Flowering project step-by-step

Transcription

1 Flowering project step-by-step

2 Flowering time regulation in apex step-by-step Profiles of expression for all genes in the flowering time course experiments TSNI algorithm approach: Per experiment/gene time course data interpolation Per gene Multiple Regression table across all experiments SVD of multiple regression table Multi-Regression Solution: network and dynamic model Model based data simulation and impact of perturbations

3 Affymetrix array: hypothesis testing with probe voting Name avn_pm avr_pm WN_PM WR_PM WNnumPMWRnumPMWN_PMpv WR_PMpv avn avr SD_N SD_P Tt _x_ me tallothione in 1H _x_ me tallothione in 1H-like prote in _x_ E me tallothione in 2A

4 Available time course data Time course: day0, day3, day5, and day7 Each day is either in two or three replicates

5 Wald table per data set Gene information (22,000 genes) Averaged profile of time course for every gene For each gene, the probe-vote based significance of differentiations between time-points

6 Flowering data: Wald table (1) Gene information and average log-signals

7 Flowering data: Wald table (2) Cumulative Ttest (Wald) and minus Log(pvalue) of Ftest

8 Flowering data: Wald table (3) Gene annotations and max of the Ftest s Log(pvalue)

9 Selection of genes: (1) the highest time course profile variability and (2) the trustable signal

10 Selection of genes highest profile variability genes 972 genes

11 A portion of 262 selected genes

12 Wt_ColX1 time course behavior: Cycle (or max expression on day 3) Average probe log-signal Statistical significance of induction VRN2 (VERNALIZATION 2) NFYB3/HAP3C (NUCLEAR FACTOR Y, SUBUNIT B3/HEME ACTIVATED PROTEIN 3C) NFYB2/HAP3B (NUCLEAR FACTOR Y, SUBUNIT B2/HEME ACTIVATED PROTEIN 3B) VRN2 (VERNALIZATION 2) NFYB3/HAP3C (NUCLEAR FACTOR Y, SUBUNIT B3/HEME ACTIVATED PROTEIN 3C) NFYB2/HAP3B (NUCLEAR FACTOR Y, SUBUNIT B2/HEME ACTIVATED PROTEIN 3B) ColX1_0d_F ColX1_3d_F ColX1_5d_F ColX1_7d_F 5.30 ColX1_0d ColX1_3d ColX1_5d ColX1_7d

13 Wt_ColX1 time course behavior: Max expression on day 5 Average probe log-signal Statistical significance of induction AGL24 (AGAMOUS-LIKE 24) RGA (REPRESSOR OF GA1-3 1) AGL24 (AGAMOUS-LIKE 24) 6.10 SAP18 (SIN3-ASSOCIATED POLYPEPTIDE 18) 5.00 RGA (REPRESSOR OF GA1-3 1) SAP18 (SIN3-ASSOCIATED POLYPEPTIDE 18) 6.00 ColX1_0d ColX1_3d ColX1_5d ColX1_7d ColX1_0d_F ColX1_3d_F ColX1_5d_F ColX1_7d_F

14 Wt_ColX1 time course behavior: Upregulation across time Average probe log-signal Statistical significance of induction AP1 (APETALA1) TFL1 (TERMINAL FLOWER 1) LHY (LATE ELONGATED HYPOCOTYL) CCA1 (CIRCADIAN CLOCK ASSOCIATED 1) LFY (LEAFY) SPL5 (SQUAMOSA PROMOTER BINDING PROTEIN-LIKE 5) FUL (FRUITFUL) SPL3 (SQUAMOSA PROMOTER BINDING PROTEIN-LIKE 3) SPL4 (SQUAMOSA PROMOTER BINDING PROTEIN-LIKE 4) SOC1 (SUPPRESSOR OF OVEREXPRESSION OF CONSTANS) AP1 (APETALA1) TFL1 (TERMINAL FLOWER 1) LHY (LATE ELONGATED HYPOCOTYL) CCA1 (CIRCADIAN CLOCK ASSOCIATED 1) LFY (LEAFY) SPL5 (SQUAMOSA PROMOTER BINDING PROTEIN-LIKE 5) FUL (FRUITFUL) SPL3 (SQUAMOSA PROMOTER BINDING PROTEIN-LIKE 3) SPL4 (SQUAMOSA PROMOTER BINDING PROTEIN-LIKE 4) SOC1 (SUPPRESSOR OF OVEREXPRESSION OF CONSTANS) ColX1_0d ColX1_3d ColX1_5d ColX1_7d ColX1_0d_F ColX1_3d_F ColX1_5d_F ColX1_7d_F

15 Wt_ColX1 time course behavior: Downregulation across time Average probe log-signal Statistical significance of induction ColX1_0d_F ColX1_3d_F ColX1_5d_F ColX1_7d_F PRR5 (PSEUDO-RESPONSE REGULATOR 5) CDF3 (CYCLING DOF FACTOR 3) GA2ox6 (GIBBERELLIN 2-OXIDASE 6) TOE1 (TARGET OF EAT 1) PRR5 (PSEUDO-RESPONSE REGULATOR 5) CDF3 (CYCLING DOF FACTOR 3) GA2ox6 (GIBBERELLIN 2-OXIDASE 6) TOE1 (TARGET OF EAT 1) FLC (FLOWERING LOCUS C) also called FLF (FLOWERING LOCUS F) and AGL25 (AGAMOUS LIKE 25) GI (GIGANTEA) FLC (FLOWERING LOCUS C) also called FLF (FLOWERING LOCUS F) and AGL25 (AGAMOUS LIKE 25) GI (GIGANTEA) ColX1_0d ColX1_3d ColX1_5d ColX1_7d

16 Network inference from gene expression perturbations and time course measurements

17 Network from time course: TSNI Algorithm (1) Equation (1) at time t k : Unknown impact on a gene If the perturbation (treatment) was applied can be rewritten in a more compact form using matrix notation: Bansal et al (2006) Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, 22:

18 Network from time course: TSNI Algorithm (2) k = 1 N*1 = N*N N*1 + N*P P*1 k = M N*1 = N*N N*1 + N*P P*1 Bansal et al (2006) Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, 22:

19 For gene i: i 1*M Across time = 1*N + 1*P P*M Across genes i Across genes N*M Across perturbations 1*M Across time i = 1*(N+P) i Across genes N*M Element (i, l) of B will be different from zero if the i-th gene is a direct target of the l-th perturbation TSNI Algorithm (3) Bansal et al (2006) Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, 22: Across genes Across perturbations Across time P*M Across time

20 Network from time course: Singular Value Decomposition Y H U X A B X i i = = ] [ Performing SVD of matrix Y we will get: Bansal et al (2006) Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, 22:

21 Estimation of parameters across all experiments For gene i and experiment k: i 1*M Across time = 1*(N+P) i Across genes Across genes Across perturbations N*M P*M Across time Across all experiments: For experiment k after matrix transposition: X T ik = (X kt U kt )*(A i B) T (M*P) x 1 (M*P) x (N+P) X T i1 (X 1T U 1T ) =. * X T ip (X PT U PT ) Bansal et al (2006) Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, 22: (N+P) x 1 A i B

22 Network from time course: The gene space of reduced dimension D space: time courses of two genes 1D space for the principal gene Y H U X A B X i i = = ] [ Bansal et al (2006) Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, 22:

23 Singular Value Decomposition M*N M*N N*N N*N Press et al Numerical receipts in C

24 SVD: Orthogonality of matrix U and V Press et al Numerical receipts in C

25 Measurements (y) Multiple Deviation from the model ( noise ) Regression Factor_1 Factor_2 measurements in the model (estimated through factors) y=x*a + e Theorem: P x = X(X T X) -1 X T is the orthogonal projection operator Why? Because: 1. P X *z = z if z is from S(X) (z = X*b) 2. PX*z = 0 if z is orthogonal to S(X) i.e. X T z=0 1. P x *z= X(X T X) -1 X T *(X*b) = X*b = z Therefore: P x *y= X(X T X) -1 X T *y = X*a 2. P x *z= X(X T X) -1 X T *z= X(X T X) -1 *(X T *z) = X(X T X) -1 *0 = 0 (X T X) -1 X T *y = a

26 Multiple Linear Regression via SVD Measurements (y) Deviation from the model ( noise ) Factor_1 Factor_2 measurements in the model (estimated through factors) y=x*a + e P x = X(X T X) -1 X T is the orthogonal projection operator (X T X) -1 X T *y = a^ According to SVD: (X T X) = U*w*V T ; where U*U T =I, V T V=I Multiplying by V from the right: (X T X)*V = U*w Multiplying by w -1 from the right: (X T X)*V*w -1 = U Multiplying by U T from the right: (X T X)*V*w -1 U T = I Thus: (X T X) -1 = V*w -1 U T

27 Data Interpolation Data interpolation is needed in order to estimate the derivatives accurately enough. The interpolation by polynomials could be performed by the multiple regression procedure. Indeed, for each gene in every experiment the logexpression in time-points 0, 3d day, 5 th day, and 7 th day gives a vector y of length 4. The coefficients (vector a )of the polynomial of power 3 can be estimated from the equation: y=x*a The four columns of matrix X consist of values of t 0, t 1, t 2, and t 3 polynomials in time-points 0, 3, 5, 7