Algorithms for Smoothing 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) Estimated time: < 1 minute

Tumor Cell Chromosomes of tumor cell: Normal cells 23 pairs of chromosomes. Each chromosome is built from two strands of bases labeled (ag tc) Two connected bases are referred to as basepair Male: XY Female: XX Some pieces of chromosomal DNA form genes. Processes in the cell ultimately form proteins from genes, which determine the way a cell behaves. Each chromosome labeled with different color (SKY experiment). You can see this cell does not at all have 2 copies of each chromosome… Pieces are lost, pieces have attached to other chromosomes, some pieces have more than 2 copies.

CGH Data  C o p y # Clones/Chromosomes  Estimated time: 1 minute Explain axis. About the data: Normalized (average 1) Log2 (why?) “1It is preferable to work with logged intensities rather than absolute intensities for a number of reasons including the facts that: (i) the variation of logged intensities and ratios of intensities is less dependent on absolute magnitude; (ii) normalization is additive for logged intensities; (iii) taking logs evens out highly skew distributions; and (iv) taking logs gives a more realistic sense of variation.” Point out some gains, losses and amplifications Clones/Chromosomes 

Naïve Smoothing

“Discrete” Smoothing Copy numbers are integers

Why Smoothing ? Noise reduction Detection of Loss, Normal, Gain, Amplification Breakpoint analysis Estimated time: about 1 minute Explain and point at breakpoints: place where value changes. Smoothing also makes it possible to do analysis on breakpoints. Recurrent (over tumors) aberrations may indicate: an oncogene or a tumor suppressor gene

Is Smoothing Easy? Measurements are relative to a reference sample Printing, labeling and hybridization may be uneven Tumor sample is inhomogeneous Estimated time: 2 minutes Unfortunately there are some problems with the experiment just described that prevent us from simply reading the copy number from the ratio’s. 1) - Not every spot on the slide has the same amount of clones on it. 2) In a tumor not every cell is the same, normal cells may occur, tumor cells of a previous stadium of the tumor may occur. The amounts of cells for a type of cell is unknown. - The amount of DNA material taken from the tumor cells may not be exactly equal to the amount taken from normal cells. - The “green” and “red” molecules do not stick equally well 3) Not every labeled fragment can stick stick to a clone, because it is on a part of the chromosome that does not occur on a clone. Not every part that can stick to a clone, does actually stick 4) Every clone has three spots on the slide. If we assume that red and green fragments same distribution over the spots, the three spots should have the same value. Which is almost the case, since there is only a small standard deviation. Problem: Assume lowest line is normal What is the real situation? 1 Cell type, then the middle line is single gain, the highest line is a double gain 2 Cell types, then possibly Type 1 occurs twice as much as type 2 and has a gain corresponding to the upper line Type 2 has a gain corresponding to the middle line vertical scale is relative do expect only few levels

Smoothing: example Estimated time: < 20 seconds.

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. Estimated time: Details of function to optimize later. What are we trying to model? Do we want to find smoothings that resemble the expert or remove experimental noise? Unfortunately not much known about the properties of the processes that take place during the experiments. We assume… We do not have tumors with detailed data about the real composition of cell types, but the model seems to resemble the expert quite well as we will see. We do have CGH data of a normal-normal experiment. A normal probability plot of that data shows that the noise in that experiment follows a normal distribution quite well.

Smoothing breakpoints variance levels

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 Estimated time: Details of function to optimize later. What are we trying to model? Do we want to find smoothings that resemble the expert or remove experimental noise? Unfortunately not much known about the properties of the processes that take place during the experiments. We assume… We do not have tumors with detailed data about the real composition of cell types, but the model seems to resemble the expert quite well as we will see. We do have CGH data of a normal-normal experiment. A normal probability plot of that data shows that the noise in that experiment follows a normal distribution quite well. We could use better models given insight in tumor pathogenesis

Fitness Function (2) likelihood: CGH values: x1 , ... , xn breakpoints: 0 < y1< … < yN < xN levels: m1, . . ., mN error variances: s12, . . ., sN2 Estimated time: ? Assume independence of all observations… likelihood:

Fitness Function (3) Maximum likelihood estimators of μ and s 2 can be found explicitly Need to add a penalty to log likelihood to control number N of breakpoints penalty Estimated time: ? * Goal: Add penalty to make sure not every index becomes a breakpoint.

Algorithms Maximizing Fitness is computationally hard Use genetic algorithm + local search to find approximation to the optimum Estimated time: ?

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 Estimated time: ?

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 Estimated time: ? Encoding: 0/1 per clone. Label a clone as breakpoint. Termination criterion (both GAs): best fitness in pool does not improve and “individuals” in pool are very similar (there is no pair of individuals that have (after smoothing) a pair of clones that differs at least epsilon in smooth cgh value (average of section)) The mutation is as follows: flip coin to either: remove the breakpoint between 2 consecutive sections with best score afterwards. Insert a breakpoint in the middle of the section with the highest standard deviation.

Experiments Comparison of GLS GLSo Multi Start Local Search (mLS) Multi Start Simulated Annealing (mSA) GLS is significantly better than the other algorithms. Estimated time: ? Data used: about 25 tumors, about 2000 clones/tumor. 23 chroms/tumor. Run the algorithm for each chromosome separately. Explain mLS Explain mSA Test: sign test.

Comparison to Expert algorithm expert

Relating to Gene Expression

Relating to Gene Expression

Algorithms for Smoothing 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) Estimated time: < 1 minute

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 Estimated time: 1 minute

Example of a-CGH Tumor  V a l u e Clones/Chromosomes  Estimated time: 1 minute Explain axis. About the data: Normalized (average 1) Log2 (why?) “1It is preferable to work with logged intensities rather than absolute intensities for a number of reasons including the facts that: (i) the variation of logged intensities and ratios of intensities is less dependent on absolute magnitude; (ii) normalization is additive for logged intensities; (iii) taking logs evens out highly skew distributions; and (iv) taking logs gives a more realistic sense of variation.” Point out some gains, losses and amplifications Clones/Chromosomes 

a-CGH vs. Expression a-CGH DNA DNA on slide 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 Estimated time: <= 2 minutes This slide explains the difference between Array CGH and Microarray gene expression experiments. Explain Copy Number: The copy number of a piece of DNA on the genome is the number of times this piece occurs in the nucleus of a cell.

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 Estimated time: ?

Experiments Both GAs are Robust: Both GAs Converge: 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 Estimated time: ?

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 Estimated time: ? Encoding: 0/1 per clone. Label a clone as breakpoint. Termination criterion (both GAs): best fitness in pool does not improve and “individuals” in pool are very similar (there is no pair of individuals that have (after smoothing) a pair of clones that differs at least epsilon in smooth cgh value (average of section)) The mutation is as follows: flip coin to either: remove the breakpoint between 2 consecutive sections with best score afterwards. Insert a breakpoint in the middle of the section with the highest standard deviation.

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 Estimated time: ? join: Repeatedly select breakpoint whose removal results in biggest improvement of fitness, until fitness does not decrease anymore.

Fitness function (2) likelihood: CGH values: x1 , ... , xn breakpoints: 0 < y1< … < yN < xN levels: m1, . . ., mN error variances: s12, . . ., sN2 Estimated time: ? Assume independence of all observations… likelihood: