Bayesian model-based inference of transcription factor activity

Abstract

Background: In many approaches to the inference and modeling of regulatory interactions using microarray data, the expression of the gene coding for the transcription factor is considered to be an accurate surrogate for the true activity of the protein it produces. There are many instances where this is inaccurate due to post-translational modifications of the transcription factor protein. Inference of the activity of the transcription factor from the expression of its targets has predominantly involved linear models that do not reflect the nonlinear nature of transcription. We extend a recent approach to inferring the transcription factor activity based on nonlinear Michaelis-Menten kinetics of transcription from maximum likelihood to fully Bayesian inference and give an example of how the model can be further developed.

Results: We present results on synthetic and real microarray data. Additionally, we illustrate how gene and replicate specific delays can be incorporated into the model.

Conclusion: We demonstrate that full Bayesian inference is appropriate in this application and has several benefits over the maximum likelihood approach, especially when the volume of data is limited. We also show the benefits of using a non-linear model over a linear model, particularly in the case of repression.

Figure 1

Expression of SEP is uncorrelated with its targets. Non-periodic expression of SEP and periodic expression of its gene targets

Figure 2

Model description. Cartoon depicting the model of [13]

Figure 3

Synthetic Example. Synthetic data example. (a) shows the true and inferred η¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWF3oaAgaqeaaaa@2E77@ profiles (note, in all figures, dashed lines correspond to the 5th and 95th percentiles). (b) Expression data and inferred profile for a typical gene. (c) Posterior for σ², true value was 0.05. (d)–(g) Posteriors for kinetic paramaters, β, γ, δ, α respectively. The boxes represent the region between the 25th and 75th percentiles with the median shown. Dotted lines give the range of the data, with outliers shown as crosses. Gray circles correspond to the true values.

Figure 4

Synthetic Example – problem genes. Dependency between β and γ and high noise in data for two genes in the synthetic example. (a) Posterior samples for β and γ for gene 10 in the synthetic example, (b) μ for gene 7 and expression data. The high level of noise leads to the poor parameter inference in this case.

Figure 5

Fission yeast example. Example of inference with real microarray data from fission yeast. (a), Inferred mean η¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWF3oaAgaqeaaaa@2E77@ profile with 5th and 95th percentiles, (b) μ for gene 1 when η¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWF3oaAgaqeaaaa@2E77@ is inferred (black lines) and when it is fixed (gray lines), (c) Posterior distribution for σ².

Figure 6

Adding delays. Posteriors from the delay example. (a) shows the posterior distributions for replicates 2 and 3 (note that ρ₁ has been fixed at 0). (b) shows the distribution of τ for three particular genes. (c) shows the difference in noise levels for the three replicates.

Figure 7

Linear versus non-linear. Benefits of the nonlinear model. (a) shows the significant improvement in likelihood over a linear model for the fission data. (b) a gene from the E. coli dataset that is modeled reasonably well by the linear model. (c) a gene from the E. coli dataset that is modeled badly by the linear model.