A simple statistical modelfor deciphering the cdc15-

synchronized yeastcell cycle-regulated genes expression

data

Ker-Chau Li , Robert Yuan Statistics, UCLAMing Yan Biochemistry , UCLA

The goal of this study is to demonstrate how simple statistical

models can be employed for helping the organization and

explanation ofcomplex gene expression patterns

Outlines

• Introd : Micro-array and cell-cycle• Data : cdc15 experiment• A statistical model• Phase determination• Comparison with Spellman et al(1998)• Regularly oscillated genes• Further discussion

MicroArray

• Allows measuring the mRNA level of thousands of genes in one experiment -- system level response

• The data generation can be fully automated by robots

• Common experimental themes:– Time Course– Mutation/Knockout Response

A B C D E …..

A -- 2.1 0.8 1.3 0.5

B 0.2 -- -0.5 2.3 0.22

… -1.2 -- 0.3 -1.1

Expression level

Time0

1

Change of Condition

Or:

Time Course:

MicroArray Technique:

Synthesize GeneSpecific DNA Oligos

Attach oligo toSolid Support

Tissue or Cell

extract mRNA

Amplificationand Labeling

Hybridize

Scan and Quantitate

Yeast Cell Cycle(adapted from Molecular Cell Biology, Darnell et al)

Getting a homogeneous population of cells:

cell cycle

Cells at variousstages of cell cycle

Synchronization conditions:-Temperature shift to 37 C for CDC15 yeast ts-strain-add pheromone-Elutriation

Release back into cell cycle

Take sampleas cells progressthrough cyclesimultaneously

The data set available at http:cellcycle-www.standford.edu

We focus on one experiment in which a strain of yeast(cdc15-2) was incubated at a high temperature(35 degrees C) for a long time, causing cdc15 arrest. Cells were then shifted back to a low temperature( 23 degrees C) and the monitoring of gene expression is taken every 10 min for 300 min.

Data from some chips are not availableWe concentrate on those from the 19 Consecutive time points from 70 minsTo 250 mins

24 Time points: (mins)

10 30 50 70 80 ..... 240 250 270 290----------> 10 mins apart

Use of full data will be discussed later.

Genes with missing values are also Deleted

There are 4530 genes remainingThe data can be represented by a4530 by 19 matrix

Example of the time curve:

Histone Genes: (HTT2)ORF: YNL031CTime course:

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YKL164C

YNL082W

Preliminary study with two-way anova

This is to investigate the constancy of average expression Level over the time for each gene and the constancy ofThe average expression level over all genes at each timePoint.

> cdc15Factor df SS MS Fgene | 4529 | 5.2408E+2 | 1.1572E-1 | 6.4169E-1time | 18 | 2.9745E+2 | 1.6525E+1 | 9.1638E+1residual |81522 | 1.4701E+4 | 1.8033E-1total |86069 | 1.5522E+4

Gene insignificantTime appears statistically significant; But …………(next slide)

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Column mean (Time) from Anova result The values are small

The expression level is log_2 of ratio of red/greenRed = light intensity for red channel - “noise”Green = light intensity of green channel - “noise”Red channel = mRNA from cells at one time pointGreen channel =mRNA from unsynchronized cells.5 fold increase = log_2 1.5=.585 ; 2^.15 =1.11=.11 fold increase

A statistical model• Motivation : modeling each curve

with simple functions such as linear, quadratic, sine, cosine appears reasonable but inflexible;

• Parsimony and accuracy can be gained if basis curves are chosen by data themselves

• The model : each gene expression curve =c0 + c1V1 + c2V2 + c3V3 + εV1 ,1 st basis curveV2 ,2 nd basis curve

V3 ,3 rd basis curve

The model -continuedThe errors have mean zero,

uncorrelated ,same variance cross the time;

But the variance may depend on genes

(This is important)

It turns out that we can find the basis functions from an application of PCA.

(see pdf file for pca)

Enhanced PCA for curve fitting Choose the number of basis curves by eigenvalues Assess the goodness of each curve fitting by R-squared and by residual sum of squares Identify genes that comply well to the model Interactive plotting helps resetting user-specified parameters

PCA:

For a list of vectors, PCA could be used for finding the common basis based on the scaling matrix.

Covariance Matrix:

The directions found will have highest variance along those directions.

Find the directions by eigenvalue decomposition:

Model the curves by the PCA directions:

Here, we chose first three PCA directions as our basis.

Σ = ( X − μ )' ( X − μ )

Σ νi

= λ νi

X

i= a

1 iν

1+ a

2 iν

2+ K + a

k iν

k+ ε

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0 205 10 15 0 205 10 15

1st PCA direction 2nd PCA direction

3rd PCA direction Eigenvalues

1. Compliance Check:

2. Cycle Component Check:

3. Smoothness Check:

H0

: three - bases model holds

H0

: a2 i

= a3 i

= 0

H0

: a1 i

= 0

Ri

2

< 0 . 56 & RSSi

> 7 . 25

( a2 i

2+ a

3 i

2) / 2

RSSi/ 15

> F2 , 15

( 0 . 95 ) = 3 . 68

a1 i

RSSi

/ 15

> t15

( 0 . 975 ) = 2 . 131

(Corr. Coff between fit and observed < .75And error s.d. Bigger than .70 , which is equivalent to .5 fold increase.)

Reject if

Reject if

Reject if

453041448928241665714951missing valuescompletenon-compliancecomplianceinsignificantcycle comonentsSignificant cyclle componentsSmoothNon-smoothFor the non-compliance group, visual examination of each curve pattern is done .*** of these 41 have visible cycle patterns. l 61781648

Noncompliance genes (41)

. High overall expression levels

. May or may not show cycle patterns… Recommendation : inspect each gene separately

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0 255 10 15 20

YJL159W

YLR126C

Phase determination

• The second and the third basis curves show clear cycle patterns. The third basis appears to be a 40 min-delayed version of the second basis, with an R-squared value of .78

• Linear combinations of these two basis curves show a variety of expression patterns.

Construction of A Compass plot

• Use of known cycle-regulated genes• Compliance checking with RSS/R^2 plot• Cycle- exhibition checking with projection angles• Coherent pattern checking by ANOVA

• ( A list of 104 known genes with 6 groups)

Phases of genes:Identify the phases of genes:

Prior Knowledge:There were 104 know genes whose phases were determined by traditional experiment methods.Known genes:

There are 6 groups of genes.SCB (G1 phase) MCB (G1 phase)Histone (S phase) S/G2 phaseG2/M phase M/G1 phase

The noncompliance genes and without significant cycle components are excluded The group of genes, SCB, are also excluded due to the inconsistent patterns within their expression vectors.

-1 1-0.5 0 0.5

YNL082W

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-1 1-0.5 0 0.5

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82 non-missing known phase genes

Remove genes with insignificant cycle component

Points obtained by normalizing the loading coeff. for 2nd and 3rd bases to unit length

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Late G1, SCB regulated genes:

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YBR067C

YCL055W

YDL055CYER001W

YGL225WYJL187C

YLR342WYMR307W

YPL256C

YPR159W

-1 1-0.5 0 0.5

Compass plot for phase assignment

S

S/G2

G2/M

M/G1

G1

Histone genes

Smooth

-8 4-6 -4 -2 0 2

10831

352

90

295

SG1

S/G2

G2/MM/G1

-6 4-4 -2 0 2

10327

255

239

SG1

S/G2

G2/MM/G1

90

165

Non-smooth

Phase Assignment

Comparison

• For the 800 cell-regulated genes classified by Spellman et al, we re-classified them with our method. If a gene does not comply with our model or does not have significant second or third regression coefficients, we would not assign the phase.

• Contingency tables of mismatched and unclassified cases.

6549645130515293222missing valuescompletenon-compliancecomplianceinsignificantcycle comonentsSignificant cyclle componentsSmoothNon-smooth800The group of 130 insiginicant cycle components appear quite bumpy. All but one in the non-complicance group show clear cycle patterns.

A non-compliance gene

YJL159W :

Spellman et.al’s Score : 10.86 R2: 0.36273 (M/G1)RSS: 14.15322Angle: -2.43803

Least Squares Estimates:

Constant -4.794002E-16 (0.222846)Variable 0 1.28464 (0.971364)Variable 1 -2.04016 (0.971364)Variable 2 -1.49779 (0.971364)

Black: data curveRed : fitted curve (full model)Blue : fitted curve (cyclic model)

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Locus_info: Other_name PIR2 YJL159W CCW7 ORE1 Gene_class HSP Gene_Info HSP150 Gene_product Heat shock protein, secretory glycoprotein Function cell wall structural protein Cellular_Component cell wall Process cell wall organization and biogenesis Phenotype Null mutant is viable Locus_notes 14 HSP150 has also been called gp400 Position_info: Chromosome X ORF_name YJL159W

An example of our non-compliance gene

YDR055W :

Spellman et.al’s Score : 7.266 R2: 0.30136 (M/G1)RSS: 7.94018Angle: -2.81396 (Insig. Coef.)

Least Squares Estimates:

Constant -5.428720E-16 (0.166914)Variable 0 1.47329 (0.727561)Variable 1 -1.07451 (0.727561)Variable 2 -0.316032 (0.727561)

Black: data curveRed : fitted curve (full model)Blue : fitted curve (cyclic model) 0 205 10 15

Locus_info: Other_name YDR055W Gene_class PST Gene_Info PST1 Description Protoplasts-secreted Gene_product The gene product has been detected among the proteins secreted by regenerating protoplasts Phenotype Viable Position_info: Chromosome IV ORF_name YDR055W

An example of non-compliancegene

YNL082W :

Spellman et.al’s Score : 4.843R2: 0.229191 (G1)RSS: 18.247480537500003

Least Squares Estimates:

Constant -6.087129E-16 (0.253035)Variable 0 1.51725 (1.10295)Variable 1 -1.74757 (1.10295)Variable 2 0.263945 (1.10295)

Black: data curveRed : fitted curve (full model)Blue : fitted curve (cyclic model)

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Top 10 scores and gene names from insignificantCycle component group

3.69 3.85 3.874 4.022 4.048 4.13 4.41 5.047 6.28 6.716

"YOR263C" "YOR320C" "YGR035C" "YCR042C" "YPR019W”"YJL194W" "YJR010W" "YEL068C" "YGR124W" "YKL172W"

78 genes score higher than 6.716; 188 genes score higher than 4.022213 genes score higher than 3.69

Yet these genes appear very bumpy; see next slide

An example of insignificant cyclecomponent gene

YGR124W :

Spellman et.al’s Score: 6.28 (S/G2)R2: 0.364945 (small)RSS: 0.812496 (small)Angle: 3.13118

CDC15

70 mins

250 mins

Locus_info: Other_name YGR124W Gene_class ASN Gene_Info ASN2 Description Asn1p and Asn2p are isozymes Gene_product asparagine synthetase Phenotype Null mutant is viable; L- asparagine auxotrophy occurs upon mutation of both ASN1 and ASN2 Position_info: Chromosome VII ORF_name YGR124W

EBP2: YKL172W

TSM1: YCR042C

YOR263C

Our\their G1 S S/G2 G2/M M/G1 TotalG1 59 6 0 0 0 | 65S 4 3 0 0 0 | 7S/G2 1 7 31 17 0 | 56G2/M 0 0 3 47 1 | 51M/G1 18 0 0 4 21 | 43Total 82 16 34 68 22 | 222

Non-smooth group from 800 genes

Our\their G1 S S/G2 G2/M M/G1 TotalG1 74 8 0 0 1 | 83S 7 10 1 0 0 | 18S/G2 5 11 43 17 1 | 77G2/M 0 0 1 39 1 | 41M/G1 43 0 0 3 28 | 74Total 129 29 45 59 31 | 293

Smooth group from 800 genesLow overall expression level

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CLN2: YPL256C

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HTA1: YDR225W

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YJL091C

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CLB4: YLR210W

(Phase ??)

(G1)

(S)

(S/G2)

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CLN2: YPL256C

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HTA1: YDR225W

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CLB4: YLR210W

(G1)

(S)

(S/G2)(Phase ??)

FKS1: YLR342W

From 5 cell

Least Squares Estimates:

Constant -5.706461E-16 (4.704328E-2)Variable 0 -0.170979 (0.205057)Variable 1 0.479678 (0.205057)Variable 2 0.762583 (0.205057)

R Squared: 0.571396 Sigma hat: 0.205057 Number of cases: 19Degrees of freedom: 15

YOR264W

From 1 , total SS small

Oscillated genes

• First curve basis is oscillating in a extremely regular way

• There are over 200 genes with such regular oscillating patterns

• Role unknown : Systematic error ? Common upstream promoter region ?

DIM1 (YPL266W)

Locus_info: Other_name YPL266W Gene_class DIM Gene_Info DIM1 Description Dimethyladenosine transferase, (rRNA(adenine-N6,N6-)-dimethyltransferase),reponsible for m6[2]Am6[2]A dimethylation in 3'-terminal loop of 18S rRNA Gene_product dimethyladenosine transferase Function rRNA (adenine-N6,N6-)-dimethyltransferase Cellular_Component nucleolus Process 35S primary transcript processing rRNA modification Phenotype Null mutant is inviable Position_info: Chromosome XVI ORF_name YPL266W

PRS1A (YLR441C)

Locus_info: Other_name YLR441C RP10A Gene_class RPS Gene_Info RPS1A Description Homologous to rat S3A Gene_product Ribosomal protein S1A (rp10A) Function structural protein of ribosome Cellular_Component cytosolic small ribosomal (40S)-subunit Process 0006416 protein biosynthesis Locus_notes 13 RP10A (RPS1A) and RP10B (RPS1B) are nearly identical; this gene has also been called PLC1, but should not be confused with PLC1 on chromosome XVI encoding a phosphoinositide-specific phospholipase Position_info: Chromosome XII ORF_name YLR441C

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GLN1: YPR035W

Least Squares Estimates:

Constant -6.276471E-16 (4.762055E-2)Variable 0 -2.47649 (0.207573)Variable 1 3.958405E-2 (0.207573)Variable 2 1.01860 (0.207573)

R Squared: 0.917337 Sigma hat: 0.207573

One gene from non-smooth groupNot in Spellman et. al.’s list.

Further discussion

• Others who use PCA

• Clustering

• Other data set

• Use of SIR/PHD

• Without a time scale ? B-cell lymphoma data

• Pathway study

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YGR231C

Least Squares Estimates:

Constant -5.803153E-16 (4.131369E-2)Variable 0 -0.156478 (0.180082)Variable 1 -1.59995 (0.180082)Variable 2 -0.623201 (0.180082)

R Squared: 0.859375 Sigma hat: 0.180082

Total sum of squares equals to 3.4591 which is about 71.6 percentile among all genes.The median of the total sum of squares is 2.27735.

One gene from smooth groupNot in Spellman et. al.’s list.

. Genes with overall small expression levels could have been Removed from the beginning???

THE END

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YBL002W

YDR224C

YER124C

YJL159W

YKL163WYKL164C

YKL185W

YMR003W

YMR011W

YNL160W

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YBL002W

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YDR224C

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YER124C

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YJL159W

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YKL163W

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YKL164C

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YKL185W

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YMR003W

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YMR011W

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YNL160W

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YDR055W