Inference of Sparse Gene Regulatory Network from RNA-Seq Time Series Data

Transcription

1 Inference of Sparse Gene Regulaory Nework from RNA-Seq Time Series Daa Alireza Karbalayghareh and Tao Hu Texas A&M Universiy December 16, 2015 Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

2 Ouline 1 Inroducion 2 Nework Model 3 Nework Inference 4 Simulaion resuls 5 Conclusions Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

3 Inroducion Main conceps High hroughpu RNA-Seq echnology has become more popular in gene expression analysis. We are ineresed in inferring he Gene Regulaory Neworks GRNs) from he coun-based RNA-Seq daa. Coninuous models like Gaussian) canno work well for he coun-based daa. Poisson model is discree, bu does no reflec he dispersion in he RNA-Seq daa. Temporal daa can be more beneficial for he nework inference. The GRNs in are usually sparse. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

4 Inroducion Model assumpions To address hese facs, in our model we assume: A Negaive Binomial BN) disribuion for he expression levels of he genes, which akes ino accoun he dispersion. A log-linear link funcion for he mean parameer. Adding an l 1 norm penaly o he cos funcion in order o reflec he sparseness in he nework. Using he ime series daa of all genes o infer he underlying GRN. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

5 Nework Model Negaive Binomial Model RNA-Seq Gene expression levels are assumed o have NB disribuion. Negaive Binomial Model f y i) ; µ i), σ i) ) = ) σ i) µ i) 1 σ + 1 i) Γy i) + 1 σ i) ) Γ 1 σ i) )Γy i) + 1) σ i) µ i) σ i) µ i) + 1 ) y i), i = 1,, n, = 1,, m µ i) is mean expression for gene i a ime. σ i) is dispersion coefficien of gene i he same for all imes = 1,, m. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

6 Nework Model Negaive Binomial Model RNA-Seq Gene expression levels are assumed o have NB disribuion. Negaive Binomial Model f y i) ; µ i), σ i) ) = ) σ i) µ i) 1 σ + 1 i) Γy i) + 1 σ i) ) Γ 1 σ i) )Γy i) + 1) σ i) µ i) σ i) µ i) + 1 ) y i), i = 1,, n, = 1,, m µ i) is mean expression for gene i a ime. σ i) is dispersion coefficien of gene i he same for all imes = 1,, m. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

7 Nework Model Negaive Binomial Model RNA-Seq Gene expression levels are assumed o have NB disribuion. Negaive Binomial Model f y i) ; µ i), σ i) ) = ) σ i) µ i) 1 σ + 1 i) Γy i) + 1 σ i) ) Γ 1 σ i) )Γy i) + 1) σ i) µ i) σ i) µ i) + 1 ) y i), i = 1,, n, = 1,, m µ i) is mean expression for gene i a ime. σ i) is dispersion coefficien of gene i he same for all imes = 1,, m. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

8 Nework Model Markov GLM GLM We use a Markov generalized linear model GLM) wih a log link funcion for he mean expression levels as ) log µ i) +1 = where q j) = log gene a ime. β i) j n j=1 y j) q j) β i) j, i = 1,..., n, = 1,..., m 1, ), and y j) is he expression level of he j-h is he connecion srengh or he weigh of he direced edge from gene j o gene i in he nework graph. β i) j s are ime invarian. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

9 Nework Model Markov GLM GLM We use a Markov generalized linear model GLM) wih a log link funcion for he mean expression levels as ) log µ i) +1 = where q j) = log gene a ime. β i) j n j=1 y j) q j) β i) j, i = 1,..., n, = 1,..., m 1, ), and y j) is he expression level of he j-h is he connecion srengh or he weigh of he direced edge from gene j o gene i in he nework graph. β i) j s are ime invarian. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

10 v9 v8 v7 v12 v11 v10 v7 v7 v7 Nework Model Model evoluion i = 1 q 1) 1 q 1) 2 q 1) m i = 2 q 2) 1 q 2) 2 q 2) m i = n q n) 1 q n) 2 q n) m = 1 = 2 = m Figure : Gene expression evoluion for i genes during m ime poins. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

11 Nework Inference Regularized cos funcion If we assume β i) i hen, µ i) +1 = exp = 0, and define [ β i) = [ q i) = β i) 1 q 1) q i) β i)),..., βi) i 1, βi) i+1,..., βi) n ] T ],..., q i 1), q i+1),..., q n) For he nework inference, we use ML plus an l 1 penaly for β i) in order o conrol he sparseness of he nework. Cos funcion o be minimized gβ i), σ i) ) = lβ i), σ i) ) + λ β i) 1, i = 1,, n where lβ i), σ i) ) is he log-likelihood funcion, and λ > 0 is he regularizaion parameer. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

12 Nework Inference Regularized cos funcion If we assume β i) i hen, µ i) +1 = exp = 0, and define [ β i) = [ q i) = β i) 1 q 1) q i) β i)),..., βi) i 1, βi) i+1,..., βi) n ] T ],..., q i 1), q i+1),..., q n) For he nework inference, we use ML plus an l 1 penaly for β i) in order o conrol he sparseness of he nework. Cos funcion o be minimized gβ i), σ i) ) = lβ i), σ i) ) + λ β i) 1, i = 1,, n where lβ i), σ i) ) is he log-likelihood funcion, and λ > 0 is he regularizaion parameer. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

13 Nework Inference ML Log-likelihood funcion lβ i), σ i) ) = = m 1 =1 { log m 1 =1 + y i) σ i) y i) +1 qi) Γ ) log f y i) +1 ; µi) +1, σi) ) )) 1 σ i) log Γ y i) ) ) log β i) + log σ i) expq i) β i) ) + 1 )} Γy i) ). σ i) )) y i) +1 log σi) Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

14 Nework Inference Minimizaion w.r.. σ and β Minimizaion of g wih respec o σ i) i = 1,, n): m 1 =1 { ) 1 F σ i) F y i) ) ) σ i) + log 1 + σ i) y i) exp q i) β i))) σ i) y i) σ i) exp q i) β i)) = 0 where Fu) = log Γu)/ u is he Digamma funcion. g is convex bu no differeniable wih respec o β i). So, we can use Fas Ieraive Shrinkage-Thresholding Algorihm FISTA) [Beck e al., 2009] o do minimizaion w.r. β i) i = 1,, n). Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

15 Nework Inference Minimizaion w.r.. σ and β Minimizaion of g wih respec o σ i) i = 1,, n): m 1 =1 { ) 1 F σ i) F y i) ) ) σ i) + log 1 + σ i) y i) exp q i) β i))) σ i) y i) σ i) exp q i) β i)) = 0 where Fu) = log Γu)/ u is he Digamma funcion. g is convex bu no differeniable wih respec o β i). So, we can use Fas Ieraive Shrinkage-Thresholding Algorihm FISTA) [Beck e al., 2009] o do minimizaion w.r. β i) i = 1,, n). Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

16 Nework Inference How o choose λ? Choice of λ criically affecs he performance Small λ: dense nework, and inferring many unnecessary connecions Very large λ: very sparse nework, and high probabiliy of missing he real connecions We use he popular BIC mehod, and choose he λ which minimizes Bayesian Informaion Crierion BIC) n { } 2lβ i), σ i) ) + k i) logm 1) i=1 where k i) = β i) 0 i = 1,, n). Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

17 Nework Inference How o choose λ? Choice of λ criically affecs he performance Small λ: dense nework, and inferring many unnecessary connecions Very large λ: very sparse nework, and high probabiliy of missing he real connecions We use he popular BIC mehod, and choose he λ which minimizes Bayesian Informaion Crierion BIC) n { } 2lβ i), σ i) ) + k i) logm 1) i=1 where k i) = β i) 0 i = 1,, n). Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

18 Nework Inference Nework Inference Algorihm Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

19 Simulaion resuls Simulaed daa 50 randomly generaed neworks wih n = 10, and m = 15 ROC curve for many differen λ from very small high TPR and high FPR) o very large low TPR and low FPR) Red diamond is he λ achieved from BIC, making saisfacory performance Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

20 Simulaion resuls Simulaed daa cond) a) True connecions b) False connecions c) Hamming disance d) MSE Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

21 Simulaion resuls Tes on he real RNA-Seq daa se We consider he GRN during he gu formaion of he sarle sea anemone Nemaosella vecensis [Boman e al., 2014]. This nework has four genes foxa, snail, wis, and β-caenin: β-ca foxa sna wi Figure : The GRN for he developing Nemaosella vecensis gu, adaped from [Boman e al., 2014] Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

22 Simulaion resuls Tes on he real RNA-Seq daa se cond) We applied our algorihm o he RNA-Seq ime series daa of hese four genes, available in [Helm e al., 2013] This daa se has he expression levels a 6 ime poins 2h, 7h, 12h, 24h, 5d, and 10d. β-ca foxa sna wi Figure : Inferred nework by our algorihm, which capures all he connecions of he real nework excep he self-loop. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

23 Conclusions Conclusions and fuure works We sudied he inference of GRNs using he RNA-Seq ime series daa. We modeled he raw coun daa by an NB disribuion. We assumed a log-linear model for he emporal evoluion of he gene expressions. We ook advanage of sparsiy in he GRNs, and incorporaed an l 1 penaly o conrol he number of connecions. The resulan algorihm demonsraes a good performance boh in he synheic and real biological daa. As a fuure work, we can consider arbirary ime-delays, and exend he curren model o infer ime-lagged regulaory ineracions. Time varian connecions among he genes can be analyzed as a fuure work as well. Alireza Karbalayghareh GRN Inference from RNA-Seq Time Series December 16, / 18

24 Thank you Any quesions?