Reconstructing promoter activity from Lux bioluminescent reporters

The bacterial Lux system is used as a gene expression reporter. It is fast, sensitive and non-destructive, enabling high frequency measurements. Originally developed for bacterial cells, it has also been adapted for eukaryotic cells, and can be used for whole cell biosensors, or in real time with live animals without the need for euthanasia. However, correct interpretation of bioluminescent data is limited: the bioluminescence is different from gene expression because of nonlinear molecular and enzyme dynamics of the Lux system. We have developed a computational approach that, for the first time, allows users of Lux assays to infer gene transcription levels from the light output. This approach is based upon a new mathematical model for Lux activity, that includes the actions of LuxAB, LuxEC and Fre, with improved mechanisms for all reactions, as well as synthesis and turn-over of Lux proteins. The model is calibrated with new experimental data for the LuxAB and Fre reactions from Photorhabdus luminescens—the source of modern Lux reporters—while literature data has been used for LuxEC. Importantly, the data show clear evidence for previously unreported product inhibition for the LuxAB reaction. Model simulations show that predicted bioluminescent profiles can be very different from changes in gene expression, with transient peaks of light output, very similar to light output seen in some experimental data sets. By incorporating the calibrated model into a Bayesian inference scheme, we can reverse engineer promoter activity from the bioluminescence. We show examples where a decrease in bioluminescence would be better interpreted as a switching off of the promoter, or where an increase in bioluminescence would be better interpreted as a longer period of gene expression. This approach could benefit all users of Lux technology.


S1 Text
This document contains detailed methods including reactions mechanisms and related velocity derivations, additional figures for model data fits, as well as diagnostic figures for MCMC method for parameter estimation for all three reactions (Fre, LuxAB, and LuxEC).

An improved biochemical model for the Lux reactions
For all the reactions, we present the detailed mechanisms and derive the velocity equations using King and Altman's schematic method (King and Altman 1956).

Fre Reaction: Flavin recycling
The Fre-catalysed FMN/NADPH reaction, given by E.C.1.5.1.29, is: The mechanism used for this reaction is given by: This mechanism is used to derive a velocity equation for the reaction. In this and other velocity equations, two parameter conventions are used: k r 1 = k −1 k 1 for reversibility constants; and (3)

LuxAB Reaction: Main light pathway
The two lux genes luxA and luxB encode for luciferase, which, in the presence of aldehyde, catalyses the reaction involving oxidation of FMNH 2 to oxidised flavin (FMN) and the emission of light. The reaction, given by E.C.1.14.13.3, is: The mechanism used for the derivation of velocity equation is: The reaction velocity is: The LuxAB model given in Equation 6 has been refined in the light of the experimental findings suggesting product inhibition by FMN. Because the exact mechanism is as yet unknown, we have included a generic product inhibition term to Equation 6 to obtain the refined velocity equation: where K F is the new parameter associated with product inhibition.

LuxEC Reaction: Aldehyde recycling
The recycling fatty acid back to aldehyde is modelled as a combined reaction catalysed by the LuxEC complex. The reaction combines E.C. 6.2.1.19 and E.C. 1.2.1.50. and is: The mechanism is: The velocity equation associated with this mechanism is: 1.2 Sub-models used for model parameter inference Fre model parameter inference In equation 11, γ N is the in vitro degradation parameter for NADPH and V F re is the rate law for this reaction, given in equation 3. This model is used to simulate the time course behaviour of NADPH and FMN for given values of the parameters which we will infer using the data given in Figure A

LuxAB model parameter inference
Parameter inference for the LuxAB reaction was carried out using an MCMC scheme, with the posterior estimates for the Fre parameters as prior estimates for these same parameters in this model. Table 1 details inferred values of LuxAB reaction related parameters. Figure F shows that model fits to data are very good; importantly, the model faithfully reproduces the apparent product inhibition observed in the data. MCMC diagnostics are shown in Supplementary Figures corresponding to LuxAB reaction. The equations used for this inference are given by: LuxEC model parameter inference where the reaction velocity is as is shown in equation 10, and which contains the parameters to be inferred.

Convergence Analysis
We have carried out a convergence analysis for kinetic parameters of individual reactions, as well as of height parameters in case of promoter activity inference. Two parallel chains were run for 500K iterations for both kinetic reactions (LuxAB and LuxEC, where LuxAB also includes inference Fre reaction related parameters, as described in the manuscript), and last 100k iterations were used to make posterior estimates, as reported in the manuscript. For promoter activity inference, chains were run for 1 million iterations in total and were sub-sampled (thinned) every 10th iteration, hence we had saved 100K samples for every chains, where last 50K samples were used to make posterior estimates. Given the stored samples, we have calculated convergence diagnosticsR (Gelman 1992) for all parameters at 15 equally spaced intervals along the chains length (Supplementary Figure N). The chains have converged in the intervals used for estimation of posterior means (given the Gelman's rule of thumb on convergence, i.e.,R ≤ 1.1).

TurnOver Rates for Lux Proteins
In Supplementary Figure O, we show the log-linear fit to individual data sets, used for estimation of Lux protein turn over rates, as described in main text. In the MCMC estimation of Fre and LuxEC model parameters, a Gibbs step is introduced for the sampling of noise precision τ , as described below. For LuxAB model, we estimate the noise variance from the replicates data.
Since we are using conjugate (with respect to the form of likelihood) prior for τ , hence, given the data and other parameters, the posterior distribution for τ has a form which can be readily sampled from, see equation 14.