Mathematical Modeling ofDNA Microarray Data:

Discovery of Biological Mechanisms withTensor Decompositions, and

Definitions of Novel Tensor Decompositionsfrom Biological Applications

Orly Alter

Department of Biomedical Engineering,Institute for Cellular and Molecular Biology and

Institute for Computational Engineering and Sciences

University of Texas at Austin

DNA Microarrays Record Genomic Signals

DNA microarrays relyon hybridization t orecord the completegenomic signals thatguide the progression ofcellular processes, suchas abundance levels ofDNA, RNA and DNA-bound proteins on agenomic scale.

From Data Patterns to Principles of NatureAlter, PNAS 103, 16063 (2006);

Alter, in Microarray Data Analysis: Methods and Applications (Humana Press, 2007), pp. 17–59.

Kepler’s discovery of his first law of planetary motion from mathematicalmodeling of Brahe’s astronomical data:

Kepler, Astronomia Nova (Voegelinus, Heidelberg, 1609), reproduced by permission of theHarry Ransom Humanities Research Center of the University of Texas, Austin, TX).

Physics-Inspired Matrix (and Tensor) ModelsMathematical frameworks for the description of the data, in which themathematical variables and operations might represent biological reality.

SVDAlter, Brown & Botstein,PNAS 97, 10101 (2000).

ComparativeGSVD

Alter, Brown & Botstein,PNAS 100, 3351 (2003).

IntegrativePseudoinverse

Alter & Golub,PNAS 101, 16577 (2004).

Uncover CellularProcesses and States

Uncover ProcessesCommon or ExclusiveAmong Two Datasets

Uncover CoordinationAmong Multiple Sets

Eigenvalue Decomposition Generalized EigenvalueDecomposition

Inverse Projection

Networks are Tensors of “Subnetworks”Alter & Golub, PNAS 102, 17559 (2005);

http://www.bme.utexas.edu/research/orly/network_decomposition/.

Æ =

+ + ...

The relations among the activities of genes, not only the activities of thegenes alone, are known to be pathway-dependent, i.e., conditioned bythe biological and experimental settings in which they are observed.

A Higher-Order SVD Predicts anEquivalent Biological MechanismLinear transformation of the data tensor from genes ¥x-settings ¥ y-settings space to reduced “eigenarrays”¥ “x-eigengenes” ¥ “y-eigengenes” space.

This HOSVD is computedfrom each SVD of the datatensor unfolded around onegiven axis,

De Lathauwer, De Moor &Vandewalle, SIMAX 21, 1253 (2000);Kolda, SIMAX 23, 243 (2001); Zhang& Golub, SIMAX 23, 543 (2001).

mRNA Expression fromCell Cycle Time Coursesunder Different Conditionsof Oxidative StressShapira, Segal & Botstein,

MBC 15, 5659 (2004);Spellman et al., MBC 9, 3273 (1998).

HOSVD Integrative ModelingOmberg, Golub & Alter, PNAS 104, 18371 (2007);http://www.bme.utexas.edu/research/orly/HOSVD/.

The data tensor is a superpositionof all rank-1 “subtensors,” i.e.,outer products of an eigenarray,an x- and a y-eigengene,

The significance of a subtensor isdefined by the corresponding“fraction,” computed from thehigher-order singular values,

The complexity of the data tensoris defined by the “normalizedentropy,”

Rotation in an ApproximatelyDegenerate Subtensor Space

An “approximately degeneratesubtensor space” is defined as thatwhich is span by, e.g., the subtensors

which satisfy

This HOSVD is reformulatedwith a unique single rank-1subtensor that is composed ofthese two subtensors,

Math Variables & Operations Æ BiologyHOSVD uncovers independent data patterns across each variable and theinteractions among them Æ global picture of the causal coordinationamong biological processes and experimental phenomena:

Equivalent DNA ´ RNA Correlation

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Analysis of Synchronized Cdc6�/45� Cultureswhere DNA Replication Initiation is Prevented

without Delaying Cell Cycle ProgressionOmberg, Meyerson, Kobayashi, Drury, Diffley & Alter, Nature MSB 5, 312 (2009);

http://www/nature.com/doifinder/10.1038/msb.2009.70

HOSVD Detection and Removal of ArtifactsReconstructing the data tensor of 4,270 genes ¥ 12 time points, or x-settings ¥ 8 time courses, or y-settings, filtering out “x-eigengenes” and“y-eigengenes” that represent experimental artifacts.

Batch-of-hybridization

Culture batch,microarrayplatform andprotocols

Uncovering Effects of Replication and OriginActivity on mRNA Expression with HOSVD

1,1,1 72% >0Steady State

First, ~88% of mRNA expression isindependent of DNA replication.

2,2,1 9% >0 � M/G1 <2·10-33 Ø S/G2 <7·10-16

3,3,1 7% >0 � G1/S <2·10-77 Ø G2/M <3·10-36

Unperturbed Cell Cycle

Replication-Dependent Perturbations4,1,2 2.7% >0 � ARSs 3’ ~10-2 Ø histones <10-12

7,3,2 0.8% >0 � histones <5·10-4

DNA replication increases time-averagedand G1/S expression of histones.

Histones are overexpressed in the controlrelative to the Cdc6� condition, and to alesser extent also relative to the Cdc45�

condition (a P-value ~2·10-15).

Second, the requirement of DNAreplication for efficient histone geneexpression is independent of conditionsthat elicit DNA damage checkpointresponses.

Origin Binding-Dependent Perturbations5+6,1,3 1.9% >0 � histones <2·10-8 Ø ARSs 3’ <2·10-3

8,3,3 0.7% >0 Ø ARSs 3’ <7·10-4

Origin binding decreases time-averaged and G2/M expression of geneswith ARSs near their 3’ ends. These genes are overexpressed in theCdc6� relative to the Cdc45� condition, and to a lesser extent also relativeto the control (a P-value <4·10-7) Æ Third, origin licensing decreasesexpression of genes with origins near their 3’ ends, revealing thatdownstream origins can regulate the expression of upstream genes.

Experimental Verification of theComputationally Predicted MechanismOmberg, Meyerson, Kobayashi, Drury, Diffley & Alter, Nature MSB 5, 312 (2009);

http://www/nature.com/doifinder/10.1038/msb.2009.70

Æ These experimental results reveal that downstream origins canregulate the expression of upstream genes.

Æ These experimental results verify the computationally predictedmechanism of regulation that correlates binding of the licensingproteins Mcm2–7 with reduced expression of adjacent genesduring the cell cycle stage G1.Alter & Golub, PNAS 101, 16577 (2004);Alter, Golub, Brown & Botstein, Proc. MNBWS 15 (2004).

Æ These experimental results are also in agreement with theequivalent correlation between overexpression of binding targetsof Mcm2–7 and expression in response to oxidative stress.Omberg, Golub & Alter, PNAS 104, 18371 (2007);Cocker, Piatti, Santocanale, Nasmyth & Diffley, Nature 379, 180 (1996);Blanchard et al., MBC 13, 1536 (2002).

Æ This demonstrates for the first time that mathematical modeling ofDNA microarray data can be used to correctly predict biologicalmechanisms.

HO GSVD for Comparative Analysis of DNAMicroarray Data from Multiple Organisms

Ponnapalli, Saunders, Golub & Alter, under revision.

Yeast Spellman et al. MBC 9, 3273 (1998).

Human Whitfield et al. MBC 13, 1977 (2002).

An HO GSVD that extendsto higher orders most of themathematical properties ofthe GSVD,

D1 =U1�1VT ,

D2 =U2�2VT ,

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DN =UN�NV T .

Æ The only framework todate that is not limited tocomparison of similar genesamong the organisms.Æ Reveals universality andspecialization that are trulyon genomic scales. Alter, Brown & Botstein, PNAS 100, 3351 (2003).

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A Higher-Order GSVD

Definition:

Di =Ui�iVT , �i = diag(� i,k )

SV =V�

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Assumption: ��V �Rn�n

Interpretation:� i,k � j ,k � 1� vk of similar significance in Di and Dj

� i,k � j ,k <<1� vk of negligible significance in Di relative to Dj

Math Variables Æ Biology

Genelets of almost equal significance in all datasets Æprocesses common to all genomes:

Common Cell Cycle Subspace

Math Operations Æ Biology

Simultaneous Classification in theCommon Cell Cycle Subspace

Schizosaccharomyces pombeRustici et al. Nat. Genet. 36, 809 (2004).

Saccharomyces cerevisiaeSpellman et al. MBC 9, 3273 (1998).

HumanWhitfield et al. MBC 13, 1977 (2002).

Genome-wide correspondence among genes of the yeasts and human.

The interplay between mathematical modeling and experimentalmeasurement is at the basis of the “effectiveness of mathematics” inphysics. Wigner, Commun. Pure Appl. Math. 13, 1 (1960).

Mathematical modeling of DNA microarray data could lead beyondclassification of genes and cellular samples to the discovery andultimately also control of molecular biological mechanisms.

Andrews & Swedlow, Nikon Small World (2002).

Our mathematical models may form the basis of a future wheremolecular biological systems are modeled and controlled as physicalsystems are today.

Thanks to –Collaborators:

John F. X. DiffleyCancer Research UK, London

Gene H. GolubComputer Science, Stanford

Robin R. GutellIntegrative Biology, UT

Michael A. SaundersOperations Research, Stanford

David BotsteinGenomics Institute, Princeton

Patrick O. BrownBiochemistry, Stanford

Students:Kayta Kobayashi, Pharmacy, UT

Joel R. Meyerson, BME, UTLarsson Omberg, Physics, UT

Cheng H. Lee, BME, UTChaitanya Muralidhara, CMB, UT

Sri Priya Ponnapalli, ECE, UTDaifeng Wang, ECE, UT

Justin A. Drake, BME, UTAndrew M. Gross, BME, UT

Support:NHGRI K01 Development

Award in Genomic Research

NHGRI R01 HG004302

And, thank you!!!