A new statistical framework to infer gene regulatory networks with hidden transcription factors

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1 A new statistical framework to infer gene regulatory networks with hidden transcription factors Ivan Vujačić, Javier González, Ernst Wit University of Groningen Johann Bernoulli Institute PEDS II, June 4-6, Eindhoven Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 1 / 18

2 Central dogma of molecular biology Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 2 / 18

3 SOS system in Escherichia coli Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 3 / 18

4 The goal The goal of the analysis is to Reconstruct the activity level of the repressor LexA. Identify kinetic parameters of the system. Order the genes in terms of speed they are repressed by LexA protein. Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 4 / 18

5 ODE model η(t) - the abundance of the protein (TF) LexA at time t T. x 1 (t),..., x m (t) - mrna concentrations of genes. ODE model for expressions of the genes ẋ k (t) = p(t; θ k, η(t)) δ k x k (t), k = 1,..., m. Michaelis-Menten form for p, i.e. Here θ k = (β k, γ k, ϕ k ), where β k - production rate, γ k - half-saturation rate, p(t; θ k, η(t)) = β k 1 γ k + η(t) + ϕ k. ϕ k - basal level of gene transcription. Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 5 / 18

6 Modelling the protein abundance We assume that η is cubic spline on T : d η(t) = µ j φ j (t), j=1 µ j IR and {φ 1,..., φ d } is truncated-power basis set for cubic spline. This induces a new set of parameters µ = (µ 1,..., µ d ) T. Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 6 / 18

7 Noise model ODE models t = {t 1,..., t n } y ki - the measured expression of gene k at t i. S k = {(y ki, t i ) IR T } n i=1 Gene expression measurements of genes are independent and y ki N (x k (t i ), σ 2 k). The log-likelihood of a single observation is up to an additive constant l(x k (t i ), σk y 2 ki ) = 1 ( ) yki x k (t i ) log(σ2 k). The contribution of each gene k to the log-likelihood of the network is σ k n l k (S k ; δ k, θ k, σk, 2 µ) = l(x k (t i ), σk y 2 ki ), i=1 where it is assumed that each function x k satisfies the corresponding ODE. Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 7 / 18

8 The likelihood is up to an additive constant l = 1 2 m k=1 1 σ 2 k y k x k (t) 2 n 2 m log(σk), 2 k=1 where x k (t) = (x k (t 1 ),..., x k (t n )) T, y k = (y k1,..., y kn ) T. Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 8 / 18

9 Discretization ODE models p(t; θ k, η(t)) = (p(t 1 ; θ k, η(t 1 )),..., p(t n ; θ k, η(t n ))) T. The difference operator D = , for = diag(t 2 t 1, t 3 t 1,..., t n t n 1 ). P δk = D + δ k I, where I is the identity matrix. The discretization of the ODE for the gene k can be written as P δk x k (t) = p(t; θ, η(t)). Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 9 / 18

10 Penalizing the likelihood We want to penalize the likelihood with the term P δk x k (t) p(t; θ k, η(t)) 2. To use RKHS theory, penalty needs to be equal to P δk x k 2 for some x k. This leads to x k = x k (t) P 1 δ k p(t; θ k, η(t)), and to transformation of the observations Penalized log-likelihood is defined as l λ = 1 2 m k=1 ỹ k = y k P 1 δ k p(t; θ k, η(t)). 1 σ 2 k ỹ k x k 2 n 2 m m log(σk) 2 λ P δk x k 2. k=1 k=1 Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 10 / 18

11 RKHS theory guarantees that for some α k IR n and K δk = (P T δ k P δk ) 1 Ω( x k ) = P δk x k 2 = α T k K δk α k, x k = K δk α k. S k is transformed set of expression measurements for the gene k. The penalized log-likelihood is l λ (, Θ, Σ, µ S) = 1 2 λ m k=1 m k=1 1 σ 2 k ỹ k K δk α k 2 n 2 α T k K δk α k, S = { S 1,..., S m }, = {δ 1,..., δ m }, Θ = {θ 1,..., θ m }, Σ = {σ 2 1,..., σ2 m}. m log(σk) 2 k=1 Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 11 / 18

12 Estimation ODE models A = {α 1,..., α m } the set of parameters characterizing x. The penalized maximum likelihood estimators of, Θ, Σ, µ and A are given by ( ˆ λ, ˆΘ λ, ˆΣ λ, ˆµ λ,  λ ) = arg max,θ,σ,µ,a l λ(, Θ, Σ, µ, A S). Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 12 / 18

13 Choosing the tunning parameter To choose λ we use AIC type of criteria. The smoother matrix is S λ,k = K δk (K δk + 2σ 2 kλi) 1. The complexity of the model is defined as df k = Tr(S λ,k ). λ is chosen as λ opt = arg min λ m [ 2 l k (S k ; ˆδ k, ˆθ k, ˆσ k, 2 ˆµ) + 2df k ]. k=1 Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 13 / 18

14 Data m = 14 expression genes of the E-coli SOS system. mrna measurements in n = 6 time points: 0,5,10,20, 40 and 50 min. Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 14 / 18

15 RESULTS Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 15 / 18

16 LexA activity ODE models Master Repressor LexA TFA Time Figure: Reconstruction of the activity of the master repressor LexA scaled between 0 an 1. The smoothed LexA profile is obtained using a cubic spline. Time is given in minutes. Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 16 / 18

17 Gene Profiles ODE models sbmc umud Gene Expression Gene Expression Time Time Figure: Reconstruction of gene profiles sbmc and umudb. Points represent the data values and lines the predicted profiles. The rest of the genes are fitted in a similar way. Time is given in minutes. Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 17 / 18

18 Kinetics To rank the genes in terms of their production rate we use the values of r k = β k γ k + η for i = 1,..., 14 and η the averaged TF level, recn (r = 15.51), sula (r = 13.62), unuc (r = 16.95) are the fastest dinf (r = 0.97) is the slowest recn and umuc show the largest values for δ k ϕ k were found to be negligible for genes sula and umuc and very small for recn. Vujačić, González, Wit (rug) Inference of ODE s PEDS II, June 4-6, Eindhoven 18 / 18