In transcriptional regulation, transcription factors (TFs) are often

unobservable at mRNA level or may be controlled outside of the system being

modelled. Gaussian processes are a promising approach for dealing with these

difficulties as a prior distribution can be defined over the latent TF activity

profiles and the posterior distribution inferred from the observed expression levels

of potential target genes. However previous approaches have been based on the

assumption of additive Gaussian noise to maintain analytical tractability. We

investigate the influence of a more realistic form of noise on a biologically accurate

system based on Michaelis-Menten kinetics

Davies, V.

Husmeier, D.

English

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          Davies, V., and Husmeier, D. (2013) Assessing the impact of non-additive noise on modelling transcriptional regulation with Gaussian processes. In: Muggeo, V.M.R., Capursi, V., Boscaino, G. and Lovison, G. (eds.) Proceedings of the 28th International Workshop on Statistical Modelling. Gruppo Istituto Poligrafico Europeo SRL, pp. 559-562. ISBN 9788896251492.  Copyright © 2013 Statistical Modelling Society.  A copy can be downloaded for personal non-commercial research or study, without prior permission or charge  Content must not be changed in any way or reproduced in any format or medium without the formal permission of the copyright holder(s)   When referring to this work, full bibliographic details must be given    http://eprints.gla.ac.uk/85494/      Deposited on: 06 August 2014              Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk Assessing the impact of non-additive noise onmodelling transcriptional regulation withGaussian processes.Vinny Davies1 and Dirk Husmeier11 School of Mathematics and Statistics, University of Glasgow, ScotlandE-mail for correspondence: v.davies.1@research.gla.ac.ukAbstract: In transcriptional regulation, transcription factors (TFs) are oftenunobservable at mRNA level or may be controlled outside of the system beingmodelled. Gaussian processes are a promising approach for dealing with thesedifficulties as a prior distribution can be defined over the latent TF activityprofiles and the posterior distribution inferred from the observed expression levelsof potential target genes. However previous approaches have been based on theassumption of additive Gaussian noise to maintain analytical tractability. Weinvestigate the influence of a more realistic form of noise on a biologically accuratesystem based on Michaelis-Menten kinetics.Keywords: Transcriptional regulation, Gaussian processes, additive and multi-plicative noise, Michaelis-Menten kinetics1 IntroductionA particular challenge in the quantitative modelling of transcriptional reg-ulation is that transcription factors (TFs), the regulatory proteins at theheart of the process, are frequently subject to post-translational modifica-tion, which may affect their DNA binding capability. Consequently, geneexpression levels of TFs contain only limited information about their actualactivities. A promising approach to deal with these difficulties was proposedin Gao et al. (2008), inspired by the work of Barenco et al. (2006). Theauthors advocate the use of Gaussian processes to define prior distribu-tions over the latent TF activity profiles. Inference is soundly based on theprinciples of non-parametric Bayesian statistics, consistently inferring theposterior distribution of the unknown TF activities from the observed ex-pression levels of potential target genes, and inferring regulatory networkstructures after marginalizing over the unknown TF activity profiles.The choice of a non-parametric prior distribution from the Gaussian processfamily is not a restrictive modelling assumption. Somewhat more restric-tive is the assumption of additive Gaussian noise, which can be found in all2 Non-additive noise in transcriptional regulation.previous applications (Gao et al. (2008), Honkela et al. (2010), etc.). Previ-ous work by Rocke and Durbin (2001) showed that mRNA concentrationsobtained from microarray experiments are of a more complex form andthe purpose of this work is to investigate what effect this deviation fromadditive Gaussianity has on the inference in transcriptional regulation.2 MethodA linear model of gene expression was proposed by Barenco et al. (2006)dxi(t)dt= Bi + Sif(t)−Dixi(t) (1)where i ∈ {1, . . . , G} is a set of genes regulated by the same TF, xi(t)are the (unknown) true gene expression levels at time point t, f(t) is the(unknown) TF activity, Bi is the basal transcription rate of gene i, Si is thesensitivity to binding of TF, and Di is a decay rate. We assume that (noisy)measurements of xi(t) can be obtained, however TF activity is unknownand therefore f(t) is assumed to be unobservable.Eq. (1) has the analytical solution:xi(t) =BiDi+ Si∫ t0exp(−Di(t− u))f(u)du. (2)Gao et al. (2008) proposed a non-parametric Bayesian approach to inferencein this model by placing a Gaussian process prior with a squared exponen-tial covariance matrix on the unknown TF activities f = (f(t1), . . . , f(tT ))at timepoints t = (t1, . . . , tT ). The linear form of the model implies that thejoint prior distribution of the expression profiles of all regulated genes, xi,is described by a Gaussian process prior with a covariance matrix, K, thatdepends on the hyperparameters of the prior, θh, as well as the parametersthat characterise the transcriptional regulation processes via eq. (2):p(x|θ′) = N (B./D,K); K = K(θ′)θ′ = (θh, B1, . . . , BG, S1, . . . , BG, D1, . . . , DG) (3)where B./D is a point-wise vector division. See Davies and Husmeier (2013)for details.To relate the unknown true gene expression profiles xi = (xi(t1), . . . , xi(tT ))to noisy measurements yi = (yi(t1), . . . , yi(tT )), Gao et al. (2008) assumedadditive Gaussian noise of constant variance σ2. The marginalisation overy is analytically tractable and gives:p(y|θ) =∫N (y|x, σ2I)N (x|0,K(θ′))dx = N (y|B./D,K(θ′) + σ2I) (4)where θ = (θ′, σ2). Inference of the parameters θ can then be achieved ina maximum likelihood or Bayesian framework; see Bishop (2006).Davies and Husmeier 3However Rocke and Durbin (2001) showed that the noise in transcriptionalprofiling with microarrays has the following more general form:yi(t) = c+ xi(t) exp(µ) + t where j ∼ N (0, σ2j ) (5)where c is mean background noise, and σ2µ and σ2t are unknown varianceparameters. ReplacingN (y|x, σ2I) in eq. (4) by the noise in eq. (5) does notgive a closed-form solution, and this has therefore been ignored in previouswork. The objective of the present study is to quantify the effect the devi-ation from additive Gaussianity has on the inference of the transcriptionalregulation.3 DataWe combined a simple regulatory network for three genes with a proteinsignalling pathway from Vyshemirsky and Girolami (2008); see Davies andHusmeier (2013) for details. The active form of the TF is unobservable dueto post-translational modification, and the processes leading to the forma-tion of active TF is controlled outside of the subsystem being modelled. Thetranscriptional profiles of the downstream genes were generated by solvingeq. (2) with the different kinetic parameters. 18 values from these expressionprofiles were then subjected to either additive Gaussian noise, or the morecomplex noise of eq. (5). For the non-Gaussian noise the standard deviationswere chosen on a roughly log scale such that σµ, σt = (0.01, 0.03, 0.1, 0.3),with equivalent values chosen for the additive noise model to allow for afair comparison. This was repeated 10 times for each standard deviationsize and noise model.4 ResultsFor a relatively small data set, our results, given in Figure 1, have shownthat the deviation from additive Gaussian noise has little negative effectwhen σµ, σt = (0.01, 0.03). For larger standard deviations the results showa consistent deterioration in the case of non-Gaussian noise, although thiscannot be easily quantified until σµ, σt = (0.3). For this level of variance,Figure 1, as well as similar results for the kinetic parameter estimates, showa roughly four fold increase in the median error.5 ConclusionOur work has considered the implications of having non-Gaussian noisewhen using Gaussian processes for modelling transcriptional regulation.This noise model violates some of the modelling assumptions and causes adeterioration in the ability of the model to perform parameter inference. Wehave shown that the effect of this noise is not as significant as first assumedand the negative effect only becomes apparent for larger variances.4 Non-additive noise in transcriptional regulation.A01 M01 A03 M03 A1 M1 A3 M3−1.0−0.8−0.6−0.4−0.20.00.20.4(a) Gene Profile, x1A01 M01 A03 M03 A1 M1 A3 M3−0.4−0.20.00.20.40.6(b) Gene Profile, x2A01 M01 A03 M03 A1 M1 A3 M3−1.0−0.50.00.51.01.52.0(c) Gene Profile, x3A01 M01 A03 M03 A1 M1 A3 M3−0.20.00.20.40.6(d) TF Profile, fFIGURE 1: Box plots of the error of gene and TF profile predictions. Boxplots for the additive Gaussian, ‘A’, and non-Gaussian, ‘M’, noise are givenin light and dark grey respectively. The standard deviations used for σµ andσt are given under each box plot and represent the values (0.01,0.03,0.1,0.3)ReferencesBarenco, M. et al. (2006). Ranked prediction of p53 targets using hiddenvariable dynamic modelling. Genome Biology, 7(3), R25.Bishop, C.M. (2006). Pattern Recognition and Machine Learning. Springer.Davies, V. and Husmeier, D. (2013). Modelling transcriptional regulationwith Gaussian processes. Technical Report. University of Glasgowwww.maths.gla.ac.uk/∼dhusmeier/MyPapers/bookChapterVinny.pdfGao, P. et al. (2008). Gaussian process modelling of latent chemical species:applications to inferring transcription factor activities. Bioinformat-ics, 24, 70 – 75.Honkela, A. et al. (2010). Model-based method for transcription factor tar-get identification with limited data. PNAS, 107(17), 7793 – 7798.Rocke, D.M. and Durbin, B. (2001). A model for measurement error ofgene expression analysis. J. Comput. Biol., 8, 557 – 569Vyshemirsky, V. and Girolami, M.A. (2008). Bayesian ranking of biochem-ical system models. Bioinformatics, 24(6), 833 – 839.

Assessing the impact of non-additive noise on modelling transcriptional regulation with Gaussian processes

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Assessing the impact of non-additive noise on modelling transcriptional regulation with Gaussian processes

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