints

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Experiments

• Both GAs are Robust:– Over different randomly initialized runs breakpoints

are (mostly) placed on the same location

• Both GAs Converge:– The “individuals” in the pool are very similar

• Final result looks very much like (mean error = 0.0513) smoothing conducted by the local expert

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Genetic Algorithm 1 (GLS)

initialize population of candidate solutions randomly

while (termination criterion not satisfied)

- select two parents using roulette wheel

- generate offspring using uniform crossover

- apply mutation to each offspring

- apply local search to each offspring

- replace the two worst individuals with the offspring

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Genetic Algorithm 2 (GLSo)

initialize population of candidate solutions randomly

while (termination criterion not satisfied)

- select 2 parents using roulette wheel

- generate offspring using OR crossover

- apply local search to offspring

- apply “join” to offspring

- replace worst individual with offspring

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Fitness function (2)CGH values: x1 , ... , xn

breakpoints: 0 < y1< … < yN < xN

likelihood:

levels:

error variances: