Defining an additivity framework for mixture research in inducible whole-cell biosensors

A novel additivity framework for mixture effect modelling in the context of whole cell inducible biosensors has been mathematically developed and implemented in R. The proposed method is a multivariate extension of the effective dose (EDp) concept. Specifically, the extension accounts for differential maximal effects among analytes and response inhibition beyond the maximum permissive concentrations. This allows a multivariate extension of Loewe additivity, enabling direct application in a biphasic dose-response framework. The proposed additivity definition was validated, and its applicability illustrated by studying the response of the cyanobacterial biosensor Synechococcus elongatus PCC 7942 pBG2120 to binary mixtures of Zn, Cu, Cd, Ag, Co and Hg. The novel method allowed by the first time to model complete dose-response profiles of an inducible whole cell biosensor to mixtures. In addition, the approach also allowed identification and quantification of departures from additivity (interactions) among analytes. The biosensor was found to respond in a near additive way to heavy metal mixtures except when Hg, Co and Ag were present, in which case strong interactions occurred. The method is a useful contribution for the whole cell biosensors discipline and related areas allowing to perform appropriate assessment of mixture effects in non-monotonic dose-response frameworks


Supplementary Material SM1
. Summary of non-linear models and estimated parameters together with relevant statistical information of the model fits for the response of Synechococcus elongatus PCC 7942 pBG2120 to 6 heavy metals and their binary mixtures (Zn, Cu, Cd, Ag, Co and Hg)  gau and lgau refers to the Normal and log-normal biphasic dose-response models described in the gaussian {drc} functions in drc R package 1 . Parameter estimates and estimated residual standard errors are calculated using the function drm {drc} which is based on the function optim {stats} 2 which relies on the minimisation of the minus log likelihood function. For a quantitative response this reduces to least squares estimation, which is carried out by minimising the following sums of squares ∑_{i=1}^N [w_i (y_i-f_i)]^2 where y_i, f_i, and w_i correspond to the observed value, expected value, and the weight respectively, for the ith observation (from 1 to N).  Table SM1.

Supplementary Material SM3 (Tutorial)
Fitting non-monotonic dose-response with differential maximal effects and analysis of departures from additivity based on multivariate Loewe additivity using R.

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A novel additivity framework for mixture-effect research in the framework of whole-cell inducible biosensors has been mathematically developed. The method proposes a multivariative extension of the effective dose (EDp) notation in order to take into account the occurrence of differential maximal effects and inhibition beyond the maximum permissive concentrations (MPCs). This allow a dimensional extension of Loewe additivity which enables its direct application in a biphasic dose-response framework. The utilities that allow to achieve this new approach have been incorporated in the (drc) package for R 1 .

#Example data:
The dataset "metaldata.csv" and the functional R-script "SM3_Script_R.R" used in the present example are freely available from http://dx.doi.org/10.6084/m9.figshare.1476176 In the present tutorial, we will explain how to use the novel method developed in order to model dose-response profiles of an inducible whole cell biosensor to individual stimuli and mixtures (in the present case, different heavy metals). We will analyse the "metaldata.csv" dataset. This dataset includes the raw data resulted from the individual exposure of the inducible whole cell biosensor Synecchoccocus elongatus PCC 7942 pBG2120 3 to six metals and their 15 possible binary combinations.
The dataset is importable as data.frame in R as follows: (from the root directory: metaldata<-read.csv(file="metaldata.csv") The data frame have 3 variables: "metal" (factor), "conc" (numeric), "IR" (induction factor, the same asBIF in the main text) (numeric). In the tutorial, we will explain the drc extension using the data for Zn, Cd and its mixture, ZnCd.

#plot:
We can visually observe all the functions used in order to have a visual overview of the models fitted.

#Effective doses (EDp) calculation:
Using the ED {drc} function, we can get the D(p) vectors at any desired fractional effect p. We selected the three representative p levels: -50 (half of the Emax, in the induction region), 99.9 (Emax) and +50 (half of the Emax, in the inhibition region) (Equivalent to the -50, 0, +50 p levels in the manuscript). For more details, see section 2.2 a multivariate extension of the effective dose notation of the manuscript). Note: to get the E(p) values, for the moment it is required to use the indicesFct {drc} function which will be described in next sections.  plot(Cd.gau, type = "obs", col= "black", log = "", ylim = c(0, 60)) plot(Cd.gau, type = "none", add = TRUE, col = "red") plot(Cd.gau2, type = "none", add = TRUE, col = "yellow") plot(Cd.lgau, type = "none", add = TRUE, col = "blue") plot(Cd.lgau2, type = "none", add = TRUE, col = "green") Based on the residuals standard errors and the visual examination, we can observed that the LogGaussian with Box-Cox modification (green line) is the model that fits better Cd experimental data, therefore it will be used for fitting Cd response in the rest of calculations. In order to predict the biosensor response to any combination of the individual metals and also, to be able to analyse departures from additivity, it is required to fit the mixture data to a nonlinear regression model equation. Zn and Cd mixture data are used as example.

#E(p) calculation
We apply the method to perform additivity predictions and to study the nature of the interaction through the indicesFct {drc} command. The first datum that has to be indicated is the ratio in which metals are present in the mixture. After that, we have to introduce the list with the model equations fitted for the mixture and the individual metals, respectively. The ratio only has to be indicated for the individual metal indicated first. (For example, in the case below, the ratio Zn:Cd is 0.355:0.645, therefore, as Zn is the first individual metal introduced, we indicated is 0.355 as ratio data). In c() vector we indicated the fractional effect (p) which we want to be considered. indicesFct(0.355,list(ZnCd.lgau2, Zn.lgau2, Cd.lgau2), c (-0.2, -0.5, -1, -2, -5, -10, -20, -30, -40, -50, -60, -70, -80, -90, -99, 99, 80, 70, 60, 50, 40, 30, 20, 10, 5, 2, 1, 0.5, 0.2)) #ECmat: Matrix with the fitted nonlinear regression model equations concentration data for the mixture (ECmix, Zn:Cd), the first individual metal indicated (EC1, Zn) and the second individual metal indicated (EC2, Cd) and the standard error, respectively. These data correspond to the D(p) data for all the fractional effect indicated in indicesFct.